Boundary Value Problems for Functional Differential Equations › 2018 › ... · With the importance of boundary value problems for functional differential equa tions in applications,

Boundary Value Problems for Functional Differential Equations

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Boundary Value Problems for Functional Differential Equations

Editor

Johnny Henderson College of Sciences and Mathematics Auburn University, Alabama, USA

l b World Scientific II Singapore· New Jersey· London· Hong Kong

Published by

World Scientific Publishing Co. Pte. Ltd.

POBox 128, Farrer Road, Singapore 9128

USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661

UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

BOUNDARY VALUE PROBLEMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS

Copyright © 1995 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented. without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, Massachusetts 01923, USA.

ISBN: 981-02-2405-2

This book is printed on acid-free paper.

Printed in Singapore by Uto-Print

v

PREFACE

The purpose of Boundary Value Problems for Functional Differential Equations is to present some of the areas of current research in such a way as to be accessible to a wide audience. In addition, it is hoped that many of these articles can serve as guides to seminars or additional topics for courses in functional differential equations.

Functional differential equations have received attention since the 1920's. Within the development, boundary value problems have played a prominent role in both the theory and applications dating back to the 1960's. Contributions herein represent not only a flavor of classical results involving, for example, linear methods and oscillation-nonoscillation techniques, but also modern nonlinear methods for problems involving stability and control as well as cone theoretic, degree theoretic, and topological transversality strategies. A balance with applications is provided through a number of papers dealing with a pendulum with dry friction, heat conduction in a thin stretched resistive wire, problems involving singularities, impulsive systems, traveling waves, climate modeling, and economic control.

With the importance of boundary value problems for functional differential equations in applications, it is not surprising that as new applications arise, modifications are required for even the definitions of the basic equations. This was the case for several researchers who for years conducted a seminar in functional differential equations at Perm State Technical University. Participants from that seminar have contributed to this volume. Also, some contributions are devoted to delay Fredholm integral equations, while a few papers deal with what might be termed as boundary value problems for delay-difference equations.

I am grateful to my technical typist, Mrs. Rosie Torbert, and to my colleague, Professor Darrel Hankerson, for their technical assistance in the preparation of this book. I also express my gratitude to Dr. Anju Goel of World Scientific Publishing.

Johnny Henderson

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CONTENTS

Preface

Right Focal Point Boundary Value Problems for Functional-Differential Equations

R. P. Agarwal and Q. Sheng

The Ideas and Methods of the Perm Seminar on Boundary Value Problems

N. V. Azbelev

On Extension of the Vallee-Poussin Theorem to Equations with Aftereffect

N. V. Azhelev and L. F. Rakhmatullina

Initial-Boundary Value Problems for Impulsive Parabolic Functional Differential Equations

D. Bainov, Z. Kamont and E. Minchev

Boundary Value Problems on Infinite Intervals J. V. Baxley

Existence of Steady-State Solutions to Some Constant-Voltage Problems

L. E. Bobisud

An Existence Theorem for Hereditary Lagrange Problems on an Unbounded Interval

D. A. Carlson

Dynamical Spectrum for Skew Product Flow in Banach Spaces S. N. Chow and H. Leiva

On Boundary Value Problems for First Order Impulse Functional Differential Equations

A. Domoshnitsky and M. Drakhlin

vii

v

1

13

23

37

49

63

73

85

107

viii

Linearized Problems and Continuous Dependence J. A. Ehme

Positive Solutions and Conjugate Points for a Class of Linear Functional Differential Equations

P. W. Eloe and J. Henderson

Boundary Value Problems for Second Order Mixed Type Functional Differential Equations

L. H. Erbe. z.-C. Wang and L.-T. Li

The Filippov Approach to Boundary and Initial Value Problems and Applications

R. B. Guenther. J. W. Lee and M. Senkyf{k

An Existence Result for Delay Equations Under Semilinear Boundary Conditions

G. Hetzer

Periodic Solutions of Functional Differential Equations of Retarded and Neutral Types in Banach Spaces

L. H. Hoa aJui K. Schmitt

Existence of Traveling Waves for Reaction Diffusion Equations of Fisher Type in Periodic Media

W. Hudson and B. Zinner

Permanence of Periodic Solutions of Retarded Functional Differential Equations

L. A. da. C. Ladeira and S. M. Tanaka

Method of Quasilinearization for Boundary Value Problems for Functional Differential Equations

V. Lakshmikantham and N. Shahzad

A Resolvent Computation Related to Completeness for Compact Operators

S. M. V. Lunel

The Study of the Solvability of Boundary Value Problems for Functional Differential Equations: A Constructive Approach

V. P. Maksimov and A. N. Rumyantsev

119

131

143

153

165

177

187

201

209

215

227

Boundary Value Problems for Neutral Functional Differential Equations

S. K. Ntouyas

Existence Principles for Nonlinear Operator Equations D. O'Regan

Sturmian Theory and Oscillation of a Third Order Linear Difference Equation

A. Peterson

Multipoint Boundary Value Problems for Functional Differential Equations

D. Taunton

Third Order Boundary Value Problems for Differential Equations with Deviating Arguments

P. Ch. Tsamatos

Periodic Solutions and Liapunov Functionals T.-X. Wang

Boundary Value Problems of Second Order Functional Differential Equations

B. Zhang

ix

239

251

261

269

277

289

301

R I G H T FOCAL P O I N T BOUNDARY VALUE PROBLEMS F O R FUNCTIONAL-DIFFERENTIAL EQUATIONS

RAVI P. AGARWAL AND QIN SHENG Department of Mathematics, National University of Singapore

10 Kent Ridge Crescent, Singapore 0511

Abstract Inequalities established in 1161 are employed to provide sufficient conditions

for the existence-uniqueness and the convergence of the Picard and the approximate Picard iterative methods for the nth order differential equations with deviating arguments together with the T-point right focal boundary conditions.

1. Introduction

Let r 2 2 be an integer and -oo < al < . .. < a, < oo be given points. Consider the nth order differential equation with deviating arguments

x(")(t) = f ( t , z 0 ~ ( t ) ) , t E [ a l , ~ , ] , ( 1 ) where xow( t ) stands for (x(wo, l( t ) ) , . . . , ~ ( w o , ~ ( o ) ( t ) ) , . . . , X ( ~ ) ( W , , ~ ( , ) ( ~ ) ) ) , and p(i) , 0 5 i 5 q are positive integers. The function f (x, ( I ) ) is assumed to be continuous on [al , a,] x R N , where ( x ) represents ( x o , ~ , . . . , X O , ~ ( ~ ) , . . - , x,,,(,)), and N = C:=o p( i ) . The functions wi,j, 1 5 j 5 p(i), 0 5 i 5 q are continuous on [al, a,], and w i j ( t ) 5 a, for all t E [al , a,]; also, these functions assume the value al at most finite number of times as t ranges over [al,ar]. Let

a = min a l , ( a15t-a,

If a < a l , we assume that a function q5 E C ( P ) [ ~ , all is given. For the,given positive integers k l , . - . , k, such that xi=, k, = n, we define so = 0 , sj = x i=, ki. Let r = min{q, sl - 1). We seek a function x E B = C( ' ) [a , a,] n C(g)[a, all n C(")[al , a,], satisfying the r-point right focal conditions:

if a < a1 and q > sl - 1, then x( ' ) ( t ) = q5(')(t), 0 5 i 5 q, t E [a,al];

if a < a1 and q < sl - 1, then x( ' ) ( t ) = q5(;)(t), 0 5 i 5 q, t E [a,

~ ( ~ ) ( a l ) = Ai.1, q + 1 5 i < sl - 1; ( 2 ) if a = al , then x( ' ) (al) = Ai,l, 0 5 i < s l - 1 ;

2

BVPs FOR FDEs

and x(i)(aj) = Ai,j, Sj-l ::; i ::; Sj - 1, 2::; j ::; r; (3)

also, x is a solution of (1) on [aI, arlo We remark that while for ordinary differential equations right focal point boundary

value problems has been a subject matter of intense study [1-3, 5, 7-15, and references therein], for the delay differential equations only little is known [4] .

2. Preliminaries

For simplicity, let the points TI ::; . .. ::; Tn be such that T. j _, +1 = ... = Ts) = aj, 1 ::; j ::; r.

Lemma 2.1 [6] The r-point right focal interpolating polynomial Pn-l(t) of degree n - 1 of the function x(t) E C(n)[al, ar ], i.e., satisfying

or, equivalently, P (i) ( ) (i)( ) n-l Ti+l = X Ti+l, 0::; i ::; n - 1, (4)

can be expressed as n-l

Pn-l(t) = L:Ti(t)X(i)(r;+d, (5)

where To(t) satisfying

1, and Ti(t), 1 ::; i ::; n - 1 is the unique polynomial of degree i

(j) Ti (r;+d = 0, 0::; j ::; i-I,

r.(i)(T·) = 1 . . , (6)

and it can be written as

1 TI T2 i-I Ti I TI I

1 0 1 2T2 (i - I)T~-2 iT~-1 Ti(t)

1!2! .. · i! 0 0 0 (i - I)! i!Ti

(7)

1 t 2 t i - l t i

[[' ['-' . . . dtidti_l ... dtb (to = t). 1"1 1"2 7,

(8)

Lemma 2.2 [6] The error function e(t) = x(t) - Pn-l(t) can be written as

e(t) = l Tn

g(t,s)x(nl(s)ds, T, (9)

RIGHT FOCAL POINT BOUNDARY VALUE PROBLEMS

where g( t, s) is the Green's function of the boundary value problem

yin) = 0,

y(i) (Ti+t) =0, O:::;i:::;n-l,

and appears as

{

~ Ti(t) ( )n-i-l _ ~ (n _ i-I)! Ti+1 - s ,

g( t, s) - n-l Ti ( t) n-i-l -L( _ · _I),(1";+l-S) ,

i=k n z • t :::; s :::; Tk+l, k = 1, ... ,n - 1.

3

(10)

(11)

Lemma 2.3 [16) For each 0 :::; k :::; n - 1 the following best possible inequality holds

(12)

where = 1 {(n-:'-l)

(n-k)! 1 ,ifO:::;k:::;n-kr -l } , if n - kr :::; k :::; n - 1 '

and k* = max{kl - 1 - k, kr' [n-;-l)} .

3. Main Results

To prove the existence and uniqueness of solutions of the boundary value problem (1)-(3) we shall convert it to its equivalent integral equation representation. For this we define functions 9 and 'IjJ as follows· :

{ 0, t E [a,ad

9(t) = 1, otherwise.

If a < al and q ~ Sl - 1, then

'IjJ(t) = { ~(t), Pn-l(t),

t E [a,al) tE[al,ar)

where Fn_l(t) is the same as Pn-l(t) with Ai,l = <t>(i)(al), 0:::; i :::; Sl - 1; if a < al and q < Sl - 1, then

'IjJ(t) = { ~(t), Pn- l (t),

t E [a,al) t E [aI, ar )

where Fn- 1(t) is the same as Pn-l(t) with Ai,l = <t>(i)(al), 0:::; i :::; q;

4

BVPs FOR FDEs

if a = aI, then 1j;(t) = Pn-l(t), t E [abar]'

It is clear that 1j; E B, and for all t E [abar] with Wi,j(t) = at, 1j;(i)(Wi,j(t)) Pl~l(al + 0). Further, the boundary value problem (1)-(3) is equivalent to the

integral equation

x(t) = 1j;(t) + B(t) l~r g(t,s)f(s,x 0 w(s))ds. (13)

Theorem 3.1 Suppose that

(i) Qk > 0, 0 ~ i ~ q are given real numbers and let I< be the maximum of If(t, {x})1 on the compact set [at, ar] x Do, where

(ii) (ar - al) ~ (Q/I<Cn,s/(n-k) , 0 ~ k ~ q;

(iii) sUP"'9~ar 11j;(k)(t)1 ~ Qk, 0 ~ k ~ q, wherever 1j;(k)(t) exists.

Then, the boundary value problem (1)-(3), or equivalently (13), has a solution in B .

Proof: Let BI = C(r)[a,ar] nC(q)[a,al] nC(q)[at,a r ] ~ B. We define an operator T from the Banach space (Bt, 11 · 11), where Ilxll = maXo~k~q{SUP"'<t<ar Ix(k)(t)l, wherever x(k)(t) exists}, into Bl by - -

(Tx)(t) = 1j;(t) + B(t) l~r g(t,s)f(s,x 0 w(s))ds . (14)

The following properties of T may be easily established:

(1) (T x) ( t) satisfies the boundary condi tions (2), (3) ;

(2) (Tx)(n)(t) = f(t,x 0 w(t)) at all points of continuity of f(t,x 0 w(t)) with t E [aI, ar];

(3) T is a completely continuous operator;

(4) fixed points of T are the solutions of the boundary value problem (1)-(3) .

Consider the closed convex subset B2 of (Bt, 11 · 11) defined by B2 = {x E BI : Ix(k)(t)1 ~ 2Qk, 0 ~ k ~ q, wherever x(k)(t) exists}. We shall show that T maps B2 into itself. For this, let x E B2• If t E [a,al], then from (1) and (iii) it is clear that I(Tx)(k)(t)1 ~ Qk, 0 ~ k ~ q. If t E [at,ar ], then since Wi,j(t) E [a,ar], 1 ~ j ~

5


p(i), 0 ~ i ~ q it follows that x 0 w(t) E Do, and hence from (2), the inequalities (2.9), and (ii) and (iii), we find

I(Tx)(k)(t)1 ~ sup Itf!(k)(t)I + Cn.k(ar - adn- k J{ al5;t:5ar

~ 2Qk, 0 ~ k ~ q.

Thus, if t E [a,ar), we have I(Tx)(k)(t)1 ~ 2Qk, 0 ~ k ~ q, wherever (Tx)(k)(t) exists, and hence T B2 ~ B2. Using the Ascoli-Arzela theorem one may easily prove that T B2 is sequentially compact. Hence by the Schauder-Tikhonov fixed point theorem T has a fixed point in B2 • From (4) this fixed point is a solution of (1)-(3). •

Corollary 3 .2 Suppose that

q p(i)

If(t, (x))1 ~ L + L: L: Li.i IXi,il"(i,i) (15) i=O i=1

for all (t, (x)) E [abar) x RN, where 0 ~ a(i,j) < 1 and L, Li ,i are nonnegative constants. Then, the boundary value problem (1)-(3) has a solution in B.


q p(i)

If(t, (x))1 ~ L+L:L:Li,ilxi ,j l, (16) i=O i=1

for all (t, (x)) E [abar) x D1 , where

D1 = {(X): Ixk.1i, " ·' IXk ,p(k)1 ~ sup 1tf!(k)(t)1 + Cn,k(ar - at}n-k Ll + c, a,9:5ar - J.!

o ~ k ~ q, wherever tf!(k)(t) exists },

and

q p(i) C L: L: Li,i sup 1tf!(i)(t)l, wherever tf!(i)(t) exists,

i=O j=1 "9:5ar q p(i)

J.! = L: L: Li,jCn,i(ar - a1t-i < 1. i=Oj=1

Then, the boundary value problem (1)-(3) has a solution in B.

Proof: Consider the closed convex subset B3 of (Bb 11 · 11) defined by B3 = {x E B1 : Ix(k)(t)1 ~ sUPa:5 t :5ar 1tf!(k)(t)I+Cn,k(ar-al)n-k~, 0 ~ k ~ q, wherever tf!(k)(t) exists} . As in Theorem 3.1 it suffices to show that the operator T defined in (3.2) maps B3

6

BVPs FOR FDEs

into itself. For this, let x E B3. If t E [a,al], then I(Tx)(k)(t)1 = 1tf;(k)(t)l, 0 ~ k ~ q, and hence TB3 ~ B3 is obvious. If t E [al , a r ), then since Wi)t) E [a,ar ), 1 ~ j ~ p(i), 0 ~ i ~ q it is clear that x 0 w(t) E Dl , and hence from the inequalities (2.9) it follows that

I(Tx)<k)(t)1

~ sup 1tf;(k)(t)1 + Cn,k(ar - adn- k sup If(t,x 0 w(t))1 ell :5t5ar a15 t51lr

~ a~~far 1tf;(k)(t)1 + Cn,k(ar - adn-k al~~~ar [L + ~E Li,; IX(i)(Wi.;(t))I]

~ a~~far 1tf;(k)(t)1 + CnAar - adn-

k [L + ~ E Li,; t~~far Itf;(i)(t) I

C ( )n-i L + c} ] + n,i a r - al --1-p

sup 1tf;(k)(t)1 + Cn,k(ar - adn- k [(L + c) + L + c p] a9:S ar 1 - P

sup 1tf;(k)(t)1 + Cn,k(ar - alt- k L + c, 0 ~ k ~ q, wherever tf; (k)(t) exists. a9:Sar 1 - p

Thus, if t E [a}, ar ) then T B3 ~ B3. Hence, for all t E [a, a r ) we find that TB3 ~ B3· •


(i) Xl E B is a solution of the boundary value problem (1)-(3) different from tf; , so that

max { sup (a r - ad

k Ix~k)(t) _ tf;(k)(t) I '

O:Sk:Sq a9:Sar Cn,k

whenever x~k)(t) and tf; (k)(t) exist } f 0;

(ii) the function f is such that

q p( i)

If(t, (x))1 ~ :L:LLi,;lxi,; -tf;(i)(Wi,;(t))1 i=O ;=1

for all (t, (x)) E [a},ar) x D2 , where

D2 = {(x) : IXk,; - tf;(k)(Wk,;(t))1 ~ Cn,k(ar - altkllXl - tf;11,

1 ~ j ~ p(k) , 0 ~ k ~ q}.

(17)

7


Then, it is necessary that 1-1 ;::: 1.

Proof: Since Xl(t) is a solution of the boundary value problem (1)-(3), and Wi,j(t) E [a,ar] for all t E [aI,ar], it follows that

Hi)(Wi,j{t)) - ¢(i)(Wi,j{t))I ::; ,,~~far (ar ;:',;d Hi)(t) - ¢(i)(t)1 Cn,i(ar - alti

::; IIXI - ¢IICn,i(ar - ad-i, 1 ::; j ::; p(i), 0::; i ::; q

and hence (t, Xl 0 w(t)) E [a!, arl x D2 • Thus, on using the hypothesis (ii) in (13), the inequalities (12) provide

q p(i)

Hk\t) - ¢(k)(t)1 ::; Cn,k(ar - adn- k sup L L Li,j Hi)(Wi,j(t)) - ¢(i)(Wi,j(t))1 al $t:5ar i=O ;=1

q p(i)

::; Cn,k(ar - adn- k L L Li,pn,i(ar - altillxl - ¢II, i=Oj=1

which is the same as

Therefore, IIxl - ¢II ::; I-IIIXI - ¢Ii, which implies that 1-1 ;::: 1. •

Remark 3.1 If (17) is satisfied, then obviously ¢ E B is a solution of the boundary value problem (1)-(3). Hence, if D2 = RN and 1-1 < 1, then ¢(t) is the unique solution of (1)-(3) in B .

For a given x E B, we define a new function ¢l E B as follows

where Fn-l(t) is the same as Pn-I(t) with Ai,1 = x(i)(al +0), so::; i::; sl-l, Ai,j = x(i)(aj), Sj_1 ::; i ::; Sj - 1, 2::; j ::; r.

Definition 3.1 A function x E B is called an approximate solution of (1)-(3) if there exist nonnegative constants 8 and f such that whenever ¢(i)(t), X(i)(t) and ¢1i)(t), 0 ::; i ::; q are defined, sUPat9Sar Ix(n)(t) - f(t,x 0 w(t))1 ::; 8, and sUP"9Sar I¢(k)(t) -¢~k)(t)1 ::; fCn,k(ar - adn- k

, 0::; k ::; q. It is clear that the approximate solution x(t) can be written as

x(t) = ¢1(t)+O(t)[r g(t,s)[f(s,xow(s))+1J(s)]ds, (18) at

8

BVPs FOR FDEs

where 1J(t) = x(n)(t) - I(t, x 0 wet»~ . The function is said to be of Lipschitz class iffor all (t, (x)) , (t, (y)) E [aI, l£T] x

D3, D3 ~ RN, q p(i)

I/(t ,(x})-/(t,(y})1 :S LLLi,;lxi,; -y;,;I · ;=0 ;=1

In what follows we shall consider the following norm in the space BI :

Theorem 3.5 Suppose that the boundary value problem (1)-(3) has an approximate solution x E B, and

(i) I is 01 Lipschitz class on [at, aT] x D4 , where

(ii) 1£ < 1, and

Then,

(1) there exists a solution x*(t) of (1)-(3) in Sex, Yo);

(2) x*(t) is the unique solution of (1)-(3) in sex, v);

(3) the Picard sequence {xm(t)} defined by

!/J(t) + B(t) l~T g(t , s)/(s , xm 0 w(s»ds,

xo(t) x(t), m = 0,1", . ,

(19)

(20)

converges to x*(t) with IIx ' - xmll :S I£m Vo , IIx' -xmll :S 1£(1-1£ )-1 IIX m - xm-lll;

(4) lor any xo(t) = x(t), where x E Sex, Yo), the iterative process converges to x*(t).

Proof: We define an operator T : Sex, v) -+ B as in (14). If x E sex, v), i.e., IIx - xII :S v then wherever x(i)(t) and x(i)(t) exist ,

sup Ix(i)(t) - x(i)(t)1 :S v (en,; )" O:S i :S q orStSa~ en,O ar - al •

9


and hence

Thus, if t E [aI, arl, then x 0 w(t) E D4 • Now let x, y E S(x, v). If t E [a, ad, then from (3.2) we have (Tx)(k)(t) - (TyP)(t) = 0, 0::; k :S q which implies that

sup Cn,o(ar - al)k I(Tx)(k)(t) _ (Ty)(k)(t)1 ::; J.Lllx - yll. (21) a$t$a, Cn,k

If t E [aI, ar ), then from (14), Lipschitz condition, and the inequalities (12) we find

I(Tx)(k)(t) - (Ty)(k)(t)1 q l'(i)

::; Cn,k(ar-adn- k sup EELi,ilx(i)(Wi,i(S))_y(i)(Wi,i(S))1 a] :5t:5ar i=O j=1

and hence

sup Cn,o(ar - al)k I(Tx)(k)(t) - (Ty)(k)(t)1 :S f.tIIX - yll. (22) a,$t$ar Cn,k

Combining (21) and (22), we get IITx - TYII :S f.tIlX - yll · Next, for all t E [a, arl equations (14) and (18) give

(Tx)(t) - x(t) = 1jJ(t) - Vh(t) - 8(t) [r g(t, s).,,(s)ds

and hence from the Definition 3.1 and the inequalities (12), we obtain

which gives

Thus, from (19) it follows that

(1 - f.ttlllTx - xII ::; (1 - f.ttl(f + 8)Cn,o(ar - alt < v.

10

BVPs FOR FDEs

Hence, the conditions of the Contraction mapping theorem are satisfied and the conclusions (1 )-( 4) follow. •

Theorem 3.5 has an important feature of being constructive: moreover, a priori as well as posteriori bounds on the difference between the iterates and the solution are available. However, in practical evaluation of the sequence {xm(t)} generated by (20) only an approximate sequence, say {Ym(t)}, is computed. To find Ym.f.l(t), the function f is approximated by fm. Therefore, the computed sequence {Ym(t)} satisfies the recurrence relation

Ym+l(t) = 1/J(t) + O(t) [r g(t,s)fm(s,Ym 0 w(s))ds (23)

Yo(t) = xo(t) = x (t), m = 0,1, · · · .

For y~)(t), Q ~ t ~ an 0 ~ i ~ q (wherever exist), obtained from (23) we shall assume that fm satisfies

sup Ifm(t, Ym 0 w(t)) - f(t, Ym 0 w(t))1 ~ 6. sup If(t, Ym 0 w(t))I , (24) al:5't:5a,. a1 :$;t5a,.

where 6. is a nonnegative constant .

Theorem 3.6 Suppose. that the boundary value problem (1)-(3) has an approximate solution x E B and the inequality (24) is satisfied. Further, we assume

(i) condition (i) of Theorem 3.5;

(ii) J1.t = (1 + 6.)J.L < 1 and 111 = (1 - J.Lt}-l(f + 5 + 6.F)Cn •o(a r - alt ~ II, where F = sUPa,9$ar If(t,x 0 w(t))l .

Then,

(1) all the conclusion (1)-(4) of Theorem 3.5 hold;

(2) the. sequence {Ym(t)} constructed from (23) remains in 5(x, lit);

(3) the sequence {Ym(t)} converges to the solution x·(t) of (1)-(3) if and only if liffim_oo am = 0, where

am = IIYm+l(t)-1/J(t)-O(t)[rg(t,s)f(s,YmOw(s))dsll ;

(4) a bound on the error is given by

IIx· -Ym+lll ~ (1-J.L)-1 [J.LIIYm+l - Ymll + 6.Cn •o(a r - at}nal~~~ar If(t, Ym 0 w(t))l] .

Proof: The proof uses the inequalities (12) repeatedly. •

11


References

1. R. P. Agarwal, On the right focal point boundary value problems for linear ordinary differential equations, Atti Accad. Naz. Lincei CI. Sci. Fis. Mat. Natur. 79 (1985), 172-177.

2. R. P. Agarwal and R. A. Usmani, Iterative methods for solving right focal point boundary value problems, J. Comput. Appl. Math . 14 (1986), 371-390.

3. R. P. Agarwal and R. A. Usmani, On the right focal point boundary value problems for integro-differential equations, J. Math. Anal. Appl. 126 (1987), 51-69.

4. R. P. Agarwal, Existence-uniqueness and iterative methods for right focal point boundary value problems for differential equations with deviating arguments, Annales Polonici Mathematici 52 (1991), 211-230.

5. R. P. Agarwal, Q. Sheng and P. J . Y. Wong, Abel-Gontscharoff boundary value problems, Mathl. Comput. Modelling 17 (1993), 37-55.

6. R. P. Agarwal and P. J. Y. Wong, Error Inequalities in Polynomial Interpolation and Their Applications, Kluwer Acad. Publ. Dordrecht, 1993.

7. J . Ehme and D. Hankerson, Existence of solutions for right focal boundary value problems, Nonlinear Analysis 18 (1992), 191-197.

8. J. Henderson, Uniqueness of solutions of right focal point boundary value problems for ordinary differential equations, J. Differential Equations 41 (1981),218-227.

9. J. Henderson, Existence of solutions of right focal point boundary value problems for ordinary differential equations, Nonlinear Analysis 5 (1981), 989-1002.

10. J. Henderson, Right focal point boundary value problems for ordinary differential equations and variational equations, J. Math. Anal. Appl. 98 (1984), 363-377.

11. J. Henderson, Existence and uniqueness of solutions of right focal point boundary value problems for third and fourth order equations, Rocky Mountain J. Math. 14 (1984), 487-497.

12. A. C. Perterson, Existence-uniqueness for focal-point boundary value problems, SIAM J. Math. Anal. 12 (1981), 173-185.

13. S. Umamaheswaram and M. Venkata Rama, Green's functions for k-point focal boundary value problems, J. Math. Anal. Appl. 148 (1990), 350-359.

14. S. Umamaheswaram and M. Venkata Rama, Existence theorems for focal boundary value problems, Differential and Integral Equations 4 (1991), 883-889.

15. S. Umamaheswaram and M. Venkata Rama, Multipoint focal boundary value problems on infinite intervals, J. Appl. Math. Stoch. Anal. 5 (1992), 283-290.

16. P. J. Y. Wong and R. P. Agarwal, Abel-Gontscharoff interpolation error bounds for derivatives, Proc. Royal Soc. Edinburgh 119A (1991), 367-372.

THE IDEAS AND METHODS OF THE PERM SEMINAR ON BOUNDARY VALUE PROBLEMS

1. Introduction

N.V.AZBELEV Perm State Technical University 614600, Perm, GSP-45, Russia

13

A large Group of Mathematicians is united around the so called "Perm Seminar on Functional Differential Equations". During 20 years this Group has been working on various problems in the theory of functional differential equations. New ideas, innovation in approach to old problems and effectiveness of some methods characterize the general picture of the theory of generalization of ordinary differential equations suggested by the Seminar. The main results of this theory are systematized in the book [1) written by the leaders of the Seminar.

2. Equations with Deviated Argument

The long developing process of the formation of our cognition (or perception), of equations with deviated argument [2,3] was accomplished recently thanks to the research work of the Perm Seminar. The initial point of this work was a natural generalization of the notion of solutions of equations with deviated argument. The essence of this generalization, its inevitability, and advantage, we shall explain using the example of the simplest scalar equation

x(t) + p(t)x[h(t)] = v(t),

x(~) = cp(e),

tE[a,b),

if e (j. [a, b) . (1)

The functions p, h, v are defined on the segment [a, b] . The so called "initial function" cp: (-oo,oo)\(a,b) -+ (-00,00) is necessary to define the superposition x[h(t)), if the function h: [a,b]-+ (-00,00) has some values outside of [a,b] .

If p, v and t - h(t) are constants and the equation is considered on the whole axis (-00,00), there is no need of the initial function and the notion of the solution is obvious. But if p, h, v are defined on a final segment [a, b), we must introduce the function cp.

14

BVPs FOR FDEs

Some authors [2,3] define the solution x : [a , b] -> (-00,00) as a continuous extension on the segment [a,b] of the given initial function (-00,00) that will satisfy (1) on [a, b]. One can see that boundary value conditions x(a) = <p(a), x(b) = <p(b) are not necessary from the point of view of definiteness of all the operations in (1).

The demand to fulfill all such conditions gave rise to much trouble in numerous endeavours to present any kind of accessible theory of equations with deviated argument . For instance, under the additional assumption x(a) = <p(a), x(b) = <p(b) we cannot consider (1) as a linear equation.

A natural theory of equations Cx = f with a linear opeartor C defined on a Banach space one can fined in [4] . The equation (1) might be rewritten in the form Cx = f with a linear operator C defined on a linear space of absolutely continuous functions x : [a, b] -> (-00,00). For this purpose we must introduce to the description of the equation, a special operator Sh , defined on linear spaces of functions by

(ShZ)(t) = {z[h(t)], ~f h(t) E [a , b], 0, If h(t) ¢ [a, b],

and a function h() _ {O, if h(t) E [a,b],

<p t - <p[h(t)], if h(t) ¢ [a, bJ .

With such notation the equation (1) assumes the form

(Cx)(t) ~ x(t) + P(t)(ShX)(t) = f(t),

where f(t) = v(t) - p(t)<ph(t). Thus the initial function Rn, II z llL = J: IIZ(S)IIRn ds, D be a Banach space of absolutely continuous functions x : [a, b)-> Rn, IIxlin = IIxIIL+llx(a)IIRn. The equation Cx = f with a linear operator C : D -+ L is said to be a linear functonal differential equation.

Let I = W, ... ,1m) : D -+ Rm be a vector-valued functional with linear independent components Ii : D -> RI. The system

Cx = f, Ix = a (2)

is said to be a linear boundary value problem [1) . Let m = n and Co : D -+ L be such a linear operator that the "model" boundary

value problem

Cox = f, Ix = a (20)

has a unique solution xED for every f ELand a ERn. Then there exist bounded operators WI : L -> D and U : Rn -+ D such that x = W,J +Ua is the solution of the problem (20). For every bounded I : D -+ Rn there exists such an ordinary differential

15

THE PERM SEMINAR IN BVPs

equation (.cox)(t) ~(:i:(t)+P(t)x(t) = f(t) that the problem (20) is solvable for every fE L.

Theorem 1. Let.c be a bounded linear operator .c: D --+ L, m = n. The problem (2) has a unique solution xED for every f E L, a ERn, if and only if there exists a bounded inverse operator [.cW,]-l :L --+ L.

The proof is founded on the fact that there exists an one-to-one mapping z = .cox, a = lx, x = W,z + Ua between the set of solutions z E L of the equation .cWz = f - .cUa and the set of solutions xED of the problem (2).

Consider as an example the scalar equation

:i:(t) + p(t)x[h(t)] = v(t),

x(o = <p(O, t E [a,b], if ~ ¢ [a,b]

with boundary value condition Ix = a under the assumption that p, h E L, v _ W h =

f E L, 1(1) # 0. As a "model" equation we can choose (.cox)(t) ~ :i:(t) = f(t). It is easy to check

that in this case (W,z)(t) = f:W,(t,s)z(s)ds, where W,(t,s) = C(t,s) - '[~f~')')l,

{I if a < s < t < b. .

C(t, s) = 0: if a ;; t <" s ;; b: As It was said above the boundary value problem

(.cx)(t) ~ :i:(t) + p(t)(ShX)(t) = f(t), Ix = a

has a unique solution xED if and only if the integral equation z = K z + f, where (Kz)(t) = f: K(t,s)z(s) ds,

K(t, s) = -p(t)u(t)W(h(t), s), (t) = {I, if h(t) E [0,1] u 0, if h(t) ¢ [0,1] ,

has a unique solution z E L for every f E L. The equation z = K z + f is uniquely solvable, if IIKIIL-L < 1.

3. Abstract Functional Differential Equation

The space D of absolutely continuous functions x : [a, b] --+ Rn is isomorphic to the direct product Lx Rn The isomorphism may be defined by

t

x(t) = jz(s)ds+{3, xED, {z,{3}ELxRn

a

This fact of the existence of the isomorphism J : Lx Rn --+ D plays a great role in the general theory of functional differential equations [1]. By replacing the space L with an arbitrary Banach space B, most of the basic statements of the theory carryover.

15


equation (.cox)(t) ~(:i:(t)+P(t)x(t) = f(t) that the problem (20) is solvable for every fE L.

Theorem 1. Let.c be a bounded linear operator .c: D --+ L, m = n. The problem (2) has a unique solution xED for every f E L, a ERn, if and only if there exists a bounded inverse operator [.cW,]-l :L --+ L.

The proof is founded on the fact that there exists an one-to-one mapping z = .cox, a = lx, x = W,z + Ua between the set of solutions z E L of the equation .cWz = f - .cUa and the set of solutions xED of the problem (2).

Consider as an example the scalar equation

:i:(t) + p(t)x[h(t)] = v(t),

x(o = <p(O, t E [a,b], if ~ ¢ [a,b]

with boundary value condition Ix = a under the assumption that p, h E L, v _ W h =

f E L, 1(1) # 0. As a "model" equation we can choose (.cox)(t) ~ :i:(t) = f(t). It is easy to check

that in this case (W,z)(t) = f:W,(t,s)z(s)ds, where W,(t,s) = C(t,s) - '[~f~')')l,

{I if a < s < t < b. .

C(t, s) = 0: if a ;; t <" s ;; b: As It was said above the boundary value problem

(.cx)(t) ~ :i:(t) + p(t)(ShX)(t) = f(t), Ix = a

has a unique solution xED if and only if the integral equation z = K z + f, where (Kz)(t) = f: K(t,s)z(s) ds,

K(t, s) = -p(t)u(t)W(h(t), s), (t) = {I, if h(t) E [0,1] u 0, if h(t) ¢ [0,1] ,

has a unique solution z E L for every f E L. The equation z = K z + f is uniquely solvable, if IIKIIL-L < 1.

3. Abstract Functional Differential Equation

The space D of absolutely continuous functions x : [a, b] --+ Rn is isomorphic to the direct product Lx Rn The isomorphism may be defined by

t

x(t) = jz(s)ds+{3, xED, {z,{3}ELxRn

a

This fact of the existence of the isomorphism J : Lx Rn --+ D plays a great role in the general theory of functional differential equations [1]. By replacing the space L with an arbitrary Banach space B, most of the basic statements of the theory carryover.

16

BVPs FOR FDEs

Thus arises the theory of the so called "Abstract functional differential equation" Cx = f with a linear operator C : D -+ B, where D is a linear space isomorphic to the direct product B x Rn (D ~ B x Rn) .

Let us denote the isomorphisms J : B x Rn -+ D and J-l : D -+ B x R n as

follows J={A ,Y}, (x=A z +Y/3, z EB, /3ERn)

J- I = [15, r], (z = hx, /3 = rx, x ED). (3)

If D is a space of absolutely continuous functions x : [a , b] -+ Rn, then (A z )(t) = J: z( s) ds, Ya = Ea (here and everywhere below E denotes the identity n x n-matrix). One can define the isomorphism for this space by

Az = W/ z , Y/3 = U/3; fix = Cox, rx = lx,

where x = W/ z + U/3 is the above considered representation of the solution of the "model problem" Co x = z , Ix = /3.

Let I = [p, ... ,1m] : D -+ Rm be a bounded, linear vector-valued functional with linearly independent components Ii : D -+ R I. The system of equations

Cx = f, Ix = a (4)

is said to be a linear boundary value problem. This problem will be written also as one equation [C,/]x = {f,a} with linear operator [C,/] : D -+ B x Rn.

The element xED has the representation x = Ahx + Yrx. Therefore we get the following decompositions

Cx = Qfix + Arx, Ix = ~hx + IIIrx,

where Q = CA : B -+ B, A = CY : Rn -+ B, ~ = IA : B -+ R m, III = CY: Rn-+ Rm are linear bounded operators. Q: B -+ B is said to be the principal part of C:D-+ B.

During the last 10 years, the effort of the Perm Seminar was devoted to various aspects of the problem (4) . We shall consider this problem under the assumption that the principal part Q : B -+ B of the operator C : D -+ B is a Fredholm operator (The operator Q : B -+ B is said to be a Fredholm one, if it may be represented as Q = P + V where P : B -+ B is an invertible operator, p-I ; B -+ B is a bounded operator, V: B -+ B is a completely continuous operator).

Theorem 2. The dimension of the space of solutions of the equation Cx = 0 is finite, but not less than n, and is equal to n, if and only if the equation Cx = f is solvable for any fEB .

The vector X = (XI, . . . , xv) , the components of which form the base of the kernel of the operator C, will be said to be the fundamental vector of the homogenous equation Cx = 0 (the components XI, • .. ,xv compose a fundamental system of solutions of homogeneous equation Cx = 0).

17


Theorem 3. The equality m = n is a necessary condition for the unique solvability of the problem (4) for every {f,o} EB x Rn .

Example. Let D be the space of absolutely continuous functions x : [0,1] -+ R 1,

(Az)(t) = fri z(s) ds, Y(3 = (3. The fundamental vector X(t) = {t , I} of the scalar

(n = 1) equation (.cx)(t) ~ x(t) + x(O) - x(I) = 0 has the dimension /I = 2> n = l. The principal part (Qz)(t) = z(t)- f~ z(s) ds for this equation is a Fredholm operator.

Theorem 4. The following assertions are equivalent. a) The equation .cx = f has solutions for every fEB . b) The dimension of the fundamental vector of the equation .cx = 0 is equal to n . c) There exists such a vector-valued functional 1 :D --+ Rn that the problem (4)

has a unique solution for every {f, o} EB -+ R n .

Let the problem (4) have a unique solution for every f, o . Then the operator [.c,l] : D -+ B x Rn has its inverse [.c, 1]-1 = {G, X} and the solution ofthis problem has the representation x = G f + X o . The operator G : B -+ D is called the Green 's operator of the problem, and the vector X = (XI, .. . ,xn ) is the fundamental vector of the equation .cx = 0 satisfying the condition IX = E .

If B is a space of summable functions f : [a, b] -+ R n, the Green's operator G is an integral one ( (G f)(t) = f: G(t, s )f(s) ds), and the kernel G(t, s) is said to be the Green's function of the boundary value problem [1] .

Theorem 5. Let W, be the Green's operator of the boundary value problem .cox = f, Ix = 0 for any "model" equation .cox = f. The problem (4) has a unique solution for every {f,o} EB x Rn, if and only if there exists the bounded inverse operator [.cW,J-IB -+ B.

The Green's operator of the problem (4) in this case has the representation G = W,[.cW,J-l.

Let U = (UI, .•. , un), Ui E D be such a vector that lU = E and det rU f- 0 (such a vector exists due to Lemma 3.2.1 [1]). Define the operator U : Rn -+ D by the vector (Ul, ... , Un) and the operator W, : B -+ D by W, = A - Uif! .

Theorem 6. Define .co :D -+ B by .cox = ox - oU(rUt1rx . Then W, is the Green's operator of the boundary value problem .cox = f , Ix = 0, and U is the fundamental vector of the equation .cox = o. Theorem 7. The problem (4) has a unique solution for every {f,o} EB x Rn, if and only if det IX =1= 0, where X is a fundamental vector of the homogeneous equation

.cx = O.

4. Examples

Let W 2 = D ~ L X R2 be a Banach space of functions x : [0,1] -+ Rl with summable derivative X, and the isomorphism J : L X R2 being defined by (Az)(t) = fri (t - s)z(s) ds, Y(3 = (31 t + (320 - t) .

18

BVPs FOR FDEs

Consider the two point boundary value problem

(.cx)(t) ~ x(t) + p(t)(ShX)(t) = f, (5)

The solution of the "model" problem x(t) = z(t), x(O) = 0, x(l) = 0 has the repre-sentation

where

1

x(t) = (W/ z)(t) = J W/(t , s)z(s) ds , o

W ( ) _ { s(t - 1), if 0::; s ::; t ::; 1, It,s - t(s-l), ifO::;t<s::;l. (6)

Then .cW/z = z + Oz, where (Oz )(t) = p(t)SdJ~ W/(t,s)z(s) ds}. Theorem 5 guarantees solvability of (4) by the inequality 1I01lL-L < l.

For every t E [0,1] the function W/(t,s) has its minimum at s = t . Therefore IW/(t,s)1 ::; t(l - t) and the inequality

1

J Ip(t)lu(t)h(t)(l - h(t)) dt < 1 (7) o

guarantees the unique solvability of (5) for every {J, a} E Lx R2. The last inequality is valid, if J~ Ip(t)lu(t) dt ::; 4. This inequality is well known in the case h(t) = t.

The inequality 1I01lL-L < 1 guarantees also the convergence of the successive ~pproximations for the equation z + Oz = f. Under the assumption p(t) ~ 0 we have z(t) = f(t) + (Of)(t) + (02 f)(t) + ... ~ f(t), if f(t) ~ O. Therefore x(t) = (W/ z)(t) ::; 0, if f(t) ~ o. Thus the Green's operator G for (5) has the property (Gf)(t) ::; 0, if f(t) ~ 0 under the assumption that p(t) ~ 0 and (7) holds.

Next consider the equation

(.cx)(t) ~ t(l - t)x(t) + p(t)(ShX)(t) = f(t). (8)

Such equations are said to be "singular". The singularity is due to the choice of the space W2 Let us choose another space W5 = D ~ L X R2 of continuous functions x : [0,1] -+ R1. Define the isomorphism J = {A, Y} : Lx R2 -+ W5 by

(Az)(t) = J~ A(t,s)z(s) ds , (Y.8)(t) = (1-t).81 +t.82, A(t,s) = ~~~':l, where W/(t,s) is defined by (6). Every element x E W5 have the representation

1

x(t) = J A(t, s)z(s) ds + .81(1 - t) + .82t . o

One can see that the last equality represents the solutions of a simple boundary value problem t(l - t)x(t) = z(t), x(O) = .81, x(l) = .82, which is the "elementary medium

19


of singularity". The operator L : W5 -+ L has the principal part (Qz)(t) = z(t) +

p(t)l K(t,s)z(s) ds, where K(t,s) = {~,(h(t),s), ~! ~m ~ ~~:~l: It is a canonical

Fredholm operator Q : L --+ L. Thus all the theorems of the previous section are applicable to the boundary value problem LX = J, Ix = 0 with bounded linear vector valued functional I : W5 -+ R2. In particular, the two point problem

t(l- t)x(t) + p(t)(S"x)(t) = J(t),

has a unique solution X E W5 for every {f, o} E L X R2, if

t

vraisup J Ip(t)K(t,s)1 dt < 1. 'E[O,t) 0

(9)

This inequality and the additional assumption p(t) < 0 guarantee for the Green's operator G : L -+ W5 the property (Gf)(t) :::; 0, if J(t) ~ O.

It is easy to see that the singular problem (9) is equivalent to a purely regular equation

t t

x(t)=- J A(t,s)p(s)(S"x)(s)ds+(1-t)ot+to2 + J A(t,s)v(s)ds o 0

in the space of continuous functions. The singularity of the problem consists of the fact that the derivative :i; of the solution is not a continuous function.

Some more complicated singular equations of this kind and boundary value problems for such equations were studied by I.Kiguradze [5], and E.Bravi [6] .

Let tt, ... , tm E (a, b) be a fixed ordered system of points. Denote by DS(m) the space offunctions y : [a, b]-+ Rl with summable derivative on [a, b] and representable in the form

t m

y(t) = J y(s) ds + y(a) + 2:X[I;,b)(t)~y(ti) ' a ,=1

Here ~y(ti) = y(ti) - y(ti - 0), and X[I;,b)(t) is the characteristic function of the segment [ti, b] . Elements of the space DS(m) are absolutely continuous functions on each of the intervals [a, tt), (tt, t2 ), ••. , (tm, b], and continuous from the right at the points tt, ... , tm . Thus , between the spaces DS(m) = D and Lx Rm there exists the isomorphism established by (3) where

t

(Az)(t) = J z(s) ds, Y = (1, X[I"b» "" X[tm,b)) , a

by = y, ry = ~y = col{y(a), ~y(td, . . . , ~y(tm)}'

20

BVPs FOR FDEs

Let Do be a space of absolutely continuous functions x : [a, bJ -+ R 1 Then y E DS( m) has the representation yet) = x(t) + L~l X[t"b](t)~(t;). Let .c : Do -+ L be a linear operator with the principal part Q : L -+ L. The linear extension of .c : Do -+ L on the space DS( m) is defined by

m

(.cy)(t) = (.cx)(t) + L A;(t)~y(td = ;=1

m

= (Qy)(t) + Ao(t)y(a) + L A;(t)~y(ti) ;=1

where Ao(t) = (.c(l))(t), A;(t) = .c(X[t"b])' i = 1, . . . , m . Thus, the principal part of .c : Do -+ L and of.c : DS(m) -+ L coincide.

For the equation .cy = J with Fredholm principal part of the operator .c : DS(m) -+ L, all the assertions of the theory of abstract functional differential equations are valid. A more detailed study of boundary value problems of such "Impulse systems" can be found in [lJ.

Consider as an example the problem (.cx)(t) ~ x(t) - x(t) = J(t), x(a) = at, x(b) = a2 ' As far as n = 1 < m = 2 the solution (if it exists) cannot be unique (Theorem 3) . But the extended problem

(.cy)(t) ~ yet) - yet) + X[t1,b](t)~y(td = J(t), y(a)=al, y(b)=a2

has a unique solution 1,' E DS(l) for every J E L, aI, a2. Indeed, the general solution _ t

of the extended equation .cy = J has the form y( t) = J et-. J( s) ds + ClYl (t) + C2Y2( t), a

where Yl(t) = et, Y2(t) = X[t1,b](t) . Thereafter

5. Abstract Functional Differential Equations and Calculus of Variations

Certain ideas of the theory of abstract functional differential equations have led the Perm Seminar to a new and very effective approach of the problem of minimization of functionals.

Let D ~ L2 X Rn be a space of functions x : [a, bJ -+ Rl, where L2 is a Banach space of quadratically integrable functions z : [a, bJ -+ Rl, J = {A, Y} : L2 xRn -+ D, J- l = [c5,rJ : D -+ L2 X Rn. Denote by To : D -+ L2, Tij : D -+ L2, I : D -+ Rft linear bounded operators. Consider the functional

21


with additional conditions Ix = a. Since F{x) = F{Az + Ya) = F1{z), the substitution x = Az + Ya reduces the

problem of minimization of the functional F(x) on D", = {x ED: Ix = a} to the problem of minimization of F1 ( z) on L 2 •

Define Q'j = T'jA, Q* - the operator adjoint to Q : L2 -+ L2, H = 2:;;'1 Qi1Q,2 +Qi2Q,t. Qo = Toll., j = -Q*(I) - 2:;;'1 Qi1T,2Y + Qi2Ti1Y, y = Ya, .c = H6, < c.p,1jJ >= J: c.p(s)ljJ(s) ds . Using these notations we obtain

1 F1(z) = 2 < Hz,z > + < j,z > +const.

Theorem 8. The functional F reaches its minimum Xo ED"" if and only if: a) Xo is a solution of the boundary value problem

.cx = f, Ix = a. (10)

b) Operator H :L2 -+ L2 is positively defined « Hz, z >~ 0 for every z E L2 and < Hz,z >= 0 only for z = 0).

Corollary. Let H = I - K , where I is an identity operator. The inequality

is sufficient for existence of a minimum Xo E D" of F. If K is an isotonic operator ((KZ1)(t) ~ (KZ2){t) for Zl{t) ~ Z2{t)) then (11) is also necessary for existence of a minimum of F.

One of the main advantages of the suggested approach to the problem of minimization of functionals consists in the wide possibilities of choosing the proper spaces D ~ L2 X Rn for "singular" functionals which do not have extremum in standard spaces. In addition Theorem 8 does not demand the tedious procedure [7] of providing the convergence of minimizing sequences.

We shall exemplify the above by the singular functional

1

F{x) = J {[s(1- .'l)x{s)]2 - [P{.'l){ShX)(SW} ds, P E L2 o

with the additional conditions x{O) = aI, x{l) = a2· It is natural to choose a space D ~ L2 X R2 such that J~ [s{1 - s)x(s)] ds <

00 if xED. Let us try W~ ~ L2 X R2 which was used in the previous section. Thus, the isomorphism J = {A, Y} : L2 x R2 -+ W~, J- 1 = [6, r] : W~ -+ L2 X

R2 is defined by (Az)(t) = J~ A(t,s)z{s) ds, A{t,s) = ~~(~':l, W/{t,s) is defined by (6), (Y,8)(t) = (1 - t),81 + t,82, 6x = t(1 - t)x(t), rx = {x(O),x(I}}. For the

1

functional considered we get Hz = 2(z - K z), (K z)(t) = J K(t, s)z(.'l) ds, K(t, s) = o

22

BVPs FOR FDEs

I { 1 if h(t) E [0 1] , . [p2(T)A(h(T),t)A(h(T),s)uh(T) dT, Uh(t) = 0; if h ¢ [0,1]'. In thIs case

(KzI)(t) 2: (Kz2)(t), if ZI(t) 2: Z2(t). Thus, the inequality IIKIIL2-L2 < 1 is necessary and sufficient for existence of the minimum of the functional on Do. = {x E W~ : x(O) = at, x(l) = a2} .

The central idea of the Seminar is to find proper space for each new problem. Some examples of constructing "unusual" spaces can be found in [8].

References

1. N.Azbelev, V.Maksimov, and L.Rakhmatullina, Introduction to the Theory of Functional Diffel-ential Equations, "Nauka" , Moscow, 1991, 280 pp. (in Russian).

2. A.Myshkis, Linear Differential Equations with Retarded Argument, "Nauka", Moscow, 1972, 352 pp (in Russian).

3. J.Hale, Theory of Functional Differential Equations, Springer-Verlag, New-YorkHeidelberg-Berlin, 1977, 400 pp.

4. S.Krein, Linear Equations in a Banach Space, "Nauka" , Moscow, 1971, 104 pp. (in Russian).

5. I.Kuguradze and B.Shekhter, Singular boundary value problems for ordinary differential equations of second order, VINITI30 (1987), 105-201 (in Russian).

6. E.Bravi, On regularization of singular functional differential equations, Differential Equations 30, No. 1 (1994), 26-34 (in Russian).

7. L.Kudryavtzsev, On existence and uniqueness of solutions of variational problems, Soviet Math. Dokl. 298, No. 5 (1988), 1055-1060 (in Russian).

8. N.Azbelevand G.Islamov, To the theory of abstract functional differential equations, Functional'no differentsial'nye uravneniya - Perm. (1983), 15-27 (in Russian) .

ON EXTENSION OF THE VALLEE-POUSSIN THEOREM TO EQUATIONS WITH AFTEREFFECT

1. Introduction

N.V.AZBELEV AND L.F.RAKHMATULLINA Perm State Technical University 614600, Perm, GSP-45, Russia

The theory of boundary value problems for the ordinary differential equation

x(t) + q(t)x(t) + p(t)x(t) = f(t), t E [a, b),

23

often uses the Sturm Theorem on the separation of zeros of nontrivial solutions of the homogeneous equation. This Theorem asserts [1] that between two adjacent zeros of one nontrivial solution of the homogeneous equation there exists one and only one zero of any independent nontrivial solution. From the Sturm Theorem follows the criterion of non-oscillation which is as follows:

Any nontrivial solution of a linear homogeneous equation of second order has at most one zero on [a, b), if and only if there exists on (a, b] a positive solution of this equation.

The Vallee-Poussin Theorem [2,3] asserts that there exists a positive solution on (a, b], if and only if a function v with absolutely continuous derivative v satisfies the inequalities

v(t) ~ 0, v(t) + q(t)v(t) + p(t)v(t) ~f cp(t) :S 0, t E [a, b),

b

v(a) + v(b) - J cp(s) ds > O . •

The property of non-oscillation of a fundamental system is a basis of some important properties of solutions [3,4]. In this paper we establish some connections between the non-oscillation of a fundamental system and properties of Green's functions of boundary value problems for linear functional differential equations of the second order.

Our investigation is founded on the general theory of functional differential equations thoroughly treated in [5].

24

BVPs FOR FDEs

2. The Vallee-Poussin Theorem

Let L be a Banach space of summable functions z : [a, b] -+ (-00,00) with the norm IIzllL = f: Iz(s)1 ds, W 2 be a Banach space of functions x : [a, b] -+ (-00,00) with absolutely continuous derivative x and the norm

IIxllw2 = IIxllL + Ix(a)1 + Ix(a)l·

Any equation Cx = / with a linear operator C : W 2 -+ L is said to be a linear functional differential equation of the second order [5] . A solution of Cx = / is an element x E W2 such that the equality (Cx)(t) = /(t) holds almost everywhere on [a,b].

We shall assume below that C : W2 -+ L is a Volterra operator, or in other words, the equation Cx = / is an equation with aftereffect . (A linear operator A : X -+ Y where X and Yare linear spaces of functions defined on [a, b] is said to be a Volterra operator [5,6], if for any c E (a,b) (Ax)(t) = 0 on [a ,c], if x(t) = 0 on [a,c]) .

We shall naturally understand equalities and inequalities between summable functions as being satisfied almost everywhere without additional stipulation.

Due to the representation

t

x(t) = J (t - s)x(s) ds + x(a) + (t - a)x(a) a

of x E W2, we have a decomposition of C : W 2 -+ L in the form

(Cx)(t) = (Qx)(t) + A(t)x(a) + B(t)x(a).

The "Principal part" Q : L -+ L of this decomposition is the product CA, where A : L -+ W2 is defined by

t

(Az)(t) = J (t - s)z(s) ds. a

We shall assume that the principal part Q : L -+ L of C : W2 -+ L has a bounded inverse Q-l : L -+ Land Q-l is a Volterra operator. As it is well known [5] under the above assumption the Cauchy problem

Cx = /, x(a) = ai, x(a) = a2 (1)

has a unique solution for any / E L, al, a2, and this solution has the representation

t

x(t) = J C(t, s)/(s) ds + alxl(t) + a2x2(t), a

25

EXTENSION OF THE VALLEE-POUSSIN THEOREM

where Xl, X2 are solutions of the homogeneous equation .ex = 0 such that Xl (a) = 1, xl(a) = 0, x2(a) = 0, x2(a) = 1. The operator C : L -+ W 2 defined by

t

(Cf)(t) = J C(t,s)J(s) ds a

is said to be the Cauchy operator of .ex = J and the kernel C (t, s) is called the Cauchy function. We assume that the Cauchy function C(t , s) is defined on the whole square [a,b] X [a,b] and C(t,s) = 0, if a ~ t < s ~ b.

We remark that the invertibility of Q is necessary and sufficient for unique solvability of the problem (1), and the fact that Q-l is a Volterra operator is necessary and sufficient for the representation of the solution of (1) in the form X = C J +alxl +a2x2.

The system of equations

(2)

with linear bounded functionals h, 12 on the space W 2 is said to be a linear boundary value problem. This problem is uniquely solvable for any J E L, aI, a2, if and only if the "Determinant of the problem"

In the case ~ =I 0 the solution X of the problem (2) has the representation

b

x(t) = J G(t, s)J(s) ds + u(t), a

where u is the solution of half homogeneous problem .ex = 0, hx = a1, 12x = a2. The kernel G(t,s) of the Green's operator

b

(GJ)(t) = J G(t,s)J(s) ds a

of the problem (2) is said to be the Green function of the problem. The following equality

(3)

holds, where the measurable and essentially bounded functions 01 and 02 are defined by Cauchy function C(t,s) and the functionals 11, 12 [5] .

For the two-point boundary value problem

.eX = J, x(a) = a1, x(b) = a2 (4)

26

BVPs FOR FDEs

we have

~ = I ::~~~ ::~~? I = I xl~b) x2~b) 1= x2(b) .

The determinant of the Cauchy problem (1) is equal to the value w(a) = 1 of the Wronskian

w(t) = I ~l(t) ~2(t) I Xl(t) X2(t)

of the fundamental system. In contrast to ordinary differential equations the Wronskian can obtain zeros at t > a. In case w(t) =f. 0 occurs, t E [a, b], the homogeneous equation Cx = 0 is equivalent to an ordinary differential equation. In this case the Sturm Theorem is valid for Cx = o.

The fundamental system of solutions of the equation Cx = 0 is said to be nonoscillatory, if any nontrivial solution of this equation has at most one zero on [a, b] considering a multiple zero twice.

We shall say that the Green's function G(t,s) of the problem (4) is strictly negative in the square (a,b) x (a,b) if at every t E [a,b] the function g(s) = G(t,s) assumes negative values almost everywhere on [a , b]. It is relevant to remark that the inequality G(t,s) < 0, (t,s) E (a,b) x (a,b) is necessary and sufficient for validity of the "Principle of maximum": the solution of the problem (4) for al = a2 = 0, J(t) ~ 0 reaches its maximum value only on the boundary of the domain of definition of the solution (of the segment [a, b]).

For the equation Cx = J, which we consider below, we impose the assumption that C = Co - T and also that the following conditions are fulfilled.

1. Co : W 2 -+ L is a linear bounded operator such that the Cauchy problem

Cox = J, x(a) = aI, :i:(a) = a2

has unique solution for any f E L, aI, a2 and this solution x has the representation

t

x(t) = J Co(t,s)J(s) ds + al zl(t) + a2z2(t), •

where z}, Z2 are such solutions of Cox = 0 that ZI (a) = 1, ZI (a) = 0, Z2( a) = 0, z2(a) = 1.

2. For each JL E (a, b] the boundary value problem

(Cox)(t) = J(t), t E [a,b], x(a) = 0, x(JL) = 0 (5)

has unique solution for any J ELand the Green's function GI;(t, s) of this problem is strictly negative in the Dquare (a, JL) x (a, JL). If JL = b we denote the Green's function Go(t, s) .

3. For each v E [a, b] the boundary value problem

(Cox)(t) = J(t), t E [a,b], x(v) = 0, :i:(v) = 0 (6)

27


has unique solution for any f ELand this solution has no negative values if f(t) ~ O. 4. The linear bounded operator T : W 2 -+ L admits a continuous extension

T : C -+ L on the space C of continuous functions x : [a, b] -+ (-00,00) with the norm IIxlic = max Ix(t)1 and has the property of being antitonic: (TX1)(t) :::; (Tx2)(t),

tela,b)

if Xl(t) ~ X2(t), t E [a, b].

Remark 1. From the Condition 2, it follows in part that the solution y of the Cauchy problem .cox = 0, x(a) = 0, x(a) = 1 has no zeros on (a, b], and from the Condition 3, it follows that the Wronskian of the fundamental system of solutions of .cox = 0 has no zeros on [a, b]. Thus, Conditions 2 and 3 guarantee that the fundamental system is nonoscillatory.

Remark 2. Conditions 1, 2 and 3 are fulfilled, if .cox = f is an ordinary differential equation with nonoscillatory fundamental system.

Define the operator H : C -+ C by

b

(Hx)(t) = J Go(t,s)(Tx)(s) ds . (7) a

Then, for x E C, Hx belongs to W2 Therefore the problem (4) is equivalent to the equation

x=Hx+r+uo (8)

in the space C. Here r(t) = I; Go(t,s)f(s) ds, '11.0 is the solution of the half homogeneous problem .cox = 0, x(a) = al, x(b) = a2. The operator H is isotonic: (HX1)(t) ~ (HX2)(t), if Xl(t) ~ X2(t), t E [a,b].

Under the above assumptions the Vallee-Poussin Theorem for ordinary differential equations admits an extension to the equation .cx = f as next stated.

Theorem. The following assertions are equivalent. a) The fundamental system of solutions of the homogeneous equation .cx = 0 is

nonoscillatory. b) There exists a v E W2 such tha,t

v(t) ~ 0, (.cv)(t):::; 0, t E [a, b],

b

v(a) + v(b) - J (.cv)(s) ds > O .

• c) The boundary value problem (4) has a unique solution for any f E L, a1 , a2

and the solution b

x{t) = J G(t,s)f{s) ds •

28

BVPs FOR FDEs

of this problem for at = a2 = 0 has no positive values, if f( t) ~ 0, t E [a, b] . d) The Green's function G( t, s) of the problem (4) is strictly negative in the square

(a, b) X (a , b) . e) The spectral radius p(H) of the operator H : C -+ C is less than unit (p(H) <

1) . Before proving the Theorem we shall prove the following auxiliary statements.

Lemma 1. Let the Condition b) of the above Theorem be fulfilled. Then v(t) > 0, t E (a, b) . If in addition v(a) = 0 (v(b) = 0) then v(a) =f. 0 (v(b) =f. 0) .

Proof. Denote.Lv = cpo The function v is a solution of the problem (4), where f = cp, at = v(a), a2 = v(b). Therefore v satisfies the equation (8):

b

v(t) = j Go(t,s) [(Tv)(s) + cp(s)] ds + uo(t), (9)

where Uo is a solution of the problem .Lox = 0, x(a) = v(a), x(b) v(b) . If f:cp(s) ds < 0, then r(t) = f: Go(t , s)cp(s) ds > 0, t E (a, b). If f:cp(s) ds = 0, then v(a) + v(b) > O. In this case uo(t) > 0, t E (a, b), because the solution Uo of L.oX = 0 cannot have two zeros. Thus, r(t) +uo(t) > 0 and hence v(t) - (Hv)(t) > 0, t E (a , b) . Therefore v(t) > 0, t E (a,b) because v(t) ~ 0 and consequently (Hv)(t) ~ O.

Let us take v(a) = 0, v(a) = O. Then v is a solution of the Cauchy problem for the equation .Lx = f. This problem is equivalent to the equation

t

v(t) = j Co(t,s) [(Tv)(s) + cp(s)] ds, (10)

where Co(t,s) is the Cauchy function for L.OX = O. Under such an assumption uo(t) == O. Indeed, both summands in the right side of (9) are non-negative and are equal to zero at the point t = a. Therefore their derivatives at the point t = a are nonnegative. As far as v(a) = 0 we conclude that uo(a) = O. Thus Uo is a solution of the homogeneous Cauchy problem and consequently uo(t) == O. The equality (9) obtains the form

b

v(t) = j Go(t, s) [(Tv)(s) + cp(s)] ds . a

The termwise substraction of the last equality from (10) yields

b

0= j[Co(t , s)-Go(t , s)][(Tv)(s)+cp(s)] ds . a

This identity is impossible because of the inequality Co(t , s) - Go(t,s) > 0 in the square (a, b) X (a, b) due to Conditions 2 and 3 and the inequality

t [(Tv)(s) + cp(s)] ds < O.

29


The assertion for the point t = b is proving analogously by changing Co( t, s) for Gg(t, s) and taking into account that Gg(t, s )-Go(t, s) > ° in the square (a, b) X (a, b) .

Lemma 2. For any v E (a,b) the Green's function C~(t,s) of the problem (6) has the property: C~(t,s) = ° in the trapezium v < s:S b, a:S t < s.

Proof. Due to the representation (3), for the solution u of the problem (6), we have

b

u(t) = J C;(t,s)f(s) ds = a

t b

= J Co(t,s)f(s) ds + J [zl(t)61(s) + z2(t)62(s)]f(s) ds, a a

where ZI, Z2 is a fundamental system for .cox = 0. Let f(t) = 0, t E [a, v] . For such an f the solution u is defined on [a, v] by

b b

u(t) = Zl(t) J 61(s)f(s) ds + Z2(t) J 62(s)f(s) ds. v v

Thus, u(t) coinsides on [a, v] with the solution of the problem .cox = 0, x(v) = x(v) =

0, and consequently

b J [zl(t)61(s) + z2(t)62(s)]f(s) ds = 0, t E [a,b] . v

Since f is an arbitrary summable function on [a, v], we get from the last equality that

C~(t, s) = zl(t)61(s) + z2(t)62(s) in the triangle a :S t < s :S b. Therefore C~(t, s) = 0, s E (v,b], t E [a,s).

Lemma 3. Let the solution y of the Cauchy problem

.cx = 0, x(a) = 0, x(a) = 1

be strictly positive on (a, b] . Then the fundamental system of.cx = ° is non-oscillatory.

Proof. As it was said above, the Sturm Theorem holds for the equation .cx = 0, if the Wronskian w(t) has no zeros on [a, b]. By virtue on this Theorem the existence of a positive solution y yields the property of non-oscillation of the fundamental system. Thus it is sufficient to prove that the Wronskian w(t) has no zeros on [a , b].

30

BVPs FOR FDEs

Thanks to the unique solvability of the Cauchy problem, w(a) -I- 0. Let us take w(v) = 0, for some v E (a, b] and where wet) -I- ° on [a, v). The boundary value problem

.ex = 0, xCv) = 0, xCv) = ° (11)

has a non-trivial solution x. We shall show that this solution has no zeros on (a, v). Let us assume x(e) = 0, e E (a, v), x(t) > ° on (C v) . The functions x(t) and z(t) = x(t) -- ""yy(t), ""Y -I- ° constitutes a fundamental system of .ex = 0. Let us take arbitrary r E (Cv) and fix a ""Y > ° in such a way that z(r) = = x(r) - ""yy(r) > 0. Since z(e) < 0, z(v) < 0, the solution z has at least two zeros on (e, v). Besides all the zeros are simple because v is the first zero of Wronskian. Denote by tIt t2 E (e, v) two adjacent zeros of z. The Wronskian

_I z(t) x(t) 1 wet) - i(t) x(t)

has zeros on (tit t2) because W(tl)W(t2) x(tl)i(tl)X(t2)i(t2) and i(tt}i(t2) < 0. This contradicts the assumption that v is the first zero of the Wronskian.

The solution x of the problem (11) satisfies on [a, v] the equality

v

x(t) = J C;(t,s)(Tx)(s) ds . a

Operator T is anti tonic and Volterra. Thus (Tx)(s) SOon [a, v]. Since CC;(t, s) ~ 0, (t,s) E [a,b] x [a,b] due to the Condition 3, the right-hand and left-hand sides of the last equality have opposite signs. This contradiction completes the proof of the Lemma. .

For the proof of the theorem, we shall follow the scheme

a) ~ b) ~ e) ~ c) ~ d) ~ a).

Implication a) ~ b). In the capacity of v we may take the solution of the Cauchy problem .ex = 0, x(a) = 0, x(a) = l.

Implication b) ~ e). By virtue of Lemma 1, we have vet) > 0, t E (a, b) and, if v(a) = 0, then iI(a) > 0, while if v(b) = 0, then iI(b) < 0. The substitution x = vz in the equation x = H x gives the equation

1 b

z(t) = vet) J Go(t,s)(Tvz)(s) ds a

with respect to z. Denote (Hoz)(t) = vit) I; Go(t,s)(Tvz)(s) ds. The operator Ho is a mapping from C into C. Indeed, define y = Hoz for z E C. If v(a) -I- 0, then

31


y(a) = O. If v(a) = 0, then

b 1! fGo(t,s)(Tvz)(s) ds y(a) ~ lim y(t) = lim a .( ) < 00.

l_a+O l_a+O V t

Analogously y(b) < 00.

There exists a one-to-one mapping x = vz , z = Hox between the set of solutions x E C of the equation x = H x and set of solutions z E C of the equation z = Hoz. Thus the spectrums of Hand Ho coincide and it is sufficient to prove that IIHollc-c < 1. Since the norm of an isotonic operator acting from C into C is equal to the norm of the value of this operator on z(t) == 1,

1 b (Hv)(t) IIHollc_c = max -( ) jGo(t,s)(Tv)(s) ds = max -(-)-. te[a,b) v t te[a,b] v t

a

The inequality (Hv)(t) < v(t), tEla, b], has been established above in the process of proving Lemma 1. Thus, to complete the proof we need to estabilish the inequalities (~(Wl < 1 and (~(Wl < 1. If v(a) =I- 0 then (~(!\al = o. Suppose v(a) = O. Then

v(a) = 1!(Hv)(t)lt=a + r(a) + u(a) > O. If cp(t) == 0, then uo(t) > 0, t E (a, b), and

consequently, uo(a) > O. In this case v(a) > 1!(Hv)(t)lt=a If cp(t) == 0, then r(a) > O. b

Indeed, if r(a) = 0, then r(t) = fCo(t,s)cp(s) ds. Termwise substraction from this a

b b equality the equality, r(t) = f Go(t, s )cp(s) ds, gives J(Co(t, s) - Go(t, s)] cp(s) ds == O.

But this identity is impossible, because Co(t,s) - Go(t,s) > 0 in the square (a,b) X

(a,b) (thanks to Conditions 1 and 2) and f;cp(s) ds < O. Thus r(a) > 0 and,

consequently v(a) > 1!(Hv)(t)lt=a' Therefore

(Hv)(a) ~ lim (Hv)(t) = lim 1!(Hv)(t) < 1. v(a) t_a+O v(t) t_a+O v(t)

The proof for the point t = b is similar by changing Co(t,s) for C8(t,s). Since IIHollc-c < 1, the spectral radius of Ho and consequently of H : C -+ C is

less than unit. Implication e) => c). The problem (4) is equivalent to the equation (8) . Therefore

the assertion about the spectral radius guarantees the unique solvability of the problem (4) for any f, 0'1, (t2. If 0'1 = 0'2 = 0, then x = r + Hr + H2r + ... is a solution of (4). If f(t) :5 0 then r(t) ~ 0 due to Condition 2 and, consequently x(t) ~ r(t) ~ O.

b Thus x(t) = f Go(t, s )f(s) ds ~ 0 for any f(t):5 o.

a

32

BVPs FOR FDEs

Implication c) => d). If 0'1 = 0'2 = 0, then the solution x(t) f: G( t, s )J( s) ds of the problem (4) satisfies the equation

b b

x(t) = J Go(t,s)(Tx)(s) ds + J Go(t,s)J(s) ds . a a

Suppose J(t) ~ 0, t E [a,b] . Then f:Go(t,s)J(s) ds s: 0, x(t) s: 0, (Tx)(t) ~ 0, t E [a,b] . Thus f:G(t,s)J(s) ds s: f:Go(t,s)J(s) ds . Since the last inequality holds for any J(t) ~ 0, G(t,s) s: Go(t,s) < ° in the square (a,b) x (a,b).

Implication d) => a) . The solution x of (4) for 0'1 = 0'2 = ° is defined by

I b b

x(t) = J C(t,s)J(s) ds + Xl(t) J 81(s)J(s) ds + X2(t) J 82(s)J(s) ds a a a

by virtue of (3). Here xI, X2 are such solutions of ex = ° that xl(a) = 1, x2(a) = 0. Since x(a) = ° for any J, then 81(s) = 0, s E [a,b] . Thus G(t,s) = x2(t)82(s) in the triangle a < t < s s: b. From d) and this inequality, we have x2(t)82(s) < ° in this triangle.

Indeed, X2(t) does not change its sign on (a, b) and besides x2(b) i- 0, because otherwise the homogeneous problem (4) would have a nontrivial solution. Thus X2(t) > 0, t E (a, b] and the assertion a) holds due to Lemma 3.

3. Examples

For some applications of Vallee-Poussin Theorem, we shall demonstrate with the equation

x(t) + q(t)x[g(t)] + p(t)x[h(t)] = u(t), t E [a, b],

x(O = cp(~), x(e) = 1/;(0, if ~ < a,

h(t) s: t, get) s: t. A solution of this equation is such a x E W2 which satisfies the equation almost everywhere on [a, b]. In contrast to authors of some papers [4,6] we do not require without fail the "continuous junction condition" x(a) = cp(a) [5]. Such a generalization of the notion of solution is quite natural and permits us to rewrite the equation in the form ex = J. For this purpose we introduce to the description of the equation a linear operator Sr defined by

and a function

(SrY)( ) = {y[r(t), if ret) ~ a, y 0, If ret) < a,

0 r (t) = {O, if ret) ~ a, 0[r(t)], if ret) < a.

33

EXTENSION OF THE VALLEE- POUSSIN THEOREM

By means of this operator the equation obtains the form

(.cx)(t) *=r x(t) + q(t)(Sgx)(t) + p(t)(ShX)(t) = f(t) , (12)

where f(t) = u(t) - q(t),pg(t) - p(t)c/(t).

We shall assume 9 and h to be measurable functions and also 9 to be such that operator Sg : L -+ L is bounded [5] . Assume also pEL and that q is measurable and essentially bounded. Under such an assumption.c : W 2 -+ L is a bounded linear Volterra operator [5] . We assume also that f E L.

Let us dwell first 01'. the equation

under the assumption that p- ~ 0, t E [a, b] . It is shown in [8] that the Wronskian of a fundamental system of solutions of Cox = 0 has no zeros on [a, b], if at any rate one of the following inequalities is fulfilled,

or

where

vraisup (t - h(t))Uh(t) vraisup Uh(t)P-(t) :S ~ tela,b] tela,b] e

if h(t) ~ a , if h(t) < a.

(13)

(14)

The solution of the Cauchy problem Cox = 0, x(a) = 0, x(a) = 1 has no zeros on (a, b]. Consequently the fundamental system of Cox = 0 is non-oscillatory if (13) or (14) is fulfilled. Such a system is a fundamental one for an ordinary differential equation Coox = O. Besides, as it was shown in [8], the equation Cox = f (under the assumption p-(t) ~ 0) is equivalent to the equation Coox = P f , where P : L -+ L is a linear bounded isotonic operator and there exists a bounded inverse p-1

: L -+ L. The solution x of the problem (5) might be represented in two forms

b b

x(t) = J G~(t,s)f(s) ds = J G~(t,s)(PJ)(s) ds, a a

where G~(t, s) is the Green function of the problem (Coox )(t) = z(t), t E [a, b], x(a) = x(p) = o. Recall (Remark 2) that G~(t,s) is strictly negative in the square (a,p) x (a,p). From this and the isotonic property of P follows the inequality G~(t,s) < 0 in the square (a,p) x (a,p) .

34

BVPs FOR FDEs

The validity of the Condition 3 might be established similarly. So, if (13) and (14) is fulfilled , then the Conditions 1, 2 and 3 for .co are fulfilled. Consider now the equation

(.cx)(t) ~ x(t) + p(t)(ShX)(t) = f(t) . (15)

Let us represent p = p+ - p-, where p+(t) 2: 0, p-(t) 2: 0, t E [a, b) . In the capacity of function v of the assertion b) of the Theorem, let us take

t

v(t) = (t - a) - J (t - s)(s - a)O"h(s)p+(s) ds. a

If

(16) a

then v(t) 2: 0, (.cv)(t) ::; 0, t E [a , b), v(b) > 0. Thus (16) and one of (13) or (14) guarantee all the assertions of the Theorem for the equation (15) .

Recall that in the case of ordinary differential equations (h(t) = 0), the condition (16) is well known [1) a.s a test for a non-oscillatory fundamental system.

Next consider the equation (12) under the assumption

q(t) ::; 0, p(t) 2: 0, g(t) = t - T , T = const > 0, t E [a, b) .

Let us take (.cox)(t) = x(t) + q(t)(Sgx)(t) and prove that the Conditions 1,2 and 3 are fulfilled for .co.

The principal part Qo = .coA of .co is defined by (Qoz )(t) = z(t)+ q(t)(Sgz )(t) . So, Qo : L --+ L is a bounded Volterra operator and

Q01 f = f + Sf + .. . + sm f, (17)

where (Sf)(t) = -q(t)(Sgf)(t), m = [b~a 1 is the greatest integer of b~a . Hence,

Q(jl : L --+ L is also a bounded Volterra operator. Thus Condition 1 is fulfilled. The Green's function WI'(t , s) of the problem

x(t) = z(t), t E [a , b) , x(a) = 0, x(JI.) = 0, JI. E (a, b)

is strictly negative in the square (a, JI.) x (a , JI.) and becomes zero in the trapezium JI. < s ::; b, a ::; t < s. The substitution

b

x(t) = (Wl' z )(t) ~ J WI'(t , s) z(s) ds a

35


into equation Cox = f gives the equation Qoz = f with respect to z. There exists a one-to-one mapping z = x, x = Wl'z between the set of solutions x of the problem (5) and the set of solutions z of Qoz = f. Hence, the problem (5) is uniquely solvable. From (17) and the isotonic property of S we conclude that z(t) = (Q(jlf)(t) 2:: 0,

b

if f(t) 2:: O. The solution x(t) = (Wl'z)(t) = JG~(t,s)f(s) ds of the problem (5) is a

strictly negative on (a,Il), if f(t) 2:: 0, f(t) ¢ O. Thus, G~(t,s) > 0 in the square (a,ll) x (a,Il) .

The fulfillment of Condition 3 is established similarly by means of the substitution b

x(t) = J f"(t, s )z(s) ds, where fV(t, s) is the Green's function of the problem a

x(t) = z(t), t E [a,b], x(v) = 0, .:i:(v) = 0, V E [a,b].

Since (Tx)(t) = -p(t)(ShX)(t) is an antitonic operator, all the conditions of the Theorem for the equation (12) are fulfilled.

Let us take v(t) = (t - a)(b - t). Then

rp(t) = (Cv)(t) = -2 - 2ug(t)q(t) + p(t)uh(t)(h(t) - a)(b - h(t))

and rp(t) < 0, if

p(t)uh(t)(h(t) - a)(b - h(t)) < 2[1 + ug(t)q(t)] . (18)

Thus, all the assertions of the Theorem for the equation (12) are valid, if the inequality (18) is fulfilled. It is fulfilled, if

vraisup Uh(t)p(t) < -(b 8 )2 (1- vraisup ug(t)lq(t)l) . tEla,b] - a tEla,b]

References

1. F. G. Tricomi, Differential Equations, Blackie & Son Limited, 1961.

2. Ch. J. de La Vallee Poussin, Sur l'equation differentielle lineare du second ordre, J. Math. Pura et Appl. 8(9) (1929), 125-144.

3. N. Azbelev and A. Domoshnitskii, A de La Vallee Poussin differential inequality, Differential Equations 22, No. 12 (1986), 2041-2045 (in Russian).

4. C. Corduneanu, Integral Equations and Applications, Cambridge University Press, Cambridge, 1991, 366 pp.

5. N. Azbelev, V. Maksimov and L. Rakhmatullina, Introduction to the Theory of Functional Differential Equations, "Nauka", Moscow, 1991,280 pp (in Russian).

36

BVPs FOR FDEs

6. J . Hale, Theory of Functional Differential Equations, Springer-Verlag, NewYork-Heidelberg-Berlin, 1977, 400 pp.

7. S. Labovskii, On linear inequalities with delay, Differential Equations 10, No. 3 (1974), 426-430 (in Russian).

8. N. Azbelev, A. Domoshnitskii, On the differential inequalities, Differential Equations 27, No. 3 (1991), 923-931 (in Russian).

INITIAL-BOUNDARY VALUE PROBLEMS FOR IMPULSIVE PARABOLIC FUNCTIONAL DIFFERENTIAL EQUATIONS

DRUMI BAINOV Higher Medical Institute

Sofia - 1504, P.O.Box 45, Bulgaria

ZDZISLAW KAMONT University of Gdansk

Gdansk, Poland

EMIL MINCHEV Sofia University Sofia, Bulgaria

Abstract

Theorems on differential inequalities generated by an initial-boundary value problem for impulsive parabolic functional differential equations are considered.

1. Introduction

37

In the recent years the theory of impulsive partial differential equations began to emerge: [1], [5], [7], [8] . It gives greater possibilities for mathematical simulation of the evolutional processes in theoretical physics, chemistry, population dynamics, biotechnology, etc., which are characterized by the fact that the system parameters are subject to short term perturbations in time.

In the present paper impulsive parabolic functional differential inequalities are considered. We note that parabolic differential and functional differential inequalities with impulses are investigated in [2], [5], [6], [8] .

2. Preliminary Notes

Let E = [O , a) x (-b,b), a > 0, b = (bI, .. . ,bn ) E R~, R+ = [0,+00) and B = [-1'0,0] x [-1',1'] where TO E R+, l' = (TI, •.• ,Tn ) E R+ . We define c = b + 1', Eo = [-1'0,0] X [-c,c], ooE = [O,a) X ([-c,c] \ (-b,b)), E* = EUEoUooE. For TO> ° we put B(-) = [-1'0,0) X [-1',1'] .

38

BVPs FOR FDEs

Suppose that 0 < Xl < X2 < ... < Xk < a are given numbers. We define

Jo = [-TO, 0], J = [O,a), Jimp = {Xt, ... ,Xk}, Eimp = {(X,y) E E: X E Jimp},

80Eimp = {(X,y) E 80E:x E Jimp}, Etmp = {(x,y) E E":x E Jimp }.

Let Cimp[E", R] be the class of all functions z: E" --t R such that: (i) the restriction of z to the set E" \ Etmp is a continuous function, (ii) for each (x,y) E Eimp there exist

lim z(t,s) = z(x-,y) (t •• )-(x.y)

as t < x,

lim z(t,s) = z(x+,y) as t>x (t •• )_(x.y)

and z(x,y) = z(x+,y) for (x,y) E Eimp. In the same way we define the set Cimp[80E,Rj. If z E Cimp[E",Rj and (x,y) E

Eimp then we write L1z(x,y) = z(x,y) - z(x-,y). Suppose that z: E" --t R and (x, y) = (x, yt, ... , Yn) E E (E is the closure of E) .

We define a function z( • • y): B --t R as follows:

z( •. y)(t,s)=z(x+t,y+s), (t,S)EB.

Suppose that TO > O. For the above z and (x,y) we define also z :B(-) --t R (.- .. )

by z(z_ .• /t,s)=z(x+t,y+s), (t,s)EB(-).

Assume that we have a sequence {t l , . . . , t r } such that

Let r; = (t;,t;+d x [-T,T], i = 1, ... ,r - 1 and

ro = { 0 (-TO, td x [-T, T] if

if -To = t l ,

-TO < tt,

r {0 if tr = 0, r= (t.,O) X [-T,Tj if tr<O.

Let to = -TO if tl > -TO and tr+1 = 0 if tr < 0. We denote by C." [B R] the Imp ,

class of all functions w: B --t R such that there exists a sequence {t l , ... , tr } (r and tt, ... , tr depend on w) and:

(i) the functions wlr;, i = 0,1, ... , r are continuous,

PARABOLIC FUNCTIONAL DIFFERENTIAL EQUATIONS

(ii) for each i, i = 1, ... ,r + 1, (ti' s) E B, ti > -TO, there exists

lim w(t,y) = w(ti,s), (t.y) .... (t; •• )

as t < ti,

(iii) for each i, i = 0,1, ... ,r, (ti' s) E B, ti < 0, there exists

lim w(t,y) = w(t;,s), (t.y) .... (t; •• )

as t> ti,

39

(iv) for each (ti'S) E B, i = O,I, ... ,r -1, and for i = r if tr < 0, we have W(ti'S) = w(tt,s).

Let Ch"p[BH,R] = {wIB(-):w E Ci';..p[B,R]} in the case TO > 0. Elements

of the sets Ci';..p[B, R] and Ci';..p[BH, R] will be denoted by the same symbols. It is easy to see that if Z E Cimp[E*,R], (x,y) E E, then z(s.y) E Ci';..p[B,R] and

z(s_ .• ) E Ci';..p[BH,R] in the case TO > 0. For wE Ci';..p[B,R] we define IIwllB = sup{lw(t,s)I:(t,s) E B}. We denote by

II·IIB(-) the supremum norm in the space Ci';..p[BH,R] . Let M[n] be the class of all matrices 'Y = bijh:Si.j~n' where 'Yij E Rand 'Yij = 'Yji· Suppose that

nimp = (Eimp U OOEimp) X R x Ci:'p[BH, R]

and f: n ~ R, g: nimp ~ R, cp: Eo U ooE ~ R where CPI80 E E Cimp[ooE, RJ, are given functions.

A function Z E Cimp[E*, R] will be called a function of class Cg,~)[E*, R] if Z

possesses continuous derivatives Dxz(x,y), Dyz(x,y) and Dyyz(x,y) for (x,y) E E\ Eimp, where

DyZ = (Dy1z, .. . ,Dynz),

Dyyz = [Dy-y-z] 00 •

• J l:S:t,J.$:n

A function f: n ~ R is said to be a parabolic with respect to Z E Ci~~)[E*, R] in E \ Eimp if for (x, y) E E \ Eimp and for any 'Y, s E M[n] such that

n

"('Yo , - s .. ).U 0 < ° L..J 11J 'J' J - , 1,;=1

f(x, y, z(x, y), Z(% .• ), Dyz(x, y), 'Y) ~ f(x, y, z(x, y), z(%.y)' Dyz(x, y), s).

40

BVPs FOR FDEs

We consider the initial-boundary value problem:

D",z(x,y) = f(x,y,z(x,y),z(z .• ),Dyz(x,y),Dyyz(x,y)), (1)

(x,y) E E \ Eimp ,

z(x,y) =cp(x,y), (x,Y)EEoUooE, (2)

tlz(x,y) = g(x,y,z(x-,y),z(Z_ .• »)' (x,y) E EimpUooEimp. (3)

and

For f:!1 -+ R, g: !1imp -+ Rand z E Ci<.!.~l[E", R] we write

F[z](x,y) = Dxz(x,y) - f(x,y,z(x,y),z(z .• ),DlIz(x,y),Dyllz(x,y)),

(x,y) E E \ Eimp,

G[z](x,y) = tlz(x,y) - g(x,y,z(x-,y),z(Z_ .'»)' (x,y) E Eimp.

3. Main Results

3.1 . Impulsive parabolic functional differential inequalities

We introduce

Assumption HI. Suppose that: 1. the function f::n -+ R of the variables (x, y,p, w, q, s) is non-decreasing with

respect to the functional argument, 2. the function g:!1imp -+ R of the variables (x,y,p,w) is non-decreasing with

respect to the functional argument and for each (x, y) E E imp, w E Ci:"p[B(-l, R] the function 6(p) = p + g(x, y,P, w), pER is non-decreasing on R .

Theorem 1. Suppose that: 1. Assumption HI holds, 2. u, v E Ci<.!.~l[E·, R] satisfy the initial-boundary inequality

u(x,y) < v(x,y), (x, y) E Eo U ooE,

3. the functional differential inequality

F[u](x,y) < F[v](x,y), ( x, y) E E \ Eimp

and the inequality for the impulses

G[u](x,y) < G[v](x,y), (x, y) E Eimp

are satisfied, 4. the function f is parabolic with respect to u in E \ Eimp. Then we have

u(x,y) < v(x,y) on E".

(4)

41


Now we consider weak impulsive parabolic functional differential inequalities.

Assumption H2. Suppose that: 1. the function u: ([0, a] \ J imp ) X R+ --+ R+ is continuous and u(x , O) = 0 for

x E [0, a] \ Jimp,

2. Uo: J imp x ~ --+ R+ is continuous, uo(x,O) = 0 for x E Jimp and the right-hand maximum solution of the problem

a'(x) = u(x,a(x)), x E J \ J imp ,

a(O) = 0, tla(x) = uo(x,a(x-)), x E Jimp

is a(x) = 0, x E J, 3. the function f: n --+ R satisfies the inequality

f(x,y,p,w,q , s) - f(x,y,p,iiJ,q,s) 2: -u(x,max{p - p, lliiJ - wIlB}) on n,

where p:::; p and w :::; w, 4. for (x,y,p,w) E Eimp X R x Ci';.,p[BH,Rj we have

g(x, y,P, w) - g(x, y,p, w) 2: -uo(x, max{p - p, lliiJ - wIlB<->}),

where p :::; p, w :::; w.

Theorem 2. Suppose that: 1. Assumptions HI and H2 hold, 2. u, v E Ci~~)[EO,R] and

u(x,y):::; v(x,y), on Eo U 80E,

3. the functional differential inequality

F[u](x,y):::; F[v](x,y), (x,y) E E \ Eimp

and the inequality for the impulses

G[u](x,y):::; G[v](x,y), (x,y) E Eimp

are satisfied, 4. the function f is parabolic with respect to u in E \ Eimp . Tben we have

u(x,y):::; v(x,y) on EO .

(5)

(6)

42

BVPs FOR FDEs

Assumption H3. Suppose that: 1. the function iT: ([0, a] \ Jimp) X R_ -> R+, R_ = (-00,0]' is continuous,

iT(x,O) = 0 for x E [0, a] \ Jimp and for p ::; p we have

f(x,y,p,w,q,s) - f(x,y,p,w,q,s) ::; iT(x,p - p) on n,

2. the function iTo: Jimp X R_ -> R+ is continuous, iTo(x,O) = 0 for x E Jimp and for p ::; p we have

g(x, y,p, w) - g(x, y,p, w) ::; iTo(x,p - p) on Eimp X R x Ci:'p[BH, RJ,

3. the left-hand minimum solution of the problem

a'(x) = iT(x,a(x)) , x E J \ Jimp,

~a(x) = iTo(x,a(x-)), x E J imp,

lim a(x) = 0 x-+a-

isa(x)=O,xEJ.

Theorem 3. Suppose that: 1. Assumptions HI and H3 hold, 2. u, v E Ci~~)[E',R] satisfy the initial-boundary inequality (4) and the func-

tional differential inequality (5) hold on E \ Eimp, 3. estimate (6) is satisfied, 4. the function f is parabolic with respect to v in E \ Eimp . Then we have

u(x,y) < v(x,y) on E* .

3.2. Comparison theorems for parabolic functional differential inequalities

In this part of the paper we formulate theorems on estimates of functions satisfying impulsive parabolic functional differential inequalities by means of solutions of impulsive ordinary functional differential equations.

Let Cimp[JO U J, R] be the class of all functions a: Jo U J -> R such that: (i) the restriction of a to the set Jo U J \ Jimp is a continuous function , (ii) for each x E Jimp there exist

as t < x,

as t > x

and a(x) = a(x+) for x E Jimp' Suppose that we have defined sequence {t l , ... , tr } such that

-To::; tl < t2 < ... < tr ::; O.

43


For tl > -To we define also to = -TO and for tr < ° we put tr+! = 0. Let J(il = (ti' ti+!), i = 0, 1, ... , r.

We denote by Ci';"p[JO, RJ the class of all functions ",: Jo -+ R such that there exists a sequence {to, tt, . . . , tT) tr+!} depending on ", such that:

(i) the functions "'1;(;), i = 0,1, .. . , r are continuous, (ii) for each i, i = 2, ... ,r + 1 and for tl > -TO, there exists

as t < ti,

(iii) for each i, i = 0,1, ... ,r - 1 and for tr < 0, there exists

as t > ti,

and ",(ti) = ",(tt). For TO> ° we put JJ-l = [-TO,O) and Ci';"p[JJ-l,R] = {"'I (-):", E Ci';"p[Jo,R]).

1.

We will denote the elements of Ci';"p[JO' R] and Ci';"p[JJ-l, RJ by the same symbols. We denote by 11·110 the supremum norm in the space Ci';"p[Jo, RJ and in the space

Ci';"p[JJ-l, RJ. For z E Cimp[E·, R] we define a function Tz: Jo U J -+ R+ in the following way

(Tz)(x) = max{lz(x,y)l:y E [-c,cJ}, x E [-TO, a) .

If a: Jo U J -+ R and x E .J then we define a function a(%) : Jo -+ R by a(%)(t) =

a(x+t), t E Jo. For the above a and x we define a(%_): JJ-l -+ R by a(%_)(t) = a(x+t)

for t E JJ-l. For a function w E Ci';"p[B, R] we define T"w: Jo -+ R+ in the following way:

(T*w)(t) = max{lw(t,s)l : s E [-T,T]}.

Lemma 1. If z E Cimp[E·,R] then Tz E Cimp[JO U J,R+] . Ifw E Ci';"p[B,RJ then T·w E Ci';"p[Jo, R+].

Assumption H4. Suppose that: 1. the functions u: ([0, a) \ J imp) X R+ x Ci';"p[JO' R+J -+ R+ and u: Jimp X R+ x

ChnP[JJ-l, R+J -+ R+ are continuous and nondecreasing with respect to the functional argument,

2. for each (x,,,,) E J x Ci';"p[JJ-l, R+J the function ,,(p) = p + u(x,p, ",), p E R+ is nondecreasing on R.;..

Lemma 2. Suppose that: 1. Assumption H4 holds and 1/J E Cimp[JO U J, R],

44

BVPs FOR FDEs

2. r; E C(Jo,R+) and w('jfi):[-To,a) -+ R+ is the maximum solution of the problem

a'(x)=u(x,a(x),a(.»), xEJ\Jimp,

a(x)=fi(x), XEJo,

~a(x) = u(x,a(x-),a(._»), x E Jimp ,

3. the function 1jJ satisfies the conditions:

1jJ(x)Sfi(x), xEJo,

~1jJ(x) S u(x,1jJ(x-),1jJ(._), x E J imp ,

4. for x E P+ = {x > 0, x E J \ Jimp: 1jJ(x) > w(Xj fi)} we have

D_1jJ(x) S u(x,1jJ(x),1jJ(%»),

where D_ is the left-hand lower Dini derivative. Then we have

1jJ(x) S w(xjfi), x E [-TO , a) .

Theorem 4. Suppose that:

(7)

1. Assumption H4 holds, f E C(O,R) and for each (x,y,p,w) E (E \ Eimp) X

R x Ci:"p[B, R] we have

f(x,y,p,w,O,O)signp S u(x, Ipl,T·w) ,

where signp denotes 1 if p ~ 0 and -1 if p < 0, 2. u E Ci~;)[E·, R] and

Dxu(x,y) = J(x,y,u(x,y),u(% .• ),Dyu(x,y),Dyyu(x,y)), (x,y) E E \ E imp ,

3. fi E C(Jo, R+) and

lu(x,y)1 S fi(x), (x,y) E Eo,

4. w(.j fi): [-TO, a) -+ R+ is the maximum solution of the problem (7), 5. the boundary estimate

lu(x,y)1 S w(x j fi), (x,y) E 80 E

and the impulsive estimate

are satisfied, 6. the function J is parabolic with respect to u in E \ Eimp .

Then we have lu(x,y)1 S w(xjfi) for ( x, y) E E·.


Let us consider two problems, the problem (1)-(3) and the problem

D",z(x,y) = l(x,y,z(x,y) , z(z .• J' Dy z(x,y),Dyyz(x , y))

(x,y) E E\Eirnp ,

z(x, y) = cp(x, y), (x, y) E Eo U aoE,

45

(8)

(9)

where 1: 0 -+ R, g: Oimp -+ R, cp: Eo U aoE -+ R, where CPI~E E Cimp[aoE, R] are given functions .

We prove a theorem on the estimate of the difference between the solutions of (1)-(3) and (8)-(10) .

Theorem 5. Suppose that: 1. Assumption H4 holds, 2. !, 1 E C(O, R), g, 9 E C(OirnP, R) satisfy the inequalities

(J(x,y,p,w,q,s) - j(x,y,p,w,q,s))sign(p- p) ~ ~CT(x,lp-pl,T*(w-w)) on 0,

ig(x,y,p,w) - g(x,y,p,w)i ~ ~iT(x,lp-pl,T'(w-w)) on Oimp,

3. c.p, cp:Eo U aoE -+ R, if E C(Jo,R+) , c.pIEo' CPIEo E C(Eo,R) , c.p1~E' CPI~E E

Cimp[aoE, R] and Ic.p(x,y) - cp(x,y)1 ~ ij(x), (x,y) E Eo,

4. the maximum sciution w('j ij) of(7) is defined on [-TO, a) and u, 11: E Cl.!.~)[E*, R] are solutions of (1)-(3) and (8)-(10), respectively,

5. 1c.p(x,y)-cp(x,y)1 ~w(Xjij) on aoE, 6. the function! is parabolic with respect to u in E \ Eimp. Then we have

lu(x,y) - 11:(x , y)1 ~ w(x;ij), (x,y) E E*.

Theorem 6. Suppose that: 1. Assumption H4 holds, 2. ! E C(O, R), 9 E C(Oimp, R) and

(J(x,y,p,w,q,s) - !(x,y,p,w,q,s))sign(p- p) ~ ~CT(x,lp-pl,T*(w-w)) on 0,

46

BVPs FOR FDEs

Ig(x,y,p,w) - g(X,y,p,W)1 ~ ~ u(x, Ip - pi, T*(w - w)) on nimp ,

3. O"(x, 0, 0) = 0 for x E J \ J imp and u(x, 0, 0) = 0 for x E Jimp, where O(t) = 0 for t E Jo,

4. the maximum solution of the problem

a'(x)=u(x , a(x),a(z))' xEJ\Jimp ,

a(x) =0, xEJo,

~a(x) = u(x,a(x-),a(Z_))' x E Jimp

isa(x)=O,xEJoUJ. Then the problem (1 )-(3) admits at most one solution in the class Ci~;)[E" R] .

Acknowledgements

The present investigations was partially supported by the Bulgarian Ministry of Science and Education under Grant MM-422.

Z. Kamont wishes to express his thanks to the University of Gdansk for the support of this research.

References

1. D. Bainov, Z. Kamont and E. Minchev, On first order impulsive partial differential inequalities, Appl. Math. Compo 61 (1994), 207-230.

2. D. Bainov, Z. Kamont and E. Minchev, On impulsive parabolic differential inequalities, (to appear) .

3. D. Bainov, V. Lakshmikantham and P. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.

4. D. Bainov and P. Simeonov, Systems with Impulsive Effect: Stability, Theory and Applications, Ellis Horwood, Chichester, 1989.

5. L. Byszewski, Impulsive degenerate nonlinear parabolic functional-differential inequalities, J. Math. Anal. Appl. 164 (1992), 549-559.

6. L. Byszewski, System of impulsive nonlinear parabolic functional-differential inequalities, (to appear in Comment. Math.) .

7. C. Y. Chan and L. Ke, Remarks on impulsive quenching problems, in First International Conference on Dynamic Systems and Applications, 1993, Atlanta, USA (Conference Proceedings, to appear) .

47


8. L. Erbe, H. Freedman, X. Liu and J. Wu, Comparison principles for impulsive parabolic equations with applications to models of single species growth, J. Austral. Math. Soc., Ser. B 32 (1991), 382-400.

9. V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Vol. 1 and 2, Academic Press, New York and London, 1969.

10. V. Mil'man and A. Myshkis, On the stability of motion in the presence of impulses, Sib . Math. J. 1, no. 2, (1960), 233-237 (in Russian) .

11. J. Szarski, Differential Inequalities, Polish Scientific Publishers, Warsaw, 1965.

BOUNDARY VALUE PROBLEMS ON INFINITE INTERVALS

JOHN V. BAXLEY Department of Mathematics and Computer Science

Wake Forest University Winston-Salem, North Carolina 27109 USA

1. Introduction

We consider boundary value problems of the form

(1.1) y"=f(x,y,y'), for x;:::>: a,

(1.2) g(y(a),y'(a)) = O.

49

Our primary goal is to obtain conditions which guarantee the existence of bounded solutions. Boundary value problems of the form (1),(2) are not very well understood and only in the last twenty-five years have begun to receive wide attention. We call attention particularly to the papers of Bebernes and Jackson [4], Baxley [1], Granas, et al [8], and O'Regan [9].

We shall unify and extend the theorems in [1] and [8]. We apply recent results [2] on existence of solutions of boundary value problems on finite intervals and obtain a bounded global solution of (1),(2) by a limiting procedure. The hypotheses in [8] were sufficiently strong to produce a uniform bound on the solutions (and their first two derivatives) of the approximating finite interval problems and thus to conclude boundedness of the solution (and its first two derivatives) of (1),(2) . Our weaker hypotheses produce a uniform bound on the solutions (but not always their derivatives) of the approximating finite interval problems. Thus, in general, we do not conclude the boundedness of the derivatives of a solution of (1),(2) . However, for a subclass of problems, containing those in [8], we do obtain the bounded ness of these derivatives.

The search for radially symmetric solutions to

(1.3) fly = F(II x lI,y), II x II;:::>: a,

(1.4)

50

BVPs FOR FDEs

with a ~ 0, leads (e.g. [5,6, 7]) basically to the problem

(1.5)

(1.6)

N-1 y" + --y' = F(x,y), x ~ a,

x

g(y(a),y'(a)) = 0,

where x is now the radial coordinate. Thus we get a problem of the form (1),(2) . If a = 0, then the boundary condition (6) takes the form y'(O) = 0 and the differential equation has an additional singularity at x = O. We provide extensions of our theorems to singular problems like (5),(6), as well as the more general equation

(1. 7) " p, F( ') y + -y = x, y, y , x

where p is a positive constant and F may be quite nonlinear in y' . Our results complement those of Berestycki, Lions, and Peletier (see [5] and references there) and O'Regan [9] , as well as extending the theorems of Erbe and Schmitt [6] .

We begin by stating a finite interval result from [2], which deals with boundary value problems of the form

(1.8) y" = f(x, y, y'), a:S x :S b,

(1.9) (1.10)

g(y(a), y'(a))

h(y(b), y'(b)) 0,

O.

We first need to describe our basic assumptions on the behavior of the nonlinear conditions (9) and (10), and certain smoothness and growth restrictions on the function f(x,y,z) in (8). For (9), we assume that

SLa: the graph of g(y, z) = 0 is a (continuous) curve which can be parameterized y = p(,), z = qb), for -00 < , < 00, where p, q are continuous and

lim qb) = -00, "Y- - oo

lim q(")') = +00, '1'-+00

limsuppb) < +00, ,,),_-00

liminfpb) > - 00. /,-+00

and for (10), we assume that

SLb: h(y, z) is continuous on R2 and given Yo E R, there exists ZI, Z2 E R so that h(y,z) > 0 for y ~ Yo, z ~ ZI, and h(y,z) < 0 for y:S Yo, z:S Z2.

51

BVPs ON INFINITE INTERVALS

If h satisfies SLb, we let TI(yo; h) (resp. T2(yo; h)) denote the infimum (resp. supremum) of the set of all such values Zl (resp. Z2)' Noting that TI(y; h), T2(y; h) are nonincreasing, we may define

and

The conditions SLa and SLb are modeled on the Sturm-Liouville conditions

(1.11) (1.12)

aoy(a) - aly'(a)

boy(b) + bly'(b)

A,

B.

Note that SLa is satisfied if ao > 0 and al ~ 0; SLb is satisfied if bo ~ 0 and b1 > O. Next are some assumptions on the function f in (8), which are used to insure

the existence of solutions of a wide variety of boundary value problems on the finite interval [a, b). Our existence theorems from [2) do not require all the assumptions listed below; in fact CI (fJ) is an alternate for BI (fJ) and C2(fJ) is an alternate for B2(fJ)·

BI(fJ) : there exists YI > 0 so that f(x, y, fJ) ~ 0 for all (x, y) E [a, b) X [YI, 00);

B2(fJ) : there exists Y2 < 0 so that f(x, y, fJ) S 0 for all (x, y) E [a, b) x (-00, Y2);

CI (fJ) : there exist Yl > 0, MI > 0 so that

f(x, y, z) ~ -MIZ log Z

for all (x,y,z) E [a, b) x [Yl>oo) x [fJ,oo);

C2(fJ) : there exist Y2 < 0, M2 > 0 so that

f(x, y, z) S M21zllog Izl

for all (x,y,z) E [a,b) x (-00,Y2) x (-oo,fJ);

DI : given TJ2 < TJI, there exist QI = QI(TJ2,TJl) SO and 81 = 81(TJ2 ,TJd > 0 so that

f(x, y, z) ~ QIZ2

for all (x, y, z) E [a , b) x [TJ2, TJI) X [81> 00);

D2 : given TJ2 < TJI, there exist Q2 = Q2(TJ2,TJd ~ 0 and 82 = 82(TJ2,TJI) < 0 so that

f(x,y,z) S Q2Z2

for all (x,y,z) E [a,b) x [TJ2,TJI) x (-00,82).

52

BVPs FOR FDEs

Here is the needed result [2, Theorem 4.1], as it applies to (8)-(10).

Theorem 1 Suppose that 9 satisfies SLa, h satisfies SLb, f is continuous on [a, b] x R2, and that

(i) there exists (31 > 1 for which f satisfies C1 ((3d, or there exists (31 > t 1 (h) for

which f satisfies Bl ((3d j

(ii) there exists (32 < -1 for which f satisfies C2((32), or there exists (32 < t2(h) for which f satisfies B 2((32) j

(iii) f satisfies Dl and D 2·

Then there exist constants 1/1, 1/2, VI, V2, dependent only on the parameters of the hypotheses, and a solution ¢> of the boundary value problem (8)-(10) for which 1/2 :=; ¢>(x) :=; 1/1, V2 :=; ¢>'(x):=; VI, for a:=; x :=; b.

For the application of Theorem 1 to our problem (1 ),(2), we shall find it convenient to use for (10) the specific boundary condition

(1.13) y(b) + y'(b) = B.

Note that h(y, z) = y + z - B certainly satisfies SLb with t 1(h) = -00 and t2(h) = 00.

We specifically want to allow the Neumann condition y'(a) = 0, which does not satisfy SLa. To make this possible, we will make a change of variable, replacing x by a + b- x in (8)-(10), to obtain:

(1.14)

(1.15)

(1.16)

y"=f(a+b-x,y,-y'), a:=;x:=;b,

y(a) - y'(a) B,

g(y(b), -y'(b» O.

Since (15) certainly satisfies SLa and (16) satisfies SLb when g(y , z) = z is the Neumann condition, application of Theorem 1 will require assuming that f(x,y,-z) satisfies the hypotheses which Theorem 1 requires of f(x,y ,z) . For our application later to radially symmetric solutions of elliptic problems, this difference is a lucky event.

Note that it may often happen that (9) satisfies SLa and also that (16) satisfies SLb for the same g. In fact this happens for the Sturm-Liouville condition (11) if ao > 0 and al > o. In such cases, Theorem 2 or Theorem 3 below might apply.

Also note that the hypotheses on f involve a number of parameters: (31, Yl, M 1 , (32, Y2, M 2, Ql, sl, Q2, S2· It is clear that if f satisfies one of these hypotheses on some interval [a, b], then f satisfies that hypothesis on any subinterval of [a, b], with the same values of all these parameters. However, in passing to a larger interval, even if a

53


hypothesis is still satisfied, the relevant parameters would usually require change. We shall say that f satisfies a hypothesis uniformly on a collection of compact intervals if f satisfies that hypothesis on each interval in the collection with the same values of all relevant parameters.

With these preliminaries, we can now state our first main results .

Theorem 2 Suppose that g satisfies SLa, that f is continuous on [a, 00) X R2, and that for each compact interval [a, b] C [a, 00),

(i) there exists (31 > 1 for which f satisfies C1 «(3I) or there exists (31 > ° for which f satisfies Bl «(31);

(ii) there exists (32 < -1 for which f satisfies C2«(32) or there exists (32 < ° for which f satisfies B 2«(32);

(iii) f satisfies Dl and D2;

(iv) there exist x' > a, ih, ih for which

for x 2: x' .

f(x,y,O) 2: 0, for y 2: iil, f(x,y,O) :-:; 0, for y :-:; ih,

Then there exist real numbers U2 < U1 , dependent only on the parameters which appear in the hypotheses relative to the interval [a, x'] and not directly on either the function f or the interval [a, x'], and a solution rjJ of {1},(2} on the interval [a, 00) for which U2 :-:; rjJ(x) :-:; U1 for a :-:; x < 00 . Moreover, if f satisfies the hypotheses Dl and D2 uniformly on compact subintervals of [a, 00), then rjJ' is also bounded on [a, 00); the bound depends only on ih , ih, and the parameters Si(U2 ,U1 ), Qi(U2,UI) which appear in Dl and D2.

Here is the corresponding theorem which is a consequence of the change of variable leading to (14)-(16).

Theorem 3 Let g(y,z) = g(y,-z) and j(x,y,z) = f(x,y,-z). Suppose 9 satisfies SLb, that j is continuous on [a, 00) X R2, and that for each compact interval [a, b] C

[a, 00),

(i) there exists (31 > 1 for which j satisfies C1«(31) or there exists positive (31 > tl(g) for which j satisfies B 1«(31);

(ii) there exists (32 < -1 for which j satisfies C2«(32) or there exists negative (32 < t2(g) for which j satisfies B 2«(32);

(iii) j satisfies Dl and D2;

54

BVPs FOR FDEs

(iv) there exist x' > a, Yb Y2 , h > 0 for which

for x ~ x'.

f(x,y, z)~h, for y~Yl> z ~o,

f(x,y, z ) ~ -h, for y ~ Ih , z ~ 0,

Then there exist real numbers U2 < U1 , dependent only on the parameters which appear in the hypotheses relative to the interval [a, x1 and not directly on either the function f or the interval [a, x'), and a solution <p of (1),(2) on the interval [a, 00) for which U2 ~ <p(x) ~ U1 for a ~ x < 00 . Moreover, if f satisfies the hypotheses Dl and D2 uniformly on compact subintervals of [a, 00), then <P' is also bounded on [a,oo); the bound depends only on Yb Y2, and the parameters Si(U2,Ud, Qi(U2,U1 )

which appear in Dl and D2.

In the case that the second alternatives in (i) and (ii) hold, Theorem 2 is a generalization of [8, Corollary 2.3]; in the case that the first alternatives in (i) and (ii) hold, Theorem 2 is a generalization of [1 , Theorem 1]. Of course, the other combinations, for example the first alternative in (i) and the second alternative in (ii) , provide "hybrids" of these earlier results. Note that Theorem 2 allows a Dirichlet condition at x = a, while Theorem 3 allows a Neumann condition at x = a.

These results may be applied to the radially symmetric problem (3),(4) if a> O. Setting

we see that

N-1 f(x,y,z) = ---z +F(x, y),

x

N-1 f(x,y, -z) = --z + F(x,y) .

x

Depending on the nature of F(x, y) and the boundary condition (4) , either Theorem 2 or Theorem 3 (or both) might apply. In fact, the nonlinear term F could also depend on the radial derivative. Note that the term involving z works in cooperation with Bi({3i) when seeking to apply Theorem 3, but works against this condition when seeking to apply Theorem 2; it is irrelevant with regard to Ci({3i) since it is linear in z and x ~ a > O.

If a = 0 in (5) ,(6) , we encounter a singularity at x = O. So let us return to the problem (1),(2), without the assumption that f is continuous at x = a. We first consider a Dirichlet condition at x = a.

Theorem 4 Suppose that f is continuous on (a, 00) x R2 , and that for each compact interval [a',b] C (a, 00), f satisfies either the hypotheses (i)-(iv) of Theorem 2 or the hypotheses (i)-(iv) of Theorem 3, uniformly on each compact subinterval of (a, x'] . Then the problem

(1.17) y" = f(x,y,y'), a < x < 00,

55


(1.18) y(a) = A,

has a bounded solution </> E C[a, 00)nC2(a, 00) . Moreover, iff satisfies the hypotheses Dl and D2 uniformly on compact subintervals of (a,oo), then </>' is also bounded on (a, 00) .

Our next theorem deals with a Neumann condition at x = a and is thus designed for application to the existence of radially symmetric solutions of (3),( 4) in all of RN Note that the theorem allows (nonlinear) dependence on the radial derivative.

Theorem 5 Let g(y,z) = -Zj then g(y,z) = z satisfies SLb with tl(g) = t2(g) = O. Suppose that f is continuous on [a, 00) X R2 , and that for each compact interval [a', b) C [a , 00), f satisfies the hypotheses (i)-(iv) of Theorem 3. Suppose also that p E C[a,+oo) n C 1(a,+00), p(a) = 0, p(x) > 0 on (a,+oo), and p'(x) ~ 0 on [a, +00). Then the problem

(1.19) p(~) (p(x )y')' = f(x, y, y'), a < x < 00,

(1.20) y'(a) = 0,

has a bounded solution </> E C1 [a, 00) n C2( a, 00). Moreover, if f satisfies the hypotheses Dl and D2 uniformly on compact subintervals of (a, 00), then </>' is also bounded on (a,oo).

This last theorem should be compared to results in (9) . In order to apply it to the existence of bounded radially symmetric solutions of (3) ,(4) in all of RN, just choose p(x) = X N - 1 with a = o.

Any of our first four theorems may be combined with the argument in [1) to give an existence (and uniqueness) result with an appropriate boundary condition at 00.

We shall need some tools from [2), which we now state. The stronger form of B 1 ((3) obtained by replacing the weak inequality f(x,y,(3) ~ 0 by f(x,y,(3) > 0 will be designated Bt((3)i the analogous strengthening of B2((3) is called 8:;((3).

Lemma 1 Suppose f satisfies condition Bt ((3d for some (31 E R and the linear function m(x) = Q + (31 (x - a) satisfies m(x) ~ Yl for a ~ x ~ b. If a ~ r "(x) = f(x, </>(x), </>'(x» , T ~ X ~ b,

</>(r) ~ m(r), </>'(r) > m'(r) = (3 ..

then </>(x) > m(x), </>'(x) > m'(x) for r < x :5 b.

56

BVPs FOR FDEs

Lemma 2 Suppose f satisfies C1eB1 ) for some (31 > 1, and the function m(x) IS

defined by m(a) = a, log m'(x) = log (3 exp[Ml (b - x)],

for a::::; x ::::; b, where a ~ Y1 and (3 ~ (31 . If a::::; r < band

¢/'(x) = f(x, <jJ(x), ¢>,(x)), r::::; x::::; b,

t/>(r) ~ m(r), t/>'(r) > m'(r),

then t/>(x) > m(x), t/>'(x) > m'(x), for r < x ::::; b.

Lemma 3 Suppose f satisfies condition D1 and

t/>"(X) = f(x, t/>(x), t/>'(x)), r::::; X ::::; b,

where a ::::; r < b. If 712 ::::; t/>(r) ::::; 7J}' s ~ Sl(7J2,7J1),

t/>' (r) ~ s exp[ -Q1 (712,711)( 711 - 712)],

then either t/>'(x) > s for r ::::; x ::::; b or there exists c E [r,b) such that t/>(c) = 711, t/>'(c) > s.

Each of these lemmas have rather obvious dual results, involving the other conditions B:;((32), C2((32), and D2; see [2] for statements .

2. Proof of Theorem 2

Choose and fix fi satisfying fi2 ::::; fi ::::; fi1· For each integer n ~ "'~a' consider the boundary value problem

(2.1) y" = f(x,y,y') + ~(y - fi) , a::::; x ::::; x' + n , n

(2.2)

(2.3)

g(y(a), y'(a))

y(x' + n) + y'(x' + n)

o fi .

Since this problem satisfies the hypotheses of Theorem 1, then it has at least one solution t/>n on [a, x' + a] . We shall show that Ascoli's theorem may be applied to the sequences {t/>n}, {t/>~} on each compact subinterval of [a, (0) and obtain a solution of (1),(2) as the limit of a subsequence of {t/>n} .

If f satisfies B1((3t} (resp. B2((32)), then the right side of (21) automatically satisfies the stronger condition Bi((31) (resp. B:; ((32)).

Our first major goal is to demonstrate the existence of numbers U1 and U2 so that U2 ::::; t/>n(x) ::::; U1 for a ::::; x::::; x' + n and all n.

We need one further preliminary result which pinpoints the purpose of the additional hypotheses of Theorem 2.

57


Lemma 4 Suppose that f(x, y,O) > 0 for x' < x ~ band y > ih . If r/>(b) + ¢/(b) ~ ih, if

r/>"(x) = f(x,r/>(x),r/>'(x)), x' ~ x ~ b,

and if a local maximum of r/> occurs at c E (x', b], then r/>( c) ~ iit .

Proof. Suppose that r/> attains a local maximum at some point c E (x' , b), where r/>(c) > iiIi c = b is impossible because r/>(b) + r/>'(b) ~ ih. Thus we have r/>'(c) = 0 and r/>"(c) = f(c, r/>(c), 0) > 0, contradicting the local maximum at x = c.

The obvious dual of Lemma 4 will be called Lemma 4'. The next lemma plays a central role in the proof of Theorem 2.

Lemma 5 Suppose the hypotheses of Theorem 2 are satisfied and let r/>n be a solution of the boundary value problem (21)-(23). Then there exist constants Ul and U2 ,

dependent only on the parameters of the hypotheses relative to the interval [a, x'], and not directly on the function f or the endpoint a, such that for each n, U2 ~ r/>n(x) ~ UI , for a ~ x ~ x' + n.

Proof. Throughout this proof, we shall be using the values of the parameters which refer to the interval [a, X1i any explicit occurence of such parameters will always refer to the interval [a, x') . Also we assume, by adjusting Yl and Y2 if necessary, that Yl ~ iit, Y2 ~ iil-

Letting b = x' and fixing n , we choose a = Yl and let ml(x) be the function of Lemma 1 (resp. Lemma 2) according as the right side of (21) satisfies Bi({h) (resp. Cl (11t)). We then let PI = inf{p(-y) : / ~ O} and choose ml ~ PI and m} ~ max{ml(x) : a ~ x ~ x'} and let

{ , () ( _) ml - PI }

tl = max m l a , SI PI , ml , . x' - a

Then we choose /1 > 0 such that

where Ql = Ql(Pt,ml). Likewise, we choose m2(x) from Lemma l' (or Lemma 2'), where a = Y2. We put P2 = sup{p(-y) : / ~ O} and choose m2 ~ P2 , m2 ~ min{m2(x) : a ~ x ~ x'}, and let

and pick /2 < 0 such that

q(-y) ~ t2exp[Q2(P2 - m2)), for / ~ /2,

where Q2 = Q2(m2,P2).

58

BVPs FOR FDEs

Since rPn satisfies the boundary condition (22), then there exists a/ E R such that rPn(a) = p(!), 4>~(a) = q(!). We now show that /2 ~ / ~ /1. If we assume on the contrary that / > /1> we have 4>n(a) ~ PI> 4>~(a) > tl ~ m~(a). If 4>n(a) ~ ml(a), then Lemma 1 (or Lemma 2) implies that 4>n(x' ) ~ ml(x' ) ~ YI, 4>~(X') > m~(x') ~ f31 > o. If instead, PI ~ 4>n( a) < ml (a), then Lemma 3 implies that either 4>~ (x) > tl for a ~ x ~ x' or there exists c E [a,x1 such that 4>n(c) = ml, 4>~(c) > t l . In the first alternative,

4>n(x' ) = 4>n(a) + [ ' 4>~(x)dx > PI + tl(X' - a) ~ ml > iiI

and rP~(X') > tl ~ O. In the second alternative, we have 4>n(c) = ml ~ ml(c), 4>~(c) > tl ~ m~(c) and Lemma 1 (or Lemma 2) with r = c guarantees that rPn(x' ) > ml(x') ~ Yb 4>~(X') ~ m~(x') > O. So, in every case, we conclude that rPn(x' ) > iiI, 4>~(X') > 0, in contradiction of Lemma 4. Thus,.., ::; /1 . A similar argument shows that / ~ /2.

Now choose &1 ~ Yl such that &1 ~ p(!), for /2 ::; / ~ /1, and put ml(x) = ml(x) + (&1 - YI). Likewise, choose &2 ~ Y2 such that &2 ~ p(!), for /2 ~ / ~ /1, and put m2(x) = m2(x)+(&2-Y2) . Finally, let Ul = ml +(&I-Yl), U2 = m2+(&2-Y2). We note the important fact that Ul , U2 depend only on the various parameters which appear in the hypotheses relative to the interval [a, x']; there is no direct dependence on either the function f or the interval [a,x' + n] . With this preamble, we now complete the proof of Lemma 5 for this case.

Suppose on the contrary that 4>n attains a maximum at x where rPn(x) > Ul . Since rPn(a) = p(!) for some / E [,2,/1], then 4>n(a) ~ &1 ~ Ul and so a < x ~ x' + n. By Lemma 4, the maximum must occur at x E (a, x']. Since 4>n (a) ~ &1 = ml (a) and 4>n(x) > Ul ~ ml(x), using the mean value theorem, we find r E (a,x) so that 4>n(r) > ml(r) and 4>~(r) > ml(r) . Then Lemma 1 (or Lemma 2) implies that 4>~(x) > m~(x) = f31 > 0, contradicting the maximum at x. Thus 4>n(X) ::; UI on [a, x' + n] and a similar argument proves the inequality U2 ~ 4>n(X) on [a , x' + n].

We now turn out attention to the existence of bounds on 4>~(x) .

Lemma 6 For each b > a I there exists Vi = Vi (b) such that 4>~ (x) ~ Vi for all x E [a, b] and all n for which x' + n ~ b + l. Proof. Let 51 = SI(U2, UJ), Ql = QI(U2, Ud be the parameters in Dl associated with f on the interval [a, b + 1] . Put 5 = max{sl' UI - U2} and Vi = s exp[-Ql(UI - U2)]. Suppose there exists n with x'+n ~ HI and r E (a, b) for which 4>~(r) > VI. Applying Lemma 3, we conclude that 4>~(x) > s for r ~ x ~ b + 1, since the alternative of Lemma 3 is clearly impossible. Therefore

lb+l

4>n(b+ 1) - 4>n(r) = r 4>~(x)dx > s ~ Ul - U2,

a contradiction since U2 ~ 4>n(x) ~ U1 on [r,b+ 1].

59


Lemma 7 Suppose f satisfies D1 uniformly on all compact subintervals of [a , 00) . Then there exists a constant VI. such that 4>~ (x) :::; VI. for all x E [a , x' + n] and all n .

Proof. By hypothesis, we can choose parameters Sl = Sl(U2 , U1) and Q1 = Q1(U2 , U1) for which f satisfies D1 on every compact subinterval of [a, 00). Put s = max{ Sl , yU2} and VI = s exp[-Q1(U2-U1)]. Supposing that there exists nand r E [a, x'+n) for which 4>~ (r) > VI. , Lemma 3 gives a contradiction since 4>~ (x' + n) = y - ¢n (x' + n) :::; y - U2 and ¢n(x) :::; U2 for a :::; x :::; x' + n.

The dual of Lemma 6 (resp. Lemma 7) will be called Lemma 6' (resp. Lemma 7').

We now complete the proof of Theorem 2. Given a compact subinterval [a, b] of [a, 00), Lemma 5,6 (and their duals) imply that for x' +n :2: b+ 1, the sequences {¢n}, {¢~} are uniformly bounded on [a, b]. Since f(x, y, z) is continuous on [a, b] x R2, the differential equation (21) shows that the sequence {¢~} is also uniformly bounded on each such interval [a , b]. The familiar diagonalization argument using Ascoli's theorem may now be used to obtain a subsequence of {¢n} which converges (together with the first and second derivatives) uniformly on each compact subset of [a, 00) to a solution ¢ of (1),(2) on [a,oo). From Lemma 5, it follows that U2 :::; ¢(x):::; U1 on [a, 00), so ¢ is bounded. If Lemmas 7, 7' apply, then we also have constants VI., li2 for which li2 :::; 4>'( x) :::; VI. on [a, 00) and so ¢' is also bounded.

3. Proof of Theorem 3

Since it is clear from the hypotheses that Y2 < YI, we choose and fix y satisfying Y2 :::; Y :::; Y1. As before we consider the problem (21)-(23), but we change variables, replacing x by a + x' + n - x to obtain

(3.1)

(3.2)

(3.3)

y" = j(a + x' + n - x,y,y') + ~(y - y) , a:::; x:::; x' + n, n

y(a) - y'(a) y, g(y(x' + n) , y'(x' + n)) o.

Since this problem satisfies the hypotheses of Theorem 1, it has at least one solution tPn on [a,x' + n] . Then ¢n(x) = tPn(a + x' + n - x) is a solution of (21)-(23) . We proceed as in the proof of Theorem 2 to see that Ascoli's theorem may be applied.

As before our first goal is the analogue of Lemma 5 for this situation. Until we have achieved this goal, any explicit reference to parameters in the hypostheses will refer to the interval [a,x'] . We may assume that Y1 and Y2 in C1(,81) (or B1(31)) and C2((32) (or B2((32)) satisfy Y1 :2: Y1 and Y2 :::; Y2. In the case that j satisfies B1((3t} and (31 > t1(g), then there exists 111 and Zl < (31 such that g(y, z) > 0 for y :2: 1111 Z :2: Zl- By increasing Y1 if necessary we may assume that Y1 :2: 111. Similar steps

60

BVPs FOR FDEs

should be taken in the case that j satisfies B2(f32) ' Since these parameters are chosen with respect to [a, x'], they are satisfactory on any subinterval. Define q = f31 if j satisfies BI (f3I) and define q by

log q = log f31 exp[MI(x' - a)]

if j satisfies CI(f3I). We now show that dE (a, x') and tPn(d) 2 YI imply tP~(d) > -q. Replacing x by a + d - x, we observe that tPn (x) = tPn (a + d - x) satisfies

y"=j(a+d-x,y,y'), a~x~d.

Defining m(x) as in Lemma 1 (or Lemma 2), with a = YI and b = d, we see that tPn(a) = tPn(d) 2 YI implies tP~(a) ~ m'(a) since otherwise Lemma 1 (or Lemma 2) gives tP~(d) > m'(d) = f31 > ZI and thus

g(tPn(a), tP~(a)) = fJ(tPn(d), tP~(d)) > 0,

a contradiction. Now let UI = YI + q2/0 + q(x' - a) . Note that the value of UI depends only on the

parameters of the hypotheses relative to the interval [a, x']; there is no dependence on the function f or the interval [a, x' + n]. We next show that tPn (x) ~ UI for a ~ x ~ x' + n and all n such that n 2 q/o. We argue by contradiction. If some tPn attains a maximum at some point c where tPn(c) > UI, then Lemma 4 implies that c ~ x'. We assert that tPn(x') 2 YI + q2/0 (from which also tP~(x') > -q) i for if not, then the mean value theorem implies the existence of a point d E (c, x') at which tPn(d) > YI + q2/0 and tP~(d) ~ -q, a contradiction of the result of the previous paragraph. Let [x', x"] be the largest subinterval of [x', x' + q/o] on which tPn(x) 2 YI. We claim that there exists a point in [x', x"] at which tP~ is positive. If we suppose the contrary, then our hypothesis gives

tP~(x) 2 f(x, tPn(x), tP~(x)) 2 0,

and thus tP~ is increasing on [x',x"] . Then integration gives

2 2

tPn(X") > tPn(x') - q(x" - x') 2 YI + ~ - ~ = YI .

lt follows that x" = x' + q/o. Then a second integration gives

tP~(x") 2 tP~(x') + o(x" - x') = tP~(x') + q > 0,

a contradiction. Thus tPn attains a maximum value greater than YI on the interval (x',x' + n], contradicting Lemma 4 and completing the proof that tPn(X) ~ UI. The construction of a lower bound U2 is similar.

Having achieved the analogue of Lemma 5, bounds on the derivatives are provided without change by Lemmas 6 and 7, and the proof is then completed just as in the proof of Theorem 2.

61


4. Proofs of Theorems 4 and 5

To prove Theorem 4, we begin with the problem

(4.1) y" = f(x,y,y'), a + lin ~ x < 00

(4.2) y(a + lin) = A

Applying, as appropriate, either Theorem 2 or 3, we see that the problem (27),(28) has at least one bounded solution ¢>n on [a + lin, 00). Moreover, since the hypotheses are satisfied uniformly on closed subintervals of (a, x'], there exist constants VI, V2 ,

independent of n, for which V2 ~ ¢>n(x) ~ VI for a + lin ~ x < 00. Moreover, the bound provided by Lemmas 6 and 6' for ¢>~ on [a + lin, b] is independent of n. Since f is continuous, the differential equation (27) then provides a uniform bound on ¢>~ on any compact subinterval of (a, 00). Ascoli's theorem may then be used to obtain a subsequence which converges uniformly (together with first and second derivatives) on each compact subinterval of (a, 00) to a solution ¢> of (17) on (a, 00). Let un(x) = ¢>~(x) for a + lin ~ x ~ 1; un(x) = 0 for a ~ x < a + lin. Then

¢>n(x)-A= [un(s)ds, a+l/n<x~a+l.

Since Un --+ ¢>~ and ¢>n --+ ¢> on (a, a + 1], then the uniform bound on ¢>~ and the Lebesgue bounded convergence theorem imply that

¢>(x)=A+ [¢>'(s)ds, a<x~a+l

and thus ¢>(x) --+ A as x --+ a, completing the proof of Theorem 4.

To prove Theorem 5, we begin with the problem

(4.3) " p'(x) , f( ') II < y = - p( x) y + x, y, y , a + n _ x < 00

(4.4) y'(a + lin) = O.

This problem satisfies the hypotheses of Theorem 3 because the term - ~y does nothing to change the parameters in the hypotheses satisfied by f(x, y, -z}. Therefore (29),(30) has at least one bounded solution on [a + lin, 00), and as before we use Ascoli's theorem to get a solution ¢> of the differential equation (19) on (a, 00) which is bounded and for which ¢>' is bounded on any bounded interval (a, b]. Let u(x) = p(x)¢>'(x). Then u(x) --+ 0 as x --+ a. Furthermore, if we multiply (19) by p(x) and integrate we obtain

lu(x)1 ~ [Ip(s)f(s ¢>(s),¢>'(s))lds.

BVPs FOR FDEs

Since 4, # are bounded on (a, a + 11 and p(x) is monotone nondecreasing, the continuity of f gives us a constant M for which

Hence IQ(x)I 5 M(x -a) , a < x 5 a + 1,

so #(x) + 0 as x -t a, and the boundary condition at x = a is satisfied.

References

1. J. V. Baxley, Nonlinear second order boundary value problems on [0, a), Quali- tative Properties of Differential Equations, Proc.1984 Edmonton Conference, W. Allegretto and G. J. Butler, eds., University of Alberta Press, Edmonton, 1986, pp. 50-58.

2. J. V. Baxley, Existence theorems for nonlinear second order boundary value problems, J. Differential Equations 85 (1990), 125-150.

3. J. V. Baxley, A singular boundary value problem in the theory of economic growth, Differential Equations, Proc. 1988 Conference in Athens, Ohio, R. Aftabizedah, Ed., Marcel Dekker, 1989, pp. 40-46.

4. J. Bebernes and L. Jackson, Infinite interval boundary value problems for y" = f(x, Y), Duke Math. Journal 34 (1967), 39-47.

5. H. Berestycki, P. L. Lions, and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in RN, Indiana Univ. Math. J. 30 (1981), 141-157.

6. L. Erbe and K. Schmitt, On radial solutions of some semilinear elliptic equations, Differential and Integral Equations 1 (1988), 71-78.

7. J. Gatica, G. Hernandez, and P. Waltman, Radially symmetric solutions of a class of singular elliptic equations, Proc. Edinburgh Math. Soc. 33 (1990), 169-180.

8. A. Granas, R. B. Guenther, J. Lee, D. O'Regan, Boundary value problems on infinite intervals and semiconductor devices, J. Math. Anal. Appl. 116 (1986), 335-348.

9. Dona1 O'Regan, Singular boundary value problems on the semi-infinite interval, Libertas Math. 1 2 (1992), 109-119.

EXISTENCE OF STEADY-STATE SOLUTIONS TO SOME CONSTANT-VOLTAGE PROBLEMS

L. E. BOBISUD Department of Mathematics and Statistics, University of Idaho

Moscow, Idaho 83844-1103 USA

1. Introduction

63

Consider a thin resistive wire stretching along the interval 0 ::; x ::; 1. By "thin" in this context it is understood that cooling of the wire by lateral radiation is accomplished with negligible variation of temperature over a cross section of the wire. It is possible to have a large lateral gradient in the temperature, and hence large heat radiation from the wire, with only small lateral change in temperature of the wire if the thickness orthogonal to the surface is small, by the definition of the derivative. Note that the wire need not have circular or even regular section for this to occur to a satisfactory degree of approximation. Correspondingly, we shall also assume that heat radiation from the wire is governed by its temperature (as in Newton's law of cooling) rather than the normal derivative of the temperature.

Application of a (constant) voltage differential if> to the ends of the wire will cause a current 1= 4>1 R, where R is the total resistance of the wire in ohms, to flow and generate heat in the wire as a whole at the rate 12 R = 4>21 R. Unless completely insulated, the wire will dissipate heat through its lateral boundary. If the electrical resistance per unit length r 2: 0 of the wire depends on the temperature u and if we allow spatial dependence as well, the total resistance of the wire is given by

R = l r(x, u(x)) dx .

Let k(x, u) denote the heat conductivity of the wire and g(x, u) the rate of lateral heat radiation per unit length; both may depend on position and temperature. Because we wish to include in our formulation such physical systems as a heater element and its power leads, we assume that g, k, and r are merely piecewise continuous functions . Under these conditions we have the steady-state heat conduction equation

(k(x,u(x))u'(x))' = g(x,u(x)) - I 2r(x,u(x)).

64

BVPs FOR FDEs

But Ohm's law yields at once that

, , 4h(x,u(x)) (k(x , u(x))u (x)) = g(x, u(x)) - ( )2 '

J~ r(s, u(s )) ds (1)

Although other boundary conditions could be considered by our methods, we shall restrict ourselves to the Dirichlet boundary condition

u(O) = Q u(l) = (3. (2)

The steady-state problem Eqs. (1)-(2) will not in general admit a solution, for thermal runaway in the corresponding time-dependent system may occur if too great heat generation occurs at high temperatures. Thus our goal is to determine conditions on g, k, and r that will guarantee existence of a steady state. This is similar in spirit to [1), where constant-current devices are considered. In these problems I is fixed, so the functional differential equation (1) is replaced with the ordinary differential equation (ku')' = 9 - Pro A different treatment of a class of constant-voltage devices is given in [4) and several subsequent papers.

The specific fixed-point technique we use is application of the topological transversality theorem [5-7) . We exploit the idea, introduced in [2) and refined in [3], of a "forbidden value" of the maximum of a solution; so doing results in substantially weaker hypotheses that involve a supremum instead of a limit (d. hypothesis H3ii below).

Eqs. (1)-(2) do not take into account possible variation of density with temperature. In the final section we consider a problem that differs from Eqs. (1)-(2) only in allowing the material density p to be a piecewise continuous function of position and temperature. Because of the temperature dependence, the material will change its length (e.g., the location of its right end point) when the voltage is applied. The new length is not known a priori but must be determined as part of the problem. This additional complication is handled by a device (Lagrangian coordinates) due to Quilghini [8] in this context and used in [1) : each point is mapped back to where it would have been under some arbitrary but fixed temperature distribution. Determination of the appropriate map is part of the new problem, which is a fixed-domain problem of the type specified in Eqs. (1)-(2). Thus existence of solutions for the temperature-dependent-density problem is thrown back to the question of existence for a problem with fixed density.

In applications the change in length is likely to manifest itself in sag rather than in change in location of endpoints, but the formulation considered here may be regarded as a one-dimensional model of a physical system with sag.

2. Preliminaries

Since we wish to consider physical systems with abrupt spatial changes in material properties, we suppose there is a fixed partition 0 = Xo < Xl < . .. < Xn = 1 of [0, 1]

65

STEADy-STATE SOLUTIONS

such that

HI. 9 and r coincide on each subinterval (Xi-I, Xi) with a function that is continuous on [Xi-I, Xi); k coincides on each (Xi-I, Xi) with a function that is positive and Lipschitz continuous on [Xi-I, X;J.

Functions with these properties will be called, respectively, piecewise continuous and piecewise Lipschitz (with respect to the given partition); it seems reasonable to seek a solution which is continuous and whose first and second derivatives are piecewise continuous (with respect to the same partition) and such that the differential equation (1) is satisfied on each (Xi-I, Xi) (equivalently, on each [Xi-I, Xi)).

The following lemma is proved in [1) .

Lemma 1 Let HI be satisfied and let there exist a constant (J" > ° such that k(x, u) 2: (J" on [0, 1) x (-00,00). Then for any piecewise continuous function h the boundary value problem

(k(x,u(x))u'(x))' = h(x) u(o) = Q u(I) = (3

has a unique solution u.

By a solution of the equation (k(x,u(x))u'(x))' = h(x) we mean a function u E C[O,I), coinciding on each (Xi-I, Xi) with a function continuously differentiable on [Xi-l>Xi), such that k(x,u(x))u'(x) E C[O,I) n C1[Xi_I,Xi) for each i, and such that the differential equation is satisfied on each subinterval (Xi-l, Xi).

3. Constant Density

As usual in fixed-point arguments, we must first obtain suitable estimates, uniform in the parameter>. E [0,1), for solutions u>. of a suitable one-parameter family of problems, namely

(k(x,u>.(x))u~(x))' = >. [g(X,u>.(X)) - ¢} ( r(x,u>.(x)) rl (3) J~r(s,u>.(s))ds

u>.(I) = (3. (4)

Note that for >. = ° this nonlinear problem has a unique solution by Lemma l. Integration over an interval (x, y) yields the basic equalities

k(y-,u>.(y))u~(y-) - k(x+,u>.(x))u~(x+) = >. t g(s,u>.(s))ds

</>2 fY ->.( )2J. r(s,u>.(s))ds (5)

J~r(s,u>.(s))ds :z;

66

BVPs FOR FDEs

and

(6)

H2. There exists a number, such that g(x, z) :::; 0 on [0,1) x (-00, ,); r(x, u) > o.

Lemma 2 u~ exists end is zero at any interior local extremum of u.!. and, if H2 holds, then u,\(x) ~ min(a,,8,,) on [0,1).

Proof. First suppose that u.!. has a local extremum at a point x in (0,1). From Eq. (5) in the limit as x and y approach x we have that k(x+,u.!.(x))u~(x+) = k(x-, u,\(x))u~(x-)), implying that u~(x+ )u~(x-) ~ O. But this and the fact that k -:/; 0 shows that both u~ (x+) and u~ (x-) must be zero.

Suppose next that for some .x E [0, 1) u,\ satisfies miIlo<x<l u,\( x) < min( a, ,8,,); let the minimum occur at x E (0, 1). By the first part of the Te~ma we have u~ (x) = O. From Eq. (5) and H2 we have, as far to the right of x as u,\(s) :::; min(a,,8,,) holds, that

k(x-, u,\(x))u~(x-) = .x { g(s, u,\(s)) ds

(j>2 {X -.x ( )2 iT r(s, u,\(s)) ds :::; O.

J~r(s,u,\(s))ds i;

Thus u,\ is nonincreasing On [x, 1), contradicting the boundary condition at 1. It is convenient to set Jl. = min(a,,8, ,); J.L is an a priori lower bound on solutions

of Eqs. (3)-(4) for .x E [0,1).

H3. Either

I . there exists a constant I< such that

g(x,u) ( ¢> )2 r(x,u) > J~minl'<.<ur(t,s)dt

holds for all u ~ I< and x E (0,1), or

ii. there is a constant 8, a nonnegative function G, and positive functions I<, R, and p, all continuous on [8,00), such that

g(x,u) ~ G(u), k(x,u) ~ J«(u), k(x,u)r(x,u):::; R(u), p(u):::; r(x,u)

67


hold on [0,1) x [6,00). Moreover, with w = max(a,,8, 6), these functions are such that

q ( q [ 1jJ2 R( ) ] ) -1/2 I sup l ( . u 2 - G(u)1«u) du 1«v)dv> In

q w }" (m1llw5z5q p(z)) v2

m. and such that for all real numbers 8 satisfying 8 > w we have

. 1jJ2 2 R(8) _ G(8)1«8) :2: O. (m1llw5z58 p( z ))

For suitable r, g, and k, the conditions of H3ii may be regarded as setting upper and lower limits, respectively, on 1jJ. The inequalities imposed on 9 in H2 and H3 force 9 to be negative for low temperatures and positive for high temperaturesj this is a reflection of the physically reasonable condition that the wire cool at high and warm at low temperatures in the absence of internal heat generation.

Lemma 3 Let HI-H3 hold. Then there is a constant Mo such that any solution u>. of Eqs. (3)-(4) for>. E [0,1) satisfies maX[O,lj u>. i- Mo and there is a constant M1 such that lu~(x±)1 ~ M1 for all solutions satisfying J1- ~ u>. ~ Mo·

Proof. Suppose H3i holds, and let u>. for some>. > 0 have a maximum greater than 1< at x E (0,1). Then u~(x) = 0 and, from Eq. (3),

(k(x,u>.(x))U~(X))'(X):2: >.[g(X,u>.(X))

1jJ2 2 r(X,U>.(X))] >0, (J~ minI'5u:5u,(x) r(t,u) dt)

where an appropriate one-sided derivative may be involved. Then there exists an f > 0 such that either k(x,u>.(x))u~(x) > 0 on [x,x + f) or k(x,u>.(x))u~(x) < 0 on (x - f,X). Either condition contradicts the maximality of u>.(x). It follows that JL ~ u>. ~ max(a,,8,1<).

Suppose now that H3ii holds. It is convenient to let r denote the absolute maximum of u>. on [0, I)j there is nothing to prove unless r > w = max(a,,8, 6). Suppose that u>. has an isolated local maximum exceeding w at x E (0,1). Then, by Lemma 2, u~(x) = OJ also, there exists a nonempty interval (X,X) on which u~ > 0 and u>. > w. For x E (X, x) we have from Eq. (6) that

_ (k(x+, u>.(x ))u~(x+)) 2

1U ,(x) 1jJ2 (",(x)

:2:2>' G(U)1«U)dU-2>'( )2}. R(u)du u~(:<) J~ p( u>.( s)) ds u,(:<)

:2: 2>' [f"~(:i:) G(u)1«u)du- (. IjJ )2 (",(x) R(U)duj. }U~(:<) mmu~(:<):5z:5u,(x) p(z) }",(:<)

68

BVPs FOR FDEs

From this we get that

K(u,\(x))u~(x)

~V2rf"'(X){( . </1 ())2 R(0')_G(0')K(0')}d0'1

1

/

2

J",(x) mIIlw:Sz:sr P z

~ V2 [fr {(. </1 ())2 R(O') _ G(O')K(O')} d0'11/2, J",(x) mIIlw:Sz:sr P z

where the final inequality is a consequence of the last part of H3 integrated from u,\(x) to r. On division and a further integration it follows that

f f. 2 R(O') - G(O')K(O') dO' K(v)dv ~ V2(x - x). ",(x) ( r [</12 1 ) -1/2

J",(x) Jv (mIIlw<z<r p( z)) - - (7)

In particular, this inequality must hold for u,\( x) the value of u,\ at the local minimum immediately preceeding x or for the value max(a, /1, 6), whichever is greater.

Let v be any number satisfying max(a,/1,6) ~ v ~ r. By the intermediate value theorem the closed set u;:l(v) is not empty; let Xv > 0 be its least member. u,\ being continuous, we must have u,\(x) < v for x < xv, so u~(xv-) 2:: 0; that is, Xv lies in some interval on which u,\ is increasing or else u~(xv) = O. With Z = {x : u~(x) = OJ, it follows that the interval [max(a,/1,6), r] is contained in Z U U{[u,\(q),u,\(p)] : u,\ has a local maximum at p and a local minimum at q < p and is increasing on [q,p] ; or u,\ has a local maximum at p, u,\(q) = max(a, /1, 6), and u,\ is increasing on [q ,pJ} . Regarding the set Z, we obviously have that

( r [</1

2 1 ) -1/2 f f (. ( ))2R(O') - G(O')K(O') dO' K(u,\(x))u~(x)dx = O. Jz J",(x) mlIlw:Sz:sr P z

As a consequence of this equality, the inequality of Eq. (7), and the containment [max(a, /1, 6), r] c Z U U[u,\(q), u,\(p)], we have on summation that

r ( r [</12 1 ) -1/2 1 1 ( . ( ))2R(0') - G(O')K(O') dO' K(v) dv ~ V2i,

w v mIIlw:Sz:sr P z

where u,\ attains its absolute maximum at i. Symmetry of the problem under the change of variable x t-> 1 - x shows that we also have

r( r[ </12 1 )-1/2 1 1 ( . ( ))2R(0') - G(O')K(O') dO' K(v)dv ~ V2(1 - i) .

w v mIIlw:Sz:sr P z

On adding these inequalities we get that

r ( r [</12 1 ) -1/2 1 1 f ( . ( ))2R(0') - G(O')K(O') dO' K(v) dv ~ /0'

w Jv mIIlw:Sz:sr P Z v 2

69


Let Mo be chosen such that

M, ( M, [4>2 ] ) -1/2 1 l or 0 ( • 2R (U) - G(u)K(u) du K(v)dv> /OJ OJ iv mlll.,:;;z:;;Mo p( z)) v 2

then we have shown that no solution of Eqs . (3)-(4) for)' E [0,1] satisfies SUP[O,IJ u), = Mo.

To get an a priori bound M1 on lu~(x±)1 for u:. satisfying II. S u:. S Mo. observe first that

l(k(x,u),(x))u~(x))'1 S max (Igl + 4>2 21T I) = D I . [O,IJX[I',MoJ (min[O,IJX[I',MoJ T)

At an interior maximum or minimum of u), we see from Lemma 2 that u~ = OJ if u), is monotone, there must exist a point xo such that lu~(xo)1 S 1.8 - al. In either event, there exists Xo E [0,1] such that lu~(xo) 1 S 1.8 - al. Therefore, for any x > Xo

and thus

k(x-, u),(x))lu~(x-)1 S 1.8 - al max k + DI = D2 [O,IJX[I'.MoJ

lu~(x-)I S D2 / min k = MI' [O,IJ x [I',Mo]

Of course, by (piecewise) continuity the same bound holds for lu~(x+ )Ij by symmetry, this bound holds on the whole interval (0,1).

Theorem 1 Assume that H1-H3 hold; then Eqs. (1)-(2) have a solution.

Proof. The conclusions of Lemmas 2 and 3 are valid. Let B be the Banach space of functions in C(O,l] that coincide on each (Xi-I,Xi) with a function continuously differentiable on [Xi-I, x;]j the norm on B is

where-

IIYllo = max IYI, [O,lJ Ilyllt = max max ly(x)l -

1 [X __ 1I.1:';)

By the Ascoli-Arzela theorem, the injection j : B --+ C(0,1] is completely continuous. Let K be the closed convex subset of those functions y in B satisfying the boundary conditions y(O) = a, y(l) =.8. Let PC denote the class of functions defined on Ui=1 (Xi-I, Xi) and coinciding on each (Xi-I, X;) with a function continuous on [Xi-I, Xi]j PC is equipped with the norm II -Ill-

Define the map Q on C(0,1] by

(Qy)(x) = g(x,y(x)) - ( I )2 r(S,y(s))j fo r(s,y(s))ds

70

BVPs FOR FDEs

it is easily seen that Q is a continuous map into PC. Let k denote k altered outside the rectangle [0,1] X [11, Mo] so that k(x, u) 2': min(t,.)E[O,I]X[I',Mo] k(t, s) > 0 on [0,1] X

(-00,00); a solution of Eqs. (3)-(4) with k replaced by k continues to satisfy the a priori estimates of Lemma 3. Let N : PC -> K be defined by N h = y precisely if y E K with (k(x,y(x))y'(x))' = hex) on each (Xi-I, X;). Lemma 1 guarantees that N is well-defined. The continuity of N as a map into B is straightforward to show; see [1) for details.

Let

u = {y E K : JI. - 1 < y(x) < Mo on [0,1] and lIy'lh < MI + I};

U is a bounded open subset of K. Any solution u>. E Vof Eqs. (3)-(4) with k replaced by k and oX E [0,1] satisfies JI. ::; u>. < Mo and lu~(x)1 ::; MI by Lemma 3 and thus (i) lies in the interior of U and (ii) is a solution of Eqs. (3)-(4) with k instead of k. H>. : U -> K defined by

H>'y= (No(oXQ)oj)(y)

is then a compact homotopy whose fixed points in V are solutions of Eqs. (3)-(4) and conversely; H is therefore fixed-point free on au. Moreover, Hoy = N(O) is the unique solution (by Lemma 1) of the problem

(k(x,y(x))y'(x))' = 0, yeO) = a, y(l) =,8.

Thus Ho is a constant map, and so essential. It follows from the topological transversality theorem [5-7] that HI has a fixed point, and therefore that Eqs. (1)-(2) have a solution.

4. Temperature-Dependent Density

We now consider a wire that, when zero voltage (or any other convenient voltage) is applied across its ends, occupies the region 0 < ~ < 1 and has temperature distribution T(~); it is not necessary to impose the boundary conditions T(O) = a, T(I) = ,8. We suppose the material property functions k(~,z), g(~,z), r(~,z) and the linear density p(~, z) are known. If now the constant voltage </> is applied across the ends of the wire, a (different) current will flow, causing the wire to expand or contract to some domain 0 < x < a (we assume for convenience that the left end of the wire is fixed) and the temperature distribution to become ultimately some u(x), 0 < x < a. Consider the map x ...... ~ which sends a point back to where it was before the new voltage </> and the boundary conditions u(O) = a, u(a) = ,8 were applied. Since mass is conserved, we must have

f p(s,u(s))ds = t p(t,T(t))dt (8)

71


where p{x,u{x» is the density at point x when the temperature distribution is u{·). Clearly, p(x,u(x» = p(e(x),u(x». Using this in the derivativeofEq. (8) we get that

d~ p(~(x), u(x» dx p(~(x), T(~(x)))'

~(O) = 0 (9)

to determine the function ~(x). In terms of the function ~(x), u satisfies the problem

[k(~(x), u(x»u'(x)l' = g(~(x), u(x)) </>2

- (J;'r(~(s),u(s))ds)2r(~(x),u(x)) (0 < x < a), (10)

u(O) = Il!, u(a) = (3 (11)

where ~(a) = 1 determines a uniquely. Eqs. (9)-(11) plus ~(a) = 1 define the problem with variable densitYj it is this problem that we want to show has a solution under suitable hypotheses. In this problem we now make the change of independent variable x 1--+ ~ and set z(~(x» = u(x)j we get that

f r(~(s),u(s»ds = loa r(~(s),z(e(s)))ds

11 p(v,T(v))_ = r(v,z(v» ( ()) dv = I(z)

o p v, z v (12)

and so we get in place of Eqs. (10)-(11) that

.!!.-[k(~'Z)P(~,Z)dZ] =p(~,T(m[ (t )_L (t )j (0<t<1), (13) d~ cl~,T(m d~ p(~,z) 9 \"z I(z)2

r \"z \,

z(O) = Il!, z(1) = (3. (14)

Conversely, suppose z(O satisfies Eqs. (13)-(14) with I(z) given by Eq. (12), and define x and a by

x = r{ p(s, T(s» ds, Jo p(s,z(s))

a= r1p(s,T(s))ds Jo p(s,z(s» .

It is then easy to see that u(x) == z(~(x)) solves Eqs. (10)-(11). Thus it suffices to show that there is a solution to the problem (12)-(14), which is on a fixed interval. By defining

we can write Eq. (13) c:.s

k(~,z) = p(~,z)k(~,z)/p(~,T(m, g(~,z) = p(~,T(~))g(~,z)/p(~,z), r(~, z) = p(~, T(~))r(~, z)/ p(~, z),

72

BVPs FOR FDEs

where J(z) is given by . fl I( z) = 10 r(v,z(v))dv.

But the problem posed by Eqs. (14)-(15) has exactly the form of Eqs. 1-2; thus by Theorem 1 sufficient conditions for existence of solutions are H1-H3 with k, g, r replaced by k, g, and r, respectively.

References

1. L. E. Bobisud, Existence of steady-state solutions for some one-dimensional conduction problems, J. Math. Anal. Appl., to appear.

2. L. E. Bobisud, J. E. Calvert, and W. D. Royalty, Existence of biological populations stabilized by diffusion, to appear.

3. L. E. Bobisud and D. O'Regan, Existence of positive solutions for singular ordinary differential equations with nonlinear boundary conditions, to appear.

4. G. Cimatti, Remark on existence and uniqueness for the thermistor problem under mixed boundary conditions, Quart. Appl. Math . 47 (1989), 117-121.

5. A. Granas, R. B. Guenther, and J. W. Lee, Nonlinear boundary value problems for ordinary differential equations, Dissertationes Math. (Rozpmwy Mat.) 244 (1985).

6. A. Granas, R. B. Guenther, and J. W. Lee, Some general existence principles in the Caratheodory theory of nonlinear differential equations, J. Math . Pures Appl. 79 (1991), 153-196.

7. D. O'Regan, Theory of Singular Boundary Value Problems, World Scientific, Singapore, 1994.

8. D. Quilghini, Una analisi Fisico-Matematica del processo del cambiamento di fase, Ann. Mat . Pum Appl. 67 (1965) , 33-74.

AN EXISTENCE THEOREM FOR HEREDITARY LAGRANGE PROBLEMS ON AN UNBOUNDED INTERVAL

DEAN A. CARLSON Department of Mathematics, The University of Toledo

Toledo, Ohio 43606-3390 USA

1. Introduction

73

In this work we investigate the existence of optimal controls for optimal control problems defined on [0, +0::» with states governed by a functional differential equation of neutral type. The objective functional is described by an improper integral and contains a nonlinear state and control dependent discount term (i.e., a "recursive utility"). Our results rely on the classical convexity and seminormality conditions found in the works of Tonelli, McShane, Cesari, and others but requires the use of a recent lower closure theorem given in Balder [4] . In addition we require certain growth conditions to insure the compactness of admissible trajectories

We concern ourselves with the existence of optimal solutions for the problem of minimizing

J(X,u) = 1+00

f(t,x,u(t))F (t, l r(s,xt,u(s))ds) dt (1)

over all pairs of functions {x, u} satisfying the neutral differential equation

(2)

the initial condition

X(s) = t/J(s), for -h::;s::;Oj (3)

state constraints

(t,Xt) E A, for t::; 0; (4)

and control constraints

u(t) E U(t, Xt) a.e. 0::; t . (5)

74

BVPs FOR FDEs

In the above we assume that h > 0 is fixed; x(·) is a continuous function defined on [-h,+oo) into n-dimensional euclidean space ffi.n ; u(·): [0,+00) -+ ffi.m is Lebesgue measurable; A = [0, +00) x X, where Xc C == C( [-h , 0]; ffi.n) is closed and bounded (here C denotes the space of continuous functions from [-h, 0] ~to ffi.n endowed with

the supremum norm, 1I</!lIoo == SUP_h<.<O 1</!(s)l)j U: A -+ 2ffi. is a set-valued map with closed graph - -

M = {(t,</!,u): (t,</!)EA,uEU(t,</!)};

and 1/J E C is a given fixed function. With regard to the the functions, we assume that (f°,r,J) : M -t ffi. x ffi. x ffi.n is continuous with JO( ., ., .) and r(-, ·,· ) both nonnegative and F : [0, +00) x ffi. -+ [0, +00) is continuous and such that F(t ,· ) is nondecreasing for each t ~ O. To insure that the differential equation Eq. (2) is well defined we assume that the operator D : [0, +00) x C -+ ffi.n satisfies appropriate regularity assumptions. Specifically we assume that D(·,·) is continuous and- atomic at zero. We refer the reader to the monograph of Hale and Lunel [16] for the relevant definitions (in particular see Section 2.7) of these well known concepts. Finally, if x : [-h, +00) -+ ffi.n is as above, for each t > 0 we define Xt E C by Xt(s) = x(t + s) for s E [-h ,O].

The study of optimal control problems defined on the infinite time interval [0 , +00) is an ongoing area of research with applications primarily arising in the area of optimal economic growth. For a survey of this theory we refer the reader to the monograph of Carlson, Haurie, Leizarowitz [9] which discusses systems of this type for a variety of systems. In the above model we have introduced do time delay into the state variable. The role of such delays in economic models is well documented beginning with the 1935 work of Kalecki [17] who used differential-difference equations to model fluctuations in a business cycle. In addition, we also see the use of delays in models considered in Frisch and Holme [15] . More recently, delay models have been encountered in fishery economics, dynamic advertising, as well as several other areas of mathematical economics. For the most part, the dynamics in these models have been of retarded type. However in Chukwu [12] a generalization of Kalecki 's model is given in which a nonlinear functional differential equation of neutral type is utilized. In [12] the primary concern is controllability and not optimality. With regards to optimal economic growth the study of such models has just recently begun to be addressed.

In the model described above, the objective functional differs from those typically encountered in economic growth models as we have included a "discount term" which depends on the past history of the trajectory. Such a factor introduces a time preference toward present (or future) generations. In more classical models this term takes the simple form of F(t, r) = exp( -rt) in which r > 0 is a positive fixed constant. In the model considered here, we have considered a generalization of this idea that permits this time rate of preference to fluctuate over time and additionally depends on both the past history of the trajectory as well as the control. For discussions

75

HEREDITARY LAGRANGE PROBLEMS

concerning the merits of discount factors of the type included here we refer the reader to the works of Becker, Boyd, and Sung [8], Epstein [13], and Epstein and Hynes [14] .

In this work we provide conditions under which the above model has a minimum. This existence theorem extends known results in several directions. Firstly, our work provides an extension of the finite interval works of Angell [1], [2] to the infinite horizon case. Secondly, it provides an extension of the existence results of Baum [7], Bates [6], and Balder [3] to a new class of dynamical systems. Finally we remark we also extends the work of Carlson [10], who apparently was the first to consider optimal control models with state and control dependent discount factors, and the improvements to [10] made in the 1993 work of Balder [4].

With these remarks the plan of our paper is as follows. In Section 2 we carefully define the notion of an admissible pair and discuss the compactness of the set of admissible trajectories under appropriate growth restrictions. In Section 3 we describe the related orientor field problem and present an appropriate lower closure theorem. Finally in Section 4 we give our existence result and an example for which the result is applicable. We end our discussion with some brief concluding remarks in Section 5.

2. Compactness of Admissible Trajectories

Under the general hypothesis listed in Section 1 for each Lebesgue measurable function u(·) : [0, +00) --+ lRm it is well known that the initial value problem described by Eq. (2) and Eq. (3) has a local solution x(·;u(·)) that is continuous on some interval, [-h, T), and additionally is such that the map t --+ D(t, Xt) is locally absolutely continuous (see Hale and Lunel [16]). This motivates the following definition.

Definition 1 A pair of functions {x(·), u(·)} is called an admissible pair if:

1. x(·) : [-h, +00) --+ IK' is continuous with t -+ D(t, xd absolutely continuous on [0, T] for each T > 0,

2. u(·) : [0, +00) --+ IRm is Lebesgue measurable;

3. The pair of functions {x(·),u(·)} satisfies equations Eq. (2) to Eq. (5); and

4. J(x, u) given by Eq. (1) is finite.

As is usual in this setting we will call x(·) an admissible trajectory and u(.) and admissible control. For brevity we let n denote the set of all admissible pairs and we assume that n 1- 0.

To apply direct methods to establish the existence of an optimal solution we

require that a minimizing sequence of admissible trajectories, say {xk(.)} +.:', be relatively compact in 3.n appropriate topology. In ordinary control proble~s: this topology is the weak topology in AG,oc , the space of locally absolutely continuous

76

BVPs FOR FDEs

functions. For neutral systems this has to be modified since a solution to such a differential equation is not necessarily differentiable. To achieve our results it is natural to assume that certain growth conditions be satisfied which insure that the corresponding sequence

l(t) = D(t, x~) k = 1,2, ...

is weakly compact in AC,oc and as a consequence we can extract a subsequence, still called {gkO} t~, that converges uniformly on compact subsets of [0, +00). This will

of course affect the behavior of the corresponding sequence of trajectories, {xk (.) }+oo . k=l

In particular we will n~ed that the latter sequence, or at least some subsequence of it, converges pointwise ,in C to an admissible trajectory, xOO. Fortunately, since we have a fixed initial condition and since the function D( .,. ) is atomic at zero, any condition which insures the convergence of the sequence {gk (.)} t~ is sufficient to

guarantee the convergence of the corresponding sequence {xkO}+oo to an admissible k=l

trajectory x*(·) . To establish this fact we refer the reader to Angell [2] and remark that the only modification to Angell's discussion is that a diagonalization procedure must be performed to generalize his result to be applicable here. With these brief comments we give the following definition and theorem .

Definition 2 We say that the growth condition (-y) holds if for any t > ° and T > ° there exists a nonnegative Lebesgue integrable function 1/Jl,T (.) defined on [0, T] such that

If(t,</>,u) l :S 1/Jl,T(t) + tfO(t,</>,u)F(t,O) (6)

a.e. in t E [O,T], (t,</>,u) E M.

Theorem 1 Assume that A is closed and that the growth condition (-y) is satisfied. Then if rtK denotes the set of all admissible pairs for which J (x, u) :S J( , where J( > ° is a fixed positive constant, the set

HK = {g(.): g(t) = D(t,xt} , {x,u} E rtK }

is a set of equiabsolutely continuous functions on each interval [0, T], T > 0.

Proof: Let T > ° and € > ° be given. From the growth condition (-y), for

u = ~€(I{ + 1 t1 there exists 1/J",TO E L1 ([0, T]j IR+) such that for almost all t E [0, T], (t, </>, u) E M we have

If(t, </>, u)1 :S 1/J",T(t) + u f°(t, </>, u)F(t, 0).


Then for each measurable subset E C [0, T] and {x, u} E OK we have

i 1 ~g(t)1 dt i 1 ~ D(t, Xt)1 dt = i If(t, Xt, u(t))1 dt

~ i[?jJU,T(t)+(1fO(t,xt,u(t))F(t,O)] dt

~ i [?jJU,T(t) + (1.f(t,Xt,u(t))F(t, l r(s,xs,u(s))ds)] dt

< i ?jJu,T(t) dt

+(1 JE [fO(t, Xt, u(t))F(t, l r(s, x., u(s)) ds)] dt

~ i ?jJ",T(t) dt

+(11+00

[fO(t, xt,u(t))F(t, l r(s, x., u(s)) ds)] dt

~ i?jJ",T(t)dt+(1I<

f I<t ~ iE?jJ",T(t)dt+ 2(J<+1)

~ i ?jJ",T(t) dt + ~. By the integrability of ?jJ",T(-) there exists h> 0 such that, if meas(E) < h, then

i ?jJ",T(t) dt < ~ . Thus for this choice of E, we have

J 1 ~g(t)1 dt ~ i ?jJ",T(t) dt + ~ ~ t

77

which implies that the set of functions !tg(t), for g(.) E HK is equiabsolutely integrable from which we immediately conclude that HK is an equiabsolutely continuous set of functions on [0, T] . 0

3. Orientor Field Problem and Lower Closure

In addition to compactness, the other component necessary to establish the existence of a minimum is that the objective function must be lower semicontinuous. In optimal control theory the standard procedure is to "deparametrize" the model and consider a related orientor field problem. To do this we introduce the set-valued mapping _ 1R2+n

Q+ : A x IR -+ 2 defined by

-+ { ° 1 Q (t,cP,Y) = (z ,z ,z)

.1 ~ r(t, cP, U), z = f(t, cP, U) , u E U(t, cP)}. (7)

78

BVPs FOR FDEs

Observe that if {x(·), u(·)} is admissible then we have

d d -(xO(t), dty(t), dtD(t,Xt)) E Q+(t,Xt,y(t)) a.e. t ~ 0

where for t ~ 0

XO(t) r(t, Xt, u(t))

y(t) lr(s,x.,u(s))ds .

Thus we see that we obtain a solution of the "orientor field equation"

(ZO(t),Zl(t),Z(t)) E Q+(t,xt,y(t)) a.e. t ~ o. To state the appropriate lower closure result we must assume that the set-valued mapping Q+(-,.,.) enjoys certain regularity and convexity hypotheses. We collect these in the following assumptions:

Assumption 1 (Orientor field condition) For every t > 0 the set

A(t)~{<p1 ,O: 6>0

1/> E A(t),e E [0,+00), II (1/>,e) - (<p,y) II < 8}] holds for every , +00) : 1/> E A(t), II1/> - <plloo < e}l 6>0

Here cl denotes the closure and II . II denotes the usual norm on the product product C x m, inherited from the topologies placed on C and m respectively.

Assumption 2 For each (t,<p,y) E A x [0,+00]' the set Q+(t,<p,y) is wnvex.

With these assumptions we can now state the desired lower closure theorem.

Theorem 2 (Lower Closure Theorem) Assume that Assumptions 1 and 2 and the

growth condition (-y) hold and let {x k( . ), uk

(-) }:: be sequence of admissible pairs for which the following hold:

l(t) D(t, x~) -+ g*(t) weakly in L}oc([O, +00), JRn) - liminfJ(xk(.),uk(.)) < +00.

11:-++00

79


Then there exists an admissible pair {x·(·),u·(·)} such that

g·(t) D(t,x;) a.e. t > 0

J(x·( .),u·(.)) :::; i.

Proof: The proof of this result is similar to the corresponding result, Theorem 4.1 (see also Theorem 4.3), in Balder [4] with appropriate modifications necessary to account for the fact that the set-valued mapping is defined on a subset of IR x C x IR, an infinite dimensional space. Additionally, we observe that it is also necessary to apply the work of Angell concerning the compactness of the trajectories to insure the existence of the trajectory x·( ·). We further note that these modifications are valid as the necessary theory required by Balder is completely outlined in [5], Corollary 3.6. 0

Remark 1 The use of Lower closure theorems of the type described above is now a standard tool used to establish the existence of an optimal solution to problems of optimal control. These theorems have their roots in the now classical lower semicontinuity results of L . Tonelli in the early part of this century as well as the work of E. J. McShane. In their present form these theorems were first announced by L . Cesari and we refer the interested to [11 J for a detailed treatise concerning these matters. The theorem given above is due to Balder UJ who introduced a novel technique in establishing lower closure theorems that are applicable in a wide variety of situations. We remark however, that his arguments also have their origins in the works of Tonelli and McShane. We refer the reader to UJ for the relevant discussion and references.

4. The Existence Result

We are now ready to state our existence result.

Theorem 3 Assume that Assumption 1 and 2 and the growth condition (I) hold and that there exists at least one admissible pair. Then there exists an admissible pair, {x·, u·}, such that for all other admissible pairs, {x, u} , we have

J(x·,u·):::; J(x,u).

Proof: We begin by observing that since f°(-"") and F(·,·) are nonnegative functions and since there exists at least one admissible pair we have

0:::; inf J(x, u) < +00. {x,u}EO

Thus there exists a sequence of admissible pairs {xk, uk}+oo such that k=l

lim J(x k, uk) = inf J(x, u) . k ..... +oo {x,u}EO

80

BVPs FOR FDEs

As a result of the lower closure theorem and the growth condition (-y) we can assume, without loss of generality, that there exists an admissible pair {x', u'}, such that

x~ --> x;, in C a.e. t 2: 0, and

D(t,x~) --> D(t,x;) weakly in Ltoc([O,+oo),lRn)

both hold as k --> +00. Moreover we also have,

J(x',u') ~ lim J(xk,uk) = inf J(x,u), k_+oo {", .. }eo

giving us that the pair {x', u'} is an optimal solution as desired. 0 To demonstrate the utility of our results we present the following general example.

4.1 Example

In this example we consider the problem of minimizing an integral function

J(x,u) = 10+00

fO(t,xt,u(t))F(t,lr(s,x.,u(s))ds) dt

over all pairs of functions satisfying

f(xt, u(t)), a.e. t 2: 0

</>(s), for -h~s~O

(t,Xt) E A = [0,+00) x X, for t 2: 0

u(t) E U(t,Xt), a.e. t 2: O.

In the above we assume that in addition to the regularity conditions indicated in section 1 we have the following:

1. There exists p 2: 1 and a positive locally integrable functions o{), (3( ,), a(·) from [0, +00) into [0, +00) such that

r(t, t/J, u) 2: a(t)lul p+1, and

If(t , t/J, u)1 ~ Q(t) + (3(t)luI P

holds for almost all t 2: 0, and

(t,t/J,u) EM = {(s,~,v):, (s,O E A, v E U(s,~)} .

2. The sets

ij+(t, </>,y) = ((ZO,zl,z) Zl

ZO 2: fO(t, </>, u)F(t,y), 2: r(t, </>, u), z = f(t, </>, u), u E U(t, </>)}

are convex and satisfy the upper semicontinuity conditions property (K) . (N.B. Explicit conditions for this to hold are given in Cesari [11]) .

81


3. There exists at least one admissible pair {x, u}.

To establish the existence of an optimal solution we need only show in this case that the growth condition (-y) holds. To see this we let f > 0 be fixed and define, for t 2:: 0,

(3(t)p+l ¢<.T(t) = o:(t) + ()F( ). w t t,O

Observe that if (t,¢,u) EM and lui::; (3(t)/(w(t)F(t,O)) then we have

Ij(t, ¢, u)1 ::; o:(t) + (3(t)luIP

(3(t)P+1 ::; o:(t) + w(t)F(t,O) + fjO(t,¢,u)F(t,O)

¢<.T(t) + fjO(t,¢,u)F(t,O)

On the other hand if (t,¢,u) E M and lui 2:: (3(t)/(w(t)F(t,O)) then we have

fP(t,¢,u)F(t,O) > w(t)F(t,O)luIP+1 (3(t) P

2:: w(t)F(t,O) () ( )Iu l w t F t,O (3(t)luIP

From this it follows that

Ij(t, ¢, u)1 ::; o:(t) + (3(t)l u IP

(3( t)p+l ::; o:(t) + w(t)F(t,O) + fjO(t, ¢, u)F(t, 0)

¢<.T(t) + fjO(t, ¢, u)F(t, 0).

Thus we see that the growth condition (,) holds for this problem and therefore the existence result given above allows us to conclude that there exists an optimal solution.

5. Concluding Remarks

In the above we have provided conditions for which a minimum exists for a class of infinite horizon optimal control problems of Lagrange type in which the admissible states are required to satisfy a nonlinear functional differential equation of neutral type. The objective functional, described by a convergent improper integral, is of "recursive-type" and contains a nonlinear discount factor which is influenced by not only the past history of the state, but also by the past history of the control. These results are obtained by using a new lower closure theorem of Balder [4] and uses the now classical seminormality, convexity, and growth hypotheses found in the works of Cesari, Balder, and others. These results extend the works of Angell [1] and those of

82

BVPs FOR FDEs

Baum (7), Bates [6], Balder [3], [4], and Carlson [10]. We remark that we have not considered boundary conditions at infinity but that our results are applicable for any closed class of admissible pairs (i.e., a set of admissible pairs which is closed in an appropriate topological sense). However, it is clear that results analogous to those of Baum [7] and of Carlson [10] could be established. For brevity we leave these results to the interested reader. An avenue of further research would be to consider the case in which the objective functional is a divergent improper integral for all pairs of functions {x, u} which satisfy Eqs. (2)-(5). In this case it is necessary to consider one of a hierarchy of weaker types of optimality, such as overtaking optimality. We refer the reader to [9] for a detailed treatment of these concepts weaker concepts for a variety of systems, including some of retarded type.

References

1. T. S. Angell, Existence theorems for optimal control problems involving functional differential equations, Journal Optim. Theory Appl. 7 (1971), 148-169.

2. T.S. Angell, Existence theorems for hereditary lagrange and mayer problems of optimal control, SIAM J. Control Optimization 14 (1976), 1-18.

3. E. J. Balder, An existence result for optimal economic growth, Journal Math. Anal. Appl. 95 (1983), 195-213.

4. E. J. Balder, Existence of optimal solutions for control and variational problems with recursive objectives, J. Math . Anal. Appl. 178 (1993), 418-437.

5. E. J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Optimization 22 (1984), 570-598.

6. C.R. Bates, Lower closure and existence theorems for optimal control problems with infinite horizon, J. Optim. Theory Appl. 24 (1978), 639-649.

7. R.F. Baum, Existence theorems for lagrange control problems with unbounded time domain, J. Optim. Theory Appl. 19 (1976), 89-116.

8. R.A. Becker, Boyd J .H., and B.Y. Sung, Recursive utility and optimal capital accumulation I: Existence, J . Econ. Theory 47 (1989) , 76-100.

9. D. A. Carlson, A. Haurie, and A. Leizarowitz, Infinite horizon optimal control: Deterministic and stochastic systems, 2nd ed., Springer-Verlag, New York, 1991.

10. D.A. Carlson, Infinite-horizon optimal controls for problems governed by a volterra integral equation with state-and-control dependent discount factor, J. Optim. Theory Appl. 66 (1990), 311-336.

83


11. L. Cesari, Optimization-theory and applications: Problems with ordinary differential equations, Applications of Applied Mathematics, vol. 17, Springer-Verlag, New York, 1983.

12. E. N. Chukwu, Mathematical controllability theory of the growth of wealth of nations, Japan J. Ind. Appl. Math. 11 (1994), no. 1, 87-111.

13. L.G. Epstein, A simple dynamic general equilibrium model, J. Econ. Theory 41 (1987), 68-95.

14. L.G. Epstein and J.A. Hynes, The rate of time preference and dynamic economic analysis, J. Political Economy 91 (1983), 611-635.

15. R. Frisch and H. Holme, The characteristic solutions of a mixed difference and differential equation occurring in economic dynamics, Econometrica 3 (1935), 225-239.

16. J.K. Hale and S.M. Lunel, Introduction to funct ional differential equations, Applied Mathematical Sciences, voL 99, Springer-Verlag, New York, New York, 1993.

17. M. Kalecki, A macrodynamic theory of business cycles, Econometrica 3 (1935), 327-344.

DYNAMICAL SPECTRUM FOR SKEW PRODUCT FLOW IN BANACH SPACES

S. N. CHOW And H. LEIVA School of Mathematics-CDSNS, Georgia Tech

Atlanta, Georgia 30332 USA

Abstract

In an earlier paper [1] we characterized the dynamical spectrum for Linear Skew-Product Semiflow in infinite dimensional Banach spaces. It was proved the spectrum is always a closed set, but it could be empty. Also we investigate the relation between the dynamical spectrum and the Lyapunov exponents. In this paper we shall characterize the dynamical spectrum for Linear SkewProduct

Flow 1r = «(),17) in a Banach Bundle £ = X x 0. The fact that 1r is a flow allows us to prove that the spectrum is a nonempty compact set and get more information about it, also we can tell more about the Lyapunov exponents. Finally ours results can be applied to hyperbolic partial differential equations and neutral functional differential equations.

1. Introduction

85

In an earlier paper [1] we began an investigation of the dynamical spectrum for time dependent systems in infinite dimensional Banach spaces, using the concept of skewproduct semiflow. That is the case for parabolic partial differential equations and functional differential equations; for that reason in [1] we use the concept of negative continuation and exponential dichotomy used by Sacker-Sell in [13]. In this definition of exponential dichotomy we assume that the unstable manifold has finite dimension and is contained in the set of points that have a unique negative continuation. We characterized the dynamical spectrum and proved that the spectrum is always a closed set, but it could be empty; also, we investigate the relation between the dynamical spectrum and the Lyapunov exponents.

This paper is concerned with the dynamical spectrum for time dependent linear systems whose solutions are globally defined in JR.. This is the case for hyperbolic

This research was partially supported by NSF grant DMS-9005420 and ULA

86

BVPs FOR FDEs

partial differential equations, neutral functional differential equations and abstract ordinary differential equation j; = A(t)x with bounded operator A(t). To study this problem we shall use the unified setting of a linear skew-product flow 7r : £ x lR -+ £ (see Definition 2.1), where £ = Xx0, X is a Banach space, 0 is a compact topological Hausdorff space and 7r is given by:

7r(x,O,t) = ((O,t)x,O .t), t E lR, x E X, 0 E 0

Many people have worked with skew-product flow, and semiflows in infinite dimensional Banach spaces. For example Sacker-Sell in [13) studied the existence of exponential dichotomy for the skew-product semiflow 7r = (<1>,0") on £ = X x 0 . Also, Magalha.es in (7) gave the following characterization of the dynamical spectrum for flows.

Theorem 1.1 (Theorem 2.8 in (7}). Let 7r = (<1>,0") be a linear skew-product flow on £ and 0 a compact connect invariant set, then

(A) ~(0) is a nonempty compact set, and consequently, it is the countable union of disjoint compact intervals.

(B) if 7r is a completely continuous flow, then ~(0) is the finite union of compact disjoint intervals ~(0) = U7=1 [ai, bi).

A different characterization of the dynamical spectrum for skew-product flow in infinite Banach spaces appears in R. T. Rau (9) and Latushkin-Stepin (5), (6). He associates a strongly continuous group to the skew-product flow 7r = (<1>,0") in the following way: Given a skew-product flow 7r = (<1>,0") on £ = X x 0 we can associate a family {T(t)}tElR of linear operators on the Banach space C(0,X) defined by

T(t)f(O) = <1>(0. (-t), t)f(O. (-t)), "It E lR (1.1)

for all 0 E 0 and f E C(0, X) . It is shown that the operator family {T(t)LElR given by (1.1) is a strongly con

tinuous group on C(0,X). Since 7r = (<1>,0") is a flow (two side flow) the definition of exponential dichotomy is

the same as in the finite dimensional case; this allows Rau in [9) and Latushkin-Stepin [5), [6) to give the following characterization of the dynamical spectrum

Theorem 1. 2 Let 7r = (<1>,0") be a skew-product flow on £ = X x 0 and denote G the infinitesimal generator of the evolution group {T(t)}tElR given by (1.1). Then

~(0) = InIO"(T(l}} \ {O}12 Re(O"(G))

87

DYNAMICAL SPECTRUM

In this paper we shall give a different characterization of the dynamical spectrum. Our characterization is an extension of the Sacker and Sell Theorem given in [12]. Here we proved that the spectrum can be written as a countable union of nonempty close disjoints intervals and something more. We show the relation between the spectrum and the spectral sub bundles associated with the corresponding spectral interval. Also, this spectral decomposition can be used to study invariant manifold around an invariant set.

In this paper we follow closely the work done by Sacker and Sell in [12] for the finite dimensional case and the notation used for them in [13] .

Since 11" is a flow (two side flow) the definitions of exponential dichotomy is very simple and we don't have to worry about negative continuation, like in [13], [1] and [7] . This allows us to prove that the spectrum is a nonempty compact set and we can give a simple characterization of the Lyapunov exponents in terms of the dynamical spectrum.

Finally we present some examples of Skew-Product Flow arising from hyperbolic partial differential equations and neutral functional differential equations.

2. Preliminaries

2.1 . Linear Skew-Product Flow

In this section we shall present some definitions, notations, and results about Skew Product flows on Banach Bundles that we will use in the next sections.

Definition 2.1 Let £: = X x 0 be given where X is a fixed Banach space (state space) and 0 is a compact Hausdorff Space. Assume that 0'(8, t) = 8.t is a flow on 0, i.e., the mapping (8, t) -+ 8· t is continuous, 8.0 = 8, and we have 8.(s + t) = (8 . s). t for all s,t E JR. Then we shall call a Linear Skew-Product Flow 11" = (iP,O') on £: as a mapping

1I"(x,8,t) = (iP(8,t)x,8. t), Vt E JR.

with the following properties:

(1) iP(8,0) = /, the identity operator, for all 8 E 8.

(2) limt-+o iP(8, t)x = x, and this limit is uniform in 8. This means that for every x E X and every f > 0 there is a 6 = 6(x, f) > 0 such that lIiP(8, t)x - xII ::; f, for all 8 E 0 whenever 0 ::; t ::; 6.

(3) iP(8, t) is a bounded linear mapping from X into X that satisfies the co cycle identity:

iP(8,s+t)=iP(8.t,s)iP(8,t) 8E0; s,tEJR (2.2)

(4) For each t E JR the mapping of £: into X given by (x, 8) -+ iP(8, t)x is continuous.

88

BVPs FOR FDEs

Properties (2) and (3) imply that for each (x, 0) E £ the solution operator t -+ ~(O, t)x is continuous for t E JR. Indeed one has

II~(O, t + h)x - ~(O, t)xll = II[~(O . t, h) - Il~(O, t)xll

which goes to zero as h -+ O. The cocycle identity (2.1) implies that ~(O,t) is an isomorphism with inverse

~-1(0, t) = ~(O . t, -t) Vt E JR

Proposition 2.1 Let 7r = (~, (7) be a linear skew-product flow on £ . Then there exist constants M ~ 1, a > 0 such that

II~(O, t)11 :::; Mea1tl , 0 E 0, t E JR. (2.3)

2.2. Projectors and Subbundles

A Banach bundle £ with fiber X over a base space 0 with projection P is denoted by (£, X, 0, P), or £ for short, and is defined as follows:

(1) X is a fixed Banach space and 0 is a compact Hausdorff space.

(2) The mapping P : £ --+ 0 is a continuous mapping.

(3) For each 0 E 0, P-l(O) = £(0) is a Banach space, which is referred to as the fiber over O.

(4) For each 0 E 0, there is an open neighborhood U of 0 in 0 and a homeomorphism

T : P-l(U) -+ XxU such that for eachN E U, P-l(N) is a mapped onto XX {N} and T : P-1(N) --+ X x {N} is a linear isomorphism.

(5) The norms 11·11 = 11·110 on the fiber P-l(O) vary continuously in O.

One can use the local coordinate notation (x, 0) to denote d. typical point in a Banach bundle £. By this we mean that (x,O) E £ . This is a shortened way to refer to property (4) above. For any subset F C £ we define the fiber F(O) := {x EX: (x,O) E F} . So £(0) := X x {OJ. Also, we define £0 = {(x,O) E £ : x = O} the zero fiber and for U C 0, we shall defin~ the set F(U) := UoEuF(O).

A mapping P : £ -+ £ is said to be a projection if P is continuous and has the form P(x, 0) = (P(O)x,O) where P(O) is a bounded linear projection on the fiber £(0). For any projector P we define the range and the null space by

'R = 'R(P) = {(x, 0) E £: P(O)x = x}

and N = N(P) = {(x, 0) E £: P(O)x = OJ.

89

DYNAMICAL SPECTRUM

Since P is continuous, this means that the fibers R(fJ) and .N(fJ) vary continuously in fJ. This also means that P(fJ) varies continuous in the operator norm. The following result can be found in Sacker- Sell [13].

Lemma 2.1 Let P be a projector on £. Then Rand.N are closed in £, and one has

R(fJ) n.N(fJ) = {OJ, R(fJ) +.N(fJ) = £

for all fJ E e. Definition 2.2 A subset v is said to be a subbundle of £ if there is a projector P on £ with the property that R(P) = V

In this case W = .N(P) is a Complementary subbundle, i. e., £ = V + W as a Whitney Sum.

Lemma 2.2 Let V C £ with the properties:

(A) V is closed.

(B) V(fJ) is a linear subspace of £(fJ) for all fJ E e. (C) codim V(fJ) is finite for all fJ E e. (D) codim V(fJ) is locally constant on e.

Then V is a sub bundle of £ .

Proof See [13].

2.3. Stable, Unstable and the Initial Bounded Sets

o

Let 7r = (cI>, u) be a given linear skew-product flow defined on £ = X x e. For A E JR we define the shifted flow as follows:

7r" = (cI>>"u), cI>,,(fJ,t) = e-"tcI>(fJ,t) for t E JR, fJ E e.

u" = {(x, fJ) E £ : lIe-"tcI>(fJ, t)xll -+ 0, t -+ -oo}

The set U" is the unstable set, S" is the stable set , and B" is the initial bounded set corresponding to 7r". If A = 0 we shall denote B = Bo, U = Uo and S = So .

We are interested in knowing when S" and U" are complementary invariant subbundles of £. The answer of this question can be formulated in terms of dichotomies.

90

BVPs FOR FDEs

Definition 2.3 A project P on £ is said to be invariant if we have

P(fJ.t)~(fJ, t) = ~(fJ, t)P(fJ), t E JR., fJ E 0

which is equivalent to:

P(8.t) = ~(fJ, t)p(fJ)~-l(fJ, t), t E JR., fJ E 0

(2.4)

(2.5)

Definition 2.4 We shall say that a linear skew-product flow 'Ir = (~,O') on £ has an exponential dichotomy (ED) over an invariant set 0, were 0 C 0, if there is an invariant Projector P on £ and constants k ~ 1, (3 > 0 such that

(2.6)

11~(fJ,t)[I - P(fJ)l~-I(fJ,s)11 :S ke{3(t-sJ, s ~ t (2.7)

for all fJ E 0

Remark 2.1 (1) If 0 = {fJ}, then E.D corresponds to the usual concept of dichotomy

(2) If 0 = 0, then E.D over 0 is equivalent to the splitting of £.

(3) P(fJ) varies continuously over 0. (4) k,(3 depend of 0.

Proposition 2.2 If'lr is a linear skew-product flow on £ = X x 0 admits an exponential dichotomy over 0, then one has that the initial bounded set B = £0 and the correspondent Projector P is such that:

R(P) = 5(0), N(P) = U(0)

£ = R(P) E9N(P) = 5(0) E9U(0)

(The whitney sum of two bundles)

3. The Dynamical Spectrum

Let 0 be an invariant subset of 0 under the flow 0'. Then the resolvent p(0) of 0 under 'Ir is defined as follows:

p(0) := P E JR. : 'irA admits an exponential dichotomy over 0}. The spectrum 2:(0) of 0 under 'Ir is defined as follows

Our main results are the following Theorems:

91

DYNAMICAL SPECTRUM

Theorem 3.1 Let 7r = (~, u) be a skew-product flow on £ = X x 0 and 8 a compact connected invariant subset of 0. Then the following statements are valid:

(A) There is a > 0 such that

1I~((J,t)1I ~ Mealtl , V(J E 0 and t E JR,

and E(8) =f. 0, E(8,1 C [-a,a]

(B) For each set Po < Al < .. . < Am} C p(8) with AO < -a and a < Am such that E(8) n (Ai_I, Ai) =f. ¢ we get that

Vi := Vi(8) = S.d8) n U).;_" i = 1,2,· .. , m

are invariant subbundles of £(8) .

(C) Let 7ri be the restriction of 7r to Vi and Ei(8) the spectrum of (Vi, 7r

i) over 8. Then one has that

(D) Ei(8) = E(8) n (Ai-I, Ai), i = 1,2, · ··, m

(E) E(8) = U~I Ei(8)

(F) Vi(8) n Vj (8) = £0(8), i =f. j

(G) £(8) = VI(8) + V~(8) + ... + Vm (8) (Whitney sum)

In order to get more information about the spectrum we shall put some restriction on the unstable manifold U). for some A E p(8) . Also we will need the following notation:

For A E p(8) we shall define:

E).(8) := E(8) n (--00, A) (3.8)

Theorem 3.2 Assume that dim£(8) = 00, A E p(8), a> 0 is as in part (A) of Theorem 3.1. Then the following statements are valid :

(A) If dimU). = n(A) < 00, then A ~ -a

(B) If dimS). = m(A) < 00, then A ~ a

(C) If A ~ -a and U). =f. £0(8), then A ~ a

(D) If A ~ a and S). =f. £0(8), then A 2: -a

(E) If dimU). --+ n( -a) < 00, as A --+ -a+ then

-a E E(8) and dimU). = n(A) < 00 VA E [-a + 00) n p(8)

92

BVPs FOR FDEs

(F) If dimS" -+ m(a) < 00, as A -+ a+ then

a E ~(0) and dimS" = m(A) VA E (-00, a] n p(0)

(G) If 1::; dimU"o = n(Ao) < 00, then AO E [-a, a] and

m

~(0) = ~"0(0) u (U [ai, bi]) 1:;:::1

m ::; dimU"o = n(Ao)

Moreover:

(H) ~"0(0) = [-a,Ao) n ~(0) (I) If ~(0) C [Ao,a]=> ~(0) = U~l[ai,b;].

(J) &(0) = S"0(0) + Vl(0) + ... + Vm (0),

where V;(0) = U";-l n S,,;> i = 1,2, .. . , mj

and

with

3.1 . Lemmas

(3.9)

Here we shall derive a number of properties of the spectrum and the resolvent set which will be used in the proof of the main theorems.

Lemma 3.1 Let 0 be a compact invariant set in e and A E JR. Then the following statements are valid:

(A) If II~,,(O, t)lI-+ 0 as t -+ +00 for each 0 E 0, then A E p(0), ~(0) ~ (-00, A), and SI"(0) = &(0) for all p. ~ A.

(B) If II~,,(O, t)11 -+ 0 as t -+ -00 for each 0 E 0, then A E p(0), ~(0) ~ (A, +00),

and UA0) = &(0) for all p. ::; A.

Proof We shall prove (A). The proof of (B) is similar. For each 0 E 0, the is T(O) > 0 such that II~,,(O, t)1I <!, t ~ T(O).

93

DYNAMICAL SPECTRUM

Consider x E X fixed with IIxll = 1. By the continuity of ~A(O,t)X with respect to 0 there exist a neighborhood Nx(O) of 0 such that II~A(B,T(O))xll < ~, for all BE Nx(O). Then by the compactness of B we have the following:

e C U~lNx(Oi ) ' T(Ol):::; T(02) :::; . . . :::; T(Om)

We shall put Tj = T(Oi), J = 1,2,· · · m.

We claim the following:

k = sup{II~A(O, t)1I : 0 E e, 0:::; t :::; Tm} < 00 .

In fact . Assume that K = 00. Then there are sequences {On} C e, {tn} C [0, TmJ such that

II~A(On) ' tnll > n , n = 1, 2, 3 . .. .

Since e and [O,TmJ are compact sets we can assume that {On} converges to 00 E e and {t n } converge to t* E [0, T mJ . Then by the Banach-Steinhaus Theorem there must be an element Xo E X so that the set:

is unbounded. On the other hand the definition of skew-product flow implies that

which is a contradiction.

Now fix t ~ 0 and let 0 E e. Then 0 E Nx(Oj,) for some J 1 and II~A(O,Til)xll < (!)2 . In the same way O· til E Nx(Oh) for some J2 and

Now continue this process until one has

1 T = Til + .. . + Ti,:::; t < T + TiCIH ) and II~A(O, t)xll :::; (2f

Since ITl :::; T :::; t and 0:::; t - T :::; Tm, we get the following

1 1 II~A(O, t)xll = II~A(O.T, t - T )~A(O, T )xll :::; k( 2")1 :::; k( 2" )t/TI = ke-ext

where a = -,f-In(!) > o. Therefore we have gotten the following

94

BVPs FOR FDEs

Since k and Q do not depends of x we get that

II~>.(O, t)1I ~ ke-at, 0 E 13, t ~ o.

From here we get that the skew-product flow 7r>. = (~>.,q) has ED over 13, with projections P(O) = I, i.e., A E p(e). On the other hand, if II ~ A, then

II~I'(O , t)1I = e(>.-I')tll~>.(O, t)1I ~ ke-at, 0 E 13, t ~ o.

Therefore,

o

Lemma 3.2 Let 13 be a compact invariant set in 0 . Then the resolvent p(e) is open. Moreover

if A E p(e), then S>. = SI' and U>. = UI'

for all II in a neighborhood of A.

Lemma 3.3 Let 13 be a compact invariant set in 0 . Then the spectrum E(e) is compact. More specifically, there exists an a > 0 such that, if A > a, then A E p(e), and S>. = E and if A < -a then A E p(e) and U>. = E.

Proof Because of Lemma 3.2 we need only to prove that E(e) is bounded. Thanks to Proposition 2.1 we get k ~ 1 and a > 0 such that

II~(O, t)1I ~ ealtl , 0 E 13, t E JR.

Then if A > a we get

1I~>.(O,t)1I ~ ke{a->.)t -+ 0, as t -+ +00,

for all 0 E e. Therefore, by Lemma 3.1 one has that

(a, 00) C p(e) ¢:} E(e) C (-oo,a]

Similarly, if A < -a one has II~>.(O , t)1I -+ 0 as t -+ -00 for all 0 E e. Consequently

(-oo, -a) C p(e) ¢:} E(e) C [-a, 00)

Hence E(e) C [-a, a].

o

Lemma 3.4 Let 0 be a nonempty set in 0 and assume that dim E > 1. Then the spectrum E(e) is nonempty. -

95

DYNAMICAL SPECTRUM

Proof Pick 80 E 8 and set Mo = H(80 ) where

H(80 ) = cl{80 .t : t E JR}

Then Mo is a compact invariant set and clearly E(Mo) ~ E(8). It will be sufficient to show that E(Mo) is nonempty. From Proposition 2.1, we have k ~ 1 and a > 0 such that

114>(8, t)1I ::; ke'>ltl, 0 E Mo, t E JR.

By Lemma 3.1 we get the following:

(a) If A > a, then A E p(Mo), SA(Mo) = &(Mo) and UA(Mo) = &o(Mo) .

(b) If A < -a, then A E p(Mo), UA(Mo) = &(Mo) and SA(Mo) = &o(Mo) . Therefore E(Mo) C [-a,a) .

Next define Ao = infp E p(Mo) : SA(Mo) = &(Mo)}

Then -a ::; A ::; a.

For the purpose of contradiction, let us assume that Ao E p(Mo). Then there are two cases to consider:

(i) SAo(Mo) = &(Mo)

(ii) SAo(Mo) i: &(Mo).

For the case (i) the Lemma 3.2 implies that SA(Mo) = &(Mo) for A in a neighborhood of Ao, which contradicts the definition of Ao.

For the case (ii) one must have UAo(Mo) i: &o(Mo). From Proposition 2.2 we get that

&(8) = UAo (8) + SAo(8), VO E Mo .

Then by using Lemma 3.2 once again, we have UA(Mo) i: &o(Mo) in a neighborhood of Ao. This contradicts the fact that UA(Mo) = &o(Mo) for A > Ao close enough to Ao. Therefore Ao E E(Mo).

o

Lemma 3.5 Let 8 be a compact invariant set in 8. Consider AI, A2 E p(8) with Al < A2·

then [AI, A2) C p(8) and SA(8) = SAl (8), UA(8) = UAI (8)

for all A E [AI, A2).

96

BVPs FOR FDEs

Proof In the same way as the proof of Lemma 8 in [12].

o The following Propositions are easy to prove.

Proposition 3.1 Let A, Band C be subspaces of X. If C ~ A then

An (B + C) = (A n B) + (A n C)

Proposition 3.2 If.xl>.x2 E p(0) and .xl < .x2, then

£(0) = U>.. (0) n S>'2(0) + U>'2(0) + S>.. (0) (3.10)

Lemma 3.6 Let 0 be a compact invariant set in 0 and .xl>.x2 E p(0) with .xl < .x2 .

Then the following statements are equivalent:

(A) There is a J.l E (.xl, .x 2) n ~(0)

(B) U>.. (0) n S>'2(0) # £0(0).

Moreover,:F = U>..(0) nU>.,(0) is an invariant subbundle of £

Proof (A) => (B)

From Proposition 3.2 we have the following

£ = U>.,(0) + S>.. (0) = u>.. (0) + S>..(0) = U>.,(0) + S>'20)

Since U>'2(0) ~ U>.. (0) and S>.. (0) ~ S>'2(0)

then S>..(0) = S>'2(0) and U>..(0) = U>'2(0)

Now we can apply Lemma 3.5, it means that

U>.. (0) = U>.(0) and S>.,(0) = S>.(0)

for .x E [.xl> .x2] C p(0) . Therefore (.xl> .x2 ) n ~(0) = 0 which contradicts (A)

(B) => (A). Define

J.l:= infp E p(0) : S>.(0) = S>'2(0)}.

Then .xl < J.l <.x2 and J.l E ~(0) .

In fact. Lemma 3.2 implies that J.l < .x2.

For the purpose of contradiction, let us assume that p E ~ ( 0 ) . Then there exists a neighborhood of p such that for all X in that neighborhood ~ ~ ( 6 ) = ~ ~ ( 0 ) . Hence, /' E ~ ( 0 ) .

Assume that p < XI. Then we get

Then applying Lemma 3.2 cnd the definition of p we get that (6) sA2 (6). From (B) we have that S ~ , ( Q ) ~ U ~ , (6) # ~o(6). So Ux1 (6) nsx, (6) # Eo(0). Which is a contradiction to the fact that X1 E ~ ( 6 ) . Thus p E (A1, X z )

Finally, since both Ux,(6) and S A , ( ~ ) are invariant subbundles of ~(61, it follows that

F = ~x,(6) n s~,(Q) is also invariant subbundle.

0

Lemma 3.7 Let Q be a compact invariant set in O and let X I , X 2 be chosen so (XI, X z ) fl Z(Q) # 4. Let

?nd + the restriction of T to F. Let k(6) := C ( F ) denote the spectrum of (F,?) over 0. Then

k(6) = c(6) n (A,, A,).

Proof We shall give the proof in three steps.

Step 1. Consider X E p(6) and define

Fs(X) :=3nsx(@) and FU(X) := FnUx(6).

Then Fs(X) and FU(X) are invariant subbundles and

In fact, suppose that A < X I . Then

~ x ( 6 ) c S A , ( ~ ) and ~x,(6) c uA(Q)

From now on we shall omit the argument if it is neccesary, in order to simplify the computation. So

98

BVPs FOR FDEs

and Fu(A) = FnU>. = U>., nS>'2 nU>. = U>., nS>'2 = F

Hence F = F.(A) + Fu(A).

Similarly, if A > A2, then

For all A E [AI, A2J we shall use Proposition 3.2 and the fact that £ = U>'2 + S>'2 = U>., + S>.,. So we have:

Therefore

u>., n (U>'2 + S>'2) u>., n (U>. + S>.) U>. n (U>'2 + S>.,)

U>'2 +F U>. + Fs(A) U>'2 + Fu(A)

U>., = U>'2 + F = U>'2 + Fu(A) + F.(A)

Since Fu(A) + F.(A) ~ F, then F = Fu(A) + Fs(A)

Step 2. p(0) ~ p(0) = 1R \ t(0). In fact, if A E p(0), then there is a projector P : £(8) -+ £(8) and positive constants k and (3 such that

and 1I~>.(e,t)p(e)~~I(e,s)xll:::; kllxlle-.B(t-s), t 2: s, e E 0

1I~>.(e,t)(I - p(e))~~l(e,s)xll:::; kllxlle.B(t-s), t:::; s, e E 0 If i> is the restriction of P to F, then

Therefore, by restricting the above inequalities to all (x, e) E F we obtain

lI(h(e,t)p(e)ci>~I(e,s)xll :::; kllxlle-.B(t-s), t 2: s

1lci>>.(O,t)(I - P(O))ci>~I(O,s)xll :::; kllxlle.B(t-s), t:::; s

Thus *>. admits an exponential dichotomy over 0, i.e., A E p(0). So p(0) C p(0) and therefore t(0) C ~(0) . This completes the step 1.

Since F ~ S>'2(0), then

1lci>>'2(O,t)xll-+O as t-+oo

99

DYNAMICAL SPECTRUM

for all (x, 8) E :F uniformly in x . Hence, applying Lemma 3.1 we obtain that E(0) ~ (-00, A2) and :F.(A) =:F, A ~ A2. In the same way we obtain :F ~ U).., (0) . This implies that

II~).., (8, t)1I -+ 0, as t -+ -00

Using Lemma 3.1 again, we get that E(0) ~ (A}, 00) and :Fu =:F. Then

E(0) ~ (AI, A2) n E(0)

In order to prove the opposite inclusion, it is sufficient to show that : if A E ,0(0) n (A}, A2), then A E p(0) . In fact, suppose that

which is equivalent to :

On the other hand, we already know that E(0) C (A}, A2) ' Therefore,

E(0) n (A}, A2) c E(0) .

Step 3. ,0(0) n (A}, A2) C p(0) . In fact , if A E ,0(0) n (AI, A2), then there is a projector Q : :F -+ :F and positive constants k and (3 such that

1I~)..(8,t)(I - Q(8))~:x1(8 , s)xll::; kllxlleP(t-sl , t::; s (3 .12)

and for all 8 E 0 Consider the projectors :

and 'R.(P) = :F = U).., n S)..2 ' N(P) = U)..2 + S).., .

From the Proposition 3.2 we have

P).. = PI + QP is a. projector on & such that

'R.(P)..) = S).., + S).., N(P)..) = U).. + U)..2

100

BVPs FOR FDEs

where S~ = R(Q) and U~ = N(Q) .

Using (3.11) and (3.12) and the fact that A E (A1> A2), A1> A2 E p(0) one can show that there are positive constants I and a such that

noindent which implies that A E p(0) .

3.2. Proof of Main Theorems

Proof of Theorem 3.1 The statements (A) , (B) and (C) follow from Lemma 3.3, 3.4 and 3.6 respectively. The statement (D) follows from Lemma 3.7.

(E) From Lemma 3.3 we have that

E(0) c [-a, a]

Therefore

m m

E(0) = E(0) n (U(Ai- 1> Ai)) = U E(0) n (Ai-I, A;) = U Ei(0) . i=l i=l i=l

(F) Consider i + 1 ::; j Vi = U),;-l n S~; :::} Vi ~ S),,,

o

by the monotocity of U), we get that Vj C U~; . On the other hand we know that U),; ns),; = £0.

(G) From Lemma 3.3 we have that :

if A > a :::} A E p(0), S),(0) = £(0) and U),(0) = £0(0).

if A < -a:::} A E p(0), U~(0) = £(0) and S~(0) = £0(0).

Also we know that

Therefore

£(0) U),O = U),o n (S~, + U),,) u),o n S)" + U~,

Proof of Theorem 3.2

DYNAMICAL SPECTRUM

VI +U>., = VI +U>., n (S>'2 +U>.,)

VI +U>., ns>., +U>.,

VI + V2 +U>., n (S>'3 +U>'3)

VI + V2 + V3 + .. . + Vm + U>'m VI + V2 + ... + Vm . (U>'m = £O(S))

101

o

(A) - (F) follow easily from Lemmas 3.2 and 3.3. Proof of (G), from Lemma 3.3 we have the following:

If AO < -a, then U>'o = £ and dimU>.o = 00

If Ao> a, then U>'o = £0 and dimU>.o = 0

Therefore Ao E [-a, a).

Now consider the set

such that a < Am, (Ai-I , Ai) n ~(S) =J ~

From Proposition 2.2. we get that: £(S) = S>'o(S) +U>'o(S). Then let us denote by 11">'0 the restriction of 11" to U>'o and ~>'o(S) the spectrum of (U>'o, 11">'0) over S . SO by using Lemma 3.7 we obtain that

Therefore dim Vi ~ 1, which implies that m :=::; dimU>.o ' Now we shall show that the resolvent P>'o(S) of (U>'o, 11">'0) consist of (k + 1) intervals where k :=::; n(Ao) . If P>'o(S) consists of (n(Ao) + 2) intervals:

n(>'0)+2 P>'o(S) = U (e;,d;), di < Ci+l'

i=l

So we get that

Then

102

BVPs FOR FDEs

has dimV, ~ 1. Since 0 is connected, then the dimension dimV, = n, is constant over 0 . On the other hand, V, n Vj =J £0(0). Therefore

dimU>.o ~ nl + n2 + ... + nn+l ~ n(Ao) + 1,

which is a contradiction. Hence p>.0(0) consists of k + 1 intervals with k :s; n. So E>.0(0) is the union of k compact intervals. The remainder of the proof is easy 0

4. Lyapunov Exponents

In this section we shall investigate the relation between the Dynamic Spectrum and the Lyapunov characteristic exponents. For this purpose we shall assume that there exists AO E p(0) such that 1:S; dimU>.o < 00 . •

Consider Po < ... < Am} C p(0), such that 0 < Am and E(0)n(A'_1,A,) =J cPo Then from Theorem 3.2 we get:

(4.13)

( 4.14)

m:S; dimU>.o, Ao E [-a, a] ( 4.15)

( 4.16)

v, = s>.; nU>.._l, i = 1,2, ... ,m ( 4.17)

(4.18)

The spectral intervals [a" b,] have been ordered so that b, :s; ai+l> i = 1,2, ... , m-1.

Let P,: £(0) -+ £(0) denote a projector corresponding to the descomposition (4.4) such that Range (Pi) = n(Pi) = V, and the null space being the sum of the remain Vj and S>'o for j =J i.

Then if P>'o : £(0) -+ £(0) is the projector on £ such that

Hence

103

DYNAMICAL SPECTRUM

Given a point (x,O) E £, x # 0, we shall define the four Lyapunov exponents of (x,O) as follows:

+ - 1 A. (x, 0) = lim -In lIiP(O, t)xll

t .... +oo t ( 4.19)

At(X,O) = lim ~InlliP(O,t)xll t .... +oo t

(4.20)

1 A;(X,O)= lim -InlliP(O,t)xll

t .... -oo t (4.21 )

( 4.22)

Theorem 4.1 (A) If(x,O) E Vi where Vi, is the spectral subbundle associated with Ao and [ai,bi), and x # 0, then the/our Lyapunov exponents (4.7)-(4 .10) lie in [ai,bi]. In particular, if ai = bi then the four Lyapunov exponents agree and the limits

lim ~ In lIiP(O, t)xll = lim ~ In lIiP(8, t)xll t .... +oo t t .... -oo t

exist and are equal to ai.

(B) If (x, 0) E S>'o, x # 0 then the two Lyapunov exponents (4.7) - (4.8) agree and the limits

lim ~ In lIiP(O, t)xll = lim ~ In lIiP(O, t)xll = Ao. t .... +oo t t .... +oo t

Proof The proof of (A) is similar to the prove of Theorem 3 in [12]. In order to prove (B) let us consider (x,O) E S>'o with x # O. Then

lim II iP>'o (0, t)xll = lim lIe->'otiP(8, t)xll = 0 t~oo t ...... +oo

Therefore, there is a constant M > 0 such that

Then o = lim~lnlle->.otiP(8,t)xll = lim [-Ao + ~InlliP(8,t)xll] t t .... oo t

So . 1

hm -In lIiP(O, t)xll = Ao t .... oo t

o

104

BVPs FOR FDEs

Definition 4.1 For all 0 E 0 we define the upper Lyapunov exponent >';-(0) and the lower Lyapunov exponent >.t (0) as follows:

>':;(0) := sup{>.;(x, 0) : x E X, x =1= O}

>.t(O) := inf{>.t(x,0) : x E X, x =1= O}

Theorem 4.2 The upper Lyapunov exponent >';-(0) and the lower Lyapunov exponent >.t (0) associated to 0 E 0 are given by :

( 4.23)

-1 >.t(O) = - lim -In II<J>(O.( -t), t)1I

1_00 t (4.24)

References

1. S.N.Chow, and H.Leiva, Dynamical spectrum for time dependent linear systems in banach spaces, Japan J. of Industrial and Applied Math. 11 (1994), 379-415.

2. S.N.Chow, K.Lu, and J.Maliet-Paret, Floquet bundles for scalar parabolic equations, to appear.

3. J.K.Hale, Asymtotic Behavior of Dissipative Systems, Math.Surveys and Monographs, Vo1.25, Amer.Soc., Providence, R.I, 1988.

4. D.Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, 1981.

5. Y.D.Latushkin, and A.M.Stepin, Linear skew-product flows and semigroups of weighted composition operators, to appear.

6. Y.D.Latushkin, and A.M.Stepin, Weighted Translation Operators and Linear Extension of Dynamical Systems, Russian Math . Surveys 46 (1991) .

7. L.T.Magalhaes, The Spectrum of Invariant Sets for Dissipative Semiflows in Dynamics of Infinite Dimensional Systems, NATO AS! series, No.F-37, Springer Verlag, New York, 1987.

8. A.Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vo1.44, Springer Verlag, New York, 1983.

9. R.T.Rau, Hyperbolic evolution groups and dichotomic evolution families, J. Dynamics Differential Equations, to appear.

105

DYNAMICAL SPECTRUM

10. D.Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Annals of Math. 115 (1982), 243-290.

11. R.J.Sacker and G.R.Sell, Existence of dichotomies and invariant splittings for linear differential systems III, J.Differential Equations 22 (1976 B), 497-552.

12. R.J.Sacker and G.R.Sell, A spectral theory for linear differential systems, J. Differential Equations 27 (1978), 320-358.

13. R.J.Sacker and G.R.Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Dynamics Differential Equations, to appear.

107

ON BOUNDARY VALUE PROBLEMS FOR FIRST ORDER IMPULSE FUNCTIONAL DIFFERENTIAL EQUATIONS'

1. Introduction

ALEXANDER DOMOSHNITSKY Department of Youth Activities

Technion, Israel Institute of Technology Haifa 32000, Israel

MICHAEL DRAKHLIN The Research Institute

The College of Judea and Samaria Kedumim - Ariel, D. N. Efraim 44820, Israel

In the present paper we investigate the following scalar equation with impulses

m+l t .

(.ex )(t) == x'(t) + L l' x(s )d.ri(t, s) = f(t), i=l ti-l

t E [O,b], (1.1)

(3j > 0, j = 1,2, ... ,m. (1.2)

where to = O,tm+l = b,f : [O,b] -+ Rl, is a locally summable function, ri(·,s) : [O,b]-+ R\ are measurable, ri(t,·): [ti_bti) -+ R\ has bounded variation R;(t) = Var!~t;_lri(t,s), where R; are locally summable functions, Var!~t;_lri(ti's) = 0, ri(t, s) = ri(t, t) for t > s. W.l.o.g. set ri(t, ti-d = 0 for i = 1, .. . , m + 1.

The following equation with delayed argument

(Mx)(t) == x'(t) + p(t)x(t - T(t)) = f(t),

x(~) = 0 for ~ < 0,

is a particular case of (1.1) for

{ p(t)

ri(t,s) = 0 for t - T(t) ~ s, for t - T(t) > S,

ti-l ~ s < ti, ti-l ~ s < ti·

(1.3)

(1.4)

The research is supported in part by the Israeli Ministry of Science and by the Ministry of Absorption, Center for Abeorption in Science.

108

BVPs FOR FDEs

Ordinary differential equations with impulses are intensively studied by many authors, see for example the recent monographs by V. Lakshmikantham, D. D. Bainov and P. S. Simeonov [1] and the bibliography there.

Various comparison theorems for solutions of the Cauchy and periodic problems for ordinary differential impulse equations have been obtained in [2] .

An extension of the results of such type to the impulse equations with delay meets essential difficulties, implied by the following reasons. In the case of ordinary impulse equations (T = 0) the graph of the solution consists of the the graphs of nonimpulsive ordinary differential equation on the intervals (ti' t i+1 ). Evidently, the solution x(t) of the homogeneous equation in this case preserves its sign. This fact leads to the conclusion about invariance of sign of the Green's function G(t,s) for different problems, since the cross-sections of G(t, s) (for fixed s) are just the solutions of the homogeneous equation.

The solutions of the equations with deflecting argument in contrast to the ordinary ones may change their sign. This makes their properties essentially different. The oscillation properties of such equations have received the attention of many researchers. Let us remind in this connection the recent monographs by G. S. Ladde, V. Lakshmikanthan, B. G. Zhang [6], and I. Gyori, G. Ladas [7] . The "addition" of impulses in those equations makes the properties of the solutions even more complicated.

The problem of existence of a nonoscillatory solution for the impulse equation with delay

x'(t) + ax(t - T) = 0,

x(t j ) = f3jx(tj - 0),

t E [0, + (Xl ),

j = 1,2, ... ,

x(O = cp(O for e < 0,

(1.5)

(1.6)

where a, T are positive constants, was studied in the well known paper by K. Gopalsamy and B. G. Zhang [3]. They have obtained the following result: there exists a nonoscillatory solution for (1.5) if the two conditions

hold.

1 aT <;,

00

L 11 - f3jl < (Xl, (1 .7) j=l

It is evident that for this statement, it is at least required that f3j --> 1 for j --> (Xl,

i.e. "vanishing" of impulses. One of our results makes it possible to eliminate this essential restriction. Moreover, the proposed approach does not assume that the cofficients p, T are positive constants.

Let us point out that in [3], as well as in many other investigations, the equation (1.5), (1.6), (1.2) is considered. It does not lead to any additional generality in comparison with the equation (1.3), (1.4), (1.2), since the first can be presented in

FIRST ORDER IMPULSE EQUATIONS

our form, where the right hand side r is defined as follows

ret) = J(t) - p(t)rp(t),

{

rp(t - ret)), where rp(t) = 0

t - ret) < 0,

t - ret) ~ O.

109

The idea of considering just the condition (1.4) is that the space of the solutions of the homogeneous equation (1.8), (104), (1.2), where

(Mx)(t) = 0, t E [0,+(0), (1.8)

is one-dimensional. Note that the general solution of (1.1) , (1.2) has the following representation [4]

t

x(t) = X(t)x(O) + J C(t, s)J(s) ds, o

where X(t) is the solution of the homogeneous equations £x = 0, (1.2) , satisfying the initial condition x(O) = 1, and C(t,s) is the Cauchy function of the equation (1.1), (1.2).

In section 2 we propose a theorem on equivalence of several statements in the case that ri(t, .), i = 1, ... , m + 1, are non decreasing functions for t E [0, b] . Some of the statements are nonoscillation of solutions of the homogeneous equation, positivity of the Cauchy function C(t,s), positivity of the Green's function of the periodic problem, existence of a positive function v, satisfying inequality £v ~ 0 and conditions (1.2). Choosing a function v, we get effective tests for nonoscillation and positivity of different Green's functions.

In conclusion let us point out that the positivity of the Green's function could be a base for construction of special monotonic techniques for obtaining comparison results for nonlinear impulse equations with delay.

2. On Sign Properties of Green's Functions

Consider the equation (1.1), (1.2) . Its general solution has the following representation

t

x(t) = C(t,O)x(O) + J C(t,s)J(s), (2.1) o

where C(s, t) is the Cauchy function of (1.1),(1.2). For a fixed s E [0, b] the function C(·,s) is a solution to the homogeneous "s-truncated" equation

J.t. m+l 1t .

(C"x)(t) == x'(t) + • x(r)drk(t,r) + 2~1 ti~l x(r)dr;(t,r) = 0, t ~ s, (2.2)

110

BVPs FOR FDEs

x(tj) = (3jx(tj - 0), j=k,k+l, . .. ,m. (2.3)

where k is an integer such that tk-l ~ s < tk, while C(s,s) = l. Denote by D = D[O,bj(tl,"" tm ) the space of functions x : (0, b] -+ Rt, absolutely

continuous on each interval (ti-I, ti) having a finite left limit at ti. If the boundary value problem (1.1), (1.2) and

x(b) = 0, (2.4)

is uniquely solvable in the space D for every function J E L = L[O,bj, then its solution can be represented in the form

b

x(t) = J G(t, s)J(s) ds, o

where the Green's function G( t, s) of this problem is

G(t )=C( )_ C(b,s)C(t,O) ,s t,s C(b,O) (2.5)

and C(t,s) = ° for t < s. If the periodic problem (1.1), (1.2) and

x(O) - x(b) = ° (2.6)

is uniquely solvable in the space D for every f E L, then its solution can be represented in the form

b

x(t) = J P(t, s)J(s) ds, o

where P(t s)=C(t s)- C(b,s)C(t,O)

, , l-C(b,O) ' (2.7)

The boundary value problem

x'(t) = J(t), t E (0, b],

x(tj) = (3jx(tj - 0), j = 1,2, .. . , m, (2.8)

x(b) = 0,

has the unique solution for every summable f. Denote by Go(t, s) the Green's function of problem (2.8). Note that Go(t,s) = ° for ° ~ s ~ t ~ band Go(t,s) < 0 for o ~ t < s ~ b.

111


Define the operator K : D f-+ D by the equality

b m+l t -

(Kx)(t) = - ! Go(t,s) '£[1' x(r)dTr;(s,r)]ds. o ,=1 t._ 1

(2.9)

Theorem 2.1. Let r;(t,·),i = 1, .. . ,m + 1, be nondecreasing. Then the following assertions are. equivalent: (I) the Cauchy function C(t, s) of (1.1), (1.2) is positive for 0 ~ s ~ t ~ b, (2) a nontrivial solution of the homogeneous equation LX = 0, (1 .2) has no zeros on [0, b], (3) the spectral radius of the operator K is less than one, (..f) problem (1.1), (1.2), (2 ... 1) is uniquely solvable for every f ELand its Green's function G(t,s) is negative for 0 ~ t < s ~ band nonpositive for 0 ~ s ~ t ~ b, (5) (only in the case (31 < 1, . . . , (3k < I) the periodic problem (1.1), (1 .2), (2.6) is uniquely solvable and its Green's function P( t, s) is positive for t, s E [0, b], (6) there. exists a nonnegative function v E D such that

(LV)(t) ~ 0, v(b) -l(Lv)(s)ds > 0, t E [O,b] .

Proof. We prove Theorem 2.1 according to the following scheme:

(6) =? (3) =? (4) =? (6), (3) =? (1) =? (5) =? (2) =? (6).

(6) =? (3). The function v satisfies the integral equation

v(t) - (Kv)(t) = t/J(t), t E [0, b],

where b

t/J(t) = v(b) - ! Go(t, S)(LV)(S) ds. t

Since 1/J(t) > 0, the spectral radius p(I<) of the operator K is less than one [5] .

(3) =? (4). The equation X = Kx + g, where

b

g(t) = - ! Go(t,s)f(s)ds, t

is equivalent to problem (1.1), (1.2), (2.4) . The condition p(K) < 1 implies that this problem is uniquely solvable and its solution can be represented in the form

b

x(t) = g(t) + ![G(t,s) - Go(t, s)lJ(s) ds. o

112

BVPs FOR FDEs

If f:::; 0, then 0:::; 9 :::; x. Consequently, G(t,s) :::; Go(t,s) .

(4) =? (6). In order to prove this assertion we set

b

v(t) = - J G(t,s)ds.

° (3) =? (1) . Define an operator ICJj: D[v,Jj] t--> D[v,Jjj, where [v , J.Lj ~ [0, b], by the equality

where G~Jj(t, s) is Green's function of the problem

x'(t) = f(t), t E [v,J.Lj,

k is an integer such that tk-l < s < tk, t8 = J.L, where D[v,Jj] is the space of functions x E D[O,b](t1, ... ,tm ) with support [v,J.Lj .

The proof is based on the following assertion. Claim. If p(K) < 1, then p(I{VI') < 1 for [v, J.Lj ~ [0, bj. To verify the claim, by virtue of (3)=?( 4) problem (1.1), (l.2),(2.4) is uniquely solvable and its Green's function G(t,s) is nonpositive for 0 :::; s :::; t :::; b and negative for 0:::; t < s :::; b.

The function b

v(t) = - J G(t,s)ds

° is a positive solution of the boundary value problem

(.cx)(t)=-l, t E [0, b],

j = 1,2, ... ,m,

x(b) = o.

It is clear that

v'(t) + J~' x(s)dsrk(t, s) + L:f=k+1 JiLl x(s)d.r;(t, s) :::; -1 , t E [v , J.L]'

113


where k is an integer such that tk-l < s < tk, to = J.L . Now, following the proof of the assertion (6)=>(3), we obtain that the spectral

radius p(Kv ,,) is less than one. This establishes the claim. We continue the proof of the assertion (3) => (1). Let us assume the contrary. Then there exist v and J.L ( v < J.L) such that

C(v,J.L) = o. In this case u(t) = C(t,v) is a characteristic function of the operator K v" and Lemma 2.1 implies that p(/{v,,) < 1.

(1) => (5). The periodic problem (1.1), (1.2), (2.6), is uniquely solvable if and only if C(b,O) #- 1. Since C(t,s) > 0 for 0::; s::; t::; b, then obviously C(b,O) is also positive. Since 131 < 1, . .. , 13k < I, C(·, 0) is nonincreasing and C(b,O) < C(O,O) = 1. Now (2.7) implies positivity of P(t,s).

(5) => (2) . Setting t < s in (2 .7), we obtain that the function C(t,O) can not have a zero on [0, b] .

In order to prove (2) => (6) we set v(t) = C(t,O). Note that the assertion (1) => (2) is obvious. Theorem 2.1 is proved. Denote R.;(t,s) = Vare=ori(t,e),

+( )_ R.;(t , s)+ri(t,s) ri t, s - 2 '

_( )_R.;(t,s)-ri(t,s) ri t, s - 2 .

Let C+( t, s) be the Cauchy function of the equation

x'(t) + E l~, x(s)d.rt(t, s) = f(t), t E [0, b],

X(ti) = f3iX(ti - 0), i = l, .. . ,m.

(3.10)

Theorem 2.2. IfC+(t,s) > 0 forO::; s::; t::; b, then C(t,s) ~ C+(t,s) > 0 for o ::; s ::; t ::; b. Proof. Using the substitution

t

x(t) = J C+(t, s)z(s) ds, o

we obtain the following equation in the space L,

z(t) - (Hz)(t) = f(t), t E [0, b],

where the operator H : L f-+ L is defined by the equality

(Hz)(t) = % l~,[1' C+(S,T)Z(T) dT]d.r;-(t, s) .

114

BVPs FOR FDEs

The spectral radius of the Volterra operator H: L t-+ L is equal to zero. Using the Neumann series, we obtain

z = (I - Htl f = (I + H + H2 + ... )f

and t

x(t) = J C+(t, s)[(I + H + H2 + . .. )f](s) ds . o

The positivity of C+(t , 3) and of the operator H implies that C(t, s) ~ C+(t, s) > 0. Remark 2.1. Obviously, letting b tend to infinity, we can obtain, with the help of Theorem 2.1 and Theorem 2.2, the properties of C(t, s) and nonoscillation of the solutions on the infinite interval [0, +00) .

Consider the following impulse equation of neutral type

(.cx)(t) == x'(t) + p(t)x(t - r(t)) - q(t)x'(g(t)) = f(t) , t E [0, +00),

x(tj) = (3jx(tj - 0) , (3j > 0, j = 1,2, . .. ,

x(o = x'(O = 0, ~ < 0,

(2.11)

where g,p,J : [0,+00) --+ Rl are locally summable, r : [0,+00) --+ [0, 00) is measurable, g(t) ~ t . Let q and 9 be such that the spectral radius p(s) of the operator S : L--+L

(Sy)(t) = {q(t)y(g(t)) , g(t) ~ 0, 0, g(t) < 0,

is less than one unifromly for every b E (0, +00) . For the necessary and sufficient conditions for the inequality p(S) < 1 to be valid see [8] . In the case p(S) < 1 the neutral equation can be rewritten as follows

(Cx)(t) == x'(t) + ((I - StITx)(t) = (I - S)-I f(t) , t E [0, 00),

where

(Tx)(t) = { p(t)x(t - r(t)), t - r(t) ~ 0, 0, t-r(t)<O.

Using the general representation of linear bounded operators acting from C to L we can rewrite this equation in form (1.1). _ Let q ~ 0, then (1 - st l is a positive operator and the inequality .cv ~ ° implies

.cv :::; ° for every v. Now it is easy to prove Theorem 2.1* Let q ~ 0, p ~ 0, then for the neutral equation {2.11} assertions 2,3,4,5,6 of Theorem 2.1 are equivalent, and each of them implies assertion 1.

Denote


h(t) = min{g(t), t - r(t}}, p+(t) = max{p(t),O},

d_(t) = min{i: tj E (h(t),t)} ,

d+(t) = max{i: tj E (h(t),t)} ,

d+(t)

B(t) = IT /3j. j=d_(t)

Corollary 2.1. Let q ~ 0 and

t J p+(s)dsS; 1+lnB(t)~ln(qe+1) h(t)

115

(2.12)

for almost all t E [0, +00) . Then the nontrivial solutions of the homogeneous equation {2.11} have no zero on [0, +00) .

Note that the inequality (2 .12) cannot be improved even in the case of equation without impulses and neutral part [6,7]. Proof. In order to prove the inquality C+(t, s) > 0 for 0 S; s S; t < + 00, we set

exp ( -e I p+(s) dS),

v(t) = /31 exp ( -e I p+(s) dS),

(2.13)

Theorem 2.2 implies that C(t,s) ~ C+(t,s) > 0 for 0 S; s S; t < +00. Obviously, that nontrivial solution x(t) = C(t,O) is positive on [0,+00). The corollary has been proved, since all nontrivial solutions to (2.11) are proportional. Remark 2.2. From the proof of Corollary 2.1 it is clear that inequality (2.12) implies existence of a nonosciliating solution to equation (1.5), (1.6), (1.2), considered by K. Gopalsamy and B. G. Zhang [3J. On the other hand their result (see inequalities (1. 7)) obtains new applications. For example, inequalities (1.7) guarantee that the Cauchy function C(t,s) is positive. Remark 2.3. In the case /3j > 1, for all j, impulses can improve nonoscillation properties of solutions. Let us formulate a result for the equation (1.3).

Denote ~ = sup(tj+l - tj),

j /3 = inf /3 . j J, h = In/3.

116

BVPs FOR FDEs

Corollary 2.2. Let there exist a natural k such that

7" > k6., t

J +() d 1 + kh p s s~--, e

t E [0, +00). t-T(t)

Then the homogeneous equation (1.3), (1 .4), (1 .2) has a nonoscillating solution. Consider equation (1.3), (1.4) and

j = 1, 2, ... (2.14)

Theorem 2.3. Let m = 1, Q; 2: 0 for j = 1,2, . .. and inequality (2. 12) be fulfilled. Then the inequalities f(t) 2: 0 for t E [0, +00) and x(O) > 0 imply that x(t) > 0 for t E [0,+00). Proof. The proof of this theorem is based on the following idea:

Let a function x be a solution of the equation (1.3) , (1.4), (2 .14), and let bI be the first zero of x, i.e. x(t) > 0, t E [0, bI)' x(bd = O. Then there exist 1;, j = 1,2, ... , such that x is also a solution of the the equation (1.3), (1.4) and

j = 1,2, ... (2.15)

for the interval [0, bI)' The inequalities Q; 2: 0, j = 1,2, . .. , imply that 1; 2: (3; > 0 for j = 1,2, .. .. Obviously, for the equation (1.3) , (1.4) , (2.15) the conditions of Corollary 2.1 are

fulfilled for the interval [0, bd . Corollary 2.1, Theorem 2.1 and Theorem 2.2 imply that the Cauchy function C-y{t,s) of the equation (1.3), (1.4), (2. 15) is positive for o ~ s ~ t < +00.

The solution x of the equation (1.3), (1.4), (2.14) can be represented in the form

t

x(t) = J C-y(t, s)f(s) ds + C-y(t,O). (2.16) o

Now the inequalities C-y (t , s) > 0, f(t) 2: 0, x(O) > 0 imply that x(t) > 0 for t E [0, btl . This contradicts the assumption x(bI ) = O. Remark 2.4. It is clear from the proof of Theorem 2.3 that nontrivial solutions of the homogeneous equation (1.8) , (1.4), (2.14) have no zero on [0, +00) .

References

1. V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.

2. S. Hu and V. Lakshmikantham, Periodic boundary value problems for second order impulsive differential systems, Nonlinear Analysis, Theory, Methods and Applications 13 (1989), 75-89.

117


3. K.Gopalsamy and B.G.Zhang, On delay differential equations with impulses, J.Math. Anal. and Applications 139 (1989), 110-122.

4. A.Domoshnitsky and M.Drakhlin, Nonoscillation of first order impulse differential equations with delay, submitted.

5. P.Zabreiko, et al., Integral Equations, Moscow, Nauka, 1968.

6. V. Lakshmikantham, G. S. Ladde and B. G. Zhang, Oscillation Theory of Differential Equation!; with Deviating Arguments, Dekker, New York, 1989.

7. I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford Mathematical Monographs, 1991.

8. M.Drakhlin, The inner superposition operator in the spaces of summable functions, Izvestiya VUZov, Matematika 30 (5) (1986), 18-24 (in Russian).

LINEARIZED PROBLEMS AND CONTINUOUS DEPENDENCE

JEFFREY A. EHME Department of Mathematics, Spelman College

350 Spelman Lane S. w., Atlanta, GA 30314 USA

1. Introduction

119

In this paper, a technique for establishing the continuous dependence of solutions on boundary data will be considered. This technique can be used for many types of different problems. The types of problems considered in this work include functional boundary value problems, boundary value problems for ordinary differential equations, along with boundary value problems for quasi-differential equations and difference equations.

The techniques used in this paper possess two properties that are very useful. The first property is no Green's function is required. This means the problems do not need to be converted into integral equations before this technique can be applied. Since the existence of a Green's function is equivalent to uniqueness for the homogeneous problem, it is not necessary to make this assumption in order to obtain continuous dependence. However, it will be necessary to make a uniqueness assumption on a linearized problem.

There also will be no need to assume solutions are bounded a priori. If one makes use of a priori bounds, then similar continuous dependence theorems can be proved using topological transversality methods. For related ideas, see [4,7].

Beginning in section 2, a series of preliminary definitions will be considered. In order for these definitions to apply to the four diverse areas to be considered, these definitions will necessarily have to be abstract. Following these abstract definitions, concrete examples will be considered.

The third section contains the proof of the general continuous dependence theorem. In section 4, the general theorem is applied to the specific cases previously considered.

120

BVPs FOR FDEs

2. Definitions and Examples

In this section, the spaces and functions used in the proof of the main theorems are defined. After the abstract definitions are completed, several specific examples are considered.

Let P denote a finite or infinite dimensional Banach space. This Banach space will represent the space of parameters. Let X, Z, and W also denote Banach spaces. In the most of the applications, these spaces will be function spaces. The space X is the solution space, while the spaces Z and Ware test spaces used to verify that potential solutions satisfy the necessary properties. The norms on these spaces are denoted I . Ip, I · Ix, I · Iz , and I . Iw respectively.

The space P x X is a Banach space under the product norm

I (p,x) Ipxx= max{1 pip, I x Ix} .

Assume

F,L: P X X -t Z

are n-times continuously differentiable functions. Also assume for fixed pEP, L(p,·) is a linear transformation. Further assume

B:PxX-tW

is n-times continuously differentiable. Define

G:PxX-tZxW

by G: (p,x) -t (L(p,x) - F(p,x),B(p,x)) .

It is important to note that zeros of the function G are equivalent to solutions of the abstract system of equations

L(p,x) B(p,x)

F(p,x) O.

(1) (2)

Also note there is no loss of generality in taking all the parameters from the same Banach space. Before we continue the development, we point out that there are several special cases which have been of interest to various researchers.

121

CONTINUOUS DEPENDENCE

2A. Boundary Value Problems for Ordinary Differential Equations

The first special case to be considered involves boundary value problems for the nth-order differential equation

yIn) = f(t,y,y', ... ,y(n-I),A), (3)

where f(t, y, y', .. . , y(n-I), A) is n-times continuously differentiable on I x Rn+l, and I = (a - E, b+ E) is an interval of the reals containing [a, b]. Let il , . .. , im be positive integers with 2 :::; 2:i:,1 i l = n and assume tl < t2 < ... < tm are contained in the interval (a, b), then a boundary condition of the form

(4)

where Ij = O, ... ,ij -l,j = 1, ... ,m is called an (il, ... ,im ) conjugate boundary condition. If instead we are given points tl :::; t2 :::; ... :::; tn, then boundary conditions of the form

(5)

are called right focal boundary conditions. Various properties of these boundary value problems have been studied extensively. See [11,15,16,17] .

For this special case, X = C(n)[a, b], Z = C[a, b] and W = Rn. The norm on C(n)[a,b] is the usual norm I y 1= maXo$'$n{maXtE[a,bj{1 y{i)(t) I}}. The space of parameters is given by P = R x Rn x Rn The parameters in this example represent boundary points, boundary values, and A. The maps Land F, which map

L, F : R x R n x R n x C(n)[a, b]--> C[a, b]

are defined by L(A, t l , ... , tn, YI, ... , Yn, y(t)) = yIn)

and F(A, t l , ... , tn, yl, ... , Yn, y(t)) = f(t, y, y', .. . , y(n-I), A).

For conjugate boundary conditions, the mapping B can be defined by

B(A, t l , ... , tn, yl, ... , Yn, y(t)) =

( (t) C ('I-I)(t )-C . 'm-I(t )-C· ) y 1 - 1,O, ···, Y 1 1,ll-t,· ·" Y Tn m,lrn-1'

For a right focal boundary condition, the mapping B is defined by

B(A, tl, ... , tn, YI,···, Yn, y(t)) = (y(tl) - YI,···, y(tn)(n-I) - Yn).

Nonlinear boundary value problems for nth order differential equations have also been considered in [2,18]. These involve problems with boundary conditions of the form

9.(A, y(t.), y'(t.), ... , y(n-I)(t;)) = Y" (6)

122

BVPs FOR FDEs

where 9i(XO' Xt, ... , xn) : Rn+l --+ R is n-times continuously differentiable. For boundary value problems of this form, the mapping B is defined by

B(A, tt, . .. , tn, yt, . .. , Yn, y(t)) = (91 (A, y(td, y'(t1 ), . .• , y(n-l)(t1 )) - Yl,· ·· , 9n(A, y(tn}, y'{tn), .. . , y(n-l)(tn)) - Yn).

2B. Functional Differential Equations

For this case, we consider a problem of the form

y'(t) = f(t,y(t),y(tP(t))),

My(r(t)) + My(u(t)) = f(t),

(7)

(8)

where f(t,yt,Y2) : [a,b] x R 2n --+ Rn is a continuously differentiable function, tP E C([a, b], R), r, 17 E C([Q, a], R), f E C([Q, a], Rn), and M, N are real constant n x n matrices. These boundary conditions generalize the usual initial delay conditions. The boundary conditions also generalize the functional boundary conditions given by Hale in [8]. Problems of this sort were considered by Ehme and Henderson in [3].

For this case, X = C 1([Q, b], Rn) . The space of parameters is given by P = C([a,b],R) x C([Q,a],R)2 X C([Q,a],Rn). Notice that tP,r,u, and f are being considered parameters. The test spaces are given by Z = C([a, b], Rn) and W = C([Q, a], Rn). The functions Land F, which map

are defined by L(tP,r,u,f,y(t)) = y'(t)

and F(tP,r,u,f,y(t)) = f(t,y(t),y(tP(t))) .

The functional boundary condition becomes the map

B(tP,r,u,f,y(t)) = My(r(t)) + Ny(u(t)) - f(t).

It is easily verified that the functions L, F and B are well defined and map the correct domain spaces into the correct range spaces.

2C. Quasi Differential Equations

Quasi differential equations of the nth order are equations of the form

123


where pi(t) > 0 and Pi E en-i([a, b], R) for 0 :::; i :::; n - 1. These problems have been studied by [5,12, 14J. Notice that if Pi(t) = 1, for 0 :::; i :::; n - 1, then these problems reduce to ordinary boundary value problems. The boundary conditions considered in this work are conjugate conditions. Let il, ... ,im be positive integers with 2 :::; E~l i, = n and assume tl < t2 < ... < tm are contained in the interval (a, b), the boundary conditions at tj become

(10)

For these problems the parameter space becomes a rather complicated looking Banach space. Set

The spaces X,Z, and Ware the same as for the example with ordinary differential equations. The maps L, F, and B are defined by

L(Po, ... ,Pn-l, tl, . . . , tm, Yl, ... ,Ym,im-l, y( t)) = (Pn-l (. .. PI (Poy)')' ... )'

F(Po,··· ,pn-l, tl, ... ,tm,Yl, .. . ,Ym,im-l,y(t)) = - f(t,PoY,Pl(PoY)',· . . , Pn-2( .. . PI (poy)')' ... )')

B(Po, ' " ,Pn-l,tl, . . . , tm,yl,·· · ,Ym,im-l,y(t)) = (POy(tl) - Yl,O,··· ,Pim-l('" PI (POy(tm))')' ... ) - Ym,im-I)'

Notice that although L is not linear, it is linear in y(t) for fixed Po,·· · ,pn-I.

2D. Vector Difference Equations

For this example, we consider an n-dimensional vector difference equation of the form

u(m + 1) = f(m, u(m))

where u satisfies the multipoint boundary conditions

k

EM,u(m,) = r, '=1

(11)

(12)

where ml < m2 < ... < mk belong to the integers Z, r = (rl, ... , rn) E Rn, and Ml, ... ,Mk are constant real n x n matrices We will use the notation Mr to denote

124

BVPs FOR FDEs

the 8th row and tth column entry of the matrix Mi . We assume f(m, x) : ZxRn -+ Rn is continuously differentiable for each fixed value of m. Problems of this type have been considered in [13,17]

We are only interested in solutions on finite length discrete intervals of the form [mo, mk+l] where mo < ml < . . . < mk < mk+l ' In this case, our solution space X = Rnm x··· x Rnm where each Ri is a copy of R. The notation u(i) will be used

o k+l

to denote a vector in Rf, where mo ~ i ~ mk+l' The solutions are being viewed as a sequence of finite dimensional vectors. The parameter space P is defined by P = Matnxn(R)k x Rn. Here the parameters are the components of the matrices M; and the components of r.

The first test space is given by Z = MatnX (mk+l-mo)(R). That is, each column of the test space is an n vector and there are mk+l - mo columns, each representing the solution at a different time. Also, W = Rn. The functions L,F are defined by

L, F : Matnxn(R? x R n x R::'" x . .. x R::'k+l -+ MatnX (mk+1-mo)(R)

with

L(M11, ... ,Mrn

,rl " 'rn ,

u(mo), u(mo + 1), ... , u(mk+t}) (u(mo + 1), .. . , u(mk+l))

and

F(Mf\ ... , Mrn, rl ' " r n , u(mo) ,

u(mo + 1), ... , u(mk+l)) = (f(mo, u(mo)), ... ,f(mk+l - 1, u(mk+l - 1))

The boundary function B is defined by

B : Matnxn(R)k x R n x Rmon x··· x R n -+ R n mk+l

with

k

u(mo), u(mo + 1), .. . , u(mk+l)) LMu(mt}-r 1=1

3. Proof of the General Theorem

In this section we will prove the main theorem. The theorem is an application of the Implicit Function Theorem, see [1] . We assume that the spaces considered have the properties defined in section 2. Suppose

G:PxX-+ZxW

125


is defined by G(p, x) = (L(p,x) - F(p,x),B(p,x)).

Lemma 3.1 The function G is continuously differentiable.

Proof: Each of the component functions of G is assumed to be continuously differentiable, thus G is continuously differentiable. QED

Lemma 3.2 If the linear problem

dLx(Po, xo)( x) - dFx (Po , xo)( x) = z dBx(Po, xo)( x) = w.

has a unique solution for each choice of z E Z, w E W, where Xo is a fixed solution to (2.1),(2.2) with parameter Po, then dGx(Po, xo)(') is a linear homeomorphism.

Proof: From Lemma 3.1, G is differentiable at the point xo, hence the partials exist. We compute the partial with respect to the x slot as follows. Let hEX, then using the definition of the partial derivative coupled with Taylor's Theorem, we obtain

G(po, Xo + h) - G(po, xo) (L(po, Xo + h) - F(po, Xo + h), B(po, Xo + h))

(L(po, xo) - F(po, xo), B(po, xo)) (dLx(Po, xo)(h) + el (h) - dFx(Po, xo)(h) + e2(h),

dBx(Po, xo)(h) + e3(h)) (dLx(Po,xo)(h) - dFx(Po,xo)(h),dBx(Po,xo)(h))+

(el(h) + e2(h),e3(h)).

The first expression is the partial of G at Xo with respect to the x slot. By definition it is a continuous linear function. The second expression is easily verified to be an error term of the correct order. Thus the partial of G is given by

dGx (Po , xo)(h) = (dLx(Po, xo)(h) - dFx(Po, xo)(h), dBx(Po, xo)(h)).

Upon inspection, this expression is seen to be closely related to the linearized problem. From our assumptions on the linearized problem, this partial is seen to be one to one and onto. The Open Mapping Theorem yields dGx(Po, xo) is a linear homeomorphism. QED

The two lemmas above are used to prove the main theorem.

126

BVPs FOR FDEs

Theorem 3.1 Assume Xo is a solution to (2.1),(2.2) corresponding to the parameter Po. Assume also the hypothesis of Lemma 3.2 concerning the linear problem are satisfied. Then solutions of (2.1),(2.2) depend continuously on Po. That is, given f > 0, there exists Ii > 0 such that 1 po - PI Ip< Ii implies there exist a solution XI satisfying L(PI,xI) = F(PI,XI) and B(Pt,XI) = 0, and moreover, 1 Xo - Xl Ix< f .

Proof: It has already been established that G(po, xo) = 0, G is continuously differentiable, and dGx(Po, xo) is a linear homeomorphism. These are the hypotheses necessary for the Implicit Function Theorem. The Implicit Function Theorem gives the existence of a continuous function X defined on an open set U C P, containing Po, such that X : U -+ X and G(p, X(p)) = 0 for all P E U. This yields X(p) is a solution to the problem

L(p,X(p)) = F(p,X(p)) B(p, X(p)) = o.

Since X is a continuous function of p, we obtain the continuous dependence result required. QED

4. Applications

The abstract continuous dependence theorem proved in section 3 will now be applied in succession to the four concrete examples considered in section 2. The benefit of the abstract approach taken in section 2 is that the proofs of Theorems 4.1-4.4 are all applications of Theorem 3.1. Because of this fact, we will only give ,the proof for the first theorem.

Theorems 4.1 and 4.2 have appeared in print [2,3] although they were not presented as a special case of a more general construction. Theorems similar to Theorem 4.4, but involving a different method of proof, have appeared in print [13]. The application to quasi-differential equations, Theorem 4.3, is a completely original application of the techniques used in this paper.

The first case considered is that of boundary value problems for ordinary differential equations. Here we consider only the case of nonlinear boundary conditions, as the other two types considered are special cases of these conditions.

Theorem 4.1 Assume f,g and the ti are as in section 2A. Let y(t) be a solution to the BVP (2.3),(2.6). Assume the linearized problem

Z (n) - "n !!L(t -y """y -y(n-I) ') z (i-I) - L.Ji=i 8 Yi ~, , , ... , , A

Ei'=l ~(A, y(tk)' y'(tk)' . .. , y(n-l)(tk))z(i-l)(tk) = 0, 1 ::; k::; n,

has only the trivial solution. Then given f > 0, there exists Ii > 0, such that if 1 ti - Xi 1< Ii, 1 Yi - Zi 1< Ii, and 1 A - N 1< Ii, then there exists a solution y(t) to the

127


BVP yIn) = f(t, y,y', ... , y(n-I), N) 9i(N, y(Xi), y'(Xi), ... , y(n-I)(Xi)) = Zi, 1 :::; i :::; n .

Moreover, 1 y<i)(t) - y<i)(t) 1< f, for 0 :::; i :::; n - 1, and x E [a, b).

Proof: Define L, F, and B as is section 2A. Using Taylor's Theorem, it can be shown that L, F, and B are continuously differentiable. The function G defined in section 2 is thus continuously differentiable at the point (A, t l , ... , tn, YI, ... , Yn, y(t)). Applying Theorem 3.1 yields the desired conclusion. QED

As the proofs of the following theorems are almost identical to the proof of the above theorem, the theorems will be stated without proof.

Theorem 4.2 Let y be a fixed solution satisfying equations (2.7),(2.8). Assume f,,p, 7", u, r, as well as M, N are as in section 2B. Assume there exists a unique solution to the linear problem

Z' = fy,(t,y(t),y(,p(t)))z(t) + fY2(t, y(t), y(,p(t)))z((,p(t)) + w(t), a:::; t :::; b,

MZ(7"(t)) + Nz(u(t)) = p(t), a:::; t :::; a,

for each w(t) E C([a, b), Rn) and p E C([a, a), Rn). Then given f > 0, there exists 6 > 0, such that if 1 ,p - ~ 1< 6,1 7" - 7' 1< 6,1 (1' - u 1< 6,1 r - f 1< 6, then there exists a solution y to (2.7),(2.8) corresponding to the data ~, 7', u, and f. Moreover, 1 y(t) - y(t) 1< f for all t E [a, b).

To simplify the statement of the next theorem, we define a series of operators by Qo(z) = PoZ,QI(Z) = PI (PozY, Qi(Z) = P2(PI(POZYY, and so on, where the Pi are as in section 2C.

Theorem 4.3 Assume f, the Pi, and the ti are as in section 2C. Let y(t) be a solution to the quasi-boundary value problem (2.9),(2.10). Assume the linearized problem

(Pn-I ( ... , PI (poz)'Y . . . , )' = E;=I ~(t, Poy, PI (poYY, ... , (Pn-2( . .. PI (Po (y)')' ... Y)Qi-1 (z(i-I»)

Poz(t l ) = O, ... ,Pim-I(···Pl(POZ(tm))')'···)'=O

has only the trivial solution. Then given f > 0, there exists 6 > 0, such that if 1 ti - Xi 1< 6,1 Yl,O - Zl,O 1< 6, ... ,1 Ym,im-I - Zm,im-l 1< 6, 1 Po - PI 1< 6, ... , and

128

BVPs FOR FDEs

1 Pn-l - Pn-l 1< 6, then there exists a solution y(t) to the quasi-boundary value problem

(Pn-l ( ... PI (Poz )')' ... )' = I(t, Poy, Pl(POZ)', . . . , (Pn-2( .. . PI (poz )')' . . . )')

satisfying the quasi-boundary conditions

Moreover, 1 Qiy(t) - Qiy(t) 1< €, for ° ~ i ~ n - 1, and tEla, h).

For the following theorem, we will use the notation Ix to denote the matrix of partials of I with respect to the components of x.

Theorem 4.4 Assume I, the matrices Mi , the integers mi, and the vector r are as in section 2D. Let u(m) be a solution to (2.11),(2.12). Assume the linearized problem

z(m + 1) = Ix(m, u(m))z(m)

k

LM1z(ml) = 0, 1=1

has only the trivial solution. Then given € > 0, there exists 6 > 0, such that if 1 MIl - M!1 1< 6, ... ,1 Mrn _1Vf';n 1< 6,1 rl - 1'1 1< 6,···,1 rn - Tn 1< 6, then there exists a solution u( m) satisfying

u(m + 1) = I(m, u(m))

k

L M1u(ml) = T. 1=1

Moreover, 1 u(m) - u(m) 1< € for all integers in 1m}, mk+l).

References

1. M. Berger, Nonlinearity and Functional Analysis, Academic Press, 1977.

2. J. Ehme, Differentiation of solutions of boundary value problems with respect to nonlinear boundary conditions, J. Diff. Eqns . 101 (1993),139-147.

3. J. Ehme and J. Henderson, Functional boundary value problems and smoothness of solutions, Nonlinear Anal. Theory, Methods & Appl., to appear.

129


4. P.W. Eloe, and J . Henderson, Nonlinear boundary value problems and a priori bounds on solutions, SIAM J. Math Anal. 15 (1984), 642-647.

5. P. W. Eloe and J. Henderson, Singular boundary value problems for quasidifferential equations, Int. J. Math. & Math. Sci. 18 (1995), 571-578 .

6. P.W. Eloe, and J. Henderson, Multipoint boundary value problems for ordinary differential systems, J. Diff. Eqns. 114 (1994), 232-242.

7. A. Granas, R. B. Guenther and J . W. Lee, Nonlinear boundary value problems for some classes of ordinary differential equations, Rocky Mountain J. Math 10 (1980), 35-58.

8. J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York-Berlin, 1977.

9. J. K. Hale and L. A. C. Ladeira, Differentiability with respect to delays, J. Diff. Eqns. 92 (1991), 14-26.

10. D. Hankerson, An existence and uniqueness theorem for difference equations, SIAM J. Math . Anal. 20 (1989),1208-1217.

11. J. Henderson, Disconjugacy, disfocality, and differentiation with respect to boundary conditions, J. Math. Anal. Appl. 121 (1987), 1-9.

12. J. Henderson and E. Kaufman, Integral conditions for disfocality of .. linear differential equation, Dyn. Sys. & Appl. 3 (1994), 405-412.

13. J . Henderson and B. Lawrence, Smooth dependence on boundary matrices, J. Difference Eqns. Appl., to appear.

14. Z. Nehari, Disconjugate linear differential operators, Trans . Amer. Math . Soc. 129 (1967), 500-516.

15. A. Peterson, Existence-uniqueness for ordinary differential equations, J. Math. Anal. Appl. 64 (1978),166-172.

16. A. Peterson, Existence-uniqueness for focal-point boundary value problems, SIAM J. Math. Anal. 12 (1981), 173-185.

17. A. Peterson, Existence and uniqueness theorems for nonlinear difference equations, J. Math. Anal. 125 (1987), 185-191.

18. G. Vidossich, On the continuous dependence of solutions of boundary value problems for ordinary differential equations, J. Diff. Eqns. 82 (1989), 1-14.

131

POSITIVE SOLUTIONS AND CONJUGATE POINTS FOR A CLASS OF LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

PAULW. ELOE Department of Mathematics , University of Dayton

Dayton, Ohio 45469-2316 USA

JOHNNY HENDERSON Department of Discrete and Statistical Sciences Auburn University, Alabama 36849-5307 USA

Abstract

The first conjugate point for a class of second order linear functional differential equations is characterized by the existence of a nontrivial solution that lies in a cone. To apply the cone theoretic arguments, linear, monotone, compact maps are constructed. To construct such maps, a standard representation theorem for Green's functions is obtained, known sign properties of Green's functions are employed and some comparison properties of Green's functions are obtained. The functional differential equation is restrictive and examples are presented to illustrate the restrictions.

1. Introduction

Let m be a positive integer and let Pi E C[O, 00), i = 1, . .. , m. Let hi E C[O,oo),i = 1, .. . ,m, be such that 0 S hi(t) S t,i = 1, .. . , m. We define the linear functional differential operator, L, by

m

Lx(t) = x"(t) + q(t)x'(O) + LPi(t)x(hi(t)), Os t, i=1

1991 Mathematics Subject Classification. Primary 34K10; Secondary 34B27.

132

BVPs FOR FDEs

where q E C[O, 00). For each b > 0, we consider the homogeneous two-point boundary conditions of the form

x(O) = 0, x(b) = o. (l.1b)

We shall, at times, denote the boundary conditions, (l.lb), by TbX = o. Let r E C[O, 00). For each b > 0, we consider boundary value problems (BVPs)

of the form, Lx(t) = r(t)x(t) ,O S t S b,

nx=O.

(l.2b)

(1.3b)

We shall say that bo is the first conjugate point of (1.2b) corresponding to (1.3b) if bo = inf{b > 0 :(1.2b), (1.3b) has a nontrivial solution} .

Azbelev [1] has shown that for linear functional differential equations , uniqueness of solutions of initial value problems can be violated; in particular, bo = 0 is possible. We shall make assumptions such that if bo < 00, then 0 < bo o

In this paper, we shall assume a sign condition on q and a sign condition on r, and characterize the existence of bo by the existence of a nontrivial solution of the BVP, (1.2b), (1.3b), for b = bo, that lies in a cone. This paper is motivated by the work in [13], [8] or [9] where sign properties and comparison properties of Green's functions have been employed in cone theoretic arguments. Also, the results in this paper are closely related to some of those found in [1] . Sign properties of Green's functions for BVPs for linear functional differential equations have received considerable attention ([1-4],[6]). In this paper, we shall obtain an appropriate characterization and representation of Green's functions in order that the the integral operator is a compact map from a Banach space of continuous functions into itself. We will then obtain comparison properties for Green's functions and apply cone theoretic arguments in a now standard way to characterize the existence of bo.

In what follows, we shall provide the cone theoretic preliminaries in §2 in order that this paper be self-contained. In §3, we shall obtain the representation theorem, provide the appropriate sign properties and obtain the appropriate comparison properties of Green's functions . Finally, in §4, we shall define appropriate Banach spaces and cones and apply the cone theoretic arguments to characterize bo.

2. Cone Theoretic Preliminaries

In this section we provide the definitions and results from cone theory which we shall apply in §4. For a thorough development, we refer the reader to Krasnosel'skii [10] . Lemma 2.3 is proved in [13] .

Let B be a Banach space over the reals. A closed , nonempty set, PCB, is said to be a cone provided (i) ax + {Jy E P, for all x , yEP and all a , {J ~ 0, and (ii) x, -x E P implies x = O. P is a reproducing cone if for each x E B there exist u, v E P such that x = u - v. P is a solid cone if intP# 0 where intP denotes the interior of P. Every solid cone is reproducing [10] .

133

LINEAR FDEs

Let PCB. Define the partial ordering on B induced by P by u :S v if, and only if, v - u E P. If M, N : B -+ B are bounded linear operators, we say that M :S N with respect to P provided Mu :S Nu for all u E P . A bounded, linear operator N : B -+ B is positive with respect to P if N : P -+ P. We say that a bounded linear operator N : B -+ B is /-Lo-positive with respect to P if there exists /1-0 E P, /1-0 "# 0, such that for each u E P , there exist positive real numbers, kl(U), k2(U), such that kl /1-o :S Nu :S k2/1-0 . Finally, if N : B -+ B is a bounded, linear operator, let s(N) denote the spectral radius of N .

Lemma 2.1. Let B be a Banach space over the reals, and let PcB be a solid cone. If N : B -+ B is a linear operator such that N : P \ {O} -+intP, then N is /1-o-positive with respect to P.

Lemma 2.2. Let B be a Banach space over the reals, and let PcB be a reproducing cone. Let N : B -+ B be a compact, linear operator which is /-Lo -positive with respect to P. Then N has an essentially unique eigenvector in P, and the corresponding eigenvalue is s(N) and s(N) is simple.

Lemma 2.3. Let Nb, a :S b :S /3, be a family of compact, linear operators on a Banach space such that the mapping b -+ Nb is continuous in the uniform operator topology. Then the mapping b -+ S(Nb) is continuous.

Lemma 2.4. Let N : B -+ B be compact, linear and positive with respect to P . Assume s(N) > O. Then s(N) is a simple eigenvalue of N and there is a corresponding eigenvector in P

Lemma 2.5. Let N1 , N2 : B -+ B be compact, linear and positive with respect to P. If NI :S N2 with respect to P, then s(Nt} :S s(N2).

Lemma 2.6. Let N : B -+ B be compact, linear and positive with respect to P. Suppose there exists /1- > 0, u E B, -u f!. P such that Nu 2 /1-U . Then N has an eigenvector in P which corresponds to an eigenvalue, A 2 /1-.

3. Properties of Green's Functions

In this section we shall obtain a representation theorem and provide the sign properties and comparison properties of Green's functions which will be employed in the cone theoretic applications of §4. Azbelev and Rakhmatullina [3] have shown the existence of a Green's function for BVPs where the linear functional differential operator is far more general than our operator, L . By a Green's function, G(bj t, s), we mean that if Lx = J, TbX = 0 is uniquely solvable, then the unique solution is

given by x(t) = J: G(bjt,s)J(s)ds. The first theorem that we shall employ, Theorem 3.1, is a special case of a

result due to Bainov and Domoshnitsky [4] who require a concept of a semi-interval of nonoscillation, a concept analogous to that of disfocal intervals for ordinary differential equations [11] . An interval, I , is a semi-interval of nonoscillation for

134

BVPs FOR FDEs

the operator, L, if x is a solution of Lx = 0 on I and x(ta) = x'(tt} = 0 for some to, tl E I implies x = o. Theorem 3.1. Assume [0,00) is an interval of semi-nonoscillation for the functional differential operator, L. Assume q(t) ~ O. Assume there exists a solution of the differential inequality, Lx(t) :::; 0,0:::; t, such that x(t) ~ 0,0:::; t and x(O) > o. Then G(b;t,s) < 0 on (0, b) X (O,b) for each 0 < b.

Remark. The condition, q ~ 0 is a necessary condition as seen by the example [4], x"(t) + qx'(O) = 0,0:::; t :::; b, x(O) = x(b) = o. Then

(t(2-qt)(b-s)/b(qb-2) + (t - s),O:::; s < t:::; b, G(b·ts)=~

" It(2-qt)(b-s)/b(qb-2),0:::;t<s:::;b.

If q < 0, then G < 0 for t < s and G > 0 for s = 0, t > o. Note that G changes sign regardless of the length of the interval, [0, b] . Recall that in the case of disconjugate ordinary differential equations, there exists fj > 0 such that G maintains constant sign on (0, b) x (0, b) if 0 < b < fj [5] .

The next result is a characterization theorem for G(b;t,s). Azbelev [1] first observed the representation (3.2). Domoshnitsky [6] points out that the construction of Green's functions for functional differential equations is difficult . It is here that we employ the restrictive delay conditions, h;(t) :::; t, i = 1, ... , m. It is also here that we require L to be so restricted in x'. It is for the purpose of defining Banach spaces and cones in §4 that we employ the other inequalities, 0 :::; h;(t).

Theorem 3.2. Assume [0,00) is a semi-interval of nonoscillation for the functional differential operator, L. Assume 0 :::; h;(t) :::; t, i = 1, . . . ,m. Then for each 0 < b, there exists a uniquel),' determined, G(b; t, s), defined on [0, b] x [0, b] satisfying the following properties:

(i) G is continuous on [0, b] x [0, b], Gt and Gtt are continuous on triangles, o :::; t :::; s :::; band 0 :::; s :::; t :::; bj

(ii) Gt(s+,s) - Gt(s-,s) = l,s E (0, b); (iii) as a function of t, LG = 0 for 0 :::; t < s, and s < t :::; bj (iv) as a function of t, nG = 0,0 < s < b.

Moreover, if f is continuous on [0, b], then the unique solution of Lx = f,O :::; t :::; b, TbX = 0 is given by

x(t) = 1b G(b;t,s)f(s)ds . (3.1)

Outline of Proof. Let u( t, s) be the Cauchy function satisfying (as a function of t), Lu(t,s) = 0 for s:::; t,u(t,s) = 0 for t < s,u(s,s) = O,Ut(s,s) = 1. Let {Xl,X2} denote a fundamental system of solutions of Lx = 0 on [0,00). Azbelev [1] has obtained the existence of u and {Xl, X2}. Define

135

LINEAR FOEs

(3.2)

on [0, b] x [0, b], where Cl and C2 are chosen to satisfy (iv). Note that Cl and C2 are uniquely determined since [0,00) is an interval of semi-nonoscillation. Clearly, G satisfies (ii)-(iv) and direct substitution (keeping in mind that u = ° for t < s) shows that (3.1) satisfies Lx = j, 0:::; t :::; b, TbX = 0.

To see that G satisfies (i), we first argue that G is continuous in s. Note that each of Cl and C2 are continuous in s if u(b, s) is continuous in s. Thus, the continuity of Gin s follows from the continuity of u in s. This requires the continuous dependence on parameters of solutions of initial value problems. To see that u is continuous in s, note that

u(t, s) = (t - s) + /.t (t - a)q( a )Ut(O, s )da (3 .3)

It mo

+ (t - a)(LPi(a)u(hi(a),s)da. 8 i=l

Since Ut(O, s) = 0, the Gronwall inequality can be applied in a standard way to lIu(t,sl) - u(t,s2)11 to obtain the continuity of u in s, where the norm denotes the supremum over t E [0,1] .

The Gronwall inequality also implies that Ilu( t, s) II is uniformly bounded for s E [O,b] . It now follows by differentiating (3 .3) with respect to t that Ilut(t,s)11 is uniformly bounded for s E [O,b]; since u is continuous in sand Ilut(t,s)1I is uniformly bounded in s, it follows that u is continuous in (t,s). Finally, since Cl (b, s )Xl (t) + C2 (b, s )X2 (t) is separable in s and t, it follows that G is continuous in (t, s). Similarly, Ut and Utt are continuous on triangles s < t and t < s.

Remark. It is interesting to note that in the case where

m

Lx(t) = X"(t) + q(t)x'(g(t)) + LPi(t)x(hi(t)),O :::; t, i=l

a Green's function satisfying (ii)-(iv) exists. However, the Gronwall inequality cannot be employed to obtain (i) and in general, the Green's function does not satisfy (i). Condition (i) is sufficient, although not necessary, to imply by the Arzela-Ascoli Theorem that the integral maps defined in §4 are compact operators. Condition (i) is a usual condition that is satisfied in the case of ordinary differential equations. The following two examples show that Condition (i) is not to be expected in the case of functional differential equations.

Example 1. Let

{t,0:::;t:::;1/2,

get) = 1/2,1/2:::; t :::; 1.

136

BVPs FOR FDEs

Consider X"(t) + x(g(t» = 0, ° :S t :S 1. Then

where

{

0, t < s,

sin(t - s),s < t < 1/2, u t s -(,)- v(t,s),s<1/2<t:S1,

t - s, 1/2 < s < t :S 1,

v(t, s) = (-1/2)(t - 1/2)2 sin(1/2 - s) + (t - 1/2) cos(1/2 - s) + sin(1/2 - s).

u is readily extended to a continuous function on [0,1J x [0,1J. Example 2. Let

{ t, ° :S t :S 1/2,

g(t) = 1/2,1/2 :S t :S 1.

Consider x"(t) + x'(g(t» = 0, ° :S t :S 1. Then

where

{

0, t < 8,

1-exp(-(t - 8»,S < t < 1/2, u(t s) =

, v(t,s),s < 1/2 < t :S 1,

t - s, 1/2 < s < t :S 1,

v(t,s) = (-1/2)(t _1/2)2 e-(1/2-.) + (t _1/2)e-(1/2-.) + 1- e-(1/2-.).

For t > s, u is not continuous at s = 1/2. We point out that the operator that would be generated in the case of Example

2 is a compact operator in the setting in §4i for simplicity, we restrict our operator, L, to depend only on x'(O) in order that Condition (i) of Theorem 3.2 holds.

Corollary 3.3. Assume the hypotheses of Theorems 3.1 and 3.2. Assume ° < b1 < b2 . Then

(3 .4)

and (3.5)

Proof. Due to the representation of G in the proof of Theorem 3.2, G(b2 it,S)G(b1it,S) satisfies the BVP, Lx = 0,0 < t < b1,x(0) = O,x(bI) = G(b2ibl,S). By Theorem 3.1, G(b2 ib1,s) < 0. The comparison inequalities in (3.4) and (3.5) now follow since [0,00) is a semi-interval of nonoscillation. To obtain the first inequality in (3.5), it follows from Theorem 3.1 and Taylor 's Theorem that ° ~ Gt(b1iO,S) . The strict inequality now follows from the comparison inequality in (3.5) .

137

LINEAR FDEs

4. Conjugate Points and Positive Solutions

In this section, we shall employ the properties of §3 to construct cones in Banach spaces and apply the results of §2 . The techniques employed here were developed in (13) and refined in (8) and (9) . Let B = {x E BC[O,oo) : x(O) = OJ , where B is equipped with the usual supremum norm, and define the cone P = {x E B I x(t) :5 O,t 2: OJ .

Remark. Throughout this paper we have assumed that [0,00) is a semi-interval of nonoscillation for Lx = O. The results of this section remain valid if for some T > 0, [0, T) is semi-interval of nonoscillation for Lx = 0 and we set B = {x E C[O,T): x(O) = O} and P = {x E B I x(t):5 0,0:5 t:5 T}.

For each 0 < b, define B(b) = {x E C 1 [O,b): TbX = O} with

Ilxllb = max{llxll. Ilx'll}

where 11·11 denotes the supremum norm on C[O,b) . Define the cone, PCb) = {x E

B( b) : x( t) :5 0,0 :5 t :5 b} . For the remainder of this paper, assume the hypotheses of Corollary 3.3 and assume ret) :5 0,0 :5 t :5 b, is continuous and does not vanish identically on any compact subinterval of [0, b). Define Nb . B(b) -+ B(b) by

NbX(t) = lb G(b;t,s)r(s)x(s)ds,O:5 t:5 b.

Lemma 4.1. (i) int PCb) = {x E P(b): x(t) < 0,0 < t < b,x'(O) < O,x'(b) > O} =I 0;

(ii) Nb: PCb) \ {OJ -+ intP(b).

Proof. Eloe and Henderson have proved (i) (7) . (ii) is an easy consequence of Theorem 3.1 and Corollary 3.3.

Lemma 4.2. Nb : B(b) -+ B(b) has an eigenvector, u E int PCb), and the corresponding eigenvalue, A, is positive, simple and larger than the norm of any other eigenvalue.

Proof. From Theorem 3.2, it follows from standard applications of the Arzela Ascoli Theorem that Nb is compact. Apply Lemmas 4.1, 2.1, and 2.2.

We shall now define further operators, Nb, Banach spaces, B(b), and cones, PCb) . For each b > 0, define Nb : B -+ B by

{

fb G(b;x,s)r(s)x(s)ds, 0 :5 t:5 b, NbX(t) = Jo

0, b :5 t.

For each b > 0, define Banach spaces B(b) = {x E Cl[O,b) : x(O) = O} with IIxllb = max{lIxll, IIx'lI} and define cones PCb) = {x E Bb : x(t) :5 OJ. Now note

138

BVPs FOR FDEs

that int P(b) = {x E P(b) : x(t) < 0,0 < t:S b,x'(O) < O} . In the discussion that follows, we shall also restrict the operator Nb to B(b) ; that is, define Nb : B(b) ->

B(b) by

NbX(t) = 1b G(b;t,s)r(s)x(s)ds ,O:S t:S b.

We shall specify the domain, B(b) , B or B(b) , when referring to the operator Nb to avoid confusion. Note that by Theorem 3.2 and using standard applications of the Arzela-Ascoli Theorem, Nb, defined on B(b), B or B(b) is a compact operator for each b> O. Finally, by S(Nb), we mean the spectral radius of Nb defined on B .

Theorem 4.3. For b > 0, S(Nb) is strictly increasing as a funct ion of b.

Proof. From Lemma 4.2, it will follow that S(Nb) > 0 for each b > O. To see this, let>. and u be as in Lemma 4.2. Then NbU(t) = >.u(t), 0 :S t :S band u E P(b) \ {O} . Extend u by u(t) = 0 for b :S t . Then for 0 :S t, NbU(t) = >.u(t) and S(Nb) 2 >. > O. Hence, we can apply Lemma 2.4.

Let 0 < bI < b2 • Since S(Nb1 ) > 0, it follows from Lemma 2.4 that there exists u E P \ {O} such that Nbl u = S(Nb.}U . Set Xl = Nb 1 U = S(Nb.}U and X2 = Nb,U . Then for 0 :S t :S bI ,

Since u(s) "I- 0 for some s E [O, bI ], it follows by Corollary 3.3 that the restriction of X2 - Xl to [0, bI ] is an element of the interior of P(bI ) . Thus, there exists a > 0 such that X2 - Xl 2 au, where this inequality is with respect to the cone, P(bI) . Since XI(t) = u(t) = 0 for t > bI and X2 E P, it follows that X2 - Xl 2 au where the inequality is now with respect to the cone P Thus, X2 2 Xl + au = (S(Nb1) + a)u . It follows from Lemma 2.6 that S(Nb,) 2 S(Nb1) + a > S(Nb1 ).

We now state and prove the main result of the paper.

Theorem 4.4. The following are equivalent:

(i) bo is the first conjugate point of (1.2b ),(1.3b); (ii) there exists a nontrivial solution, x, of the BVP, (1.2b),(1.3b), for b = bo,

such that X E P(bo); (iii) S(Nbo) = 1.

Proof. (iii) implies (ii) follows from Lemma 2.4. To see that (ii) implies (i), let u E P(bo) \ {O} satisfy (1.2b) ,(1.3b) , for b = bo o

Extend u by u(t) = O,bo < t. Then s(Nbo ) 2 1. If S(Nbo) = 1 the proof is complete by Theorem 4.3. See the details provided in the proof of Theorem 4.2 [8] . So, assume S(Nbo) > 1. Let v E P \ {O} be such that NboV = s(Nbo)v. Note the restriction of v to [0, bo] is in the interior of p( bo). Thus , there exists a > 0 such

139

LINEAR FDEs

that u ;:::: liv where the partial order is with respect to F(bo). Since, u = v = 0 for t > bo, u ;:::: liv where the partial order is now with respect to P Assume now that li is maximal. Then

U = NboU ;:::: Nbo(liv) = lis(Nbo)V

which contradicts the maximality of li. Hence, S(Nbo) = 1. (i) implies (iii) follows from Lemma 2.3, Theorem 4.3 and the fact that IINbll-+ 0

as b -+ 0+ . Details are provided in [7] or [8] . Of course, to apply Lemma 2.3, we require that the map b -+ Nb is continuous. Let Ilull = 1. Assume b2 - bl > O. Then (Nb. - NbJU =

lb

' lb. (G(b2 it,S) - G(blit,s))r(s)u(s)ds + G(b2 i t , s)r(s)u(s)ds = o b,

lb

' lb. v(t,s)r(s)u(s)ds + G(b2 ;t,s)r(s)u(s)ds, o b,

where vet,s) is the solution of Lx = 0,0 ~ t ~ bl,x(O) = O,x(bd = G(b2 ;bl ,s). Note that Ilv(t,s)1I = -v(bl,s) = -G(b2 ibl ,s), since [0,00) is a semi-interval of nonoscillation. G is uniformly continuous on compact domains and so v( t, s) is uniformly small in s for b2 - bl sufficiently small. Thus, I J:' (G(b2 i t, s ) -G(blit,s))r(s)u(s)dsl is uniformly small if b2 - bl sufficiently small. The second term is uniformly small since G is uniformly bounded on compact domains .

Remark. Azbelev [1] points out that Wronskians offundamental systems of linear functional differential equations can vanish and so conjugate points of (1.2b) , (1.3b) can generate multiple zeros. He proves (Lemma 2 [1]) that the first conjugate point is simple. Theorem 4.4 offers another proof of this observation. The nontrivial eigenfunction, x, that lies is a cone satisfies x E intF(bo). In particular, x'(bo) > 0 and bo is a simple zero.

Example. Consider, again,

x"(t) + qx'(O) = 0,0 ~ t ~ b,x(O) = x(b) = O.

Then

. _{t(2-qt)(b-S)lb(qb-2)+(t-S),0~S<t~b, G(b, t,s) - t(2 _ qt)(b - s)lb(qb - 2),0 ~ t < s ~ b.

Xl = 1, X2 = t(2 - qt) form a fundamental system of solutions for x"(t) + qx'(O) = o 0 < t and the Wronskian of Xl and X2 vanishes at 1/q· It is readily shown that if 0 "< ; < lib, then [0, b] is a semi-interval of nonoscillation and so the results of §3 apply. One can show directly that G satisfies the conditions of Corollary 3.3. If

140

BVPs FOR FDEs

0< q < lib, the results from §4 apply to x"(t) + qx'(O) + x(t) = 0, 0 ~ t ~ b, x(O) = x(b) = O. The solution of

x"(t) + qx'(O) + x(t) = 0,0 ~ t,x(O) = O, x'(O) = 1

is x(t) = qcos(t) + sin(t) - q. Note that bo < 7r and so if q < 1/7r the results of §4 apply.

Remark. This paper has dealt only with delay equations in order to obtain the representation in Theorem 3.2. Then the comparison inequalities, given in Corollary 3.3, follow readily. A representation theorem, analogous to Theorem 3.2, is valid for a two-point conjugate BVP for a linear functional differential equation with only advances if one employs a Cauchy function which vanishes for s < t . Thus, one can define and characterize the first conjugate point, ao, for the BVP, Lx = O, x(a) = 0, x(T) = 0, where the righthand endpoint T is fixed and L represents a linear functional differential equation with advances.

References

1. N. V. Azbelev, On the zeros of the solutions of a second order linear differential equation with retarded argument , Differential Equation.'! 7 (1971) , 1147-1157.

2. N. Azbelev, V. Maksimov, L.Rakhmatullina, Introduction to the Theory of Functional Differential Equation.'!, " Nauka" , Moscow, 1991. (Russian)

3. N. V. Azbelev and L. F . Rakhmatullina, Functional-differential equations, Differential Equation.'! 14 (1978), 771-797.

4. D. Bainov and A. Domoshnitsky, Theorems on differential inequalities for second order functional differential equations, Gla.'!nik M atematiCki (to appear).

5. W. Coppel, Disconjugacy, Lecture Notes in Mathematics, vol. 220, SpringerVerlag, Berlin and New York, 1971.

6. A. Domoshnitsky, Conservation of the sign of Green's function of a two-point boundary-value problem for an nth order functional-differential equation, Differential Equation.'! 25 (1989), 934-937.

7. P. Eloe and J . Henderson, Comparison of eigenvectors for a system of two-point boundary value problems, Recent Trends in Ordinary Differential Equations, vol. 1, World Scientific Publ. Co., River Edge NJ USA, 1994.

8. P. Eloe and J . Henderson, Focal points and comparison theorems for a class of two point boundary value problems, J. Differential Equation.'! 103 (1993), 375-386.

9. P. Eloe and J. Henderson, Focal point characterizations and comparisons for right focal differential operators, J. Math. Anal. Appl. 181 (1994) , 22-34.

10. M. A. Krasnosel'skiz, Po.'!itive Solution.'! of Operator Equation.'! , Fizmatgiz, Moscow, 1962 (Russian); English transl., Noordhoff, Groningen , The Netherlands, 1964.

11. Z. Nehari, Disconjugate linear differential operators, Tran.'!. Amer. Math. Soc. 129 (1967), 500-516.

141

LINEAR FOEs

12. R. Nussbaum, Periodic solutions of some nonlinear integral equations, Proceedings Internatl. Conf. on Differential Equations (Gainesville, 1976), Academic Press, Inc., New York, 1977, pp. 221-249.

13. K. Schmitt and H. Smith, Positive solutions and conjugate points for systems of differential equations, Nonlinear Anal 2 (1978), 93-105.

BOUNDARY VALUE PROBLEMS FOR SECOND ORDER MIXED TYPE FUNCTIONAL DIFFERENTIAL EQUATIONS

L. H. ERBE Department of Mathematical Sciences

University of Alberta Edmonton, Alberta, Canada T6G 2G1

WANG ZHICHENG AND LI LONGTU Department of Applied Mathematics, Hunan University

Changsha, Hunan 410082, P. R. China

Abstract

In this paper we consider the application of the topological transversality theorem along with suitable a priori estimates to obtain existence of solutions to certain boundary value problems for mixed type functional differential equations of second order.

1. Introduction

143

Let r, h ~ 0 be given real numbers, denote by ]Rn the ordinary n-dimensional Euclidean space with norm I . I and let C([a, bl, ]Rn) denote the Banach space of continuous ]Rn-valued functions defined on [a, bl c ]R with the topology of uniform convergence. If [a, bl = [-r, hI, we let C := C([-r, hI, ]Rn) and for 'P E C its norm is I'PI = sup 1'P(s)1 If a, b E ]R and x E C([a - r, b + hI, ]Rn), then for t E [a, bl

-r<.<h we let Xl E -C-be defined by xl(lI) = x(t + lI), -r ~ II ~ h. If DC ]R x C x ]Rn and f : D ~ lRn is a given continuous function, we shall consider the following second

Research supported by NSERC Canada and National Science Foundation of China.

144

BVPs FOR FDEs

order mixed-type fum tional differential equation

x"(t) = f(t,x\x'(t)). (1.1)

Equation (1.1) is quite general and includes second order ordinary differential equations (r = h = 0)

x"(t) = f(t,x(t),x'(t)),

retarded functional differential equation (RFDE, h = 0),

x"(t) = f(t, Xl, x'(t)), (1.2)

as well as equations of the form

x"(t) = f(t,x(t - rl(t)), .. . ,x(t - rn(t)),x'(t)), (1.3)

where -h::; ri(t)::; r, i = 1, ... ,n. In this paper we shall consider the following boundary value problem for the

second order mixed-type functional differential equation

{

x"(t) = f(t,xl,x'(t)),

ax(t) - {3x'(t) = a(t),

,x(t) + 6x'(t) = b(t),

-r ::; t::; 0 (I)

where r, h ~ 0, T > 0 are given, and a, {3, " 6 are nonnegative constants with p = ,{3 + a, + a6 > o. We assume further that a(t), b(t) are given continuous functions defined on [-r, 0] and [T, T + h], respectively, and f: [0, T] x C x IRn -+ IRn

is continuous. We shall use the topological transversality theorem (cf. [1, 2]) along with suitable a priori estimates to obtain the existence of solutions of (I). The topological transversality theorem is based on the notion of an essential mapping and, in many cases, is easier to apply than methods based on topological degree. Granas et al. in [1-4] used this method to obtain existence results for various types of BVPs for ordinary and delay differential equations. We shall consider the further extension to mixed-type functional differential equations for problems of the form (I). Our considerations are motivated by [5] wherein BVPs for second order mixed-type equations are considered for singular functions using iterative techniques and the paper [6] which considers global existence for singular functional differential equations.

2. Topological Transversality and Existence Principle

In this section we shall establish the existence principle for the BVP (I). We first introduce some notation:

IIxllo:= sup Ix(t)1 O~I~T

MIXED TYPE FDE's

IIxlh := max{lIxllo , IIx'lIo}

KlIO, T) := {X E C2[O, T): O'x(O) - f3x'(O) = a(O), 'Yx(T) + t5x'(T) = b(T)}

K2[-r, T + h) := {x E C[-r, T + h); X[o,T) E C2[0, T)}

145

K1[-r, T + h) := {x E K2[-r, T + h): O'x(t) - f3x'(t) = a(t), t E [-r, 0) } 'Yx(t) + t5x'(t) = b(t) , t E [T, T + h)

CO,l[-r, T + h) := {x E C[-r, T + h): xl[o,T) E Cl[O, T)} with norm

IIxllr,h := max { sup Ix(t)l, sup Ix(t)l, IIxlh} -r:::;t:s;O T:::;t:S;T+h

where C l [0, T) := Cl([O, T), IR R), C2[0, T) := C2([0, TJ, IRn).

By a solution to (I) we mean a function x(t) E K2[-r,T + h) which satisfies the differential equation b (I) along with the boundary conditions.

In order to apply the topological transversality theorem, we need to formulate (I) into an appropriate fixed point problem. To this end, we consider the following diagram

K1[-r,T+h) ~ CO,l[-r,T+h)

K1[O,T) L

-------+

where i(x) = x for x E K1[-r, T + h) is the natural inclusion and for x E K1[O, T) we define the mapping R by

{

Xl(t),

(Rx)(t) = x(t),

X2(t),

tE[-r,O)

t E [O,T)

tE[T,T+h).

(2.1)

Here Xl (t), X2 (t) are obtained by solving the linear equations in (I) and are given explicitly by

tE[-r,O) (2.2)

t E [T,T + h). (2.3)

146

BVPs FOR FDEs

The mapping F>.. is defined by

(F>..x)(t):= >'f(t,xt,x'(t)), x E CO,l[-r, T + h), >. E [0,1] (2.4)

and the mapping L is given by

(L:.:)(t) = X"(t), t E [0, T), x E K1[0, T]. (2.5)

In the following lemma we describe some of the properties of the above mappings.

Lemma 2.1. For the mappings defined above we have: (i) the inclusion map i is completely continuous;

(ii) the extension map R is continuous; (iii) L -1 exists and is continuous.

Proof: (i) Let B be a bounded subset of K1[-r, T + h]. Then the set i(B) = {i(x): x E B} is bounded in CO ,l[-r,T + h] since IixIiCo.1[-r,T+h] S; Iixlir,h' We need only show that i(B) is equicontinuous so that i(B) is relatively compact by the Ascoli-Arzela Theorem. Suppose then that sup{lixll r,h: x E B} S; M for some M > 0. Then for any c: > ° and x E i(B) we consider five cases:

Case 1: s, t E [-r,O]. Since x(t) = X1(t) for t E [-r,O] and since X1(t) is uniformly continuous on [-r ,O), there exists 61 = 61 (c:) such that Ix(t) - x(s)1 = IX1(t) - x1(s)1 < c:/2 whenever It - sl < 61 .

Case 2: s, t E [T, T + h]. Since x(t) = X2(t) for t E [T, T + h), an argument as in Case (i) gives 6~ = 62 (c:) such that

It - sl < 62 ===? Ix(t) - x(s)1 = IX2(t) - x2(s)1 < c:/2.

Case 3: s, t E [O,T] . In this case we have Ix(t) - x(s)1 = I I: x'(u)dul S; I I,t Ix'(u)ldul S; Mit - sl so that there is a 63 = 63 (c:) > ° such that It - sl < 63 ===? Ix(t) - x(s)1 < C:2·

Case 4: If s E [-r,O] and t E (O,T), then xes) = X1(S) and

Ix(t)-x(s), = Ix(t)-x1(0)+x1(0)-x1(s)1

S; Ix(t) - Xl (0)1 + IX 1(0) - x1(s)1 < c:/2 + c:/2 = c:

if It - 81 < min(61 ,63):= 64 •

Case 5: If 8 E [0, T) and t E [T, T+ h] and if It - sl < min(62 , 63 ) = 65 , then as in Case 4 we have

Ix(t) - x(s)1 S; Ix(t) - x2(T)1 + IX2(T) - x2(t)1 < c:/2 + c:/2 = c:.

147

MIXED TYPE FDE's

Consequently, whenever It - Sl < 6 = min{61 ,62 ,63 } we have Ix(t) - x(s)1 < E for all x E iCE) which shows that i is completely continuous.

(ii) The fact that the extension map R is continuous follows since if x, y E K1 [0, TJ then we have

We note IXI(t) - YI(t)1 = le}t(x(O) - y(O»1 ~ IIx - ylll for t E [-r, OJ , if (3 > 0 (IXI(t) - YI(t)1 = 0, if (3 = 0, t E [-r, sJ). Similarly, for t E [T, T + h] we have IX2(t) - Y2(t)1 = le- t f( x2(T) - Y2(T»1 ~ IIx - yilt. Therefore, IIRx - Ryll ~ IIx - yilt so it follows that R is continuous.

(iii) Since the general solution of Lu = 0 is given by u = CI + C2t, an application of the boundary condition gives

(2.6)

Since p = ,(3 + ory + 016> 0, this system has a unique solution for CI, C2 so that L is one-to-one on K1 [0, Tj and hence L -1 exists, In order to show that the domain of L-I is C[O, Tj, we consider the following case:

For any 9 E C[O, Tj, let

U(t)=CI+C2t+ iT G(t,s)g(s)ds

where CI, C2 are the solutions of (2.6) given by

{

CI = a(0)(6 + ,T) + (3b(T) 016 + (3, + a,T

abeT) - ,a(O) C2 =

016 + (3, + a,T

and G(t,s) is the Green's function for the BVP

u"(O), au(O) - (3u'(O) = 0 = ,u(T) + 6u'(T).

(2.7)

(2.8)

(2.9)

From (2.8) and (2.9) it follows that u E K1[O, Tj and Lu = 9 on [0, Tj and so L is onto C[O,TJ . By the open mapping theorem, L-I is continuous.

To prove the cor..tinuity of F). we need to consider the restriction of F). to an appropriate closed subset of CO,I[-r, T+hJ. Suppose then that f: [0, TJ X C X IRn -.

IRR is completely cont~nuous and introduce the following notation:

148

BVPs FOR FDEs

For any M > 0 we define

MI := max {M, sup IXI (t)l, sup IX2(t)l, -r::;t::;O T::;t::;T+h

sup {If(t,xi,x'(t)l: IIxllr,h :s; M}} tE[O,T]

C~/[-r, T + h) := {x E CO,I[-r, T + h) :

G'x(t) - /3x'(t) = aCt), t E [-r,O),

,x(t) + 6x'(t) = bet), t E [T, T + hI}

C~~[-r, T + h) := {x E C~l[_r, T + h) : IIxllr,h :s; MI + I}

K1M[-r, T + h) := {x E K1[-r, T + h): IIXllr,h :s; MI + I}

K1M[O,TI := {x E K1[O,T): IlxliI :s; MI + I}

We consider the commutative diagram

C~~[-r, T + h) C[O,T)

K1[O,T)

(2.10)

(2.11)

(2.12)

(2.13)

(2.14)

(2.15)

where RM, iM, and F), are the restrictions of the corresponding maps defined in (*). By the choice of .'111 > 0, the above restricted maps are all well-defined.

We may now pr:>ve the continuity of F),.

Lemma 2.2. Let I: [0, T) x C x JR." -+ JR." be completely continuous. Tben for any M > 0, tbe following statements bold:

(i) iM is compact and RM is continuous, (ii) F), is continuous for each >. E [0,1).

Proof: Part (i) follows from Lemma 2.1 so we need only prove (ii). To this end, let>. E [0,1) be fixed. Note that for every x, y E C~~[-r, T + h) we have

IIF)'x - FwllB = sup I(F),x)(t) - (F),y)(t)1 O::;t::;T

= sup IV(t,xt,x'(t)) - >'f(t,yi,y'(t))1 O::;t::;T

:s; sup II(t, xt, x'(t)) - f(t, yt, y'(t))I. O::;t::;T

149

MIXED TYPE FDE's

This implies the continuity of F>. since xt is continuous with respect to t, f is continuous and {xt : x E C~~[-r, T + h]} is a precompact subset in C by the Ascoli-Arzela theorem. 0

We may now state and prove the existence principle.

Theorem 2.3. Assume that f : [0, T] x C x Rn ~ Rn is completely continuous. H there is a constant M > ° such that for every solution x of the BVP

{

X"(t) = >..f(t, xt , x'(t)),

ax(t) - (3x'(t) = aCt),

,x(t) + 6x'(t) = bet),

t E [0, T], A E [0,1]

tE[-r,O]

tE[T,T+h]

we have IIxllr,h $ M, then the BVP (I) has at least one solution.

Proof: Let Ml be defined as in (2.10) and set

V:= {x E K1[O,T]: IIxlll $ Ml + I} C K1M[O,T].

We define a map H: [0,1] x V ~ K1[O, T] by

(2.16)

(2.17)

Then for every A E [0,1], H(A,.) is a compact map by Lemma 2.2. Moreover, the fixed points are precisely the solution to (I>.). Further, H( A, .) has no fixed points on BV by the choice of V and the map H(O, ·) = Cl +C2t where Cl, C2 are given by (2.8) is essential. Therefore, the topological transversality theorem shows that H(l,·) has at least one fixed point in V which gives a solution to (I) in K1[-r, T + h]. This completes the proof. 0

3. Applications of the Existence Principle

The applicability of Theorem 2.3 depends on finding suitable a priori bounds for the solutions of the BVP (I>.), independent of A E [0,1]. We next state two lemmas whose proofs are essentially the same as those in [7] .

Lemma 3.1. Suppose that the function f satisfies the following conditions:

(HI) There exists a constant M > ° such that for every (t,u,v) E [0, T] xCxRn with lu(O)1 > M and u(O)· v = ° implies u(O) · f(t, u, v) > ° where te." denotes the dot product.

150

BVPs FOR FDEs

Then every solution X of(l>.),'\ E [0,1] satisfies max Ix(t)l:::; Mo = max{M,AI, -r9$T+h

Bd where

AI=SUp{XI(t): -r:::;t:::;O, Ix(O)I:::;M} and

BI = SUp{X2(t): T:::; t :::; T + h, Ix(T)I:::; M}

and XI(t) , X2(t) are given in (2.2) and (2.3).

Lemma 3.2. Assume that there exists a constant Mo > ° such that max Ix(t)1 -r<t<T+h

:::; Mo for every solution of the BVP (1).), ,\ E [0,1] . Also, suppose that the continuous function f : [0, T] x C x IRn --+ IRn satisfies the following conditions:

(H2 ) u(O)· f(t,u,v):::; k1 lvl 2 + k2 (H3) v ' f(t,u,v):::;(k3IvI 2 +k4)lvl for any (t,u,v) E [O,T] x C x IRn with lIull:::; M o, where kl' k2, k3 , k4 are positive constants satisfying

and k (1 - kd2

3 < ,,r CJY'o

Then there exists a constant M independent of,\ such that max Ix'(t)1 :::; M for 09$T

every solution of the BVP (1).) .

By combining the previous two lemmas, we may state the following theorem.

Theorem 3.3. Let f: [0, T] x C x IRn --+ IRn be completely continuous and assume that the hypotheses (HI)' (H2), (H3) are satisfied. Then the BVP (1) has at least one solution.

References

1. A. Granas, R. B. Guenther and J. W. Lee, Some general existence principles in the Carathe6dory theory of nonlinear differential systems, J. Math. Pum. et Appl. 70 (1991), 153-196.

2. A. Granas, R. B. Guenther, and J. W. Lee, On a theorem of S. Bernstein, Pacific J. Math. 74 (1978), 67-82.

3. J. W. Lee and D. O'Regan, Existence results for differential delay equations I, J. Diff. Eqns. 102 (1993), 342-359.

4. J. W. Lee and D. O'Regan, Existence results for differential delay equations II, Nonlinear Analysis 17 (1991), 683-702.

5. L. H. Erbe and Qingkai Kong, Boundary value problems for singular second order functional differential equations, J. Compo Appl. Math. 53 (1994), 377-388.

151

MIXED TYPE FDE's

6. Huaxing Xia and T. Spanilly, Global existence problem for singular functional differential equations, Nonlinear Analysis 20 (1993), 921-934.

7. S. K. Ntouyas, Y. G. Sficas and P. Ch. Tsamatos, An existence principle for boundary value problems for second order functional differential equations, Nonlinear Analysis 20 (1993), 215-222.

153

THE FILIPPOV APPROACH TO BOUNDARY AND INITIAL VALUE PROBLEMS AND APPLICATIONS

RONALD B. GUENTHER 1, JOHN W. LEE AND MARTIN SENKYRiK 1

Department of Mathematics, Oregon State University, Kidder Hall 968, Corvallis, OR 97991 USA

1. Introduction

In this paper we prove a general existence principle for the second order differential equation (or system of equations)

y" = f(t, y, y') (1)

on a semi infinite interval with no requirements on the continuity of f. Many results for ordinary differential equations with discontinuities only in the

time variable were proved by using Caratheodory's definition of a solution. Filippov's definition of a solution is more general than that of Caratheodory and includes it as a special case. A standard approach to differential equations with discontinuities in the spatial variable is to solve the problem on each side of the discontinuity separately and and then try to match these solutions. A totally different approach is used here. Using Filippov's theory, we reformulate differential equations as differential inclusions and then prove an existence principle for differential inclusions on a semi infinite interval to obtain existence results for the original equation. An application to a pendulum with dry friction is given.

2. Existence Theory for Semi Infinite Intervals

Consider an initial value or a boundary value problem for a differential equation

X' = f(t,x), (2)

where f(t,.) may be discontinuous and x E R,M. Based on the idea that E C R,M

with p(E) = 0 should play no role, Filippov defined solutions of (2) as solutions of

lpartially supported by ONR N00014-92J 1226

154

BVPs FOR FDEs

the differential inclusion constructed as a convexification of f with respect to x E RM in the following way:

x'(t) E n n konvf(t, U(x(t), 0) - E), 6>0 ,.8=0

for almost every t, where U(x, 0) = {y : Ix - yl < o} and konvY is the closed convex hull of Y.

Definition 1 Let x be absolutely continuous on the interval 1. If x satisfies

x'(t)E n n konvf(t,U(x(t),o)-E) =K{J(t,x)} =kt(x) 6>0 ,.8=0

for almost every t E I, we say x is a Filippov solution of (2) .

We denote the collection of nonempty, compact, convex, subsets of a topological space X by Kv(X).

Definition 2 Let p ~ l.ThenF : [0,(0) X RN X R,N --+ K v(R,N) is locally L1'CaratModory if (a) the map z --+ F(t, z) is upper semi continuous (u.s.c.) for every x E [0,(0); (b) the map t --+ F(t,z) is measurable for every z E R,N; (c) for each compact interval I C [0,(0) and for each r > 0, there exists hr E £1'(1) such that

Izl $ r => IF(t,z)1 $ hr(t) for a.e. t E I.

We shall use the following lemma; see [8] .

Lemma 3 Let F : [0,(0) X (tN X R,N --+ K V (R,N) be locally £1' - Caratheodory and 1= [0, b] with b > 0 be a compact interval. Define Co(1) = {y E C(I) : y(O) = O} and

(NFy)(t) = l F(s,y(s),y'(s))ds.

Then NF : C 1(1) --+ 2c,,(l) and a) NF : C 1(1) --+ Kv(Co(I)); b) NF is u.s.c.; c) NF is completely continuous.

The next result helps establish a link between Filippov solutions of differential equations and Caratheodory solutions for differential inclusions.

155

THE Fn.IPPOV ApPROACH

Theorem 4 Suppose f(t,::) is measurable in [0,(0) x R 2N, I is a compact interval, Ie [0,(0) and further suppose that for any bounded closed domain Del X R 2N

,

there uists an integrable function B(t), which can depend on D, such that

If(t,x)1 ~ B(t) (3)

almost everywhere in D. Moreover assume that for all (to, xo) e I x R2N there uists 61, 62 > 0 and a function Crt): [to - 61, to + 61]-+ R such that

If(t,x)1 ~ C(t) (4)

for (t, x) e [to - 151, to + 151] x cl(U(xo, 152» = el(U(to, Xo, 151, 152» (elY is the closure of Y) , where at the endpoints, we consider appropriate one-sided neighborhoods. Then the function

n n konvf(t,U(x(t),c5)-E) = K{f(t, x)} = kt(x) (5) 6>0 ,.E=O

is locally L'- Caratheodory.

Proof. The proof is a straight forward modification of the proof given in [13], where the proof is done for I = [0,1].

A proof of the next lemma may be found in [3].

Lemma 5 Let r : X -+ 2Y be a point-compact, u.s.c. map. Let {Xa} be a net in X and Ya e r(Xa) for each a. If Xa -+ x and Ya -+ y, then y e r(x).

Theorem 6 Let F : [0,(0) x R,2N -+ RN be locally L"- Caratheodory. Suppose y .. e JV2"[O, n] solves

y: e F(t, Yn, Y:), aYn(O) + {3y:(O) = 1

and that there are bounded sets B" and BZ in RN for k = 1,2, ... such that

n ~ k ~ Yn(t) e B", y~(t) e Brr for 0 ~ t ~ k.

Then there exists y e W;!-:'[O, (0) such that

{ y" e F(t, y, y~ a.e. and ay(O) + {3y'(O) = 1,

y(t) ,e 13", y'(t) e ~ for 0 ~ t ~ k for every k.

Proof. First we show that

{for each k = 1,2, ... , {Yn}n>" and {y~}n>" are

uniformly bounded and equicontinuous on [0, k]. (6)

156

BVPs FOR FDEs

This is clear for {y .. } .. ~k. We need to show that {y~} .. ~k is equicontinuous on [0, k]. Let t, t" e [0, k]. Then for n ~ k, we have for some w" e F(t, y", y~) and for almost all t e [0, k] that

Iy~(t) - y~(t*)1 = It w,,(s)dsl ~ J: hk(s)ds,

where hI. e Ll[O, k] is determined from the bounded sets in the theorem and the fact that F is locally LP- Carathoodory. The equicontinuity of {y~}">k is now a direct consequence of the absolute continuity of the Lebesgue integral. The existence of the function w" used in the preceding argument is a consequence of Kuratowski-RyllNardzewski theorem; see [1].

From (6) and the Arzela-Ascoli theorem it follows that there exists Nl" C N+, the set of all positive integers, and Zl e Cl[O, 1] such that

Y(;) -+ z(j) J" - ° 1 as n -+ 00 through N" n 1, - , 1·

H Nl = Nt - {I}, then (6) and Arzela-Ascoli imply that there exists N; C Nl, the set of all positive integers, and Z2 e C l [0,2] such that

Y(;) -+ z(j) J" = ° 1 as n -+ 00 through N." " 2 , , 2·

Evidently, Z2 = zion [0,1]. Proceed inductively to obtain

Nl 2 N2 2···2Nk 2···, Nd;;{k+1,k+2, ... }, ZkeCl[O,k] with

z" e Cl[O, k] with y~) -+ zf) uniformly on [0, k] as n -+ 00 through Nk •

Clearly, Z/o+l = Zk on [0, k] for k = 1,2, .... Define y : [0,00) -+ R" by yet) = Zk(t) on [0, k]. For any k,

{ y e ClIO, k], ay(O) + .By'(O) = ,"(,

yet) e 11;., y'(t) e B;' for t e [0, k].

Finally, we show that y satisfies the differential inclusion y" e F(t, y, y') almost everywhere on [0,00). To this end, fix k. For ° ~ t ~ k and n e Nk we have

y~(t) -y~(O) e [F(s,y,,(s),y~(s))ds = (NFy,,)(t)

where NF : Cl[O, k]-+ 200[0,101. Consequently,

y~ - y~(O) e NFY".

FUrthennore, NF is point-compact and u.s.c. by Lemma 3. Since

y~(t) - y~(O) -+ z~(t) - zHO) and y,,(t) -+ Zk(t)

157

THE FILIPPOV ApPROACH

as n -+ 00 through N Jr, it follows from Lemma 5 that

Since z" = y on [0, k], we conclude that

y'(t) -:- y'(O) e l F(s, y(s), y'(s))ds for t e [0, k].

Thus, y e W'l,p[O, k] ~d y"(t) e F(t, y(t), y'(t)) for a.e. t e [0, k] and for each k= 1,2, ....

Remark 1: As the proof of Theorem 6 reveals, the Sturm-Liouville boundary condition ay,,(t) + py~(t) = '"Y can be replace by any boundary condition of the form tp(y" (0) , y~(O)) = 0 with tp : R2N -+ ~ continuous. Furthermore, if each y,,(t) satisfies several such boundary conditions, the global solution y(t) in the conclusion of the theorem also satisfies the same conditions. In particular, if y,,(O) = A and y~(O) = B for all n, then y(t) will satisfy the same initial conditions.

Remark 2: Combining the results of Theorems 4 and 6 gives an existence principle for Filippov solutions to certain initial and or boundary value problems.

Remark 3: The proof of Theorem 6 follows lines of reasoning first used in [9], where existence theorems are established for classical solutions to boundary value problems on infinite intervals, especially for semi-conductor devices. The results in [9] are restricted to differential equations. Further work [5] and [11] establishes similar results for Caratheodory solutions for certain classes of differential equations, including the Thomas-Fermi equation.

3. Applications

In this section, the foregoing existence results are applied to a pendulum that experience both viscous damping and dry friction and is driven by an external force. Periodic solutions for such pendulums were established in [13]. Here we establish global solutions in time for the initial value problem. After existence is established, we present and discuss some interesting numerical results.

Consider the pendulum equation

y" + by' + ksgn(y') + csin(y) = e(t), y(O) = Yo, y'(O) = y~, (7)

BVPs FOR FDEs

where b, k, and c are positive constants, e(t) E L~,[o, oo) , and e(t) is further restricted according to the cases treated below. Of course, yo and y; are given initial data.

For the moment restrict t to the interval [O, TI. The following reasoning establishes the p e n ' bounds need to guarantee the existence of a solution to (7) on [0, T ] and also to establish the estimates needed to apply Theorem 6 and hence obtain global solutions to the initial value problem for (7). Let X E (O,1] and y = y(t) be a solution to

and define

yn Y? E(t) = - 2 + Xc(l - cosy), Eo = - 2 + c(1 - cos yo).

Note that E(0) 5 &. Multiply the diffktential equation in ( 7 ) ~ by y' to obtain

E1(t) = -Xbyn - Xk lyll+ Xyle(t). (8) Now we consider three dXerent cases corresponding to t hee interesting physical situations.

Case 1. le(t)l 5 k; dry Mction predominates. Then the right member of (8) is nonpositive, E(t) is decreasing, and hence,

E(t) 5 E(0) 5 Eo.

It follows that

1y1(t)l 5 c = (yf + 241 - aa for 0 5 t T

Notice that the constant C does not depend on X E (O,1] or on T. Case 2. e(t) E LZIO, oo); the driving force has finite energy. Observe that, for any E > 0,

1 t - Lt byndr + y'edr 5 - /i by'% + it yadr + 5 1 e2dr. 0 2

NOW take E = b, use the foregoing estimate in (8), and integrate to find that

It follow that there is a constant C independent of A E (0,1] and of T sueh that

lyl(t)( 5 Cfor 0 St ST. (10)

159


Case 9. le(t)1 :5 k for t ~ T1 ; dry friction ultimately dominates. Since e(t) is locally in L2, a bound for Y'(t) as in (10) holds on [O,T1] and, if

T ~ TI , the bound for Y'(t) as in case 1 holds on [TI' TJ. Thus, once again, IY'(t)1 :5 C on [0, TJ for any T, where C is a constant independent of T.

Under the assumptions in cases 1, 2, or 3, there is a constant C independent of A E (0,1] and of T such that

ly'(t)1 :5 C for 0 :5 t :5 T. (11)

It follows that

Iy(t) I :5 Iyol + 1o'IY'(r)dr l :5 Iyol + CT for 0:5 t :5 T, (12)

ly(t)l:5 IYol + Ck for 0 :5 t :5 k for any k :5 T. (13)

Apply (11) and (12) with T = n and the existence results in [5] or [11] to obtain the existence of a solution y,,(t) to (7) on [0, n] . Furthermore, it follows from (11) and (13) that the hypotheses in Theorem 6 hold with BIc = [-lyol- Ck, Iyol + Ck] and B: = [-C, C). Consequently, (7) has a globally defined solution y(t) such that

ly(t)l:5 IYol + Ck for 0:5 t:5 k for any k , Iy'(t) I :5 C for 0:5 t.

Further important qualitative information about the solution y(t) is provided by the following estimates. Equation (8) with>' = 1 holds for the solution y(t) . Now consider case 1 in which le(t)1 :5 k and, hence, E'(t) :5 _byt2. Integrate to obtain

E(t) :5 -l byt2dr + Eo.

It follows that, for any t ~ 0,

yt2 ft 2 :5 - 10 by

t2dr + Eo,

yt2 + 2l byt2dr :5 2Eo,

~ (e2bt lot yt2dr) :5 2Eoe2bt,

10' yt2dr:5 ~.

Consequently, in case 1, y' E L2[O, 00).

160

BVPs FOR FDEs

Finally, we consider the case of a bounded forcing term that may not be dominated by frictional terms.

Case -/. le(t)1 ~ Kj bounded forcing. As in cases 1, 2, and 3 assume first that 0 ~ t ~ T. From (8),

E'(t) ~ K Iy'l ,

and, hence,

E(t) ~ Eo + K lIY'(r)1 dr,

1 2Y'(t)2 ~ Eo + Ktma.x {ly'(r)1 : 0 ~ r ~ t}.

It follows that there is a constant CT independent of ). such that

ly'(t)1 ~ CT for 0 ~ t ~ T, (14)

and that for any 0 ~ k ~ T there is a constant Ck (dependent on k but not on T ) such that

ly'(t)1 ~ Ck for 0 ~ t ~ k.

Then

ly(t)1 ~ Iyol + CTT for 0 ~ t ~ T,

ly(t)1 ~ Iyol + Ckk for 0 ~ t ~ k for any k ~ T.

(15)

(16)

(17)

Just as in cases 1-3, use (14)-(17) to obtain the existence of solutions y,,(t) to (7) on [0, n] and which satisfy the hypotheses in Theorem 6 with Bk = [-IYol-Ckk, Iyol+ Ckk] and BZ = [-Ck, Ck]. Consequently, just as in the other cases, (7) has a global solution y(t) defined on 0 ~ t < 00.

Remark 4: Figures 1 and 2 show particular pendulum motions. In Figure 1 the forcing term dominates the frictional effects and the pendulum rotates counterclockwise with ever incrasing angular coordinate. The plot of the angular velocity indicates sticking pericds where no motion occurs due to the dry friction. In Figure 2 the pendulum starts near its unstable equilibrium and is driven by a force with variable sign. The pendulum oscilates about its unstable equilibrium and then falls doum and oscillates about its stable equilibrium. As in Figure 1 we can observe periods of sticking.

8

7

6

5

3

2.

a

-1 a 0.5


angls

1. 5 2.

angular velocity

2.5 timet

3 3.5 4 4.5 5

Figure 1: A solution of the equation y" + 2y' + 3siny + 2signy' = 7Isin2?rt/51

3.5,---------.--------.----------,

2.5 angle

1.5

timet

Figure 2: A solution of the equa.tion y" + 2y' + 3siny + 2signy' = 5sin2?rt

161

162

BVPs FOR FDEs

References

1. J.P. Aubin, A. Cellina, Differential inclusions, A series of comprehensive studies in mathematics, No. 264, Springer-Verlag, Berlin - Heidelberg - New York, (1984).

2. C. Castaing, M. Valadier, Convex analysis and measumble multifunctions, Lect. notes in math., N 0.580, Springer-Verlag, Berlin - Heidelberg - New York, (1977).

3. J. Dugundji, A. Gra.na.s, Fi1:ed Point Theory, Vol 1, Monographie Matematyczne, PNW, Warsza.a.wa, 1982.

4. A.F.Filippov, Differential equations with discontinuous right-hand side, Translat. Amer. Math. Soc. 42 (1964), 199-230.

5. M. Frigon, Application de la thforie de la tmnsversaliU topologique Ii des problemes non lineares pour des equations differntielles ordinaires, Dissertationes Math. 296, (1990).

6. M. Frigon, A. Granas, Theoremes d'existence pour des inclusions diff&eD.tielles sans convexite, C.R. Acad. Sci. Paris, Serie 1, (1990)., 819-822.

7. M. Frigon, A. Granas, and Z.E.A. Guennoun, Sur l'intervalle ma.ximal d'existence de solutions pour des inclusions diff ere ntielles, C.R. Acad. Sci. Paris 306, Serie I, (1988), 747-750.

8. A. Granas, R.B. Guenther and J . W. Lee, Some existence results for the differential inclusions y(k) E F(x, y, •.. , y(k-l)), y E 8, C.R . Acad. Sci. Paris 307, Serle I (1988), 391-396.

9. A. Granas, R.B. Guenther, J.W. Lee, D. O'Regan, Boundary value problems on infinite intervals and semiconductor devices, Journal for Mathematical Analysis and Applications 116(2) (1986), 335-349.

10. J. Kurzweil, Ordinary Differential Equations (Translation from the Czech edition by Michael Basch), Elsevier, Amsterdam - Oxford - New York - Tokyo, (1986).

11. J.W. Lee, D. O'Regan, Existence principles for differential equations and systems of equations, Proceedings, Topological Methods in Differential Equations and Inclusions, NATO ASI Series C, Kluwer Academic Publishers, Dordrecht, NL, (in press).

12. M. Senkytik, A topological approach to dry friction and nonlinear beams, Ph.D. Thesis, Oregon State University, Corva.llis, Oregon, (1995).

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13. M. Senkytik and R.B. Guenther, Boundary value problems with discontinuities in the spa.tial varia.ble, Journal of Mathematico1 Analysis and Applications (in press).

AN EXISTENCE RESULT FOR DELAY EQUATIONS UNDER SEMILINEAR BOUNDARY CONDITIONS

1. Introduction

G. HETZER Department of Mathematics, 304 Parker Hall

Auburn University, AL 36849-5310, USA

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Let us begin with describing an example from applications that leads to a "nondegenerate" boundary value problem in the sense of Waltman and Wong (d. [9]).

Example 1.1. Diaz has studied in [4] the controllability of the climate in the framework of a Budyko-type energy balance climate model. The resulting equation is a reaction-diffusion equation with a discontinuous reaction term. If one additionally accounts for the long response times within the climate system, memory terms arise (cf. [8]). In particular, the continental ice-sheets have memory spans of tens of thousands of years. For the purpose of illustration, let us restrict ourselves here to the "global case" and denote by u = u(t), say, the ten-year mean of the globally averaged surface temperature in Kelvin. We account then for expansion or retreat of these ice-sheets by employing a long-term mean f~Tj3(S)U(t +s,x)ds with T:=::;j 104 ys., the memory span of the system, and j3 E COO 0 - T, 0]) satisfying j3( s) > 0 V s E [-T, 0], j3(j)(-T) = 0 Vj E Z+ and f~T j3(s)ds = 1. The albedo a will be a function of u(t) as well as of J~T j3( s )u( t+s, x )ds. Thus, the absorbed radiation flux at time t > 0 is given by Q(t)[l- a( u(t) , f-T j3(s )u(t+ s, x )ds )] , where Q = Q(t) is the ten-year mean of the globally averaged solar radiation flux. Finally, the outgoing terrestrial radiation flux is denoted by g(>., u(t)), e.g. g(>., u) = e(>., u)u4 according to the Stefan-Boltzmann law with e an infrared opacity function and>' ERa control parameter which models the human impact on the emitted radiation flux (keywords: greenhouse effect, CO2 ,

trace gases). The energy balance equation yields

u(t) = Q(t)[l- a(u(t), lTj3(s)u(t+ s,x)ds)]- g(>.,u(t)) (1.1)

for t > O. Selecting a climate history {) E C([-T, 0], R+), a time b > 0 and a temperature p E (0,00) one asks e.g. for a >. E R and a solution u of Eq. (1.1) satisfying Uh-T,O) = {) and u(b) = p in order to study, whether human impacts could drive the

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BVPs FOR FDEs

climate system to the temperature p at time b. Of course, the outlined setting is just that of a "toy-model", and e.g. the framework considered in [8] would immediately lead to a corresponding boundary value problem (in time) for a functional reactiondiffusion equation on the sphere. In this note, we intend to describe some basic existence results for first order semilinear functional differential equations subject to semilinear boundary conditions of the form

x(t) + l(t,xt) = f(t,xt) t E [O,b] , 00xo + 0bXb = g( XO , Xb),

(1.2)

where I: [O,b] x C([-I,O],RD) --> Rn is continuous, l(t ,· ) is linear and 0 0 , 0b are bounded linear mapping from C([-I, 0], Rn) into a Banach space V. Throughout, we also require f : [0, b] xC([-I, 0], RD) --> R n to be continuous and 9 : C([-I, 0], RD)x C([-I, 0], Rn) --> V to be completely continuous and write Ut for s t--+ u(t + s) E C ([-1, 0], R n) . The modifications necessary to deal with the par ameter-dependent problem Eq. (1.2) are mostly technical, and hence we are not going to dwell upon this aspect, all the more so as realistic modeling anyway requires to consider functional reaction-diffusion equations, a topic beyond the scope of this paper. In the special case, where g is a constant, Eq. (1.2) has been studied by Waltman and Wong [9] using Granas 's version of the Schauder fixed point theorem and a shooting method and by Hale [6], who employed a reduction process that rewrites Eq. (1.2) as an operator equation via the variation of constant formula. Let us briefly indicate the latter. Consider the homogeneous linear equation

Ii + l(·,y.) = 0. (1.3)

We denote by 3 = 3(t,s)if> (s E [O,b], t E [s,b], if> E C([-I,O],Rn)) the solution operator of Eq. (1.3), i.e. 3(t, s)if> = Yt with y the solution of Eq. (l.3) satisfying y. = if>, and write Y = Y(t,s) for the fundamental solution of Eq. (1.3), i.e. Y(t,s) is the zero matrix for -00 < t < sand s E [0, b], and given z E Rn and s E [0, b], t t--+ Y(t, s)z is the solution of Eq. (1.3) on [s, b], which satisfies y(s) = z. Assuming that the initial value problem

{x(t)+I(t,Xt) =f(t,xt) Xo = if>

has a unique solution x = x(t,if» for all if> E C([-I , O],RD) (we write x. (if» for the mapping T t--+ X(T + Sj if» on [-1,0]), one obtains for the special case 9 E V, b 21 that Eq. (l.2) is equivalent to

(1.4)

which can be studied by methods from functional analysis. It was shown by Waltman and Wong [9] for b 2 1 and in [6] for the general case that Eq. (1.4) is solvable

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SEMILINEAR BOUNDARY CONDITIONS

provided that 9 0 + 9b 0 3(b,0) is onto and liIIltltPlioo-oo IIfl~~t~l,", = ° uniformly for t E [0, b]. For Eq. (1.2) it appears to be more convenient to write the boundary value problem as an operator equation Lx = Nx from a subspace of C([-l,b],RD) into C([-l,O],RD) X C([O,b],RD) and to employ solvability results for semilinear operator equations in place of fixed point theorems. Thereby, one also avoids to impose "uniqueness conditions" on f and g.

2. Boundary Conditions and Preliminaries

Throughout the following notations are used: For n E Nand b E (0,00) set C := C([-I,O],RD), Y := C([O,b],RD) and Z := C([-I,b],RD) and equip each of these spaces with the respective maximum norm 11 ·1100' We begin with some remarks about the linear initial value problem

{ u(t) + l(t, Ut) = y(t) Uo = q

t E (0, b] (2.1)

assuming y E Y, q E C and I : [0, b] x C --+ R D continuous with l( t, .) linear for t E [O,b]. It is well-known (d. [6,7] e.g.) that Eq. (2.1) has a unique global solution, which will be called z = z(q,y), and that the mapping z : C X Y --+ Z is continuous. Operators S : Y --+ C and T : C --+ C are defined by Sy = z(O, y h and Tq = z(q,Oh, respectively. Clearly, Sand T are bounded linear operators, S is completely continuous and T is completely continuous provided that b ~ 1. Actually, Tit)+! is completely continuous ([p) := sup{m EN: m ::; p} for pER). Before formulating standing basic hypotheses for Eq. (1.2) let us take a look at another example.

Example 2.1. (Perturbed periodicity) Consider u(t) = f(t,ud 0::; t ::; b subject to Ub = uo+g( uo), where f : [0, b) xC --+ R D is continuous, 9 : C --+ C is completely continuous and limll",lIoo_oo IIg(cp) 1100 / IIcplioo = 0. The homogeneous linear problem is UlIo,b) == 0, Ub = uo,which has the constants as solutions. The nonhomogeneous problem UlIo,b) = Y E Y, Ub = Uo + p with p E C is solvable, iff p( b) = - J; y( s )ds, thus the classical Fredholm alternative holds. What we are going to suggest is to rewrite the original semilinear problem as an operator equation Lu = Nu, where L : Z :::> dom( L) --+ Y x C defined by Lu := (u IIo,b], Uo - Ub) is a Fredholm operator of index ° and Nu := (J(·,u.),g(uo)) is continuous. An extended version of this example arises from a simple economic model.

Example 2.2. (Controllability of inflation rate) Kehong Wen proposed the following model to describe monetary growth cycles (d. [10) e.g.).

{ x(t) = -ax(t - T) + u(t - 0") + (x(t - T)) u(t) = (J - 6x(t). (2.2)

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BVPs FOR FDEs

Here, notations have been adapted for our purposes, Q , f3 and fI are positive reals, and u, T belong to (0,1] . The functions x and u stand for, respectively, the deviation of inflation rate and money supply rate from their growth trend and cI>(1]) := 1]e-~2. Economists sometimes consider downsizing the inflation trend by controlling the money supply rate. A first step in dealing with controllability in the framework of Eq. (2.2) is to study this system subject to boundary conditions

Xb = Xo -1'1(XO , uo) Ub = Uo -1'2(XO, UO, Xb,Ub)

(2 .3)

(b> 1). Here,1'1 would be a target reduction function for the period [b-l , b] , which should of course depend on the histories Xo and Uo, whereas 1'2 would stand for the change in the money supply rate. Clearly, this leads to the same type of operator equation as in example 2.1. These and other examples fall into the following scope:

(HI) I: [0, b] x C ~ R D continuous, I( t, .) linear for t E [0, b] ;

(H2) f : [0, b] x C ~ RD continuous and bounded on bounded subsets;

(H3) V Banach space, 0 p : C ----t V bounded linear operator for p E {O, b} ,

(H4) g : C X C ----t V completely continuous.

We are going to employ the following results about quasi bounded perturbations of Fredholm mapping of index 0, which can be found in [5]. Recall that a mapping r : E ----t F between Banach S'ttili E and F is said to be quasibounded, iff its quasinorm IIlflll := lim sUPIi'Ii_co ~.i, < 00. Another tool which we will use later is the so-called topological degree, which goes back to Brouwer. This concept associates with triples (F,O,z) an integer deg(F,O,z), which is a measure for the number of solutions of F(x) =Z in O. Here, 0 is an open bounded subset of, say, RD, z E RD and F : cl(O) ~ RD ;s continuous with z ¢ F(aO). We have

deg(F,O, z ) = L: sgn(detJF(x)) x EF-'( . )

in the special case where F E C2(cl(O) , RD) and the Jacobian hex) is invertible for all x E F-1(z). Of course, the set F-1(Z) is finite under these assumptions. The extension to the general case occurs by means of approximation and Sard's theorem. We refer to [3] for details. Moreover, recall that a continuous mapping F from a subset M of a normed space B into another normed space E is called completely continuous, iff F maps bounded subsets of M into relatively compact subsets of E . Now, let L : B :J dome L) ~ E be a Fredholm-operator and P : B ~ Band Q : E ~ E be projections with ran( P) = ker( L) and ker( Q) = ran( L) , respectively. We say that F is L-completely continuous, iff QoF is continuous and (Llker(p)ndom(L))-1 0 (Id- Q)o F

169


is completely continuous. The following degree continuation theorem for semilinear operator equations with quasi bounded nonlinear parts is due to J . Mawhin.

Proposition 2.3. (d. [5; Theorem VIII. I. , p. 135)) Let E, F be Banach spaces, L : E :J dom(L) --+ F be a Fredholm operator of index 0 and N : E --+ F be L-completely continuous. Moreover, denote by P : E --+ E and Q : F --+ F continuous projectors with ran(P) = ker(L) and ker(Q) = ran(L), respectively, and by J: ran(Q) --+ ker(L) a linear isomorphism. Assume that

(a) (Llker(p)ndo771(L»)-l 0 (Id - Q) is quasibounded;

(b) There exist a, r E (0,00) such that each solution of Q 0 N x = 0 fulfils IIPxll < a lI(Id - P)xll + r;

(c) (1 + a) 111(Llker(p)ndo771(L»)-l 0 (Id - Q) 0 NIII < 1;

(d) deg(J 0 Q 0 Nlker(L)' B(r) n ker(L), 0) = 0, where B(r) the open ball in E with radius r and center

Then, (L - N)(dom(L)) :J ran(L) , hence in particular there exists an x E dom(L) with Lx = Nx .

Now assume that E is a subspace of B(S, R D), the Banach space of bounded

functions from S into RD, and that IIxllE ~ Ilxlioo for all x E E . We will use the following version of Proposition 2.3.

Proposition 2.4. (d. [5; Theorem VIII.2., p. 136)) Let E satisfy the assumptions described before, F be a Banach space, L: E :J dom(L) --+ F be a Fredholm operator of index 0 and N : E --+ F be L-completely continuous. Moreover, denote by P : E --+ E and Q : F --+ F continuous projectors with ran(P) = ker(L) and ker(Q) = ran(L), respectively, and by J : ran(Q) --+ ker(L) a linear isomorphism. Assume that

(1) There exists..\ E (0,00) such that for each x E ker(L) and each s E S IIxllE $ ..\lx(s)l;

(2) There exists rl E (0,00) such that Q 0 N x "I- 0 for each x E dom( L) that fulfils Ix(s)1 ~ rl Vs E S;

(3) 111(Llker(p)ndo771(L»)-l 0 (Id - Q) 0 NIII < (1 + ..\)-\

(4) deg(J 0 Q 0 Nlker(L)' B(..\rl) n ker(L), 0) = O.

Then, (L - N)(dom(L)) :> ran(L) , hence in particular there exists an x E dom(L)

with Lx = Nx.

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BVPs FOR FDEs

3. An Equivalent Operator Equation

In this section the boundary value problem (1.2) will be rewritten as an equivalent operator equation Lx = N x and some properties of Land N will be established. Throughout, (HI) - (H4) are assumed. We set :

Ru := UhO,b] for u E Z,

dom(L):= {u E Z: Ru E C'([O,b],RD)},

Lu = ((Ru) · + 1(· , u .),00uo + 0bUb) for u E dom(L),

Nu = (fC- , u.), g(UO,Ub)) for u E Z.

Obviously, u solves Eq. (1.2), iff u E dom(L) and Lu = Nu. Moreover, L is a closed linear operator from Z into Y xV.

ker(L) = {z (q , O)lq E C, (00 + 0 b 0 T)q = OJ, (3.1)

and

ran(L) = {(y,cp) E Y x V: 3q E C : (00 + 0 b 0 T)q + 0 b 0 Sy = cpl. (3.2)

The following fact can readily be established:

Lemma 3.1. Let (H1) and (H3) be fulfilled and 0 0 + 0 b 0 T be a ~+-operator, then

(i) L is a ~+-operator,

(ii) dim(V/ran(00 + 0 b 0 T)) :::: dim((Y x V)/ran(L)) .

We obtain as a consequence that L is a Fredholm operator in case that 0 0 + 0 b 0 T IS one.

Proposition 3.2. Suppose that (H1) (H4) are satisfied and that 0 0 + 0 b 0 T is a Fredholm operator. Let P : Z ---+ Z be a continuous linear projector on ker(L) , Lp' := (Llker(p)ndom(L)t' and Q : Y x V ---+ Y x V be a continuous linear projector

with ker(Q) = ran(L) . Then Lp' 0 (Id - Q) 0 N is completely continuous.

Proof. Let P : C ---+ ker(00 + 0 b 0 T) be defined by Pq = p z (q,O)o for q E C. Pis a continuous linear projector on ker(00 +0b oT) , hence J( := ((00 +0b oT)lran(p)f' exists and is bounded, and a simple calculation shows

(3.3)

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Setting Qj := prj 0 Q for j E {V, Y}, where prj denotes the natural projector on j, we obtain from Eq. (3.3) for u E Z:

L;,1 0 (Id - Q) 0 Nu = (Id - P)z(I{ 0 pry u Nu - K 0 Qv 0 Nu

- K 0 Qy 0 S(pry 0 Nu - Qy 0 Nu),pry 0 Nu - Qy 0 Nu)

= (Id - P)z(K[prv 0 Nu - Qv 0 Nu],O) - (Id - P)z(K 0 Qy 0 S(pry 0 Nu - Qy 0 Nu),O)

+ (Id - P) z(O,pry 0 Nu - Qy 0 Nu].

Since S and z(O,·) are completely continuous and N maps bounded subsets of Z into bounded subsets of Y xV, the last two terms of the previous equation map bounded subsets of Z into relatively compact ones. Moreover, Qv 0 N is completely continuous thanks to (H4) , hence it remains to show that L-;,1 0 (Id - Q) 0 N is continuous. To this end, let (U)jEN E ZN, U E Z and IIUj - ull oo --t O. (H4) implies IIprv(Nuj - Nu)lIoo --+ 0, and we know pry(Nuj - Nu)(t) --+ 0 thanks to the continuity of f. Since (NU)jEN is bounded in Y x V, hence (pry(Nuj» in Y, a well-known result from functional analysis implies that (pry(Nuj» converges weakly to (pry(Nu», thus (Nuj» is weakly convergent to Nu, consequently L,,1 0 (IdQ) 0 NUj converges weakly to L,,1 0 (Id - Q) 0 Nu. Therefore, the compactness of L,,1 0 (Id - Q] 0 NUj shows norm-convergence to Nu.

4. An Existence Result

In this section we shall present a general existence result for Eq. (1.2) in case that f and 9 are quasi bounded with quasi norm O. As a special case we obtain the WaltmanWong result mentioned in the introduction.

Throughout we assume that (HI) - (H4) are fulfilled and understand L as defined in the previous section. Lemma 3.1. shows that L is a Fredholm operator provided that 8 0 + 8 b 0 T is one, but it is far too technical to derive the Fredholm index ind(L) := dim ker(L) - dim Y /ran(L) from ind(80 + 8 b 0 T) . In fact, we have found for all applications we know of that it is much easier to compute ind(L) directly. Therefore Vfe assume

(H5) 8 0 + 8b 0 T is a Fredholm operator and L has index O.

We have

Theorem 4.1. Let (Hi) - (HS) be fulfilled, liIIlJlqlloo-oo If(t,q)1 / IIqlloo = 0, uniformly for t E (0, bJ, and IIg(q,p) 1100 /(lIqlloo + IIplloo) --+ 0 as IIqlloo + IIpli oo --+ 00.

Further assume:

(i) Let k := dim ker(L) and 7], ( : [-1,0] --+ Mk,n be matrix-valued functions of bounded variation such that ran(L) = ((y,1/» E Y x C,J; d7](s) y(s) + J~1 d( 1/>( s) = O} . Here, Mk,n denotes the vector space of real k x n matrices .

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BVPs FOR FDEs

(ii) Thereisap>O withf;d1/(s)f(s,u.) + f~ld((s)g(UO,Ub)(S) f; o for each u E dom(L) which satisfies lu(s)l:::: p for all s E [-1, b].

(iii) d:= min{lu(s)1 : u E ker(L), lIulioo = I} > O.

(iv) There is an expansive linear isomorphism w from Rk onto ker(L) such that deg(G, int(Bpd-')' 0) f; 0, where B pd-' is the closed ball in Rk of radius pd-1

and center 0 and G : Rk --+ Rk is given by

G(x):= l d1/(s)f(s,w(x).) + II d((s) g(w(x)o, w(x)b)(s)

for x E Rk.

Then there exists at least one solution of Eq. (1.2) .

Proof. We want to apply Propositions 2.4. to Land N as defined in Section 3 and observe that (HI) (H5) and Proposition 3.2. ensure that N is L-completely continuous. Thus, we only have to realize (1) (4) of Proposition 2.4. The growth assumptions made for f and 9 immediately imply (3) . Moreover, we obtain from (iii) for u E ker(L)\{O} that d-1 Iu(s)1 = d- 1 lIulioo ,',:\i~ :::: lIull oo, hence (1) is fulfilled with A = d- 1 •

Now, let 1 .:5 j .:5 k and 1/j and (j denote the j-th row of "I and (, respectively, then rj(y,tP) = f;d1/j(s)y(s) + f~ld(;(s)tP(s) for (y,tP) E Y X C defines a continuous linear functional rj : Y X C --+ R. Choosing (Yj , tPj) E Y x C with f;(Yj, tPj) = 6i,i for 1 .:5 i, j .:5 k we can define a bounded linear projector Q in Y x C by Q(y, tP) = Ej=lri(Y,tP)(Yi , tPj) . It follows from (i) that ker(Q) = ran(L) and from (ii) that Q 0 Nu f; 0 for u E dom(L) with lu(s)1 :::: p Vs E [-1, b]. Thus (2) is derived. Setting B := {u E ker(L) : Ilulloo.:5 pd-1

} we get w-1 (B) C B pd-' thanks to w being expansive. Furthermore, B contains all the zeroes of Q 0 Nlker(L)' since u rt B yields lIulioo > pd-t, i.e. lu(s)1 :::: d lIulioo > p for all s E [-1, b] by definition of d, thus f;d1/(s)f(s,u.) + f~ld((s)g(UO,Ub)(S) f; o in view of(ii), which leads toQNu f; o. Now, choose linear isomorphisms J : ran(Q) --+ ker(L) and A : Rk --+ ker(L) with J 0 Q 0 Nlker(L) = A 0 G 0 w-1

, then all zeroes of G are contained in Bpd-l . Therefore additivity and product formula for the Brouwer degree yield

Ideg(J 0 Q 0 Nlker(L)' int(B), 0)1 = Ideg( G, int(Bpd- 1 ), 0)1 f; 0,

which implies (4) .

Remarks 4.2.

(i) There exist always matrix-valued functions "I and ( satisfying hypotheses (i) of Theorem 4.1., since ran(L) has finite codimension in Y x C. One can determine them by observing that Eq. (3.2) implies (y, tP) E ran(L) , iff 0 b 0 Sy - tP E

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ran(00 + 0b 0 T) . Hence following the arguments of the proof of [7; Theorem 4.1., p. 181] one obtains (y,1jJ) E ran(L), iff

l v(t)y(t) dt + /1 dcp (s)1jJ(s) = ° for all solutions v, cp of an adjoint problem that we are not stating here.

(ii) If 9 is a nonzero constant in C, i.e. nonhomogeneous boundary conditions are of concern, then it is necessary for the solvability of Eq. (1.2) that 9 E ran(00 + 0b 0 T) + 0b 0 S. Suppose now that 9 E ran(00 + 0 b 0 T), then J~l d((s)g(s) = 0, and hypotheses (ii) reduces to a condition for f alone of the type one would get in the homogeneous case when using the approach described in the introduction.

(iii) Hypotheses (iii) can be equivalently formulated as a "unique continuation property" for ker(L), i.e. u E ker(L) and u(s) = ° for some s E [-1,0] imply u == 0.

A direct consequence of Theorem 4.1 . is the following extension of the Waltman-Wong result:

Corollary 4.3. Let (HI) (H4) be satisfied and 0 0 + 0 b 0 T be bijective. Further suppose liIllJlqlioo-+oo If(t, q)1 / IIqlloo = 0, uniformly fort E [0, b], and IIg(q,p)lIoo /(lIqlloo + IIplloo) --° as IIqlloo + IIplioo -- 00.

Then Eq. (1 .2) has at least one solution.

Proof. L is an injective Fredholm operator in view of Eq. (3.1), hence all assumptions of Theorem 4.1. are fulfilled .

Now we are going to illustrate the applicability of Theorem 4.1. to problems with "perturbed periodic boundary conditions" .

Corollary 4.4. Let k : [0, 1] x R -- [-1, 1] be a continuous function with k( t, x) ::; t for all t E [0,1] and x E R, F: [0,1] x R -- R be continuous, and 9 : C x C -- C be completely continuous and bounded. Further suppose:

(i) limlxl-+oo IF(t, x)1 / Ixl = ° uniformly for t E [0,1],-

(ii) (3v E {-1, I} )(3r ~ O)(Yt E [0, 1])(Yx E R : Ixl ~ r) : vF(t, x)sgn x ~ 0,

(iii) There exists a continuous function h : [0,1] -- R+ with J~ h(t)dt > > sup{g(q,p)(O) : q, p E C} and liminflxl-+oo IF(t,x)1 ~ h(t) uniformly for t E [0,1].

Then u(t) = F(t, u(k(t, u(t)))

Uo = U1 + g(uo, ud (4.1)

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BVPs FOR FDEs

has at least one solution.

Proof. We set 0 0 = Ide, 0 1 = -Ide and f(t ,q):= F(t,q(k(t,q(O» - t) \1O:::; t:::; 1, \lq E e and apply Theorem 4.1. We have (00 - 0 1 0 T)q(s) = q(s) - q(O) \1-1 :::; s :::; 0, hence ker(00 - 0 1 0 T) consists of the constant functions on [-1,0]' thus ker(L) = {u E Z : u constant} in view of Eq. (3.1). Eq. (3 .2) yields that (y,p) E YxC belongs to ran(L), iff there exists a q E e with q(s)-q(O)- J~-' y(T)dT = p(s), which is equivalent to - J~ y(T)dT = p(O), i.e. ran(L) has codimension 1. Consequently, L is a Fredholm operator of index 0. Obviously, f satisfies (H2) and is quasi bounded with IIlflll = 0. We set TJ = -t for t E [0,1], (It-l,O) == -1 and ((0) = 0, and obtain (i) of Theorem 4.1. Now choose t: = ~[J~ h(t)dt - sup{lg(q,p)(O)1 : q, pEe}] and find 6 according to (iii) such that IF(t , x)1 ~ h(t) - dor t E [0,1] and x E R \ [-6, 6]. Select r > ° according to (ii) and set p := max{h,r} , then (ii) implies for all U E dom(L) with lu(s)1 ~ p that

fl dTJ(t) f(t, Ut) + 10 d((s) g(uo, ud(s) = - fl F(t, u(t , k(t, u(t))) dt + g( Uo , ud(O) Jo -1 Jo

is either smaller than -;< or greater than ~ . Thus (ii) of Theorem 4.1. is fulfilled and clearly d = 1 in (iii) of that theorem. Finally, let w be the inverse of the identification map between constants on [-1,1] and their value, then G(u) := J~ F(t , w(u) dt + g(w(u), w(u»(O) for u E R, and (ii) and our previous considerations show that vuG(u) > ° for lui = p. Therefore deg(G, (-p, p), 0) f 0, and (iv) of Theorem 4.1. is also established. Consequently, Eq. (4.1) has a solution.

Remarks 4.5.

(i) Corollary 4.4. does not apply to Example 2.2., but it should be clear from the proof how to proceed in that case. The linear operator L would be given by

L(x,u) = ((x+Qx( ' -T)-u(. -u), idhx),(XO-Xb,UO-Ub»)

for (x, u) E Z, R(x, u) continuously differentiable, which is a Lo-completely

continuous perturbation of Lo : (x,u) ...... ((x , u),(xo - Xb,UO - Ub»), hence a. Fredholm operator of index O. The new difficulty consists in determining the kernel of L, which depends on the choice of the parameters involved.

(ii) There are many other existence results for semilinear Fredholm operator equations, and most of them provide further useful existence theorems for Eq. (1.2) via the approach outlined in Section 3. We refer to [5] for a collection of such abstract solvability assertions.

(iii) Our approach applies also to various classes of neutral functional differential equations. It may be necessary, though, to replace C by another space of "initial conditions" and to use related existence results for L-condensing nonlinearities.

175


(iv) We mention [1) and [2) for other results on problems with nonlinear boundary conditions. Most applications in these papers focus on ~point boundary" conditions, though.

References

1. A. R. Abdullaev, Solvability of boundary value problems for functional-differential equations with Lipschitz nonlinearities, Differential Equations 25 (1989), 1283

1287.

2. A. R. Abdullaev and A. B. Burmistrowa, Solvability of boundary value problems in the case of resonance, Differential Equations 25 (1989), 1437 1442.

3. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag 1985.

4. I. J . Diaz, to appear in The Mathematics of Models for Climatology and Environment, Proc. NATO ASI, Tenerife, 1995.

5. R. E. Gaines and L. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Mathematics, vol. 568 Springer-Verlag, 1977.

6. J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, 1977.

7. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993.

8. G. Hetzer, S-shapedness for Energy Balance Climate Models of Sellers-Type, to appear in The Mathematics of Models for Climatology and Environment, Proc. NATO ASI, Tenerife, 1995.

9. P. Waltman and J. S. W. Wong, Two point boundary value problems for nonlinear functional differential equations, Trans. Amer. Math. Soc. 164 (1972), 39 - 54.

10. Kehong Wen, Pin Chen, and J. S. Turner, Bifurcation in a Lienard equation with delays, in Proc. Dynamic Systems Appl. 1 (1994) , eds. G. S. Ladde and M. Sambandham,. 377 - 384.

177

PERIODIC SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS OF RETARDED AND NEUTRAL TYPES IN BANACH

SPACES

LE HOANHOA Dai Hock SuPhan-School of Education

280 An Duong Vuong HoChiMinh City, Vietnam

KLAUS SCHMITT Department of Mathematics

University of Utah Salt Lake City, UT 84112 USA

Abstract

The paper presents some applications of fixed point theorems for operators of the form U + C on '" bounded closed convex subset of a locally convex space to the existence of periodic solutions of functional differential equations of retarded and neutral types in a Banach space.

1. Introduction

Let E be a Banach space and let r > 0 be given. We denote by C = C ([ -r, 0], E) the Banach space of all continuous functions from [-r,O] to E equipped with the usual norm. For each continuous function x : IR -+ E and for t ~ 0 we let Xt E C be defined as in [5].

In this paper we shall consider the following functional differential equations

X'(t) = A(t)xt + g(t,Xt), (1)

and

[X(t) - A(t)x(t - r)l' = g(t, Xt), (2)

where {A(t)} is a continuous time periodic (with period w) family of bounded linear operators from C to E or E to E and g : [0,00) xC -+ E is completely continuous and

178

BVPs FOR FDEs

time periodic with period w . We shall employ fixed point results from [9J to deduce the existence of periodic solutions of (1) and (2). We shall adopt the notations and conventions of [9J , the latter paper hence is requisite for the reading of the present one.

In case E is a finite dimensional space these problems have been extensively investigated, see e.g. [2J, [3], [4J, [6], [7], [l1J and also many references in [5J . In this case [2J employs homotopy arguments for semilinear Fredholm maps to, in fact, reduce (2) to (1). This reduction is not possible in infinite dimensions.

This paper is organized as follows . We first state the relevant fixed point theorem and provide, without proofs, some conditions for the relative compactness of subsets of some function spaces used in our discussion. We then use these criteria together with some existence and continuous dependence theorems for initial value problems and the fixed point results of [9J to obtain existence theorems for periodic solutions of (1), and (2). The proofs given here, for lack of space, will only be sketched. Complete proofs are contained in the authors' longer (unpublished) article [10J .

2. A Fixed Point Theorem

In this section we shall state a fixed point theorem from [9J for operators of the form U + C on a bounded closed convex subset of a locally convex space, where C is a completely continuous operator and U satisfies condition (A) (defined below) . This fixed point theorem will be the principal tool to establish periodicity theorems for the functional differential equations stated. Condition (A) is defined precisely as follows: Condition (A): Let X be a locally convex topological vector space and let P be a separating family of seminorms on X. Let D be a subset of X and let U : D ..... X . For any a E X, define Ua : D -+ X by Ua(x) = U(x) + a .

The operator U : D --t f{ is said to satisfy condition (A) on a subset 0 of X if: (A.l) For any a E 0, Ua(D) C D. (A.2) For any a E 0 and pEP there exists ka E Z+ with the property: for any

f > 0, there exist r E Nand 5 > 0 such that for x, y E D with Q~(x , y) < f + 5 implies

Q~(U;(X), U;(y)) < f,

where Q~(x,y) = max{p(U~(x) - U1(y)), i,j = 0,1,2, ... ,k.}, here N = {1,2,3, ... } and Z+ = NU {O} .

We have the following fixed point theorem (see [9]).

Theorem 1 Let X be a sequentially complete locally convex space with a separating family of seminorms p . Let U and C be operators on X such that

(i) U satisfies condition (A) on X .

(ii) For any PEP, there exists k > 0 (depending on p) such that p(U(x) - U(y)) ~ kp(x - y) for all x,y E X,

179

RETARDED AND NEUTRAL EQUATIONS

(iii) There exists Xo E X with the property: for any pEP there exist r E Nand A E [0,1) (r and A depending on p) such that

p(U;o(x) - U;o(Y))::; Ap(X - y).

(iv) C is completely continuous p( C (A)) < 00 whenever p( A) < 00, for A eX.

(v) lim p(C(x))/p(x) = 0 for all x EX . p(:r)_oo

Then U + C has a fixed point.

3. Sufficient Conditions for Relative Compactness

Let S be a compact metric space and X a Banach space with norm 1· 1. Let C(S) be the Banach space of all continuous maps from S to X with norm 11 · 11 = sUP. ES Ix(s)l . Relative compactness in C(S) is then given by the theorem of Ascoli Arzela (see e.g. [1)).

If S is a metric space such that

S = Ur'Sn, Sn C Sn+l, n = 1,2,··· ,

with each Sn compact and for every compact subset J( C S, there exists Sn such that K C Sn, then C(S), defined as before becomes a Fnkhet space using the family of semi norms {Pn}, Pn(x) = sUP'ESn Ix(s)l, and the metric

d(x,y) = fTn Pn(x,y) . 1 I+Pn(x , y)

Using this set-up we have the following result .

Proposition 1 A set A in C(S) is relatively compact if and only if for each n E N, A is equicontinuous in Sn and the set {x(s) : x E A, s E Sn} is relatively compact in X.

The proof follows from the theorem of Ascoli-Arzela and a diagonalization process. Let us now denote by B(S), S as above, the Banach space of bounded continuous

maps from S to X, with norm defined as above. In this case we have the following result.

Proposition 2 Let A be a subset of B(S) and let A(S) = {x(s) : x E A, s E S} .

• If A(S) is relatively compact in X, then A is relatively compact in B(S) .

• If, in addition S is totally bounded and A is equicontinuous on S, then the following are equivalent:

- A is relatively compact in B(S)

- A( S) is relatively compact in X .

180

BVPs FOR FDEs

4. Initial Value Problems, Continuous Dependence

Let E and C be as above and let X = C([-r,oo),E). Consider the initial value problem

x'(t) Xo

where

f(t,xt) + g(t,xd, t ~ 0 4> E C,

f : [0, 00) x C -+ E

is continuous and satisfies: For each n E ]'\I there exists kn such that

If(t,x) - f(t,y)1 S knllx - yll, Vx,y E C, Vt E [O , n].

Further 9 : [0,00) x C -+ E

(3)

is completely continuous. Under these conditions one has the following (global) existence theorem.

Theorem 2 Let f and 9 satisfy the above conditions and let 9 be quasibounded, i.e.

lim Ig(t,x)I_ O lixli-co IIxll - ,

uniformly with respect to t on bounded intervals. Then problem (3) has u solution x: [-r,oo) -+ E.

Problem (3) is equivalent to the integral equation

x(s) = 4>(O)+J;f(t,xt)+J;g(t,xd, Xo = 4> E C.

(4)

Since [-r , 00) = Uf[-r, n], one may think of X as a Fnkhet space with metric defined by a sequence of seminorms as considered above. For x E X one defines x E X by

_( ) = { x(s) + 4>(0) - x(O), if x s 4>(s), if

Further define U, G : X -+ X as follows:

Ux(s) = 10' f(t,xt),

Gx(s) = 10' g(t,Xt).

s ~ 0 s E [-r,O].

One then sees that a fixed point x of the operator U + G will yield x as a solution of the initial value problem. In order to see that U + G has a fixed point, one employs the fixed point theorem (Theorem 1) of [9]. The verification is technical and lengthy.

Concerning continuous dependence of solutions on initial conditions and the equations we have the following result.

181


Theorem 3 Let {A(t)} be a family of bounded linear operators from C to E depending continuously upon t. For each n E N let gn be a mapping satisfying the conditions imposed on 9 by the previous theorem and assume gn converges uniformly to go. Further let {4>n} c C with 4>n -+ 4>0 in C. For each n E N let Xn E X be a solution of

X'(t) = A(t)xt + gn(t, Xt), t ~ ° Xo = 4>n E C,

and assume that for n = ° the problem is uniquely solvable. Then Xn -+ Xo in X .

5. Periodic Solutions

(5)

Let us now consider the problem of the existence of periodic solutions of the equation

X'(t) = A(t)xt + g(t, Xt), (6)

where {A(t)} is a family of bounded linear operators from C to E depending continuously upon t which is also w- periodic in t. Further the mapping 9 satisfies the conditions imposed on 9 earlier and is also w- periodic in t. We further shall assume that initial value problems associated with this equation are uniquely solvable (existence follows from our earlier theorem, we hence assume uniqueness here) . If we denote by x( 4» E X the unique solution with initial condition 4>, then the continuous dependence theorem guarantees that the mapping

is a continuous mapping. One next establishes the existence of two-parameter families of bounded linear

operators S(t,s):C-+C, V(t,s):E-+E, t~-r, s~t

such that a solution x of (6) is given by

Xt(4))( · ) = S(t, 0)4> + l Yt( ., s)g(s, x s (4)))ds,

where S(t+w,s) = S(t+w,t)S(t,s)

and V(t + w,s) = V(t + w, t)V(t, s), t ~ s.

One then defines the operator

T : C -+ C, T = S + F,

182

BVPs FOR FDEs

by

F(4)) = 10'" Vw(-,s)g(s, x s(4)))ds

and S = S(w,O),

and one obtains periodic solutions of the equation as fixed points of the operator T. One obtains the following theorem.

Theorem 4 Under the above assumptions the equation has an w-periodic solutions, whenever the family {A(t)} is uniformly asymptotically stable.

The proof proceeds via a sequence of steps:

• One shows that the operator F is completely continuous.

• One shows that F is quasibounded with quasinorm 0, i.e.

IFI = lim sup IIF(u)1I = O. lIull-oo lIuli

• The uniform asymptotic stability of {A(t)} implies that there exist positive constants a, b such that

IIS(t,O)4>1I :::; bll4>lIe-at

and hence

One hence may apply the fixed point theorem (Theorem 1) to deduce the existence of a fixed point.

6. The Neutral Equation

Let us now consider the neutral equation

[x(t) - A(t)x(t - r)]' = g(t, Xt), (7)

under assumptions similar to the above, i.e.

9 : [0, 00) x C --+ E

is completely continuous and

lim Ig(t,x)1 - 0 11"11_00 IIxll - ,

uniformly with respect to t on bounded intervals with {A(t)} a continuous family of bounded operators from E to E. Under these assumptions one has the following existence theorem.

183


Theorem 5 The initial value problem

[x(t) - A(t)x(t - r)l' Xo

(8)

has a solution.

One first shows that the initial value problem (8) is equivalent to the integral equation

x(t) = A(t)x(t-r)-A(0)x(-r)+1/>(0)+J~g(s,x8)ds, t~O Xo = I/> E C.

One then defines the operators

and

Zx(t) A(t)x(t - r) - A(O)x( -r), t ~ 0 = 0, -r ~ t ~ 0,

Gx(t) = 1/>(0) + J~ g(s, xs)ds, t ~ 0 = I/>(t), -r ~ t ~ o.

(9)

(10)

(11)

Then Z is a bounded linear operator on X and G will be compact. Furthermore one may verify that the conditions of the fixed point theorem (Theorem 1)) hold and obtains a fixed point of the operator F + G and hence a solution of the initial value problem.

One next obtains a continuous dependence theorem analogous to the one above, namely.

Theorem 6 Let {A( t)} be a family of bounded linear operators from E to E depending continuously upon t. For each n E N let gn be a mapping satisfying the conditions imposed on 9 by the previous theorem and assume gn converges uniformly to go. Further let {I/>n} c C with I/>n -+ 1/>0 in C. For each n E N let Xn E X be a solution of

[x(t) - A(t)x(t - r)l' = gn(t,Xt), t ~ 0 Xo = I/>n E C, (12)

and assume that for n = 0 the problem is uniquely solvable. Then Xn -+ Xo in X.

These results, in turn, may be used in a very similar manner to our earlier procedure to deduce an existence theorem for periodic solutions of the neutral equation. One obtains the following result assuming the w- periodicity of the operators involved.

184

BVPs FOR FDEs

Theorem 7 Under the above assumptions the equation has an w-periodic solutions, whenever the family {D(t, 4» = 4>(0) - A(t)4>} is stable.

(See [5] for the concept of stability of such a family.) A result similar to the one just stated also holds for the more general case of

operators of the form

D(t,4» = 4>(0) - L Ai(t)4>( -ri),

where t 2": 0, 4> E C, ri > O. Note that this family is stable, whenever

L IIAi(t)11 < 1,

(see e.g. [6], [11]).

References

1. J. Dieudonne, Foundations of Modern Analysis, Academic Press, New York, 1969.

2. L. Erbe, W. Krawcewicz, and. J. Wu, Leray-Schauder degree for semilinear Fredholm maps and periodic boundary value problems of neutral type, Nonl. Anal., 15 (1990), 747-764.

3. G. Gustafson and K. Schmitt, A note on periodic solutions for delay-differential equations, Pmc. Amer. Math. Soc. 42 (1974), 161-166.

4. J . Hale and M. Cruz, Asymptotic behavior of neutral functional differential equations, Arch. Rat. Mech. Anal. 5 (1969), 331-353.

5. J . Hale and S. Lunel, Introduction to Functional Differential Equations, Springer Verlag, New York, 1993

6. J. Hale and J. Mawhin, Coincidence degree and periodic solutions of functional differential equations, J. Diff. Eq. 15 (1974), 295-307.

7. G. Hetzer, Boundary value problems for retarded functional differential equations, Comment. Math. Univ. Carol. 16 (1975),121-137.

8. L. Hoa, On a fixed point theorem of Krasnosel'skii and its applications, Acta Math. Vietnamica 14 (1989), 3-17.

9. L. Hoa and K. Schmitt, Fixed point theorems of Krasnosel'skii type in locally convex spaces and applications to integral equations, Results in Math. 28 (1994), 290-314.

185


10. L. Hoa and K. Schmitt, Periodic solutions of functional differential equations of retarded and neutral types in Banach spaces, preprint.

11. O. Lopes, Forced oscillations in nonlinear neutral differential equations, SIAM J. Appl. Math. 29 (1975), 195-207.

187

EXISTENCE OF TRAVELING WAVES FOR REACTION DIFFUSION EQUATIONS OF FISHER TYPE IN PERIODIC MEDIA

W. HUDSON AND B. ZINNER Department of Discrete and Statistical Sciences Auburn University, Alabama 36849-5307 USA

Abstract

The existence of periodically varying wavefronts is established for a class of reaction-diffusion equations of Fisher type where the reaction term is periodic in the space variable.

1. Introduction

Fisher's equation (Fisher 1937)

Ut = u"'''' +u(1- u),x E (-oo,oo),t ~ 0

was proposed as a model for the propagation of an advantageous gene in a homogeneous unbounded space. Since it is one of the simplest nonlinear reaction-diffusion equation that admits traveling waves, it has been the focus of a large number of studies.

Fisher's equation and extensions thereof have been analyzed, among others, by Kolomogorov, Petrovskii, and Piskunov (1937), Aronson and Weinberger (1975), Hadeler and Rothe (1975), Murray (1977), Fife (1979). Most of the work is concerned with traveling waves propagating in a homogeneous media. The problem of propagation speed in a heterogeneous media was investigated by Gartner and Freidlin (1979), Freidlin (1986), Shigesada, Kawasaki, and Teramoto (1986) .

The existence of periodically varying traveling waves for the bistable reactiondiffusion equation with periodic coefficients was established recently by Xin (1991, 1992, 1993, 1994).

In this article we consider the equation

(1.1) Ut = U""" + f(u,x),u = u(t,x),x E (-oo,oo),t ~ 0,

where f satisfies the conditions

BVPs FOR FDEs

( C l ) f is jointly continuous and Lipschitz continuous in u .

(C2) f is periodic in x with period 1, and

(C3) f is of Fisher type, i.e., there exists a 1-periodic continuous function G(x) > 0 such that f ( u , x ) > 0 for u E (O,ii(x)) and f (u , x ) 5 0 for u > ~ ( x ) .

We will show the existence of a critical speed c, such that for all c 2 c, there is a periodically varying wavefront of (1.1) of speed c.

More precisely, let

c, = inf sup y"(x) + P ( X ) Y ( X )

"' z ~ [ O , 1 ] Y ( X )

where p ( x ) = s u p { y : u E (0 , ~ ( x ) ) and the infinum is taken over all r > 0 and y E C 2 ( R ) for which y ( x ) > 0 for all x and y ( ~ ) e - ~ ~ is periodic with period 1.

Theorem 1.1 For all c 2 c, there exists a function u ( t , x ) which is a solution of (1.1) in the distributional sense. This solution u may be written in terms of a wave profile U , namely u ( t , x ) = U ( x + ct, x ) , ( t , x ) E R2 where U ( x + ct, .) is periodic with period 1. Furthermore, u( . , x ) is nondecreasing for each x , u ( t , .) is uniformly Lipschitz continuous, liw,-, u( t , x ) exists and is zero, lim,+m inf u ( t , x ) 2 min C(x) , and lim,,+, sup u ( t , x ) 5 maxi i (x ) for all t E R .

The lower speed c, agrees with the speed of propagation obtained by Gartner and Freidlin (1979) for compactly supported initial data.

2. Existence of Approximating Wavefronts

In this section we will show the existence of approximating wavefronts V m . The wave profile U of our Theorem will be obtained by transforming a subsequential limit V of these approximating wavefronts. The wavefronts Vm satisfy the equations

. m

(2.1) c,,,V ( k t , k2-m) = 22mAmVm(c,,,t, k2-m) + f (Vm(cmt , k2Tm), k2-m),

k E Z , t E (-oo, m), m = 1,2,3,. . . where A m V ( t , x ) abbreviates V ( t , x - 2-m) - 2 V ( t , X ) + V ( t , x + 2-") and c ~ ( c t , x ) denotes $ ~ ( c t , x ) . In the remainder of this article V m represents a solution of (2.1) with the following properties:

Vm(O, 0 ) = min i i ( x ) . m

V ( t , k2-"') 2 0 , (2.2) V m ( t , k2-m + 1 ) = V m ( t + 1, k2-m),

limt,-, V m ( t , k2-m) = 0 , and min i i ( x ) 5 limt,, V m ( t , k2-m) 5 maxi i (x ) ,

189

EQUATIONS OF FISHER TYPE IN PERIODIC MEDIA

for all k E Il, t E IR. Such a solution exists for all positive Cm ::::: c.(m), where

(2.3) c.(m):= infsuplj(a,r) , r ,a i

the supremum is over j E {I, 2, . .. , 2m },

ao := a2me-r, a2m+1 := al er , and the infimum in (2.3) is taken over all r > ° and all a = (at, a2, . . . ,a2m) such that all the coordinates of a are positive. (See Corollary 4.1 in Hudson and Zinner 1994).

Lemma 2.1 Let ~:= inf{JL(x) : x E IR}. Then ~> 0.

Proof: Since f( u, .) has period 1, the definition of JL( x) implies that JL(-) has period one. Hence there exists a sequence (xn) in [0,1] such that

Let Xo be a limit point of xn. We may and do assume that Xn converges to Xo. Now by the definition of JL there exists a point Uo E (0, u(xo)) such that

° < ~JL(xo):S f(uo,xo). 4 Uo

Since f( uo, ·) and u are continuous, there exists 0 > 0 such that for all x E (xoo,xo + 0), ~JL(xo) :S J(':'"x) and Uo < u(x). So for n so large that IXn - xol < 0,

~JL(xo) :S !<:xn) :S JL(xn). Since ~= limn JL(xn), ~JL(xo) :S ~ > O.

Lemma 2.2 Let c ::::: c. . Then there exists a sequence of positive numbers {cm } such that Cm > c.(m) , lim em = c. , and system (2.1) has a solution Vni with the above

m_oo

properties.

Proof: It suffices to show that c. > 0 and that 2~ sup c.(m) :S c •. Let I!:. be as in Lemma 2.1. Then ~> O. Choose sequences {Yn} and {rn}, rn > 0,

Yn E C2(1R), Yn(x) > 0, Yn(x)e- rnX periodic with period 1, such that

y~(x) + JL(x)Yn(x) < + 1 sup ) c.-.

xe[O,I] Yn(x - n

Since Yn(O) < Yn(I), t~ere exists b E (0,1) such that y~(b) > O. Therefore we may assume without loss of generality that y~(O) > OJ otherwise consider Yn( b + x) .

Then

190

BVPs FOR FDEs

and therefore

t 1 fl y~(1) - y~(O) + 10 p(x)Yn(x)dx ~ (c. +;;) 10 Yn(x)dx .

Since

and fl p(x)Yn(x)dx ~ P fl Yn(x)dx 10 -10

it follows that Co + ~ ~ i!: Hence Co ~ e.> o. Let a'J = Yn(j2-m) for j = 1, ... , 2m. Then for an := (ai', a~, ... , a~~), x'J := j2-m,

for j = 1, ... ,2m • Note that there exists mo such that for all m ~ mo,

1 lj(an, rn) _ Y~(Xj) + p(Xj )Yn(Xj) 1< ~ Yn(xj)rn n

for j = 1, . .. ,2m. It follows that for m ~ mo, .:.(m) ~ supjlj(an,rn) ~ Co +! and therefore lim supc.(m) ~ c •.

m_oo

3. The Existence of Periodically Varying Waves

The purpose of this section is to show the existence of periodically varying wavefront solutions to (1.1) with the properties described in our Theorem. We begin byestablishing the Lipschitz continuity in x for the solutions vm to the discretized problem. Then we show that the solutions vm contain a subsequence which converges almost everywhere in JR.2 to a function V. The function V( ct, x) is a weak solution to

cV = Vu + I(V,x)

and hence u(t,x) := V(ct,x) is a weak solution to (1.1). Next, we show that V(t + 1,x) = V(t,x + 1) which implies the periodicity of U in x. Finally we show that u(t,x) = V(ct,x) satisfies the conditions lim u(t,x) = 0 and lim infu(t,x) >

x~-oo x-oo

min ii( x). We will assume in the following without loss of generality that 1 is bounded.

Lemma 3.1 For every t E JR., mEN, and k E Z

jVm(t, k2-m) - Vm(t, (k - 1)2-m) I~ M2-m

where M := lIuli oo + 11/1100 .

191


Proof: Let t and m be chosen arbitrarily and then fixed. For brevity write Wk(t) :=

V m(t,k2-m), k E Z. Since for each k and m, V m(-,k2- m) is nondecreasing, 0::; Wk and consequently

0::; 22m(Wk+1 - 2Wk + wk-d + f(Wk,k2- m).

Let j ::; r be any two integers. Then

r-l

0::; L:{Wk+1 - 2Wk + Wk-l + 2-2m llflloo} k=j

and hence

(3.1)

Sum over r = j to r = j + 2m - 1 to see that

(2m _ 1)2m 2m(wj - wi-d ::; Wj+2~-1 - Wj-l + 2 . T 2m llflloo

::; 11111100 + IIflloo

Hence

(3.2)

Similarly, use (3.1) to see that

r

::; L: {wr - Wr-l + (r - j)T2mllfIl00}

m( ) (2m - 1)2m2-2mllfll = 2 Wr - WT-I + 2 00

::; 2m(wr - Wr-l) + IIflloo.

So

(3.3)

The Lemma follows from (3.2) and (3.3).

Definition 3.2 For real numbers x such that k2-m ::; x < (k + 1)2-m, define

o:m(x) := 2m((k + I)Tm - x) and am(x) :== 1- o:m(x).

Then for such an x and s E JR, let

192

BVPs FOR FDEs

Lemma 3.3 For every t E IR, x, Y E IR,

Proof: The proof will be divided into two cases. In the first case we suppose that for some integer k, k2-m :s x, y :s (k + 1 )2-m

• In this case

wm(t, x) - vm(t, y)1 I(am(x) - am(y))(vm(t, kTm) - vm(t, (k + I)Tm))l

:s 2m{(k + 1)2-m - x - (k + I)Tm + y}MTm

:s Mix - yl

In the second case suppose that (k - 1)2-m :s x :s k2-m :s (k + r)2-m :s y :s (k + r + 1)2-m . Then by the triangle inequality and case 1,

Wm(t,y)-Vm(t,x)l :s Wm(t,y)-Vm(t,(k+r)Tm)1 T

+ L wm(t, (k + j)Tm) - vm(t, (k + j - I)Tm)1 j=l

T

:s M{y-(k+r)Tm +LTm +(k2-m -x)} j=l

Mlx-yl·

Lemma 3.4 Let x E IR be fixed. Then any subsequence Vm'( .,x) of Vm(·,x) contains a further subsequence Vm"(·,x) which converges to u unique right continuous nondecreasing function V(·,x) at each point of continuity ofV(·,x).

Proof: Each V m (., c) is a continuous nondecreasing function which satisfies the conditions lim Vm(s,x) = 0 and lim Vm(s,x) exists. The Lemma now follows by the

"_-00 5-+00 Relly Compactness Theorem (e.g. see R. G. Tucker [11], Theorem 2, p. 83).

The next task is to show that there is a subset D of IR and a subsequence Vmo(k) of vm such that

(a) Vmo(k)(.,x) converges to say V on DC for each binary rational x, and

(b) D is countable.

Let B denote the set of all binary rationals in IR and write the elements of B as a sequence (x n ). By Lemma 3.4, there is a right continuous non decreasing function V(·, xt} and there is a subsequence Vm,(k) of vm such that V(t, Xl) := limk~oo Vm,(k)(t, xtl exists for each point t of continuity of V(·, xt}. Let DI denote the countable set of discontinuities of V(·,XI) . Then for each t ft Db Vm,(k)(t,XI) converges to V(t,xt}. Now use Lemma 3.4 again to choose a subsequence V m2 (k) of Vm,(k) which converges

193


to V(·,X2) say, at each point of continuity of V(·,X2). Continue this process. At the ph step, choose a subsequence Vmj(k) of V ffli - 1 (k) such that Vmi(k)(t,Xj) converges to a limit V(t,Xj) at each point t of continuity of V(·,Xj) . Let Dj denote the set of discontinuities of V( ·,Xj). Then for t i U{D i , vmi(k)(t,Xl) converges to V(t,Xl), l = 1,2, ... ,j. Now use the diagonal argument to see that there is a subsequence Vmo(k) such that for each tiD := Ur'Di ,

lim Vmo(k)(t,x) = V(t,x), x E B. k-+oo

Since each Di is countable, D is countable.

Lemma 3.5 For each tiD and for each x E IR., the sequence vmo(k)(t, x) converges.

Proof: It is enough to show that Vmo(k) (t, x) is a Cauchy sequence when tiD and x E JR. Let I" > O. Choose a binary rational q such that Ix - ql < f/(3M). Choose k so large that if rl 2 K and r2 2 K, then Ivmo(r,)(t,q) - vmo(r2 )(t,q)1 < 1"/3. Then for such rl and r2,

IVmoh)(t,x) _ vmo(r')(t,x)l::; JVmo(r,)(t,x) _ vmo(r,)(t,q)1

+Ivmo(r')(t,q) _ vmo(r')(t,q)1

+IVmo(r»(t,q) _ v mo (r2 )(t,x)1 I"

::; Mix - ql + 3" + Mix - ql f I" I"

< 3"+3"+3"=f.

Definition 3.6 For tiD and x E IR.\B, define

V(t,x):= lim Vmo(k)(t,x). k-+oo

Note that V( ·,x) is nondecreasing for each x E JR so that right limits exist at each point. Consequently we may define

for each tED.

V(t,x) := lim V(s , x) .!t .t/.D

Lemma 3.7 For any Xl! X2 E JR and for any t E JR,

Proof: If tiD, then

IV(t, X2) - V(t, xII Ji.~ JVmo(r)(t, X2) - vmo(r)(t, Xdl ::; MIX2 - XII.

194

Now if tED,

BVPs FOR FDEs

limlV(s,x2) - V(s,xI)1 _1t

-'tD :::; Mlx2 - xII .

Lemma 3.8 For each X E JR., V(.,x} is continuous at every point in DC.

Proof: Let x E JR. and to E DC . Choose f > 0 and then q E B such that Ix - ql < f/(3M). By the definition of D, V(·,q) is continuous at to and hence there is a 6 > 0 such that if Is - tol < 6, then lV(s, q) - V(to, q)1 < f/3. Consequently, if Is - tol < 6, then

lV(s,x) - V(to,x)1 :::; lV(s,x) - V(s,q)1

+1V(s,q) - V(to,q)1

+1V(to,q) - V(to , x)1 f

:::; Mix - ql + 3 + Mix - ql f f f

< 3+ 3 + 3 = f.

Now let c ~ c. be chosen arbitrarily and then fixed . For each m choose em ~ c.(m) such that Cm ! c. Section 2 shows that such a sequence (em) exists. vm now denotes a wavefront with speed em.

Definition 3.9 For (t, x) E JR.2 let k := [2mt] be the greatest integer in 2mt and define

Lemma 3.10 vm satisfies the differential equation

em V m (emt , x) = 22m Llm vm(emt , x) + rpm(t, x), (t, x) E 1R.2.

Proof: Let k:= [2mx]; then k2-m :::; x < (k + 1)2-m. Now

and

em V m (em t , (k+l)2-m) = 22mLlm Vm(em t , (k+l)2-m)+ J(vm(em t , (k+l)2-m), (k+l)2-m).

Since vm(emt,x) = Qm(x)Vm(cmt,k2-m) + am(x)Vm(emt,(k + 1)2-m) and since Qm(x + 2-m) = Qm(x) = Qm(x - 2-m), the Lemma follows by a simple calculation.


Lemma 3.11 For every x E IR and t such that ct f/. D,

lim <pmo(k)(t,x) = f(V(ct,x),x). k_oo

Proof: Let jk := [2mo(k)x) . Then by the definition of <pmo(k),

l<pmo(k)(t, x) - f(V(ct, x), x)1 ~ amo(k)(x )Amo(k) + amo(kl(x )Bmo(k)

and

Now

IVmo(k)(akt,jkTmo(k») - V(ct,x)1 ~ IVmo(k)(akt,jkTmo(k») - V(ct,jkTmo(k»)1

+1V(ct,jk2-mo(k») - V(ct,x)1 ~ IVmo(k)(akt,jkTmo(k») _ V(ct,jk2- mo (k»)1

+MljkTmo(k) - xl-

Since V (., x) is continuous at ct and monotone in t and since ak -t C,

IVmo(k)(akt,jk2-mo(k») - V(ct,jkTmo(k»)1 -t 0

as k -t 00. Since Ijk2-mo (k) - xl -t 0 as k -t 00,

lim IVmo(k)(akt,jkTmo(k») - V(ct,x)1 = o. ,1,-00

So from the joint continuity of f, it follows that

lim Amo(k) = O. k_oo

A similar argument shows that

195

Lemma 3.12 V is a weak solution in the distributional sense to the differential equa-tion

oV 02V c7jt(ct,x) = ox2 (ct,x) + f(V(ct,x),x)

that is, if 9 is any function in Coo (1R2) with compact support, then

-[ ~g(t,x) ' V(ct,x)dtdx = JR'vt

[ 02g JR' ox2(t,x)V(ct,x)dtdx

+ [ g(t,x)f(V(ct,x),x)dtdx JR'

196

BVPs FOR FDEs

where ak := c".o(k) . Integrate by parts to see that

l ·-w l ~ W ak g(t,x)V (akt,x)dtdx = - -a (t,x)· Vmo (akt,x)dtdx . R' R' t

Also JR

, g(t, x ){vmo(k)(akt, x + 2-mo (k») - 2vmo(k)(akt, x) + vmo(k)(akt, x - 2-mo (k»)}dtdx = JR' vmo(k)(akt, x ){g(t, x - 2-mo (k») - 2g(t, x) + g(t, x + 2-mo (k»)}dtdx.

Consequently,

_ r aag(t,x)vmo(k)(akt,x)dtdx = JR' t

2mo(k) r Vmo(kl(akt, X ){g(t, x - 2-mo(k») - 2g(t, x) JR2

+g( t, x + Tmo(k») }dtdx

+ r g(t,x)<pmo(k)(t,x)dtdx. JR'

Let k ---> 00 and apply the Dominated Convergence Theorem to see that

~ ag l a2g - -a (t,x)V(ct,x)dtdx = V(ct,x)-a 2(t,x)dtdx

R' t R' X

+ r g(t,x)J(V(ct,x),x)dtdx . JR' Lemma 3.13 For all (t,x) E JR2, V(t + 1,x) = V(t,x + 1) .

Proof: We will first prove that the Lemma holds for vm when x E Bk, the set of kth

order binary rationals, k ~ m. Limiting arguments will complete the proof. Let m be any fixed positive integer and suppose x E Bk, k ~ m . Then x E Bm so for some integer n, k E {O, 1, . .. , 2m

- I}, x = n + k2-m, Tm = 1/c"., and for any t E JR,

vm(cmt + 1, x) = Vm(c".(t + Tm), x) = U;::k(t + Tm)

= U;:'+l,k(t) = vm(cmt, x + 1).

Now let DC - 1 = {t - 1 : t E DC} and G = DC n (DC - 1) . Since D is countable, so is GC and for t E G, x E B

V(t+1,x) lim Vmo(k)(t + 1,x) k .... oo

= lim Vmo(k)(t,x+l) k .... oo

V(t,x+1).

197


Next recall that V is continuous in x and B is dense in JR so V(t + l,x) = V(t,x + 1) holds on G x JR. Finally, if t E GC and x E JR,

V(t + l,x) = lim V(s + l,x) = lim V(s,x + 1) = V(t,x + 1) . • It slt

.EG sEG

Definition 3.14 For any (t, x) E JR2, define U(t, x) := V(t-x, x) . Since V(t+ 1, x) = V(t, x + 1), U(t, x + 1) = U(t, x).

Remark 3.15 From the above definition we have

U(x + ct,x) = V(ct,x) .

Consequently u(t,x) := U(x + ct,x) is a weak solution to (1.1) by Lemma 3.12. Let .,p := ~minu(x) . Since for each m, vm(o,O) = .,p E (O,u(O)), V(O,O) = .,p . But f(.,p,O) > 0 so V(-oo,O) <.,p < V(+oo,O) .

Our final task is to show that lim u(t,x) = 0 and lim infu(t,x) ~ minu(x) for x--+-oo x--+oo

all t E JR. Since for T = ~, u(t + T, x) = U(x + c(t + T), x) = U(x + 1 + ct, x) = U(x + 1 + ct, x + 1) = u(t, x + 1) for all (t, x) E JR2, it suffices to show that lim u(t, x) = 0

t-.-oo and lim infu(t,x) ~ minu(x). Let

t ..... oo

u(-oo,x) = lim u(t,x) and u(oo,x) = lim u(t,x). t--oo t_oo

These limits exist because u is monotone in t. Furthermore u( -00, x) and u( 00, x) are Lipschitz continuous because u(t, x) is Lipschitz continuous in x where the Lipschitz constant is independent of t. Also note that u( -00, x) and u( 00, x) are solutions of

(3.4) u"(x) + f(u(x),x) = O,u(x) ~ O,u(x + 1) = u(x),x E JR

in the distributional sense. In order to show that any continuous function u(x) which is a solution of (3.18) is actually of class C2 we will need two well-known elementary lemmas on distributions which are stated without proof.

Lemma 3.16 Let f and 9 be continuous real-valued functions defined over JR and let J and 9 denote the distributions determined by f and 9 respectively. Suppose l' = g. (The differentiation is in the sense of distributions.) Then f is differentiable at each point of JR and f' = 9 on JR .

Lemma 3.17 Let 8 and T be two distribution over JR and suppose that 8' = T'. Then for some constant C, 8 = T + C.

198

BVPs FOR FDEs

Since u" = - f(u(x),x) it follows from Lemma 3.17 that ul = -F(x) + c where F(x) = Jr: f(u(t), t)dt and c is some constant . By Lemma 3.16 it follows that ul(x) exists and is equal to -F(x) + c. That is,

ul(x ) = - f f(u(t), t)dt + 1.1/(0).

Repeating this argument we can show that 1.1 is actually of class C2

Lemma 3.18 Assume u(x) is a solution of (3.4) and u(xo) E [0, min u(x)) for some Xo E JR . Then u(x) = 0 for all x E JR .

Proof: Let Xl be such that u(xd = minu(x) . Then U(XI) E [O,minu(x)) and ul(x) = -J~f(u(t),t)dt . Note that u(x) is nonincreasing for x> Xl as long as u(x) < minu(x) because f(s,t) ~ 0 for all s E [O,minu(x)) . Since u(xd is the minimum of 1.1 and since 1.1 is periodic it follows that u(x) = U(XI) for all x. It follows that u(x) = 0 for all x .

Now u( -00, x) and u( 00, x) are solutions of (3.4) . Furthermore 0 :::; u( -00, 0) :::; ~minu(x) :::; 1.1(00, 0). Therefore by Lemma 3.18, u(-oo , x) = 0 and u(oo, x) ~ min u( x) for all X E JR.

References

1. D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion and nerve propagation, Lecture Notes in Mathematics 446, SpringerVerlag (1975) , pp. 5-49.

2. P. Fife, MathematIcal Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer-Verlag (1979) .

3. R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics 7 (1937) , 355-369.

4. J. Ga.rtner and M. I. Freidlin, On the propagation of concentration waves in periodic and random media, Soviet Math . Dokl. 20 (1979) , 1282-1286.

5. M. I. Freidlin, Limit theorems for large deviations and reaction-diffusion equations, Ann. of Probability 13 (1985), 639-675.

6. K. P. Hadeler and F. Rothe, Traveling fronts in nonlinear diffusion equations, J. Theor. BioI. 52 (1975), 251-263.

7. W. Hudson and B. Zinner, Existence of traveling waves for a generalized discrete Fisher's equation, Com. Appl. Nonlin. Analysis 1, No. 3, (1994), 23-46.

199


8. A. Kolmogorov, 1. Petrovskii, and N. Piskunov, Etude de l'equation de la quantite de la matiere et son application a un probleme biologique, Moscow University Bull. Math. 1 (1937), 1-25.

9. J. D. Murray, Lectures on Nonlinear Differential-equation Models in Biology, Clarendon Press, Oxford, 1977.

10. N. Shigesada, K. Kawasaki, and E. Teramoto, Traveling periodic waves in heterogeneous environments, Theor. Populo Biology 30 (1986), 143-160.

11. H. Tucker, A Graduate Course in Probability, Academic Press, 1967.

12. X. Xin, Existence and stability of travelling waves in periodic media governed by a bistable nonlinearity, Journal of Dynamics and Differential Equations, 3, No. 4, (1991), 541-573.

13. G. Papanicolau and X. Xin, Reaction-diffusion fronts in periodically layered media, J. Stat. Physics 63, Nos. 5/6, (1991), 915-931.

14. J. X. Xin, Existence of planar flame fronts in convective-diffusive periodic media, Arch. Rat. Mech. and Anal. 121 (1992), 205-233.

15. J. Xin, Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media, Journal of Stat. Physics 73 (1993), 893-925.

16. J. X. Xin, Existence of multidimensional traveling waves in transport of reactive solutes through periodic porous media, Archive for Rational Mech. and Analysis 128 (1994), 75-103.

PERMANENCE OF PERIODIC SOLUTIONS OF RETARDED FUNCTIONAL DIFFERENTIAL

EQUATIONS

LUIZ A. da C. LADEIRA AND SUELI M. TANAKA Instituto de Ciencias Matematicas de Sao Carlos

Universidade de Sao Paulo Campus de Sao Carlos 13560-970 - Sao Carlos - SP - Brazil

Abstract

For a retarded functional differential equation given by x(t) = F(x(t),x(t - (7),x(t - T)) with an w-periodic solution p(t), it is proved that if we take delay sufficiently close to (7 and T we still obtain periodic solutions near to p(t) of period close to w. An illustration of this result is given.

1. Preliminaries

201

Theorem 1.1 Let E be an open subset of a Banach space Y , F be a closed subset of a Banach space X, where interior of F is a non-empty set. Assume that T : F x E -+ F satisfies the following hypotheses: (i)T(x,.) : E -+ F is continuous. (ii) T( . , y) : F -+ F is continuous and, for each y E E , has a unique fixed point x(y) that depends continuously on y. (iii) If x(E) Fl C F, then T(x, y) is continuously differentiable in y for (x,y) E Fl X E. (iv) There is an open set F2 C X such that FeF2 and the derivative DxT(x,y) of T(x,y) with respect to x is continuous and satisfies the condition: I DxT(x,y) I::; 0<1, V(x,Y)EF2 xE.

Then the fixed point x(y) E F,y E E , of T(x,y) is continuously differentiable in y .

Proof: Use the Contr~ction Mapping Principle. o

Theorem 1.2 Consider the following retarded functional differential equation:

x(t) = f(a,t,xt) (1.1)

BVPs FOR FDEs

where x ( t ) E IRn;a E R ; x t E C = C ([-r,O],IRn), f : IR2 x C + Rn continuous. If x ( t ) is p/q-periodic solution of (1.1), for p, q positive integers, then x ( t ) has the

form: (i) x ( t ) = B,(t) + Bl ( t ) cos qwt + Bz(t) sin qwt + B3(t) cos Sqwt + B4(t)

P P sin 2qwt + . - . + BP,(i) cos(? - 1)qwt + BP-z(t) sin (- - 1)qwt + B p - ~ ( t ) cos q ~ t ; for 2 -

p even and (ii) x ( t ) = B,(t) + Bl ( t ) cos qwt + B2( t ) sin qwt + B3(t) cos2qwt

P - 1 P - 1 + B4(t ) sin 2qwt + .. . + BP-2(t) ~ o s ( ~ ) q w t + Bp-l( t) sin ( ~ ) ~ w t ; 2 for p odd.

In both case we consider w = 2nlp and Bk a llq-periodic function from IR to IRn; with k=O, . . . , p - 1.

Proof: See Ladeira and Tanaka [ I ] , and Tanaka [7].

2. S ta tement and Proof o f t h e Problem

Consider the following retarded functional differential equation:

x ( t ) = F ( x ( t ) , x( t - a ) , x ( t - 7 ) ) (2.1)

with F E C1 (R3",IRn), 0 < a 5 7 , x ( t ) E IRn, V t E R. Suppose that

( H . l ) The equation (2.1) has a nonconstant w-periodic solution p(t), w > 0.

Therefore, we have the linear variational equation associated with p(t):

aF where L ( t ) : C + IRn, L ( t ) p = -(p(t),p(t - a) ,p( t - 7 ) ) p(0) +

ax a F a~ -(p(t), p(t - 4, ~ ( t - 7 ) ) p(-a) + z ( ~ ( t ) , ~ ( t - 01 , ~ ( t - 7 ) ) l p ( - ~ ) , ay f w r > 0 , 7 I r .

Let us also suppose that (H.2) 7 = {p( t ) , t E IR) is a nondegenerate periodic orbit.

T h e o r e m 2.1 Consider

where ( E 15 E,, E, a strictly positive real number Then there is a real number &I, 0 < €1 I E,, so that i f we take e;

I E 15 € 1 the equation (2.3) has a periodic solution x ( e ) ( t ) of period w(s) , with " (0) = p, w(0) = w; and moreover, x (e) and W ( E ) depend continuously on E .

203

PERMANENCE OF PERIODIC SOLUTIONS

t Proof: Let x(t) be a solution of (2.3). For any f3 > -1, we take s = --f3 and

1+ y(s) = x(t), therefore y(s) must satisfy:

. U+E r+E y(s) = (1 + (3)F(y(s),y(s - 1 + (3)'y(s - 1 + (3)) . (2.4)

Note that if we find an w-periodic solution of (2.4) then we get a (1 + (3) w-periodic solution of (2.3)j so that we take another change of variable y(s) = p(s) + z(s), and thus z(s) satisfies:

U+E U+E r+E z(s) = (1 + (3)F(p(s) + z(s),p(s - 1 + (3) + z(s - 1 + (3)'p(s - 1 + (3)

r+E +z(s - --(3)) - F(p(s),p(s - u) ,p(s - r)) .

1+ (2.5)

1 . Let us take f3 such that 1f31 ::s 2' r = 2( r + Eo) > 0 and we consIder (2.5) as a

functional differential equation in C.

For n = {u : IR -+ IRnj 1.1 is continuous and w-periodic } with the norm 111.1110 = sUPtE IR IIu(t)IIIRn , we define

204

BVPs FOR FDEs

G . lR x 0 x (-'f. f.) X (-~ ~) --> lRn

• 0,0 2'2

G(s,U,f.,(3) =

a+f. a+f. T+f. r+f. (1 + (3)F(p(s) + u(s),p(s - 1 + (3) + u(s - 1 + (3) ' p(s - 1 + (3) + u(s - 1 + (3))

of -F(p(s),p(s - a),p(s - T)) - ox (p(s) ,p(s - a),p(s - T)) u(s)

of , of - oy (p(s),p(s - a"p(s - r))u(s - a) - oz (p(s),p(s - a),p(s - r))u(s - r)

It is shown that the equation (2 .5) is equivalent to

i (s) = L(s) z. + G(s, Z,f.,(3) (2.6)

and thus we interpret it as a perturbation of linear equation (2 .2) . Using (H.2) , we can decompose C relative to the characteristic multiplier 1 as

C=EffjK, we also take P the projection induced by this decomposition such that P takes C onto K .

Note that the equation (2.2) has a unique w-periodic I.i ., then the formal adjoint equation associated has only an w-periodic solution I.i. denoted by q(t) and assume that f;;' q( a) qT (a) da = 1 with qT the transpose of q.

We know that if hE 0, f;;' q(a) h(a)da = 0, the equation

i( s) = L( s) z. + h( s) (2.7)

has a unique solution in 0 whose (I - P)-projection is zero. Let us designate this solution by K hand K is a continuous linear operator taking 0 into O.

Consider the continuous linear transformation Q taking 0 into 0 such that Q(h) = U;;' q(a) h(a)da)qT(.), and we can obtain K(I - Q)h the unique w-periodic of i(s) = L(s) z. + (I - Q) h(s) whose (I - P)-projection is zero.

Define

1 1 H : 0 x (-f.o,f.o ) x (-2"' 2") -> 0

H(u, f., (3) = K(1 - Q) G( . , U, f., (3).

From the Mean Value Theorem, H is a uniform contraction with relation to f., {3 and then we can use the Fixed-Point Theorem to show that there are positive constants Db f.}, {31, u·(f., (3) E OJ Ilu·(f., (3)110 ~ D}, 1f.1 ~ f.}, 1(31 ~ (31, u·(f.,(3) continuous in (f.,{3) , u·(O,O) = 0 such that u·(f.,(3) is solution of

i(s) = L(s)z. + G(s,u·( f. , (3) , f., (3) - B(f.,(3)qT(s) (2.8)

205


with

B(c:,{J) = low q(a) G(a,u·(c:,{J),c:,{J) da. (2.9)

The Theorem 1.1 insures that u·(c:,{J) is continuously differentiable in (c:,{J) aI}.d au·

we can prove that o{J (c:, (J) must satisfy:

v(s) = L(s)v. + A(s,v,c:,{J) - ~;(C:,{J)qT(s), (2.10)

where

of of A(s,v,c:,{J) = - ox (p(s),p(s -17),p(s - T))V(S) - oy (p(s),p(s -17),p(s - T))

aF v(s -17) - oz (p(s),p(s - 17),p(s - T)) v(s - T)+

17+C: T+C: (1 + (J)[ a.,p(s) v(s) + b.,p(s) v{s - 1 + (J) + c.,p(s) v(s - 1 + (J) J+

• 17+C:. 17+C: T+C: F(p(s)+u (c:,{J)(s),P(s-l+{J)+u (c:,{J)(s-I+{J),P(s-I+{J)

r+c: 17+C: . 17+C: duo 17+C: +u·(c:,{J)(s - 1 + (J)) + (1 + (J) b.,p(s) (p(s - 1 + (J) + Ts(c:,{J)(s - 1 + (J))

r + c: . r + c: duO r + c: +(1 + (J)c.,p(s)(p(s - 1 + (J) + Ts(c:,{J)(s - 1 + (J))

and a.,p(s) = F,,(p(s) + u·(c:,{J)(s),

17+C:. 17+C: T+C:. r+c: P(s-l+{J)+u (c:,{J)(s-l+{J),P(s-l+{J)+u (c:,{J)(s-l+{J))

b.,p(s) = Fy(p(s) + u·(C:,{J)(s),

17+C:. 17+C: r+c:. r+c: P(s-l+{J)+u (c:,{J)(s-l+{J),P(s-I+{J)+u (c:,{J)(s-I+{J))

c.,p(s) = Fz(p(s) + u·(c:,{J)(s),

17+C:. 17+C: r+c:. r+c: p(s - 1 + (J) + u (c:,{J)(s - 1 + {J)'p(s - 1 + (J) + u(C:,{J)(s - 1 + (J))'

Thus, the equation (2.10) has an w-periodic solution and so

BVPs FOR FDEs

hence,

In particular, taking e = p = 0, we have that

Let y(t) be a solution of (2.2), i ( t ) = y(t) + tp(t) and then

Suppose that the expression (2.11) is zero, we get y(t) = i ( t ) - tp(t) solution of (2.2), with fj and p w-periodic functions, yo = go - (.)A = cp.

And so,

= cp - w up, = cp + U(-wp,) = cp + U(Ucp - cp).

Therefore cp E N(I - U)'. We can prove that cp and p, are 1.i. and thus we have a contradiction because the hypothesis (H.2) implies that 1 is a simple characteristic multiplier.

And then applying the Implicit Function Theorem for B(E,P) we get strictly positive real numbers ,f3z,~2; Pz 5 PI, 82 5 el, a continuous function P(e); I/?(e)I 5 Pz for all s satisfying IeI 5 ez such that P(0) = 0 and B(e, P(E)) = 0.

From the equation (2.8) we conclude that u*(e, P(e)) is solution of (2.6), u*(O, 0) = 0.

Thus for all e; 0 < lei 5 sz, we have x(e)(t) a solution of (2.3) of period W ( E ) = (1 + P(e))w that satisfies x(0) = p, w(0) = w, with x(e) e ~ ( e ) depend continuously on 8.

207


Remarks: It can be seen from the proof of Theorem 2.1 that the same conclusion holds if

the equation (2.3) is replaced by the equation

i(t) = F(x(t),x(t - (0- + £t}),x(t - (r + £2))) '

In this case, the periodic orbit continues to exist for £ = (£1, £2) in a full neighborhood of (0-, r) in the parameter space, and has the same continuity proprierties.

The proof also shows that the result is valid for equations with finitely many delays.

3. Application

Consider the following scalar functional differential equation:

i(t) = ~[b2 - x2(t -.!.) + x(t)x(t - .!.)] a 4 2

(3.1)

with a, b, nonzero real numbers. We can prove that p(t) = a + bcos 211"t is a I-periodic solution of (3.1) and the

variational equation associated is given by:

y(t) = ~[(a - bcos 211"t) y(t) + (a + bcos 211"t) y(t - ~) - 2(a + b sin 211"t) y(t - .!.)] a h.2)

In order to obtain a I-periodic solution y(t) of equation (3.2) we use the Theorem 1.2 and so y(t) has the form:

y(t) = Bo(t) + Bl(t) cos211"t + B2(t) sin211"t + B3(t) cos411"t (3.3)

for B t , i = 0,1,2,3 a scalar 1/4-periodic function. We can show that the function 1, cos 211"t, sin 211"t, cos 411"t are i. i. over the 1/4-

periodic functions, namely, the unique scalar 1/4-periodic functions Ql(t), Q2(t), Q3(t), Q4(t) such that Ql(t) + Q2(t) cos 211"t + Q3(t) sin 211"t + Q4(t) cos 411"t = 0 are the zero functions.

Substituting (3.3) into (3.1) and using that l,cos211"t,sin211"t,cos411"t are i.i. over the 1/4-periodic functions, we conclude that Bo(t), B1(t), B2(t), B3(t) must satisfy:

. 211"b Bo(t) = --Bl(t)

a (3.4)

(3.5)

208

BVPs FOR FDEs

(3.6)

(3.7)

Therefore, B2 k, k E ffi., Bo Bl B3 0, from (3.3), y(t) = k sin 211"t, k E ffi..

It is clear that p(t) generates the l-periodic solutions of (3.2), then, p(t) is a non degenerate solution of (3.1) .

It follows that the equation (3.1) satisfies the statements (H.l) and (H.2), hence, the equation

11" 1 1 x(t) = -W - x2(t - (- + c)) + x(t) x(t - (-2 + c)) 1

a 4

has a periodic solution x(e) of period w(e) if we take e sufficiently close to zero, x(O) = p, w(O) = I, x(e), w(e) depend continuously on c. 0

References

1. L. A. V. Carvalho, On a method to investigate bifurcation of periodic solutions in retarded differential equations, Preprint.

2. J. R. Clayessen, Effect of delays on functional differential equations, 1. Differential Eqns. 20 (1976), 404-440.

3. J.K. Hale and L.A.C. Ladeira, Differentiability with respect to delays, 1 .Differential Eqns. 92 (1991), 14-26.

4. J . K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993.

5. L. A. C. Ladeira, Resultados sobre diferenciabilidade com rela!;ao ao retardamento, Tese de Livre-docencia, ICMSC-USP, Sao Carlos, 1992.

6. L. A. C. Ladeira and S. M. Tanaka, A method to calculate periodic solutions of functional differential equations, Preprint.

7. S. M. Tanaka, Urn estudo sobre solu!;oes peri6dicas de Equa!;oes Diferenciais Funcionais (A study on periodic solutions of F.D.E.), Doctoral Thesis, ICMSCUSP, Sao Carlos, 1994.

209

METHOD OF QUASILINEARIZATION FOR BOUNDARY VALUE PROBLEMS OF FUNCTIONAL DIFFERENTIAL EQUATIONS

V. LAKSHMIKANTHAM AND N. SHAHZAD Applied Mathematics Program, Florida Institute of Technology

Melbourne, Florida 92901 USA

1. Introduction

Quasilinearization is a well known technique for obtaining approximate solutions of nonlinear differential equations [1, 2]. It provides a monotone sequence of approximate solutions that converges quadratically to a solution of the boundary value problem (BVP),

-x"(t) = f(t, x(t)) , t E [0,1]'

x(o) = a, x(1) = b,

if f is uniformly convex. This technique has recently been generalized and extended using less restrictive assumptions so as to be applicable to a large class of functions [8-10] . This generalization is also discussed for initial value problems [5-6].

In this paper, we shall extend the method of quasilinearization to a boundary value problem associated with functional differential equations, that is,

-x"(t) = f(t , x(h(t))), t E J = [0,1],

where f E e[J x JR, JR] and h E C[J, JR], subject to the boundary conditions

x(t) = ¢>(t), t E Jo = (-00,0], x(1) = b,

(1.1)

where ¢> E C 1 [Jo,JR]. This we shall do in the framework of the method of upper and lower solutions so that one can obtain a monotone sequence that converges quadratically to the unique solution of the BVP (1.1).

We have discussed a simple BVP (1.1) in order to bring out the ideas involved in the method of quasilinearization. One could however, apply our method to the more general BVP

-x" = f(t,x(t),x(h(t,x(t))),

x(t) = ¢>(t) on Jo and x(t) = tf;(t) on J1 = [1,00), since necessary existence results are available. See [3].

210

BVPs FOR FDEs

2. Main Results

A function a E B = C[lR,lR]n BC1[Jo,lR]n BC1[{b},lR]n C1[lR,lR] having a continuous second derivative on J will be a lower solution of (1.1), if

-a"(t) ::; f(t,a(h(t))), t E J,

a(t) ::; tjJ(t) on Jo, a(l) ::; b.

An upper solution fl(t) of (1.1) is defined similarly by reversing the above inequalities. For further details we refer to [3].

Next we recall a result [3] in a special form which is needed for the main result.

Theorem 2.1: Assume that (i) ao, flo are lower and upper solutions, respectively, of (1.1) such that ao ::; flo

on lR; (ii) f E C[J x n, lR] and f(t, y) is nondecreasing in y for each t, where n = {x:

ao(t) ::; x ::; flo(t), t E lR} . Then there exists a solution x(t) of (1.1) such that ao(t) ::; x(t) ::; flo(t), t E lR.

Now we are in a position to prove our main result.

Theorem 2.2: Assume that (i) ao, flo are lower and upper solutions, respectively, of (1.1) such that ao ::; flo

on lR; (ii) f E C[J x n, lR], fx, fn exists and are continuous on J x n satisfying

fn(t,x) 2: 0 on J x n; (iii) f(t,y) is nondecreasing in y for each t; (iv) K < 4, where K is a bound of fx(t,x) on J x n. Then there exists a monotone sequence {wn(t)} which converges uniformly and montonically to the unique solution of (1.1) and the convergence is quadratic.

Proof. In view of (ii), we see that

f(t,x) 2: f(t,y) + fx(t,y)(x - y),

for x 2: y. Consider the BVP

_X"(t) = f(t,ao(h(t») + fx(t,ao(h(t)))(x(h(t)) - ao(h(t)))

== g(t, x(h(t»j a(h(t))), x(t) = tjJ(t), t E Jo, x(l) = b.

Let wo(t) = ao(t) so that (i), (2.1) and (2.2) yield

-a~(t) ::; f(t, ao(h(t») == g(t, ao(h(t»j ao(h(t))),

(2.1)

(2.2)

QUASILINEARIZATION FOR BOUNDARY VALUE PROBLEMS

-f3~(t) ~ f(t, 130 (h(t)))

~ f(t,ao(h(t))) + f,,(t,ao(h(t)))(f3o(h(t)) - ao(h(t)))

= get, f3o(h(t))j ao(h(t))).

211

Then by Theorem 2.1, there exists a solution WI(t) of (2.2) such that wo(t) ::; WI(t) ::; f3o(t), t E JR.

Since WI ~ Wo, 130 ~ WI, we get using (2.1) and (2.2),

-w;'(t) = g(t,wI(h(t))jwo(h(t)))

= f(t,wo(h(t))) + f,,(t,wo(h(t)))(WI(h(t)) - wo(h(t)))

::; f( t, WI (h( t))) = g( t, WI (h(t))j WI (h( t))),

-f3~(J) ~ f(t, 130 (h(t)))

~ f(t,WI(h(t))) + fx(t, WI (h(t)))(f3o (h(t)) - WI (h(t)))

= g(t,f3o(h(t))jWI(h(t))).

Consequently, we have, as before, a solution W2(t) of the BVP

such that

-W~(t) = g(t,w2(h(t)) jWI(h(t))),

W2(t) = r!J(t), t E Jo, xCI) = b,

This process can be continued successively to obtain the monotone sequence {wn(t)} satisfying

where wn(t) is a solution of the BVP

-W~ = g(t,wn(h(t))jWn_I(h(t)))

= f(t,wn-I(h(t))) + f,,(t,Wn-I (h(t)))(wn (h(t)) - wn-I(h(t))),

wn(t) = r!J(t), t E Jo, wn(l) = b.

Employing standard arguments (as in [4]), it is easy to conclude that the sequence {wn(t)} converges uniformly to the unique solution x(t) of (1.1).

Next we shall show that the convergence of wn(t) to x(t) is quadratic. For this purpose, consider Pn+I(t) = x(t)-Wn+I(t) ~ 0 for each n and note that Pn+I(t) = 0, t E Jo and Pn+I(l) = O.

212

BVPs FOR FDEs

Hence we can write

where

Pn+1(t) = 101

G(t,S)[J(s,x(h(s») - g(S ,Wn(h(S»;Wn_1(h(s)))]ds,

G(t, X) = {OG,(t,X) , t E J t rf. J.

Here G(t,x) is the Green's function given by

Gtx ={t(1-X) OS; x S;tS;1 (,) x( 1 - t) 0 S; t S; x S; 1.

It is easy to observe that G(t,x) is nonnegative. Now

OS; Pn+1(t) = 101

G(t , s) [J(s , x(h(s») - f(s,wn(h(s»)

- fz(s,wn(h(s )))(Wn+1 (h(s» - wn(h(s)))] ds

S; 101

G(t,s) (Jx(s,x(h(s)))(x(h(s)) -wn(h(s)))

- fz(s,wn(h( s »)(Wn+1 (h(s» - wn(h( s »)] ds

= 101

G(t , s) [fzx(t,e)(x(h(s» -wn(h(s)?

+fz(s,wn(h(s))(x(h(s)) - wn+1(h(s)))] ds

= 101

G(t,s) [J%X(s,e)p~(h(s)) + fz(s, wn (h(S)))Pn+1 (h(s))] ds ,

where Wn < e < x. Then we get

A sup IPn+1(t)1 S; -1 B sup IPn(t)12 , tElR - tElR

where on J X fl, IG(t,s)1 S; t, If"z(t,x)1 S; L, Ifz(t,x)1 S; K, A = t and B = f.

213

QUASILINEARIZATION FOR BOUNDARY VALUE PROBLEMS

References

1. R. Bellman, Methods of Nonlinear Analysis, Vol. II, Academic Press, New York 1973.

2. R. Bellman and R. Kalaba, Quasilinearization and Nonlinear Boundary Value Problems, American Elsevier, New York 1965.

3. S.R. Bernfeld and V. Lakshmikantham, An Introduction to Nonlinear Boundary Value Problems, Academic Press, New York 1974.

4. G.S. Ladde, V. Lakshmikantham, and A.S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston 1985.

5. V. Lakshmikantham, An extension of the method of quasilinearization, J. Optim. Theory. Appl. 82 (1994), (in press).

6. V. Lakshmikantham and S. Koksal, Another extension of the method of quasilinearization, Proc. of Dyn. Systems and Appl., Dynamic Publishers, Atlanta (1994), 205-210.

7. V. Lakshmikantham and S. Malek, Generalized quasilinearization, Nonlinear World 1 (1994), 59-65.

8. S. Malek and A.S. Vatsala, Method of generalized quasilinearization for second order boundary value problem, Inequalities and Applications 3, World Scientific, to appear.

9. N. Shahzad and A.S. Vatsala, Improved generalized quasilinearization method for second order boundary value problem, Dyn. Systems and Appl. (1995), to appear.

10. N. Shahzad and A.S. Vatsala, An extension of the method of generalized quasi linearization for second order boundary value problems, Applicable Analysis (1995), to appear.

215

A RESOLVENT COMPUTATION RELATED TO COMPLETENESS FOR COMPACT OPERATORS

SJOERD M. VERDUYN LUNEL* Faculteit der Wiskunde en Informatica,

Universiteit van Amsterdam Plantage Muidergracht 24

1018 TV Amsterdam, The Netherlands

Abstract

In a recent paper [4) we have studied the problem whether the period map of a periodic delay equation has a complete span of eigenvectors and generalized eigenvectors. There is an abstract theory (see [4) and also [3)) that can be used to verify whether the eigenvectors and generalized eigenvectors corresponding to the nonzero spectrum of a compact operator from a given class of operators are complete. To use the abstract results one needs good estimates for the resolvent operator near infinity and to compute the resolvent explicitly one often has to solve a boundary value problem. In this paper we first give an abstract theorem and then we discuss some explicit examples.

1. A Result About Completeness

Let H be a complex Hilbert space and let T : H -+ H be a compact operator. Let ET denote the span of the eigenvectors and generalized eigenvectors corresponding to the nonzero eigenvalues of T. If ET is dense we call the system of eigenvectors and generalized eigenvectors complete. In general, the nonzero spectrum of a nonselfadjoint compact operator T can be empty and to study T, one first introduces the singular values of T using the compact positive selfadjoint operator T*T . If

denote the eigenvalues of T*T, the singular values of T are defined as follows

sj(T) := }.,j(T*T)1/2, j ~ 1.

The research of S.M. Verduyn Lunel has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.

216

BVPs FOR FDEs

For example, the integration operator V : prO, 1] ---> L2[0, 1]

(V.p)(0) = 2i l .p(s)ds

is compact but has no nonzero spectrum (such operat~rs are called Volterra operators). The singular values are given by Sj(V) = 4((2j - 1}7r)-I, j ~ 1. (See [2] for details and more examples.)

Theorem 1.1. Let T : H -t H be a compact operator sucb tbat Ker T = Ker T* = {OJ . Suppose tbere exists a r > 0 sucb tbat

00

LSj(T), < 00 . (1.1) j =l

Let

6.1 = {rei81 r ~ 0, 101 ~ f ~ ~}, 6.2 = {rei8 1 r ~ 0, 171" - 01 ~ f ~ ~},

n = C \ (6.1 U 6.2 )

and let J.Lj denote tbe poles of tbe meromorpbic function

z -t (1 - zT)-l .

If tbere exists a positive integer m , sucb tbat for every f > 0 tbere is a constant M sucb tbat

11(1 - zT)-l l1 ~ Mlzlm as Izl-t 00 in n wbile Iz - J.Ljl > f for j = 1,2, . .. , (1.2)

tben ET=H.

Proof. We argue by contradiction. If ET "I H, then Ef "I {OJ. If P is the orthogonal projection onto Ef and

V = PTIE; : E; -t E;, (1.3)

then V* = P*TIE;

and u(V*) C {OJ . Thus V* is a Volterra operator. Since L~l sj(TY < 00, we have from [2; VI.1.3-4] that

j=l

Therefore we can apply Theorem X.2.2 of [2] to show that, given 8 > 0, there exists a constant C such that

z E C. (1.4)

A RESOLVENT COMPUTATION

Fix y E E;}:, y"l O. For every x E H define F( ·;x): «:: -+ «:: by

F(z;x) = (y,(I - ZT)-lX)

= ((/ - ZV*tly, x), z E «::.

217

So F('jx) is an entire function. From assumption (1.2), it follows that F(·;x) is bounded by a polynomial of degree at most m in n. To estimate F(·; x) in the sectors ~l U ~2' we use the Phragmen-Lindelof theorem. First we conclude from inequality (1.4) that

z E «::.

Since F(·;x) is bounded by a polynomial of degree at most m along the boundaries of ~l and ~2' the Phragmen-Lindelof theorem implies that there exists a constant M such that

z E «::.

Thus the Liouville Theorem yields that there are constants ai = ai(x), 0 ~ i ~ m, such that

F(z; x) = ao + alz + .. . + amzm.

On the other hand, it follows from the definition of F(·;x)

F(zj x) = (y, (I - zTt1X ) = (y, x) + z(y, Tx) + z2(y, T 2x) + ... + zm+1(y, Tm+1x).

In particular, (y, Tm+1x) = O.

Since x is arbitrary and ImTm+1 is dense, it follows that y = O. A contradiction. D

Example 1.2. If T is injective and selfadjoint, the conditions of Theorem 1.1 are satisfied with m = 0 and, hence, one can use Theorem 1.1 to conclude that the eigenvectors of T are complete.

2. An Example From Delay Equations

An example of a nonselfadjoint operator that satisfies the conditions of the theorem is the period map corresponding to the following linear periodic delay equation (see [1])

x(t) = b(t)x(t - 1), (2.1)

where b(t) is a continuous periodic function with minimal period I, b(t + 1) = b(t) that has isolated zeros only. In order to work with a Hilbert space as state space we define H = «:: x £2([-1,0];«::). For (c,t/» E H we consider equation (2.1) with the initial condition

x(s)=c, x(s+O)=t/>(O), -1~O<O.

218

BVPs FOR FDEs

For every SEnt, (c,¢» E H, Eq. (2.1) has a solution x(s,¢» defined on [s,oo) and defines a forward evolutionary system T( t, s) by

T(t, s)(c, ¢» = (x(t), x(t + .)). The period map U : H -+ H is defined by U = T(l, 0) and explicitly given by

U(c,¢» = (c+ II b(s)¢>(s)ds,8 >-+ c+ /1 b(s)¢>(s)ds) (2.2)

Let (TC)(t) = c

and

If

(V ¢>)(t) = 1:1 b(s)¢>(s) ds,

then T = iU becomes

Therefore

1- zT =. . (1 - iz -iZT2) -tZTl 1- tzV

=(1 -iZT2(I-izV)-I)(~(iz) 0 ) o I -tZTl I - iz V '

where

Thus

(I - zTtl = ( . . ~(iz)-: 0) (1 iZT2(I - iZV)-I) tZ~(tz)-I(I - tZVt1Tl (I - tzvt1 0 I .

To compute (I - izVt1 we solve a boundary value problem. If 1jJ = (I - iZV)-I¢>, then ¢> = 1jJ - izV1jJ. Hence we have to solve

1jJ' - izb1jJ = ¢>'

1jJ( -1) = ¢>( -1). (2.3)

(2.4)

If n~(z) denotes the fundamental matrix solution of if = zby, it follows from Eq. (2.3)-(2.4) that

((I - izVtl¢>)(8) = n~l(iz)¢>(-1) + i: n:(iz)¢>'(s)ds

18 . J"b( )d = ¢>(8) + iz -1 etZ

• Cf Cfb(s)¢>(s) ds . (2.5)


A simple computation shows that

~(iz) = 1 - iz - iZT2(1 - iZV)-1izT1

= 1 - iZn~1(iz)

= 1 - izeizii , where b = 1° b(s) ds. -1

Thus we have the following representation for (1 - zTt1:

219

(2.6)

(I - zTt1(c,tP) = (~(izt111:(iz),0 1-+ iz~(izt1n~1(iz)lI:(iz) + «(1 - izVt1tP)(0)), (2.7)

where

lI:(iz) = c + iZT2(I - izVt11jl

= C + iz l1 b(s)ljl(s) ds + (iz)2 i: b(O) l1 n~(iz)b(s)ljl(s) ds dO

= c + iz l1 b(s)ljl(s) ds + (iz) i: (/ izb(O)n:(iz) dO)b(s)ljl(s) ds

= c + iz l1 n~(iz)b(s)ljl(s) ds.

(2.8)

From representation (2.6)-(2.7) we conclude that the nonzero spectrum of T is given by

..!:. E q(T) if and only if e-il'ii - iJ1. = o. J1.

If b = 0, the spectrum of T is finite and the eigenvectors and generalized eigenvectors cannot be complete.

If b '" 0, we assume that b > O. Let J1.;, j ~ 1 denote the zeros of e-izii - iz ordered

according to increasing modulus. It is a simple calculation to show that

Thus we have that

00 1 E -I .11+6 < 00 for every Ii > O. ;=1 J1.J

00

E A;(T)1+6 < 00,

;=1 where A1(T) ~ A2(T) ~ ... denote the nonzero eigenvalues of T.

In order to apply Theorem 1.1 we first show that T satisfies (1.1) with r = 2, i.e., the singular values of Tare square-summable. Such operators are called HilbertSchmidt operators. Since T = F + V, where F is a finite rank operator and V an integral operator given by

V(c,tP) = (0,0 1-+ l1 b(s)ljl(s)ds),

220

BVPs FOR FDEs

it follows from the fact that both F and V are Hilbert-Schmidt operators (see [2;Prop IX.1.1]) that the sum T = F + V is a Hilbert-Schmidt operator. Thus T satisfies (1.1) with r = 2 and 0 = 0 1 U O2 , where

0 1 = {reiO 17r/4 ~ 0 ~ 37r/4}, O2 = {reiOI7r/4 ~ 7r - 0 ~ 37r/4}.

Because le-izbl = e!.'lzb and I~zl > ¥Izl in 0 1 U O2 , it follows from Eq. (2.5) that there are positive constants Mb M2 , M3 and M4 such that

1 Mlexp{ min fO b(O") dO" .J222Izl} as Iz l-+ 00 in 0 1

1I(I- izV)-I I1~ -1~.<0~o1·0 .J2

M2exptl~~~:<;O 1 b(O") dO"T1zl} as Iz l -+ 00 in O2

and

16.(iz)-11 = 1 -v'2 . 1

M3 as Izl -+ 00 in 0 1

M4 y.;rexp{-bT 1zl} as Izl-+ 00 III O2

Together with (2.7)-(2.8), these estimates allow us to estimate (1 - zTtl in O. If b ~ 0, the first component of (I - zTtl is bounded in O. To estimate the second component we first have to rewrite the expression.

iz6.(iz)-10~I(iz)K(iz) + (1 - iZV)-I4»(O) = iz6.(iztl[O~I(iz)c + 4>+

10 O:(iz)b(s)4>(s) ds + izO~1 fO O:(iz) b(s)4>(s) ds). -I 10

If b ~ 0, it is straightforward to conclude that

11(1 - zTtl 1i ~ Mizi as Izl-+ 00 in O.

Furthermore, the explicit representation for T shows that T is injective and has dense range. Therefore the assumptions of Theorem 1.1 are satisfied and we conclude that if b ~ 0, then U given by (2.2) has a complete span of eigenvectors and generalized eigenvectors in H . Similar arguments as given in (4) show that T does not have a complete span of eigenvectors and generalized eigenvectors if b changes sign (see (3) for details).

One should note that the operator T is a two dimensional perturbation of a Volterra operator. Thus the example shows that a finite rank perturbation of a compact operator with no nonzero eigenvalues can have a complete span of eigenvectors and generalized eigenvectors. This rather surprising fact is the object of further study. As an illustration of difficulties we will give another example of a finite rank perburbation of a Volterra operator that, although the perturbed operator has infinite nonzero spectrum, does not have a complete system of eigenvectors and generalized eigenvectors.

221


3. A Noncompleteness Result

Let H be as before and define T : H -+ H by

10 1-1 18

T(c,4» = (c+ -1 4>(s)ds,O 1--+ c- -2 4>(s)ds + _24>(s)ds). (3.1)

To compute the nonzero spectrum of T suppose that T(c,4» = J.L-l(C,4» . Then

c + 1° 4>( s) ds = .!.c -1 J.L

C -1-1 4>(s) ds + 18

4>(s) ds = .!.4>(O), -2 -2 J.L

-1::; 0::; O.

The second equation yields

4>(0) = c.

So 4>(0) = ceP8 , -1 ::; 0 ::; 0 and substitution into the first equation yields that J.L

satisfies the equation J.L - el'o = O. So the nonzero spectrum of T is infinite but we shall show that T does not have a complete span of eigenvectors and generalized eigenvectors.

In order to do so, we have to construct a pair (c, 4» E H such that (c, 4» f. (0,0) and (c,4» E E;/; . Since ET = Elheup(T)ImP>., it follows that

(Here P>. denotes the spectral projection corresponding to an eigenvalue). of T.) From the Laurent series expansion, it follows that if (c, 4» E E;/; then

Z 1--+ (I - izT*tl(C, 4» is an entire function. (3.2)

First we compute T*. From

(T*(d, 'I/J), (c, 4») = ((d, 1/J), T(c, 4»)

= d(c + 1: tjJ(s) ds) + 1: 1/J(s)(c _1~1 tjJ(u) du + [2 4>(u)du) ds

= (d + l21/J(S) ds)c + 1: d4>(s) ds _1~11/J(S)(J.-l 4>(u) du) ds

+ 1: 1/J(S)(i'1 4>(u)du)ds

= (d+ l21/J(S)ds)c + 1: dtjJ(s)ds -1~\1: 1/J(s)ds)tjJ(u)du

10 ° + -1(L 1/J(s)ds)tjJ(u)du,

222

BVPs FOR FDEs

it follows that

T*(d, t/J) = (d + i: t/J(s) ds, dX[-I,Oj + r t/J(s) dSX[-I,Oj - i~ t/J(s) dSX[-2,-lj), (3.3)

where X[a,bj(O) = 1 if 0 E [a, b) and ° otherwise. Before we can construct a pair (c, 1/» =I (0,0) such that (3.2) holds we have to

compute the operator (I - izT*tl. If (d, t/J) = (I - izT*tl(C, </» then

(c, 1/» = (d-izd-iz l2 t/J( s) ds, t/J-izdX[_I,Oj-iz r t/J( s) dSX[_I,Oj+iz i~ t/J( s) dSX[-2,-lj).

Thus we have to solve the following system of equations for d and t/J given c and </>:

c = d - izd - izjO t/J(s)ds, -2

</>=t/J-izd-iZrt/J(s)ds on (-1,0),

</>=t/J+izj· t/J(s)ds on [-2,-1), -2

where at -1 we have the boundary condition

jo j-1 </>(-I)=t/J(-I)-izd-iz _1t/J(s)ds+iz -2 t/J(s)ds.

(3.4)

(3.5)

(3.6)

(3.7)

We first solve Eq. (3.6). From 1/>' = t/J' + izt/J with 1/>( -2) = t/J( -2), it follows that

t/J(O) = e-iz(8+2)</>(_2) + 1:2 e-iz(8-')I/>'(s)ds

=1/>(0)_izj8 e-iz(8-s)l/>(s)ds -2::;0<-1. -2

(3.8)

Similarly, on [-1,0),

t/J(O) = e-iz(8+1)(t/J(_I) - </>(-1)) + 1/>(0) - iz 1:1 e-iz(8-s)l/>(s)ds -1::; 0::; ° and, using boundary condition (3.7) at 0 = -1, we find

1/>(0) = e-iz(8+l) (izd + iz i: t/J(s) ds - iz i~1 t/J(s) ds) + 1/>(0)

8 (3.9) -izj e-iz(8-')I/>(s)ds -1::;0::;0

-1

At 0 = ° we must satisfy the boundary condition izd = t/J(O) - 1/>(0) or,

izd=e-iZ(izd+izjO t/J(s)ds-izj-lt/J(s)ds)-izjO e-zi(-·)</>(s)ds. (3.10) -1 -2 -1

223


Therefore, by using (3.8),

. 1° iz(e'Z -1)d - iz 'I/J(s)ds = izl(iz; ¢», -1

where

l(iz; ¢» = - i~1 ¢>(s) ds + iz i~1 l2 e-iz(8-·)¢>(S) ds dfJ

-eiZ 1° e-zi(-')¢>(s)ds (3.11) -1

10 -iz(-1-.) -1.( ) d = - e 'l'S S. -2

Together with boundary condition (3.4) this yields the following system of equations

. 1° (etz - 1)d - -1 'I/J(S) ds = l(iz; ¢»

(1- iz)d - iz i: 'I/J(s)ds = c.

This system can be solved for d and J~ 1 'I/J ( s ) ds if det ~ (z) i- 0, where

(e

iZ - 1 -1) ~(z) = l' . . - zz -zz

The solution is given by

d = det ~(z) (-izl(iz; ¢» + c) (3.12)

and

l1 'I/J(s)ds = det~(z)(-(I-iz)l(iz;¢»+c(eiZ -1)). (3.13)

Thus we found that if (d, 'I/J) = (I - izT*)-1(C, 'I/J), then

t/J(O) = 1/1(0) - iz /2 e- iz(9-')I/I(s)ds , -2 ~ 0 < -1,

10 1-1 1-119

t/J(O) = e-iz(9+1) (izd + iz -1 t/J(s) ds - iz -2 I/I(s) ds + (iz)2 -2 ( -2 e-iz(9-')I/I(s) ds) dO)

+ 1/1(0) - iz /1 e-iz(9-')I/I(s)ds

10 1-1 1-119 =ize-iz(9+1)(d+ _. t/J(s)ds)_ize- iz(9+1)( -2 I/I(s)ds-iz -2 (_2 e- iZ(9- ' )I/I(s)ds)dO)

+ l/I(fJ) - iz 1: e-iz(9-')I/I(s) ds ,

(3 .14)

224

BVPs FOR FDEs

where d and f~1 1jJ( s) ds are given by (3.12) and (3.13), respectively. To construct a pair (c, 4» i- (0,0) such that (I - izT*)-1(c, 1jJ) is an entire function

we proceed as follows. If c = 0 and 4> is such that there exists an entire function J with l(iz,4» = det .6.(iz)J(iz), then

d = -izJ(iz) i: 1jJ(s) ds = - J(iz) + izJ(iz) .

So representation (3.14) implies that (I - izT*)-1(0, 4» is entire. Thus it remains to find a function 4> and an entire function J such that l( iz, 4» = det .6.(iz)J(iz). From (3.11), we conclude that

(-iz + e-iz)f(iz) = _ fO

eizs 4>(s) ds . L2 (3.15)

If J is given by

then

10 . ' 1° , (-iz+e-iz)J(iz) = -iz e,zsa(s)ds+e-'z e,zsa(s)ds -1 -1

f O r-1 = a(O) - e-iza( -1) + L1 eizsa'(s) ds + J-2 eizSa(s + 1) ds .

Thus if we take a(O) = a( -1) = 0, a differentiable on [-1,0] and 4> equal to a' on [-1,0] and equal a(· + 1) on [-2, -1] then (3.15) holds and hence

z ...... (I - izT*t1(0, 4» is a nontrivial entire function.

This completes the proof that the system of eigenvectors and generalized eigenvect~rs of T given by (3.1) is not complete.

We conclude with a remark. From the definition of T* it is easy to see that the eigenvectors and generalized eigenvectors of T* are identically zero on the interval [-2, -1] . Therefore, the system of eigenvectors and generalized eigenvectors of T* cannot be complete. Thus in this example we see that both T and T* are not complete. For this type of operators this is a general fact (see [3]). Therefore to show that completeness fails for T, it actually suffices to prove the existence of a pair (c, 4» such that

z ...... (I - izTt1( c, 4» is entire.

It is not difficult to see that any pair (c, 4» with c = 0, 4>(8) = 0 for -1 ~ 8 ~ 0 and 4> not identically zero, has this property.

225


References

1. J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.

2. I. Gohberg, S. Goldberg and M.A. Kaashoek, Classes of Linear Operators, Birkhii.user, Basel, 1990.

3. I. Gohberg, M.A. Kaashoek and S.M. Verduyn Lunel, in preparation.

4. S.M. Verduyn Lunel, About completeness for a class of unbounded operators with applications to delay equations, J. Differential Equations (1995) , to appear.

227

THE STUDY OF THE SOLVABILITY OF BOUNDARY VALUE PROBLEMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS: A

CONSTRUCTIVE APPROACH

V. P. MAKSIMOV AND A. N. RUMYANTSEV Perm State University 614600, Perm, Russia

1. Introduction

A survey is concerned with a constructive approach to the study of the solvability of BVP's. The principal idea of this approach is the use of the computer in establishing solvability on the basis of special theorems and techniques. Broad classes of linear BVP's for FDE's as well as certain classes of nonlinear BVP's are considered.

2. Preliminaries

Here we give a brief survey of certain results in the theory of FDE's which form a basis of effective investigation of BVP's (for details see [1] as well as [2]).

A FDE is representable in the form

x(t) = (Fx)(t), t E [0, T] (1)

where F is an operator defined on some set of absolutely continuous functions x : [O,T] --+ Rn. If F is the Nemytskii operator (Fx)(t) = f(t,x(t)), then (1) is a differential equation. For (Fx)(t) = J! K(t, s, x(s)) ds this is an integral-differential equation. Equations with deviating argument

x(t) = f(t, x[h(t)], x[g(t)]) , t E [0, T], (2)

if ~ ¢ [O,T]

(here '1',"": Rl\[O,T]--+ Rn and h,g: [O,T]--+ Rl are given functions), can be rewritten in form (1) by putting

(3)

228

BVPs FOR FDEs

Here for y: [O,T]-+ RR, r: [O,T]-+ Rl and z: Rl\[O,T]-+ RR :

(s )(t) _ {y[r(t)] if r(t) E [0, T]; rY - 0 if r(t) f/. [0, T];

r {O if r(t) E [0, T]; z (t) = z[r(t)] if r(t) f/. [0, T].

If h(t) = t - TI, g(t) = t - T2, TI,T2 = const, then (2) is a differential-difference equation.

Equation (1) with aftereffect occupies a special place. It is an equation with the Volterra operator F, Le., an operator such that for any T E [O,T] and any XI,X2 from its domain, which are such that Xl(t) = X2(t) on [0, TJ, the equality (Fxt}(t) = (FX2)(t) is fulfilled on [0, T]. Equation (2) is an equation with aftereffect if h(t) :S t, g(t) :S t. The Volterra property of the operator F permits us to study behavior of the solution x to equation (1) on any segment [0, T] C [0, T] ignoring the values of x(t) and (Fx)(t) when t > T, Le., to study the so-called local-solution X'T : [O,T]-+ Rn

Let the operator F be a mapping from the space DR of absolutely continuous functions x : [0, T] -+ RR into the space LR of Lebesgue summable functions y [0, T] -+ Rn We define the norms of the spaces DR and LR by

T T

Ii Xlivn = Ix(O)1 + J 1:i:(s)1 ds , liyliLn = J ly(s)1 ds, o o

where I . I is a norm of Rn. If a continuous operator F : DR -+ Ln is compact and it has the Volterra property, then there are for equation (1) valid analogs of fundamental results of the theory of differential equations: on local solvability of the Cauchy problem :i: = Fx, x(O) = 0; on extendibility of solutions; on a solution's continuous dependence on the initial value 0.

Let TJi : DR -+ RI, i = 1, ... , N, be a given system of functionals. An operator

is called a vector-functional. A BVP for equation (1) with an operator F : Dn -+ LR is the system

:i: = Fx, TJX = O.

BVP's are a basis for the modern theory of FDE [1] . Such theory is constructed for the class of "reducible" equations (1) which contains, in particular, all the equations with a correctly solvable Cauchy problem or a correctly solvable BVP. The linear case of equation (1) we shall write in the form

I:-x =J, (4)

where I:- : DR -+ LR is a linear bounded operator. Characteristic properties of I:become clear if we rewrite this operator in a form based on an isomorphism between

229

A CONSTRUCTIVE ApPROACH

Dn and Ln x Rn, which is defined by

t

x(t) = J x(s) ds + x(O). o

By applying C to the both sides of the latter, we then obtain the representation

(Cx)(t) = (Qx)(t) + A(t)x(O). (5)

The general theory of equation (4) has been constructed under the assumption that the operator Q ("principal part" of C) is Fredholm one, i.e. it is representable in the form of the sum of a completely continuous operator and an invertible one.

The equation

x(t) - P(t)x[g(t)]- R(t)x[h(t)] = r(t), t E [0, T],

x{e} = 'P(~), x(o = 1fJ(O, for ~ ¢ [0, TJ,

takes form (4) if we put

(Cx)(t) = x(t) - P(t)(Sgx)(t) - R(t)(ShX)(t),

j(t) = r(t) + P(t)1fJg(t) + R(t)'Ph(t).

(6)

Natural assumptions concerning P, R, g, h, r, 'P, 1fJ such that this C is a bounded operator acting from Dn in Ln and j E Ln are given in [1] . In this case the principal part Q of C has the form Q = 1- S - K; where I is the identity operator, K is a completely continuous integral operator, (Sz)(t) = P(t)(Sgz)(t).

In what follows the invertibility of Q is assumed. The principal part of C defined by (5) is invertible when h(t) ~ t and there also exists T > 0 so that g(t) ~ t - T,

T E [O,T]. Under such an assumption the Cauchy problem

Cx = j, x(O) = a (7)

is uniquely solvable for any j E Ln, a E Rn, and its solution x is representable in the form

x(t) = (E- j(Q-1A)(S)ds)a+ j(Q-lj)(S)dS. (8)

The matrix X ( t) = E - fci (Q-l A)( s) ds is called the fundamental matrix of the homogeneous equation Cx = O. Each column Xi, i = 1, . . . ,n of X is the solution to the Cauchy problem Cx = 0, x(O) = ei, where ei is the i-th column of the identity matrix E.

230

BVPs FOR FDEs

Let Ii : nn -> R 1 \ i = 1, . .. , N, be a linear bounded functional. An operator I: nn ---t R N, Ix =col{/lx, ... , INX}, is called a vector-functional.

The system ex = j, Ix = /1 (9)

is called the linear BVP, and the equation Ix = /1 is its boundary conditions. If we apply the operator I to the both sides of the identity x(t) = J~ x(s) ds + x(O), we then obtain the representation

T

Ix = J ~(s)x(s) ds + I}/x(O), (10) o

where each column of the (N x n)-matrix I}/ is the value of I on the corresponding column of the identity (n x n)-matrix E, that is I}/ = I(E). The (N x n)-matrix ~ has measurable and essentially bounded elements and is determined by

I (j y(s) dS) = 1 ~(s)y(s) ds, y E Ln.

Representation (8) implies that the solvability (unique solvability) of BVP (9) is equivalent to the solvability (unique solvability) of the algebraic system

IX · Q = /1 -Ig, (11)

where each column of the (N x n)-matrix IX is the value of I on the corresponding column of X, and 9 is the solution to the Cauchy problem ex = j, x(O) = O. In the case N = n (that is, the number of boundary conditions equals the dimension of the system of FDE), BVP (9) is uniquiely solvable if and only if IX is invertible:

det IX i- O. (12)

A constructive way to verify effectively this criterion without explicit evaluation of the fundamental matrix X is described below (see the next section) .

3. The Study of the Solvability of Linear BVPs

As we have already noted, the criterion of the uniquie solvability of the linear BVP

ex = j, Ix = /1, /1 E R n (13)

with n boundary value conditions is the invertibility of the matrix IX (12). This condition cannot be verified immediately, because the fundamental matrix X(t) cannot be evaluated explicitly. In addition, even if the matrix X were known, then the elements of IX, generally speaking, could not be evaluated explicitly. We offer a method

231


of the verification of (12), which is based on the theorem about inverse operators (see, e.g. [3]). By virtue of this theorem, the matrix IX is invertible, if one can find an invertible matrix r such that

1 II/X - rll < IIr-IIi. (14)

As it has been shown in [4,5) (see also [1)), such a matrix r for the invertible matrix IX always can be found among the matrices r = T X, where T : Dn --+ Rn is a vector-functional near to I, and X is the fundamental matrix of the homogeneous equation lx = 0 with operator l : Dn --+ Ln near to C. Ways to find such a matrix r are constructive if we are able to use Computer Algebra Systems (in particular, the systems of analitical computations JLSIMP-JLMATH, Formac, Reduce, and others) guaranteeing the reliable verification of inequality (14). Use of these systems places certain demands on the operators l and T. Here we describe these demands by the so-called "property C" (computability) .

Further, we assume that the value T is rational. Let 0 < tl < ... < tm < T, where tI, ... , tm are rational numbers. Denote £1 = [0, t1 ); £i = [ti-I, t;), i = 2, ... , m; £m+I = [tm' T), £0 = (-00,0). Let Xi be the characteristic function of the set £i. We denote by 'P;:' the set of all functions y of the form

m+l

y(t) = L Xi(t)Pi(t), t E [0, T), i=1

where the components of vector-functions Pi : [0, T) --+ Rn, i = 1, ... , m + 1, are polynomials with rational coefficients.

We say that a linear bounded operator l : Dn --+ Ln satisfies the property C if it maps the set Dn n 'P;:' into 'P;:' .

If lx = x + P(· )x, then the operator l is computable, for example, under the condition that the columns of the matrix P are functions of 'P;:'.

The operator l: (lx )(t) = x(t) + P(t)(ShX)(t), t E [0, T), satisfies the property C if the columns of the ma.trix P belong to the set 'P;:' and the function h E 'P!. satisfies the property ~q: for any i = I, .. . , m + 1 there exists the unique q, 0 S; q S; j such that h(t) E £q for t E £j. _

We say that a line"r bounded functional I : Dn --+ Rn satisfies the property C if it maps the set Dn n 'P;:' into the set of rational numbers.

Our way to prove the solvability of BVPs essentially uses an approximation of operators determined by the original BVP in the class of computable operators. Constructive theorems on the solvability of BVPs are formulated in terms of parameters characterizing a precision of such an approximation. The precision of an approximation l for C is defined by the inequality

l(Cx)(t) - (lx)(t)J S; >'v(t)lIxIiDD Vx E Dn, (15)

232

BVPs FOR FDEs

where,xv E P;:. . Here for (n x m)-matrix A = {ajj}, lAJ means the (n x m)-matrix T

{Iajjl} . Further, let I: Dn -+ Rn, Ix = J 4la(s )x(s )ds + Wax(O), be a vector-functional o

T approximating the vector-functional I : D n -+ Rn, Ix = J41(s)x(s) ds + Wx(O).

o We suppose that the columns of the (n x n)-matrix 4la belong to the set P;:', and

that the elements of the (n x n)-matrix Wa are rational numbers. The precision of the approximation is defined by inequalities

l41(t) - 4la(t)J :::; 4lv, t E [O,T], (16)

where 4lv and Wv are (n X n)-matrices with rational elements. By means of estimate (15) we can construct (n x n)-matrices Xa(t) and Xv(t) with columns from P;:', and being such that the fundamental matrix X(t) satisfies the inequality

t

!lX(S)-Xa(S)J ds:::;Xv(t), tE[O,T] . (17) o

Further, by means of this inequality and inequalities (16) we can construct an (n x n)matrix Mv with rational elements and being such that

(18)

Therefore, the elements of the matrix Ma = lXa can be explicitly evaluated because of the property C.

This then enables us to verify effectively by means a real computer the conditions of the following theorem.

Theorem 1. Let the matrix Ma be invertible and

1 IIMvll < IIM;ll1·

Then problem (13) is uniquely solvable Jor any J E Ln and (3 E Rn Below we present some examples of the computer-assisted study of BVPs. These

examples show there are many situations, where the known sufficient conditions of the solvability of a BVP (in particular, conditions obtained by the technique described in N. Azbelev's survey, this issue) are inapplicable, and our constructive computerassisted approach gives the only chance to achieve the result .

The unique solvability of the following BVP's has been established by a computer-assisted check of the conditions of Theorem 1.

Example 1.

x(t) - 0 [

-2

texp(~)

2t In(1 + O.lt) ~-2

o o 1 8t "9 x x(t) = J(t), t E [0,1],

-2 -t

233


Example 2.

[XI(t)] [2t - 3 4t - 3 + 2t2] . (t) + 2t (t) 2 sin(!¥) X X2 X2 (/+2)

[

Xl (¥) J - [!I(t)] X2 C'-21/2) - !2(t) , t E [0,1],

Xi (e) =0, e!l'[O,I], i=I,2; 1

J XI(O) (2s - l)x2(s) ds + -5- = (32.

o

Here XI(-) and X2(-) are the characteristic functions on the intervals [0,2-t] and [2-t,l] respectively.

4. Nonlinear BVP's: a priori Inequalities Technique

The main point of any research of nonlinear boundary value problems, say of the form

X= Fx,

'T/X = 0,

(19)

(20)

if! the problem of constructing a priori estimates for solutions. The sentence " ... an a priori estimate IIXilDn ~ d takes place for any solutions to problem (19), (20)" means that there is d E (0,00) such that problem (19), (20) has no solution X such that ilXilDn > d. Thus a statement on a priori estimates of solutions does not include the assumption of the solution existing.

We say that for any solution X to equation (19) the a priori inequality

Ix(t)1 ~ m(t, Ix(O)I), t E [O,T] (21)

takes place if there is a function m : [0, T] x R+ ---+ R+ such that m(., T) E LI 'IT and equation (19) has no solution x such that inequality (21) is violated.

Here a priori inequalities of form (21) play the role of the source of a priori estimates for solutions to BVP's. The way of using a prio,ri inequalities (21) to obtain a priori estimates is very simple in the case that the vector functional 'T/ satisfies the condition

I'T/X - x(O)1 ~ jL(lxl, Ix(0)l) \Ix E nn, (22)

where the functional jL : LI x RI ---+ RI does not decrease with respect to the first component. In this case the existence of an a prio,ri estimate of solutions to problem

234

BVPs FOR FDEs

(19), (20) is assured by the following condition: the set B = {6 > 0 : 6 :::; Jl[m(., 6), 6)} is bounded. Indeed, let 60 be such that 6 :::; 60 V6 E B. Then for any solution x of problem (19) , (20) the estimate

IIXIlDn :::; 60 + sup IIm(·,6)IILI 6E[0.601

takes place. More detailed information of ways for studing nonlinear BVP's by using a priori inequalities can be found in the book [1] and in the paper [6] . This article is primarily concerned with the constructive aspects of our approach. Let us note that effective use of a priori inequalities needs the construction of m in an explicit form. For broad classes of FDE's, the question of the construction of m is reduced in [7] to the question of the construction of the general solution to an ancillary ordinary differential equation if = w(t,y) . This equation is defined by a majorant of F. Thus our opportunity to CO:lstruct the function m is essentially restricted to within the context of E. Kamke's book [8] . Of course, one can find w such that w(t,y) :::; w(t,y) and a new equation if = w(t,y) is integrable. Most often this way brings either a linear equation or a Bernoulli equation. Note that the roughness of the a priori inequality obtained in such way may be unacceptable. In the case of ordinary differential equations, an ingenious technique for constructing a priori inequalities was proposed by E. Zhukovskii [9] .

The difficulties associated with the integration of the majorant equation are essentially growing in the case that we use vector a priori inequalities.

We say that for any solution x to equation (19) a vector a priori inequality

L:i:(s)J :::; met, Lx(O)J) t E [o,T] (23)

takes place if there is a function m : [0, T] X R+ -+ R+ such that m(·, r) E Ln Vr and equation (19) has no solution x such that inequlity (23) is violated. Recall that, for a = col(al , ... , an), LaJ means col(lall, ... , lanl).

In this vector case, the majorant system may in exceptional cases be integrated. The possibilities of both the construction and the use of vector a priori inequalities expand essentially by modern Computer Algebra Systems (Formac, Reduce and so on). To illustrate this we consider the case that the operator F in equation (19) satisfies the condition

L(Fx)(t)J :::; A(t) (LX(O)J + j L:i:(s)J dS) +,(t) , t E [O,T], Vx E nn . (24)

Here elements of the (n x n)-matrix A(·) and vector-function ,(.) are polynomials with rational coefficients (belong to Pl). Denote by yet, r) the solution of the Cauchy problem if = A(t)y+,(t), yeO) = r, r E R+. The question of constructing a function m is assumed by

235


Theorem 2. Suppose that F satisfies condition (24). Then for any solutions to equation (19), vector a priori inequality (23) holds with m defined by

m(t, r) = A(t)y(t, r) + -.,.(t) .

Now we formulate the preposition which brings a way of constructing a function ml(t, r) such that ml(t, r) ~ m(t, r) "It E [0, T), r E R~ and ml is a little different from m.

Theorem 3. For every c > 0, there exist a (n x n) -matrix Yi(t), vector ft(t) with elements belonging to PI, and a rational number ~ such that the vector-function Yl(t,r) == Yi(t)r+~ . r+ fl(t), t E [O,T), r E R~ satisfies the inequalities Ly(t,r)Yl(t,r)J :5c · r+col(c, . .. ,c), y(t,r):5Yl(t,r).

Such a function Yl and hence ml(t,r) = A(t)[Yi(t) + ~)r + A(t)fl(t) + -.,.(t) may be found practically by using the mentioned above Computer Algebra Systems. For details of such construction see [10).

Using a priori inequality L:i:(t)J :5 ml(t, Lx(O)J) established for solutions to (19) we shall study the solvability of BVP (19), (20) . We suppose that

T

L1/X - x(O)J :5 \If Lx(O)J + J cfI(s) L:i:(s)J ds + a, "Ix E nn, o

where a E R~, \If and cfI( ·) are (n x n) -matrices with elements belonging to the set of rational numbers and to PI respectively. Denote

T

M = E - \If - J cfI(s)A(s)[Yi(s) + ~. E) ds. o

Theorem 4. Let F : nn -+ Ln be a completely continuous operator, and suppose the matrix M is positively invertible. Then BVP (19), (20) has at least one solution.

An algorithm for the study of the solvability of BVP's based on Theorems 2-4 is realized as a system of programs. In the case of solvability, the computer outputs the information of component-wise estimates of a solution.

Notice that BVP's (19), (20) with an operator satisfying condition (24) arise, in particular, in mathematical economics [11).

5. Supplementary

Techniques of the constructive study of linear BVP is developed and extended by authors to impulsive BVP's with boundary inequalities, i.e. problems of the form

.cy = f, iY :5-.,. (25)

236

BVPs FOR FDEs

with linear bounded operators.c : Dsn(m) -t Ln, I : Dsn(m) -t Rk, and an arbitrary number k of boundary conditions. Here Dsn(m) = DS(m) x . .. x DS(m), DS(m) is the space of piecewise absolutely continuous functions y : [0, T]-t Rl described in N. Azbelev's survey, this issue. One can find details concerning problems (25) and its application in mathematical economics in [12] .

For a study of certain classes of FDE, (2) in part, at times the frame work of the space Dn ~ Ln x Rn is too restricted. Thus the question arises for more suitable spaces. One of them may be a space of absolutely continuous functions with derivative belonging to a Orlicz space LM. In this case, the role of the main space of solutions is played by the space DM = LM x Rn. Certain classes of BVP's with solutions belonging to DM have been studied by L. Kultysheva [13] . For certain BVP's for FDE's with singularity, special spaces of solutions which agreed with the kind of singularity were constructed by A.Shyndyapin [14] .

A priori inequalities find applications in the study of BVP 's for ordinary differential equations :i; = f(t, x(t)) in the case that f does not satisfy the Caratheodory conditions. For this case an investigation on techniques based on a priori inequalities has been developed by E. Zhukovskii [9].

A priori inequalities have been used also by A. Bulgakov [15] and M. Karibov [16] to obtain tests of the solvability of BVP's for Functional Differential Inclusions.

6. References

1. N. V. Azbelev, V. P. Maksimov, and L. F . Rakhmatullina, Introduction to the Theory of Functional Differential Equations, Nauka, Moscow, 1991, (in Russian) .

2. N. V. Azbelev, V . P. Maksimov, and L. F. Rakhmatullina, Functional Differential Equations, World Federation Publishers Company, INC, Tampa, to appear.

3. V. Hutson and J. Pym, Applications of Functional Analysis and Operator Theory, Academic Press, New York, 1980.

4. A. N. Rumyantsev, To the study of the solvability of boundary value problems by using Computer Algebra Systems, Archives of VINITI AN SSSR, N 3373, 1987, (in Russian) .

5. A. N. Rumyantsev, A constructive study of the solvability of boundary value problems for funct ional differential equations, Doctoral thesis, Urals State University, Sverdlovsk, 1988, (in Russian) .

6. V. P. Maksimov, On certain nonlinear boundary value problems, Differential Equations 19, No. 3 (1983), 396-414, (in Russian) .

237


7. N. V. Azbelev and V. P. Maksimov, A priori estimates of solutions to the Cauchy problem and the solvability of boundary value problems for equations with delayed argument, Differential Equations 15, No. 10 (1979) , 1731-1747, (in Russian).

8. E. Kamke, Differentialgleichungen, Losungsmethoden und Losungen. v. 1, Leipzig, 1959.

9. E. Zhukovskii, Operator inequalities and functional differential equations, Doctoral thesis, Urals State University, Sverdlovsk, 1984, (in Russian).

10. A. N. Rumyantsev, The study of the solvability of boundary value problems by using vectoral a priori inequalities, Archives of VINITI AN SSSR, N 5028, 1987, (in Russian).

11. D. L. Andrianov, A priori estimates in stabilization problems for trajectories of nonlinear economic systems, Doctoral thesis, Leningrad State University, Leningrad, 1985, (in Russian) .

12. V. P. Maksimov and A. N. Rumyantsev, Boundary value problems and problems of pulse control in economic dynamics. Constructive study, Russian Mathematics (Iz. VUZ) 37, No. 5 (1993), 48-62.

13. L. M. Kultysheva, On equations with the internal superposition operator, Doctoral thesis, Alma-Ata, 1981, (in Russian) .

14. A.l Shyndyapin, To the questing on singular functional differential equations, Doctoral thesis, Tbilisi State University, Tbilisi, 1984, (in Russian) .

15. A.1. Bulgakov, V. P. Maksimov, Functional and functional differential inclusions with Volterra operators, Differential Equations 17, No.8 (1981), 1362-1374, (in Russian).

16. M. R. Karibov, Functional differential inclusions, Doctoral thesis, Gorki State University, Gorki, 1981, (in Russian) .

BOUNDARY VALUE PROBLEMS FOR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

S. K. NTOUYAS Department of Mathematics, University of Ioannina

451 10 Ioannina, Greece

1. Introduction

Let R" be the n-dimensional euclidean space and let 1.1 be any convenient norm of R". For a fixed r > 0, we define

to be the Banach space of all continuous functions 4 : [-r,O] -t R" endowed with the sup-norm

I I ~ I I = ~ U P { I ~ ( ~ ) I : -r r 0 5 01.

For any continuous function x defined on the interval [-r, 11, and any t E [O, 11 we denote by xt the element of C defined by

In the present paper, we consider neutral functional differential equations of the form

d -[zf(t) - g(t, zt)] = f (tl Xt, xf(t)), t E [O, 11 dt (El

where f : [O,1] x C x R" + R" and g : [O, 11 x C + R" are continuous functions.

This is a particular class of neutral functional differential equations which includes the usual second order functional differential equations of retarded type (g = 0). A good guide to the literature for neutral functional differential equations, is the related chapter in the book of Hale [5] and the references therein. Equations of the above type appear in hereditary control systems.

A function x : [-r, l] R" is said to be a solution of (E) with initial function d E C a t t = O , i f

240

BVPs FOR FDEs

(ii) XI E C for t E [0,1]' and

(iii) the difference x'(t) - g(t, XI) is differentiable and satisfies (~) for t E [0,1] .

Boundary value problems for neutral functional differential equations is a very interesting project and its development is much slower as compared with the other topics on neutral functional differential equations. The main difficulty is localized in approaching the relations between equation with boundary value. The main purpose of this paper is to discuss the existence of solutions for the boundary value problem (BVP for short) for neutral functional differential equations

![X'(t) - g(t,Xt)] = f(t,XI,X'(t)),

Xo = </>,x(1) = TJ

t E [0,1] (E)

(BC)

where f : [0,1] x C x R:' --+ Rn and 9 : [0,1] x C --+ Rn are continuous functions, </> E C and TJ E R:' .

The key tool in our approach is the Leray Schauder alternative [1]. This method reduces the problem of existence of solutions of a BVP to the establishment of suitable a priori bounds for solutions of these problems. We notice that this method is a modification of the well known Topological transversality method of Granas [1] . For applications of this method in boundary value problems for functional differential equations we refer to [6], [9], [12]. For some recent results on initial and boundary value problems for neutral functional differential equations see [7-9]. Also we remark that in [10-11] boundary value problems for neutral functional differential equations have been studied with the aid of the shooting method.

We notice that recently in [8], the global existence of solutions for initial value problems for neutral functional integro-diffrential equations of the form

is studied.

! [x'(t) - g(t, XI)] = L(t, Xt) + f(t, Xt),

Xo = </>, x'(O) = TJ

t E [O,T]

The paper is organized as follows. In section 2 we present some notations and preliminaries. The basic existence theorems are given in section 3. Finally in section 4 we give two applications of our results.

2. Preliminaries

If m a positive integer we denote by C([O, 1], Rm), Ck([O, 1], Rm) and Lk([O, 1], Rm) the classical spaces of continuous Rm-valued functions, k-times continuously differentiable ~-valued functions and measurable Rm-valued functions whose k-th power of the

241

BOUNDARY VALUE PROBLEMS FOR NEUTRAL EQUATIONS

euclidean norm 1.1 is Lebesque integrable on A. We introduce the following notations:

IIxlio = sup{lx(t)1 : ° :::; t ::; I}

IIx'llo = sup{ lx'(t)1 : 0:::; t :::; I}

IIxll* = max{lIxll, Ilx'lI}

IIxlh = t Ix(s)lds.

The proofs of our existence theorems in the next section are based on the theory of completely continuous mappings. We recall here that, if X, Yare normed spaces, then an operator A : X --+ Y is completely continuous if it is continuous and the image of any bounded set is included in a compact set . See [1] .

Also a function 1 : [0,1] x C --+ Rn is called completely continuous ([5]), if it is continuous and takes dosed bounded sets of [0,1] x C into bounded sets of ~.

Before stating our basic existence theorems, we need the following lemma which is known as "Leray-Schauder alternative".

Lemma 2.1 Let S be a convex subset of a normed linear space E and assume ° E S. Let F: S --+ S be a completely continuous operator, and let

£(F) = {x E S : x = )"Fx for some 0<)..<1}.

Then either £(F) is unbounded or F has a fixed point.

3. Existence Results

We present our main existence result for the solutions of the BVP (E)-(BC).

Theorem 3.1 Let 1 : [0,1] x C x ~ --+ Rn and 9 : [0,1] x C --+ Rn be continuous functions. Assume that:

(HI) There exist bounded real valued functions a, b defined on [0,1] such that

Ig(t, u)1 :::; a(t)lIull + b(t), t E [0,1], u E C. (3.1)

(H2 ) The function 9 is completely continuous and such that the operator G : C([-r,O],~) --+ C([O, 1],~) defined by (G4»(t) = g(t, 4» is compact, i.e. GB is relatively compact whenever B is bounded.

(H3) There exist p,q,r in LI([O, l],R) such that

I/(t , u, v)1 :::; p(t)lIuli + q(t)lvl + r(t), t E [0,1], u E C, v E W. (3 .2)

242

BVPs FOR FDEs

Then the BVP (E)-(BC) has at least one solution provided

Proof. Consider first the case 4>(0) = O. Then the set

Co = {x E C1 ([0, IJ,Rn) : x(O) = O}

is a subspace of C1([0, 1), ~).

For a function x E Co we define the function i : [-r, 1)--+ ~ by

i(t) = {4>(t) , t E [-r, 0) x(t), t E [0,1) .

Now, define the operator F : Co --+ C 1 ([O, 1),~) by

Fx(t) = ['7 -l g(t , xt)dt -ll f( s,X.,x'(s))dsdt)t

+ l g(s,x.)ds + l f f(r , XT) x'(r))drds,t E [0, 1) .

It is obvious that F(Co) c;:::; Co, and x is a solution of the BVP (E)-(BC) if, and only if, Fxl[O, 1) = xl[O, 1) and Xo = 4>, x(l) = '7 .

In order to prove the existence of a solution of the BVP (E) -(BC) we apply Lemma 2.1. First we obtain the a priori bounds for the functions x E Co such that x = )"Fx,).. E (0,1). In other words we shall prove that the set £(F) = {x E Co : x = )"Fx,O < ).. < I} is bounded.

Let x E £(F) . Since x(O) = 0 by the relation x(t) = x(O) + J~ x'(s)ds we have

IIxllo:::; IIx'llo.

Also using (HI) and (H3) we get

Ix(t)l:::; 1'71 + 2l[a(t)max{lIxllo, 114>1I} + b(t))dt

+ 2ll[P(s)max{lIxllo, 114>1I} + q(s)lIx'llo + r(s))dsdt

:::; 1'71 + 211alll{11xllo + 114>11) + 211 bll 1

+ 2l1plh(lIxlio + 114>11) + 2l1qllll1x'llo + 211rlh :::; 1'71 + 2(lIalh + Ilplh)lIxlio + 2l1qlhllx'llo

+ 2(lIalh + IIplh)II4>1I + 211 bll 1 + 211rlh,

(3.3)

and


Ix'(t)1 ~ 1'71 + lIalh(lIxlio + 114>11) + IIblll + 2l1pll1(lIxllo + 114>11) + 211qlllllx'ilo + 211rlh + lIallo(lIxlio + 114>11) + IIblio.

Consequently

IIx'llo ~ (lIalh + 211plh + lIalio + 21lqlh)llx'llo + 1'71 + (lIalh + 211plh + lIallo)II4>1I + IIblll + IIbllo + 211r lll

or

and therefore, by (3.3), IIxlio ~ IIx'llo ~ c.

From (3.4) and (3.5) we have that

IIxll* ~ c,

which implies that the set £(F) is bounded. This proves our first claim.

243

(3.5)

In the second step, we will prove that the operator Fx is completely continuous.

Let {hll} be a bounded sequence in Co, i.e

for all v,

where K is a positive constant. We obviously have IIhlltll ~ K, t E [O,lJ for all v. Hence we obtain easily that

IFhll(t)1 ~ 1'71 + 2Ulalll + IIplll + IIqlh]K + 2(lIalh + Ilplll)II4>1I + 2(lIb111 + IIrlll), I(Fhll),(t) I ~ 1'71 + Ulalll + lIalio + 2(lIplh + Ilqlh)]K

+ (lIalh + lIalio + 2I1plh)II4>1I + IIblll + IIbllo + IIrll!' which means that {Fhll } and {(Fhll)'} are uniformly bounded.

We rewrite the operator Fx as

Fx(t) = Hx(t) + Tx(t), t E [0, IJ

where Hx(t) = ['7 -l g(t,xt)dtJt + l g(s,x.)ds, t E [0,1]

and

Tx(t) = -t II f(s, X., x'(s))dsdt + ll' f(T, X.,., x'(T))dTds, t E [0,1] .

244

BVPs FOR FDEs

Then we have the estimates:

IThv(t1 ) - Thv(t2)1 ~ [2(lIplh + IIqlh)I< + 211plhllt/JID + 211 r lhlltl - t21

I(Thv)'(tl) - (Thv)'(t2)1 ~ [(lIplh + IIqlldI< + IIplIlllit/J11) + Ilrlll]lt1 - t21 ·

From these estimates we deduce that the sequences {Thv} and {(Thv)'} are equicontinuous. Thus, by the Arzela-Ascoli theorem th~ operator Tx is completely continuous. Furthermore, by assumption (H2) the operator Hx is completely continuous and therefore the operator Fx is completely con~nuous . Thus our second claim is proved.

Consequently, by lemma 2.1 the operator Fx has a fixed point in Co. Then it is clear that the function

z(t) = {t/J(t), t E [-r,O] x(t), t E [0,1]

is a solution of the BVP (E)-(BC).

If t/J(O) f 0, by the transformation

y = x - t/J(O)

the BVP (E)-(BC) reduces to the following BVP

! [y'(t) - g(t, Yt + t/J(O))] = f(t , Yt + t/J(O) , y'(t)) ,

Yo = .p - t/J(O) == ~,y(l) = 7J - t/J(O)

for which ~(O) = o.

t E [0,1]

Theorem 3.2 Let f : [0,1] x C x Rn -> Rn and 9 : [0,1] x C -> Rn be continuous functions. Assume that (H2 ) holds and moreover

(Hd There exist bounded real valued functions u, b defined on [0,1] and real constant k, 0 ~ k ~ 1 such that

Ig(t, u)1 ~ a(t)lIuli k + b(t), t E [0,1]' u E C.

(H3) There exist p, q, r in £1([0,1], R) and real constants f. m 0 < f. < 1 0 < m ~ 1 such that ' , - - , -

If(t, u, v)1 ~ p(t)lIuli l + q(t)lvlm + r(t), t E [0,1]' u E C, v E W .

Then the BVP (E)-(BC) has at least one solution provided

Q(kHllalh + lIallo) + 2Q(f.)lIplh + 2Q(m)lIqlh < 1

245


where

Q( ) = {O, o:s: j.l < 1 j.l 1, j.l = 1.

Proof. As in the proof of the Theorem 3.1 and using (HI), (H3)' instead of (HI)' (H3)' we obtain

IIxllo:S: 1171 + 2!1lalhmax{lIxllo, 11¢>II}k + Ilblh] + 2[lIplllmax{lIxllo, II¢>II V + Ilqlhllx'lI~ + IIrlll]

:s: 1171 + 2l1alh(llxll~ + 1I¢>lIk) + 211bll l + 2l1plh(llxll~ + 1I¢>lI i ) + 2I1qlllllx'lI~ + 211rlh

:s: 2l1alhllxll~ + 2l1plhllxll~ + IIqlhllx'lI~ + 1171 + 2l1alhll¢>lIk + 211blh + 2l1plldl¢>lI i + 211rlll

:s: 21lalhllxll~ + 2l1plIlllxll~ + Ilqlhllx'lI~ + d, (3.6)

where d = 1171 + 2l1alllll¢>lIk + 211blh + 2l1plhll¢>lI i + 2l1rlh · Also, we have

IIx'lIo:S: 1171 + lIalll(lIxll~ + 1I¢>lIk) + IIbll l + 2l1plII(llxll~ + 1I¢>lI i ) + 2I1qlhllx'll~ + 211rlh + lIallo(lIxll~ + 1I¢>ln + IIbllo

:s: (lIalh + lIallo)lIx'lI~ + 2I1plIlllx'lI~ + 211qlh Ilx'lI~ + 1171 + lIalhll¢>lIk + IIbll l + 2l1plhll¢>lI i + 211rlll + Ilalloll¢>lIk + IIbllo

:s: (lIalh + lIallo)lIx'lI~ + 2I1plIlllx'lI~ + 2l1qlllllx'llm + e, (3.7)

where e = 1171 + lIalhll¢>lIk + IIblh + 2l1plIIIl¢>lI i + 211rlh + Ilallll¢>lIk + Ilbllo. Now for the sake of simplicity we consider the case where 0 < k < 1 and f = m = 1.

Then, by (3.7) we have

or (1 - 211plh - 2I1qlh)lIx'llo :s: (lIalh+ lIallo)llx'll~ + e. (3.8)

At this point, we note that if L ~ 0, 0 :s: j.l :s: 2, h > 0 are given constants, then there exists a constant N > 0 such that

hZ2 Lzi' < - + N, x >_ O. - 2

Hence, if we put z = Vllx'lIo,L = lIalll+llallo,j.l = 2k,h = 1-21IplIl-2I1qlll inequality (3.8) imlpies

(1 - 211plh - 2I1qlh)lIx'llo :s: ~(1 - 211plh - 2l1qlldllx'llo + e + N

246

BVPs FOR FDEs

or , 2(e+N)

IIx 110 :::; 1 _ 211plh _ 211qlh = Cl·

Finally, this inequality, (3.6) and the obvious relation IIxlio :::; IIx'llo imply that

which completes the proof. The proof for the other cases is similar and is omitted.

Corollary 3.3 Let f, 9 are as in Theorem 3.2. Let also (H2 ) holds, and (Jid, (Ji3 )

with 0 :::; k < 1,0 :::; f. < 1,0 :::; m < 1.

Then the BVP (E)-(BG) has at least one solution.

4. Applications

We close this paper by giving two applications of our results. One for a BVP for neutral functional differential equations with a "mixed "boundary condition and another for a "three-point"BVP for functional differential equations.

Consider first the following BVP

~[x'(t) - g(t,xd] = f(t,xt,x'(t»,

Xo = 4>,ax'(O) + ,8x(l) =."

t E [0,1]

where f : [0,1] X G([-r,O],R) x R -+ Rand 9 : [0,1] x G([-r,O],R) -+ Rare continuous functions, 4> E G and a,,8,.,, real constants, with a + ,8 # O. As far as we know this type of BVP is new, even for ordinary differential equations, i.e when 9 = 0 and r = o. Theorem 4.1 Let f : [0,1] x C([-r,O],R) x R -+ Rand g : [0,1] x C([-r,O],R) -+ R be continuous {unctions. Assume that (Ht) , (H2 ) and (H3) hold.

Then the BVP (E) - (BCd has at least one solution provided

Proof. As in the proofs of Theorems 3.1 and 3.2 we consider only the case 4>(0) = 0, because by a transformation the BVP with 4>(0) # 0 is reduced to this case. Consider also the subspace Co and the function x as defined in Theorem 3.1. Since x(O) = 0, we obviously have

IIxllo:::; IIx'llo.

247


Therefore it is enough to prove that the IIx'llo is bounded. The BVP (E) - (BCd is equivalent to the following integral equation

1 11 lIlt x(t) = --13[77 - ag(O,<fo) - 13 g(t,Xt)dt - 13 f(s,X.,x'(s))dsdt]t a + 0 0 0

+ 19(s,x.)ds+ If f(r,x.,.,x'(r))drds, tE [0,1] .

Then we easily obtain

1 IIx'llo ~ la + 131 [1771 + lal{lIallo(lIxlio + II <folD + IIbllo} + IJ3I{lIalh(llxllo + 1I<fo1l)

+ IIblid + IJ3I{lIplll(lIxlio + 1I<fo1l) + IIqlhllx'll + Ilrlld] + lIallo(lIxlio + 1I<fo1l) + IIbllo + IIplll(lIxlio + 1I<fo1l) + Ilqlhllx'll + IIrlll

lal 1131 1131 , ~ {(I + la + .81 )llalio + la + J3l l1alh + (1 + la + 131 )(llplll + Ilqlll)} IIx 110

1771 lal 1131 + la + 131 + la + 131 (lialloll<foll + IIbllo) + la + 131 (1lalllll<fo11 + IIbll l) + lIalloll<foll + Ilbllo + IIplh II <foIl + Ilrlh

from which, by (4.1) the result follows.

For a = 1 and 13 = ° in (BCl ) we have an initial value problem, for which we have the following Corollary.

Corollary 4.2 Assume that (HI)' (H2) and (H3) hold. Then the initial value problem

![x'(t) - g(t,xd] = f(t,xt,x'(t)),

Xo = <fo, x'(O) = 77

has at least one solution provided

t E [0,1]

As a second application we consider the folowing three-point boundary value problem for second order functional differential equations (for simplicity we set 9 = 0)

x"(t) = f(t, XI, x'(t)), t E [0,1] Xo = <fo,x(l) = ax'(()

(e) (bc)

where f : [0,1] X C([-r, 0], R) x R -+ R is a continuous function, a E R, a#- 1 and (E(O,l).

248

BVPs FOR FDEs

For recent work for three or m-point BVP for ordinary differential equations we

refer to [2-4].

Theorem 4.3 Let j . [0,1] x C([-r, 0], R) x R --+ R be a continuous function and a E R, ai-I and ( E (0,1) be given. Assume that f satisfies (H3).

Then the BVP (e)-(bc) has at least one solution provided

(~1 ~I:: + 1)(lIplh + Ilqlld < 1. (4.2)

Proof. Again consider the case 1,6(0) = 0. Then, since Ilxlio :s IIx'llo, it suffices to prove that IIx'llo is bounded. From

x(t) = _1_[a [' j(s,x. ,x'(s))ds - e r j(s ,X., x'(s))dsdt]t 1 - a 10 10 10 + l' f f(T, XT, X'(T))dTds , t E [0,1]

we have

IIx'llo :s ~1 ~I:: {lIplh(llxllo + 111,611) + IIqlllllx'llo + IIrlid

+ Ilplh(lIxlio + 111,611) + IIqlldlx'llo + IIrlll 1 + lal , 1 + lal :s (11 _ al + l)(11plh + IIqlll)lIx 110 + (11 _ al + l)(lIplllll¢1I + Ilrlll).

This completes the proof.

Remark 4.4 It is obvious that the above method can also be applied to other types of BVP. For example for the BVP (E) - (BC2 ) where (BC2 ) are the usual separate boundary conditions Xo = ¢,ax(l) + .8x'(I) = TJ ·

Moreover, instead of the BVP (e)-(bc) we can study the following m-point BVP

x"(t) = f(t, Xt , x'(t)), t E [0,1] m-2

Xo = ¢,x(l) = L aix'(~i) i=l

(e)

(bc)m

where f : [0,1] x C([-r,O],R) x R --+ R is a continuous function, ~i , ai E R, i =

1, .. . , m - 2 with ° < 6 < ~2 < . .. < ~m-2 < 1. This can be done by using the a priori bounds obtained for the three point BYP, since it is well known that there exists ( E [6, ~m-2] such that x(l) = ax'(() with a = L:~12 ai . More general results of this type will be appear elsewhere.

249


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9. S. Ntouyas and P. Tsamatos, Initial and boundary value problems for functional differential equations of delay and neutral type ,Bolletino U.M.I. 8-B (1995) .

10. Y. Sficas and S. Ntouyas, A boundary value problem for neutral functional differential equations, Math . Nach . 110 (1983), 143-158.

11. Y. Sficas and S. Ntouyas,A two point boundary value problem for neutral functional differential equations, Proc. Royal Soc. Edinburgh 94A (1983), 331-338.

12. P. Tsamatos and S. Ntouyas, Some results on boundary value problems for functional differential equations, Inter. J. Math. and Math . Sci . (to appear) .

EXISTENCE PRINCIPLES FOR NONLINEAR OPERATOR EQUATIONS

DONAL O'REGAN Department Of Mathematics, University College Galway

Galway, Ireland

Abstract

Existence principles are presented for operator equations. These results are then used to establish existence theory for nonlinear integral equations. Our arguments rely on fixed point methods. In particular we use the SchauderTychonoff theorem and also a nonlinear alternative of Leray- Schauder type.

1. Introduction

This paper presents existence principles for the nonlinear operator equation

y(t) = Ly(t) + Ny(t)

251

(1.1)

where t E [0, T] or t E [0, T). Throughout L will be a linear Volterra operator [4,9] acting on some function space and N will be a nonlinear operator acting on the same function space.

Typical examples of (1.1) include the Volterra integral equation

y(t) = Ly(t) + l I«t, s, y(s)) ds (1.2)

and the Urysohn integral equation

y(t) = Ly(t) + loT I«t,s,y(s))ds. (1.3)

In [3] Corduneanu has obtained many interesting existence results for (1.1), with t E [0, T], using either the Banach contraction principle or Schauder's fixed point theorem. However in this paper we use a nonlinear alternative theorem of LeraySchauder type and as a result we are able to obtain a general existence principle for (1.1) . From this existence principle we are able to deduce easily existence results

252

BVPs FOR FDEs

for the integral equations (1.2) and (1.3) . For example our technique leads naturally to the study of dependence of the interval of existence of a solution to (1.2) upon Land Kj this technique is in the spirit of results obtained for ordinary differential equations, see [5,8] . The case when t E [0, T) , 0 < T:::; 00 has also been examined by Corduneanu [4] . In this case also we obtain some new results. In addition our theory complements and extends that in [4] .

To conclude the introduction we gather together the two fixed point theorems which will be used in this paper.

Theorem 1.1. (Schauder- Tychonoff) (2j. Let Q be a closed convex subset of a locally convex linear topological space E . Assume that f : Q -+ Q is continuous and that f( Q) is relatively compact in E. Then f has at least one fixed point in Q.

Theorem 1.2. (Nonlinear Alternative) (5,10j. A ssume U is a relatively open subset of a convex set Q in a Banach space E. Let F : U -+ Q be a compact map with 0 E U Then either (i) . F has a fixed point in Uj or (ii). there is a u E aU and a A E (0, 1) such that u = AFu .

Remark. By a map being compact we mean it is continuous with relatively compact range. For later purposes, a map is completely continuous if it is continuous and the image of every bounded set in the domain is contained in a compact set in the range.

2. Existence Principles for the Compact Interval

This section examines the operator equation

y(t) = Ly(t) + Ny(t) for t E [0, T] j here 0 < T < 00. (2.1)

We present an existence principle for (2.1) . However before we discuss the nonlinear problem we first consider the linear equation [3,4,9]'

y(t) = Ly(t) + g(t) for t E [0 , T] . (2.2)

Assume the following conditions are satisfied:

(2.3)

and

{ L : C([O, T] , Rn) -+ C([O, T], Rn) is a Volterra linear operator and L is continuous and completely continuous with (LO)(t) == O. (2.4)

Now for any 9 E C([O, T], Rn) we have from [3,4,9] that (2.2) has a unique solution x E C([O, TJ, Rn). Thus the mapping

V : g-+ x

253


is a bijection of C([O, T], Rn) onto C([O, T], Rn). Also since the inverse map

f!:x->g

is continuous then V is continuous (bounded inverse theorem) from C([O, T], Rn) onto C([O, T], Rn). Let R = V - I where I is the identity mapping of C([O, T], Rn) . The operator R is called the resolvent operator corresponding to L. Notice R is a linear continuous operator on C([O, TJ, Rn). Now

x = Vg can be written as x = 9 + Rg.

Also since the equation (2.2) can be written as (I - L)x = 9 we have that

V=(I-Lt1 and R=(I-Lt1-I=L(L-I)-1=(I-L)-lL.

Notice R : C([O, T], Rn) --> C([O, T], Rn) is completely continuous. To show (2.1) has a solution, the idea is to look at the family of problems

y(t) = Ly(t) + >"Ny(t) for t E [0 , T] (2.5}A

for ° < >.. < 1. Now by a solution to (2.5)>. we mean a function y E C([O, T], Rn) which satisfies the equation in (2 .5}A for t E [0 , T] .

Theorem 2.1. Let U be a relatively open subset of C([O,T],Rn) with ° E U. Suppose (2.4) holds and in addition assume the following condition is satisfied:

N : U -> C([O, T], Rn) is a continuous and compact operator. (2.6)

Then either (iJ. (2.1) has a fixed point in U (i .e. N + RN has a fixed point in UJ; or (iiJ . there exists a >.. E (0,1) and ayE au with y satisfying (2.5)>. (i.e. there exists a >.. E (0,1) and ayE au with y = >"(Ny + RNy)).

Proof: As above it is easy to see that solving (2.5)>. is equivalent to the fixed point problem

y = >..(Ny + RNy) == >..Fy . (2.7)>.

Notice F : U -> C([O, T], Rn) is continuous and compact since R is continuous and completely continuous and N satisfies (2.6). Thus theorem 1.2 with Q = E = C([O, T], Rn) guarantees the result.

We now use theorem 2.1 to discuss nonlinear integral equations. Consider first the Volterra integral equation

y(t) = Ly(t) + IT<(t,S,y(s))ds for tE[O,T] (2.8)

254

BVPs FOR FDEs

where L satisfies (2.4). Here 1<: [0 ,T] 2x Rn ..... R n. We assume also that 1«x,s,u) is L1-Caratheodory uniformly in x.

Definition 2.1. We say 1«x, s, u) is V-Caratheodory uniformly in x if for each x E [O,T] the function 1<x : [O ,T] x Rn ..... Rn given by 1<x(s,u) = 1«x,s,u) satisfies (i). the map u ..... 1<x(s , u) is continuous for a .e. s E [0, T] (ii) . the map s ..... 1<x(s, u) is measurable for all u E Rn j in addition we need (iii). for each r > ° there exists hr E L1[0,T] sach that lui S; r implies 11«x,s ,u)1 S; hr(s) for almost all s E [O , T] and all x E [O , T].

Theorem 2.2 . Assume (2.4) holds, 1< [0 , T]2 X Rn ..... Rn and that J«x, s, u) is L1-Caratheodory uniformly in x. Suppose for each r > ° and x E [0, T] we have

lim rz sup 11« z, s, u) - J«x , s, u)1 ds = 0, z E [0, Tj. (2.9) z-x Jo lul~r

In addition assume there is a constant Mo, independent of >., with

for any solution y to

Iylo = sup ly(t)1 =1= Mo [O,T)

y(t) = Ly(t) + >.l J«t ,s,y(s))ds for t E [O ,Tj

for each>' E (0, 1) . Then (2.8) has a solution in C([o,TJ,Rn).

Proof: Let N : C([O, T], Rn) ..... C([O, Tj, Rn) be given by

Ny(t) = l J«t,s,y(s))ds .

Also let

u = {u E C([O,Tj,Rn): lulo < Mo}.

(2.10h

The result follows im~ediately from theorem 2.1 (notice alternative (ii) cannot occur) once we show N : U ..... C([O, T], Rn) is continuous and compact . We first show that N is continuous. To see this let Yn ..... Y in C([O,TJ,Rn) . Then there exists r > ° such that IYnio S; rand IYlo S; r. There exists hr E V [0, Tj such that 1J«x,s,u)1 S; hr(s) for almost all s E [O,Tj and all x E [O,Tj and lui S; r. Also for each x E [0, Tj we have

1«x's'Yn(s)) ..... 1«x,s,y(s)) for a.e. s E [O , Tj

255


and this together with the Lebesgue dominated convergence theorem implies that NYn(x) -+ Ny(x) pointwise on [0, T). We next show the convergence is uniform. Fix x E [0, T) and let z E [0, T) with x < z. Then

INYn(x) - NYn(z)1 ::; Ie: IK(x,s,Yn(s)) - K(z,s,Yn(s))1 ds +f: IK(z,s,Yn(s))lds (2 .11) ~f; sUPlul~r IK(x,s,u)-K(z,s,u)lds+f: hr(s)ds.

Consequently {NYn} is equicontinuous at x for each x E [0, T) and hence uniformly equicontinuous on [0, T). This together with the fact that NYn -+ Ny pointwise on [0, T) implies that the convergence is uniform. Consequently Ny E G([O, T), Rn) and N : G([O, T), Rn) -+ G([O, T), Rn) is continuous. In addition the Arzela-Ascoli theorem guarantees that N is completely continuous. To see this let 0 ~ G([O, TJ, Rn) be bounded i.e. there exists r > 0 with Ivlo::; r for each v E O. Also there exists hr E Ll[O,T) with IK(x,s,v)1 ::; hr(s) for a.e. s E [O,T) and all x E [O,T) and v E O. The equicontinuity of NO on [0, T) follows essentially the same reasoning as that used to derive (2.11). Also NO is bounded since

INv(x)l::; sup I fX K(x,s,v(s))dsl::; sup fX hr(s)ds xe[O,Tj Jo xe[O,Tj Jo

for each v E O. Consequently N : G([O, TJ, Rn) -+ G([O, TJ, Rn) is completely continuous.

Essentially the same reasoning as that used in theorem 2.2 establishes the following existence principle for the Urysohn integral equation

y(t) = Ly(t) + loT K(t,s,y(s))ds for t E [O,T) (2.12)

Theorem 2.3. Assume (2.4) holds, K : [0, T)2 X Rn -+ Rn and that K(x, s, u) ~s

L1-Garatheodory uniformly in x . Suppose for each r> 0 and x E [0, T) we have

lim fT sup IK(z,s,u)-K(x,s,u)lds=O, zE[O,T) . z_x Jo lul~r

(2.13)

In addition assume there is a constant Mo, independent of A, with Iylo =I- Mo for any solution Y to

y(t)=Ly(t)+AloTK(t,S,y(s))ds for tE[O,T)

for each A E (0,1). Then (2.12) has a solution in G([O, T), Rn).

Existence theory in the spirit of [1,7,11,12) could be established for the integral equations (2.8) and (2.12). We provide one such result for the Volterra equation (2.8).

256

BVPs FOR FDEs

Theorem 2.4. Assume (2.4) holds, K : [0, T]2 X Rn --> Rn and that K(x, s, u) is Ll-Caratheodory uniformly in x. In addition assume (2.9) is true and that

{

there exists a continuous function 1j;: [0,00) --> (0,00), a continuous function tjJ: [0, T]--> [0,00), a constant 0" ::::: 1, with I(I + R) (I~ K(t, oS, u(s)) dS) I ~ tjJ(t)1j; (I~ lu(x)ICT dX) , t E [0, T] (2 .15) for any u E C([O, T], Rn)

and fT foo dx

Jo tjJCT(x)dx < Jo 1j;CT(X) (2.16)

are satisfied. Then (2.8) has a solution in C([O, T], Rn) .

Proof: To show the existence of a solution to (2.8) we apply theorem 2.2. Let y be any solution to (2.lOk Now (2.10).\ together with (2.15) yields

ly(t)1 = 1,\ (Ny(t) + RNy(t)) I ~ tjJ(t) 1j; (lIY(xW dx), t E [0, T] (2 .17)

and hence

Iy(t)r ~ tjJCT(t), t E [O,T]. 1j;CT (Ici Iy( x)lo- dX)

(2.18)

Integrate (2.18) from ° to t to obtain

J (r ly(xW dX) ~ r tjJCT(x)dx ~ fT tjJCT(x)dx < f OO ~ Jo Jo Jo Jo 1j;CT(X)

where J(z) = J~ ",:i:z;r Consequently

This together with (2.17) implies that there is a constant Mo with ly(t)1 < Mo, t E [0, T] for any solution y to (2 .10k

3. Existence Principles for the Half Open Interval

In this section we look for a C([O,T),Rn) solution to

yet) = Ly(t) + Ny(t) for t E [0, T) ; here 0< T ~ 00. (3.1)

257


Recall (4) if U E G([O,T),Rn) then for every mE {1,2, ... } we define the semi norm Pm(u) by

Pm(U) = sup lu(t)1 [O,lm]

where tm 1 T; the metric is defined by

d(x,y) = f ~ Pm(x - y) m=l 2 1 + Pm(x - y)

Note G([O, T), Rn) is a locally convex linear topological Hausdorff space (in fact it is a Frechet space). The topology on G([O, T), Rn), induced by d, is the topology of uniform convergence on every compact interval of [0, T).

Theorem 3.1. Let Q be a closed, convex subset of G([O,T),Rn) . Suppose the following condition are satisfied:

{ L: G([O, T), Rn) --+ G([O, T), Rn) is a Volterra linear operator and (3.2) L is continuous and completely continuous with (LO)(t) == 0.

N : Q --+ G([O, T), Rn) is a continuous and compact operator (3.3)

and N + RN: Q --+ Q. (3.4)

Then (3.1) has a solution in Q.

Proof: As in section 2, see [4), solving (3.1) is equivalent to the fixed point problem

y = Ny + RNy == Fy. (3.5)

Notice F: Q --+ Q is continuous and compact . Theorem 1.1 yields the result .

Consider the Volterra integral equation

y(t) = Ly(t) + l I«t,s,y(s))ds for t E [O,T). (3.6)

Theorem 3.2. Assume (3.2) holds, I<: [0, T)2 X Rn --+ Rn and that I«x, s, u) is locally L1-Garatheodory uniformly in x . Suppose for each r > ° and x E [0, b) we have

lim r sup 1I«z, s, u) - I«x, s, u)1 ds = 0, z E [0, b) z~x Jo lul~r

for any b < T . Finally assume the following conditions are satisfied:

(3.7)

{

there exists a continuous nondecreasing function 'IjJ : [0,00) --+ (0,00), a continuous function ¢>: [0, T) --+ [0,00), a constant u ~ 1, with

1(/ + R) (I~I«t,3,u(s))ds) I ~ ¢>(t)'IjJ (I~ lu(x)I" dx), t E [O,T) (3.8) for any u E G([O, T), Rn)

258

BVPs FOR FDEs

and rb roo dx 10 ¢/'(x) dx < 10 tf;"(x) for any b < T .

Then (3.1) has a solution in C([o,T),Rn).

Proof: Let N: C([O, T), Rn) -> C([O, T), Rn) be defined by

Ny(t) = l K(t,s,y(s))ds.

To show existence of a solution to (3.1) we apply theorem 3.1. Let

(3.9)

Q = {y E C([o,T),Rn): lly(s)I" ds ~ a(t) and ly(t )1 ~ </>(t)tf;(a(t)), t E [O,T)}

where

a(t) = r 1 (ll</>(x)I" dX) and J(z) = fa% tf;:~x) . (3.10)

Notice Q is convex and bounded.

Remark. Recall A ~ C([O, T) , Rn) is bounded iff there exists a positive continuous function q: [0 , T) -> R with Ix(t)1 ~ q(t) for all t E [0, T) and x E A.

Also Q is closed since if Yn E Q wi th pm (Yn) --+ pm (y) for each m = 1,2, .. . then for fixed t E [0 , tmJ we have J~ IYn(x)I" dx ~ a(t) which implies J~ ly(x)I" dx ~ a(t) . Hence y E Q.

We next claim that N + RN : Q -> Q. Let y E Q and fix t < T . Notice for x < t that

and so

Consequently

INy(x) + RNy(x)1 ~ </>(x)tf; (f Iy(s)l" dS)

INy(x) + RNy(x)I" ~ </>"(x)tf;" (f ly(s)I" dS) .

lINy(x) + RNy(x)I" dx ~ l </>"(x)tf;"(a(x)) dx = l a'(x) dx = a(t)

since r(x) ds rx

10 tf;"(s) = 10 </>"(s)ds .

Thus Ny + RNy E Q and so N + RN : Q --+ Q. It remains to show N : C([O, T), Rn) -> C([O, T), Rn) is continuous and com

pletely continuous. If this is true then (3.3) is satisfied and the result follows from theorem 3.1. We first show that N is continuous. To see this let Yn -> y in C([O, T), Rn). Now Pm(Yn) -> Pm(Y) implies that there exists r > 0 such that

259


Pm(Yn):::; r and Pm(Y):::; r. Also there exists hr E L1[0,tmJ such that IK(t , s,u)l:::; hr(s) for almost all s E [0, tmJ and all t E [0, tmJ and lui:::; r. For each t E [0, tmJ we have

K(t,s,Yn(s)) -t K(t , s,y(s)) for a.e. s E [O,tmJ

and so the Lebesgue dominated convergence theorem implies that NYn(t) -t Ny(t) pointwise on [0, tmJ. Let x, z E [0, tmJ with x < z . Then

INYn(x) - NYn(z)1 :::; J~ sUPlul$r IK(x, s, u) - K(z, s, u)1 ds + J: hr(s) ds . (3.11)

Consequently {N Yn} is equicontinuous at x for each x E [0, tmJ and hence uniformly equicontinuous on [0, tmJ . This together with NYn -t Ny pointwise on [0, tmJ implies that the convergence is uniform. Consequently N : C([O, T), Rn) -t C([O, T) , Rn) is continuous.

To show N is completely continuous let !1 ~ C([O, T), Rn) be bounded i.e. there exists r > ° with pm (v) :::; r for each v E !1. There exists hr E L1 [0, tmJ with IK(t,s,v)l:::; hr(s) for a.e. s E [O,tmJ and all t E [O,tmJ and v E!1. The equicontinuityof N!1 on [0, tmJ follows the reasoning used to prove (3.11) . Also N!1 is bounded since for t E [0, tmJ we have

INv(t)l:::; sup I rt K(t,s,v(s))dsl:::; sup r hr(s)ds tE[O,tm] Jo tE[O,tm ] Jo

for each v E!1. Consequently N : C([O, T) , Rn) -t C([O, T), Rn) is completely continuous.

References

1. H. Brezis and F.E. Browder, Existence theorems for nonlinear integral equations of Hammerstein type, Bull. Amer. Math . Soc. 81(1975) , 73-78.

2. C. Corduneanu, Integral equations and stability of feedback systems, Academic Press, New York, 1973.

3. C. Corduneanu, Perturbations of linear abstract Volterra equations, J. Int . Eq. Appl. 2(1990), 393-401.

4. C. Corduneanu, Integral equations and applications, Cambridge Univ. Press, New York, 1990.

5. A. Granas, R.B. Guenther and J.W. Lee, Some general existence principles in the Caratheodory theory of nonlinear differential systems, J. Math . Pures Appl. 70(1991), 153-196.

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BVPs FOR FDEs

6. G. Gripenberg, S.O. Londen and O. Staffans, Volterra integral and functional equations, Cambridge Univ. Press, New York, 1990.

7. R.B . Guenther and J.W. Lee, Some existence results for nonlinear integral equations via topological transversality, J. Int. Eq. Appl. 5(1993), 195-209.

8. J.W. Lee and D. O'Regan, Topological Transversality: applications to initial value problems, Ann. Polon. Math. 48(1988), 31-36.

9. L. Neustadt, Optimization (A theory of necessary conditions) , Princeton University Press, 1976.

10. D. O'Regan, Theory of singular boundary value problems, World Scientific Press, Singapore, 1994.

11. D. O'Regan, Existence results for nonlinear integral equations, J. Math. Anal. Appl., to appear.

12. D. O'Regan, Existence theory for nonlinear Volterra and Hammerstein integral equations, Dynamical systems and applications, World Scientific Series In Applicable Analysis, Vol 4, to appear .

261

STURMIAN THEORY AND OSCILLATION OF A THIRD ORDER LINEAR DIFFERENCE EQUATION

ALLAN PETERSON Mathematics Department, University of Nebraska-Lincoln

Lincoln, NE 68588-0323 USA

Abstract

We are concerned with the Sturmian properties of a third order linear difference equation. We will define a band of solutions at a point and show under certain conditions this band of solutions satisfies what we call the Sturm separation property. We use these results to prove some oscillation results.

1. Introduction

In this paper we study the third order linear difference equation

ly(t) = b.3y(t -1) + p(t)b.y(t) + q(t)y(t) = 0

for t in the discrete interval [a + 1,00) == {a + 1, a + 2, . .. } where a is an integer. We assume p(t) and q(t) are real valued functions on [a + 1,00). It is easy to see that solutions of ly(t) = 0 exist on the whole interval [a, 00).

Let to E [a, 00), then later we will define a band of solutions of ly( t) = 0 at to. We will then show under certain conditions that the solutions in this band at to satisfy what we will call the Sturm separation property. We will also be concerned with oscillation results for this difference equation. The difference equation ly(t) = 0 was the main concern in the paper [3]. Our results are motivated by the results in Gregus [1].

2. Preliminary Results

It is useful to consider the adjoint equation [3] of ly(t) = 0, namely

l+z(t) = b.3 z(t - 2) + b.(p(t - l)z(t - 1)]- q(t)z(t) = O.

In working with this adjoint equation it is convenient to consider solutions of this equation which are also defined to the left of a. Hence we define p(t) == p(a+ 1), q(t) ==

262

BVPs FOR FDEs

q(a + 1) for t ~ a. Then it is easy to see that solutions of l+z(t) = 0 are defined on the set of integers. From the Lagrange identity, proved in [3], we have if y(t) is defined on [a, 00) and z(t) is defined on [a - 1,00), then we get the Lagrange identity

z(t)ly(t) + y(t)lz(t) = fl.{z(t); y(t)} (1)

for t E [a + 1,00) where the Lagrange bracket {z(t);y(t)} of z(t) and y(t) is defined by

{z(t); y(t)} = z( t)fl. 2y( t -1) - fl.z( t -1 )fl.y( t) + {fl. 2z(t - 2) + p( t -1 )z(t -1 )}y(t) (2)

for t E [a + 1,00). We now define generalized zeros as done by Hartman in his seminal paper [2]. For

a real valued function y(t) defined in [a,oo) we say y(t) has a generalized zero at a iff y(a) = O. We say y(t) has a generalized zero at to > a provided either y(to) = 0 or there is an integer k such that to - k ~ a,y(to - k + i) = 0, 1 ~ i ~ k - 1, and

(-l)ky(to - k)y(to) > O.

Let b ~ a, then ly(t) = 0 is said to be disconjugate [2] on [a, b + 3] provided no nontrivial solution of ly(t) = 0 has three (or more) generalized zeros in [a, b + 3].

We say that a real valued function y(t) defined on [a, 00) has a generalized zero of order k at a iff y(a + i) = 0, 0 ~ i ~ k - 1. We say y(t) has a generalized zero of order k at to > a provided either y(to - 1) I: 0 and y(to + i) = 0, 0 ~ i ~ k - 1 or y(to + i) = 0, 1 ~ i ~ k - 1, and

(-l)ky(to)y(to + k) > O.

Note that in the first part of the last statement Hartman [2] would say y(t) has generalized zeros at to,· .. , to + k - 1, and in the second half of the last statement Hartman [2] would say y(t) has generalized zeros at to + 1,···, to + k. It can be easily proved that no nontrivial solution of ly(t) = 0 has a generalized zero of order three (or higher). We say that ly(t) = 0 is (2,1)-disconjugate on [a,b+ 3] provided if y(t) is a nontrivial solution with a generalized zero of order two at to E [a, b + 1] then y(t)y(t + 1) > 0 on [to + 2, b + 3]. Similarly we say 13Y = 0 is (1,2)-disconjugate on [a, b + 3] provided if y has a generalized zero of order two at to, then y(t) is of one sign on [a, to - 1] in the case where y(to) = 0 and y(t) is of one sign on [a, to] in this case when y(to) I: O. Sufficient conditions for ly(t) = 0 to be (1,2)-disconjugate and for ly(t) = 0 to be disconjugate are given in [3] .

For the self-adjoint second order differential equation (r( x )U')' + f( x)u = 0 it is indeed well known that the zeros of linearly independent solutions of this differential equation separate each other. For linear difference equations there can be two linearly independent solutions with a generalized zero at the same point so the Sturmian

263

STURMIAN THEORY AND OSCILLATION

theory is more complicated. By Theorem 6.5 [4] we have that if F is the set of all solutions of the second order self-adjoint difference equation

Do[r(t)Dou(t - 1)] + f(t)u(t) = 0, (3)

where r(t) > 0, then the family offunctions F satisfies the following property: Sturm Separation Property. A vector space of real valued functions F defined

in a common discrete interval I is said to have the Sturm separation property on I provided the following statements are true. Two linearly independent functions in F can not have a common zero in I. If a function in F has a zero at tl and a generalized zero at t2 > t l , then any second linearly independent function in F has at least one generalized zero in [tl , t2]. If a function in F has a generalized zero at tl and a generalized zero at t2 > tl , then any second linearly independent member in F has at least one generalized zero in [t l , t2].

Motivated by results in [1] we define a band of solutions of ly = 0 at to to be the set of all solutions y(t) of ly = 0 satisfying y(to) = o. One of our main results is that if ly(t) = 0 is (1,2)-disconjugate on [a, 00) and to E [a, 00), then the band of solutions of ly(t) = 0 at to has the Sturm separation property on [to + 1,00).

We also give some oscillation results. A nontrivial solution of ly = 0 is said to be oscillatory on [a, 00) provided it has infinitely many generalized zeros in [a, 00). If one nontrivial solution of ly(t) = 0 is oscillatory, then we say ly(t) = 0 is oscillatory on [a, 00). If a nontrivial solution is not oscillatory on [a, 00) we say it is nonoscillatory on [a, 00). If ly(t) = 0 is not oscillatory on [a, 00), then we say that it is nonoscillatory on [a, 00). It is easy to find examples of equations of the form ly(t) = 0 which have both oscillatory and nonoscillatory solutions on [a, 00).

3. Main Results

First we prove the kind of result that we expect relating ly = 0 with its adjoint equation l+z = o. Theorem 1. Ifu(t) andv(t) are solutions ofly(t) =0 on [a, 00), then

_I u(t) v(t) I_I u(t) v(t) I z(t) - ~u(t) ~v(t) - u(t + 1) v(t + 1)

fort E [a, 00) is solution of the adjoint equation [+z(t) = 0 on [a,oo).

Proof. Let z(t) be as in the statement of the theorem. Then

Doz(t - 1) = I Do2:(~t~ 1) Do2V(~t~ 1) I (4)

for t E [a + 1, 00 ). It then follows that

264

BVPs FOR FDEs

2 I u(t - 1) v(t - 1) I I tJ.u(t - 1) tJ.v(t - 1) I tJ. z(t - 2) = tJ.3U (t _ 2) tJ.3V (t _ 2) + tJ.2U (t - 1) tJ.2V (t - 1)

for t E [a + 2, 00). Since u(t) and v(t) are solutions of ly(t) = 0 we get that

2 I tJ.u(t - 1) tJ.v(t - 1) I tJ. z(t - 2) + p(t - l)z(t - 1) = tJ.2u(t _ 1) tJ.2v(t _ 1) (5)

for t E [a + 2,00). Taking the difference of both sides of (5) we get that

3 I tJ.u(t) tJ.v(t) I tJ. z(t - 2) + tJ.[p(t - l) z(t - 1)) = tJ.3

U(t -1) tJ.3v(t -1)

for t E [a + 2,00). Again using the fact that u(t) and v(t) are solutions of ly(t) = 0 we get that

tJ.3Z (t - 2) + tJ.[p(t - l) z(t - 1)) = q(t)z(t)

for t E [a + 2,00). Hence z(t) is a solution of l+z(t) = 0 on [a, 00). We now use Theorem 1 to prove our main separation result .

Theorem 2.1f ly(t) = 0 is (1,2}-disconjugate on [a,oo), then for any to E [a, oo) the band of solutions of ly(t) = 0 at to satisfies the Sturm separation property on [to + 1,00) .

Proof. Let Yi(t,to),i = 1,2 be solutions of ly(t) = 0 satisfying the conditions

Yl(tO,tO) = 0, tJ.Yl(tO, to) = 1,tJ.2Yl(tO,tO) = p(t)

Y2( to, to) = 0, tJ.Y2(tO, to) = 0, tJ. 2y2( to, to) = l. Use the difference equation ly(t) = 0 to extend the defintions of Yl (t, to) and Y2(t, to) to the left of a. It follows that

Set

tJ.2Yl(tO - 1, to) = 0 tJ.2Y2(tO - 1, to) = l.

(6)

for t E (-00,00). From Theorem 1, z(t,to) is a solution of l+z(t) = 0 on (-00,00). From (6), (4) and (5) with u(t) replaced by Yl(t,tO) and v(t) replaced by Y2(t,tO) we get that

z(to, to) = 0

tJ.z(to - 1, to) = 0

265


~2z(to-2,to) = l. Using J+z(to) = 0 we get that z(to + l,to) = l.

We now show that z( t) > 0 on [to + 1,00). First assume there is a tl > to + 1 such that z(tl ) = O. Then

It follows that there are constants 0, {3, not both zero, such that

aYlett, to) + {3Y2(tl , to) = 0 aYI(t1 + 1, to) + {3Y2(t l + 1, to) = o.

From this we get that yet) == aYI (t, to) + {3Y2(t l , to) is a nontrivial solution of Iy(t) = 0 with y(to) = O,y(tl ) = y(tl + 1) = O. But his contradicts the fact that Iy(t) = 0 is (1,2)-disconjugate on [a,oo). Hence z(t) of. 0 for t ~ to + l.

Next assume there is a tl ~ to + 1 such that Z(tl) > 0 but Z(tl + 1) < O. Since z( t l ) of. 0, there are constants I, h, not both zero, such that

IYI(t1 + 1, to) + hY2(tl + 1, to) = o. Let yet) = IYI(t, to) + hY2(t, to), then it follows that

yet) YI(t,tO) Y2(t,tO)

y(tt} YI(tI, to) Y2(t l , to) = o. (7) y(tl + 1) YI(t l + 1, to) Y2(t l + 1, to)

Note that y(to) = 0, y(tt} = 1, y(tl + 1) = O. Letting t = tl + 2 in (7) and expanding along the first column we get

( 2) I YI(tt, to) Y2(tt, to) I Y tl + YI (tl + 1, to) Y2(t l + 1, to)

-y(tl) I YI(tl + 2, to) Y2(t l + 2, to) 1=0. YI(t1 + 1, to) Y2(t l + 1, to)

It follows that

y(tl + 2)Z(tl) = -z(tl + 1).

It then follows that y(tl + 2) > O. But then yet) is a nontrivial solution with a zero at to and a generalized zero of order 2 at t l . This is a contradiction. Hence we get that z(t) > 0 on [to + 1,00).

266

BVPs FOR FDEs

Assume y(t) is in the band at to. By the Lagrange identity we get that

{z(t); y(t)} == C.

Since {z(to); y(to)} = 0 we get that C = 0 and hence

Z(t)~2y(t -1) - ~z(t - I)~y(t) + {~2Z(t - 2) + p(t -I)z(t - I)}y(t) = 0

for t E [to + 2,00). When this last equation is expanded out the coefficients of y( t + 1) and y(t - 1) are positive on [to + 2,00) . Hence this equation is equivalent to a selfadjoint second order difference equation. Hence by Theorem 6.5 in [4], its solutions (the band at to) satisfy the Sturm separation property on [to + 1,00) .

Corollary 3.Assume p(t) :s 0, on [a + 1,00) and q(t) 2 0 on [a + 1,00), then for all to E [a,oo) the band of solutions of ly(t) = 0 at to satisfies the Sturm separation property on [to + 1,00).

Proof. From Theorem 4 in [3] the third order difference equation ly = 0 is (1,2)disconjugate on [a, 00). Using Theorem 2 we get the desired result.

The proof of the next theorem is very similar to the proof of Theorem 2 and so we will omit this proof.

Theorem 4.1f ly(t) = 0 is (2,l)-disconjugate on [a,oo), then for any to 2 a + 3, the band of solutions of ly(t) = 0 at to satisfies the Sturm separation property on [a,to-I].

The following corollary follows from Theorem 4 and Theorem 6 in [3].

Corollary 5.Assume p(t) :s 0 on [a + I,oo),q(t) 2 0 on [a + 1,00) and for all tl E [a+ 1,00)

t-I

Ip(t)12 2 E q(s)

for tl :s t < 00, then for any to 2 a the band of solutions of ly(t) = 0 at to satisfies the Sturm separation property in [a, to - 1] and in [to + 1,00).

We can now state and prove an oscillation result.

Theorem 6.Assume ly(t) = 0 is (l,2)-disconjugate on [a, 00). If there is a nontrivial solution ofly(t) = 0 with a zero in [a, 00) that is oscillatory in [a,oo), then every solution with a zero in [a,oo) is oscillatory in [a, 00).

Proof. Assume u(t) is a nontrivial solution of ly(t) = 0 with a zero in [a, 00) such that u(t) is oscillatory on [a,oo). Let to be a zero of u(t).

Now assume y(t) is a nontrivial solution of ly(t) = 0 with a zero in [a,co) . Let tlbe a zero of y(t). If u(t) and y(t) are linearly dependent, then y(t) is oscillatory. Assume u(t) and y(t) are linearly independent. If u(t) and y(t) have a common zero, say t 2, then they are in the band at t2 and it follows from Theorem 2, that u(t) and

267


y(t) satisfy the Sturm separation property in [t2 + 1,00]. This implies that y(t) is oscillatory on [a, 00).

Finally consider the case where u(t) and y(t) do not have any common zeros. Let v(t) be a nontrivial solution satisfying

Since u(t) and v(t) are in the band at to, v(t) must be oscillatory on [a, 00). But v(t) and y(t) are in the band at t} so we get that y(t) is oscillatory on [a,oo).

Note from Theorem 5 in [3] we get that if ly(t) = 0 is (1,2)-disconjugate on [a, 00) then it has a nonnegative solution on [a, 00) . It follows that ly(t) = 0 has a nonoscillatory solution on [a, 00) .

Example. The third order difference equation

is a simple example of an equation which is (1 ,2)-disconjugate on [a, 00), has a positive solution and every solution with a zero in [0,00) is oscillatory in [0,00).

References

1. M. Gregus, Third Order Linear Differential Equations, D. Reidel Publishing Company, 1987.

2. P. Hartman, Difference equations: Disconjugacy, principal solutions, Green's functions, complete monotonicity, Trans. Amer. Math. Soc. 246 (1978),1-30.

3. J. Henderson and A. Peterson, Disconjugacy for a third order linear difference equation, Computer Math. Applic. 28 (1994), 131-139.

4. W. Kelley and A. Peterson, Difference Equations, An Introduction with Applications, Academic Press, 1991.

270

BVPs FOR FDEs

2. Existence of Solutions

Let Cr, r > 0, be the space of all continuous functins ¢l : [-r,O] -+ JR . For ¢l E Cr, define the norm

1I¢l1l[-r,O) = sup 1¢l(8)1 · 8e[-r,O)

So Cr is a Banach space. For every continuous function x : [-r, T] -+ JR, T > 0, and for every t E [0, TJ, we denote by Xt the element of Cr defined by

xt(8) = x(t+8), 8 E [-r,O].

Also, if I is a compact interval of the real line JR, we shall use c(n-l)(I, JR) to denote the set of all (n - 1) times continuously differentiable JR-valued functions on I. Also, for x E C(I, JR), we define

IIxlll = sup Ix(t)l . tel

For n ~ 2 and k ~ 2, we shall be concerned with the following FDE,

satisfying,

{

xo(s) = ¢l(s), (i.e. x(O + s) = ¢l(s», -r ~ s ~ 0 x(')(1/j) = Aj,i, 0 ~ i ~ mj - 1, 2 ~ j ~ k - 2, x(i)(T) = Ak,;, 0 ~ i ~ mk - 1,

(1)

(2)

where ¢l E Cr, L:~=2 ml = n - 1, 0 < 1/j < njH < T, 2 ~ j ~ k - 3, 1/j E JR, and Aj ,; E JR.

We now state the Leray-Schauder Alternative.

Lemma 2.1 Let C be a convex subset of a normed linear space E and assume 0 E C. Let F : C -+ C be a completely continuous operator and let £(F) = {x E C : x = )'Fx for some 0 <), < I} . Then either £(F) is unbounded or F has a fixed point.

We now present our theorem insuring the existence of a solution of (1), (2) .

Theorem 2.2 Let f : [0, T] X Cr x JRn-l -+ JR be a continuous function . Suppose that there exists a constant M > 0 such that IIxll[-r,T) ~ M and IIx(i)lho,T) ~ M, 1 ~ i ~ n - 1, for each solution x of the BVP

x(n)(t) = Af(t, xe(8), x'(t), ... , x(n-l)(t», 0 ~ t ~ T, ), E [0,1], (3.\)

271

MULTIPOINT BOUNDARY VALUE PROBLEMS

2 ~ j ~ k - 1,

where E~=2 ml = n - 1, 0 < TJj < TJj+l < T, 2 ~ j ~ k - 3, and Aj,i E IR, for each A E [0, I} . Then the BVP (3d (41 ) has at least one solution.

Proof: There are two cases: ¢(O) = 0 and ¢(O) -I 0 . Case I: ¢(O) = O. Consider the space E of all functions x E c(n-I)([O, TJ, IR) endowed with the norm

IIxlln-1 = max{lIx(i)lI[o,T] : 0 ~ i ~ n - I}.

Now let C C E be defined by

C={XEE : x(O)=O}.

C is convex and 0 E C. Now define F : C ---> E by

(Fx)(t) = loT G(t, s )f(s, x., x'(s),- .. x(n-l)(s ))ds + h(t),

where G(t , s) is the Green's function for the BVP

x(n)(t) = 0, x(O) = 0, x(i)(TJj) = 0, 0 ~ i ~ mj - 1, 2 ~ j ~ k - 1, x(i)(T) = 0, 0 ~ i ~ mk - 1.

where h(t) is the (n _1)8t degree polynomial that satisfies (2), and where

(0) _ { x(s + 0) , s + 0 ~ 0, x. - ¢(s + 0), s + 0 < 0,

for -r ~ 0 ~ O. F(C) ~ C and (Fx)(O) = 0 for all x E C. We shall now show that F is com

pletely continuous. First, from the continuity of f and properties of the integral, F is continuous. Let B be a bounded subset of C . Now there exists b ~ 0 such that IIxlln-1 ~ b, x E B . So, IIx(i)lI[o,T] ~ b, 0 ~ i ~ n - 1, x E B. Then for any t l ,

t2 E [O,T} and any x E B,

Ix(i+I)(Vlt"t,,;) I It I - t21

~ bit I - t21 , 0 ~ i ~ n - 2.

Thus, B is a (uniformly) equicontinuous family of functions, (in fact, equicontinuous with the c(n-2) norm).

272

BVPs FOR FDEs

Now define iJ C CT by iJ = {Xt : x E B} . We want to show that there exists a compact D C CT such that iJ ~ D C CT ' To do this it suffices to prove that the set iJ is uniformly bounded and equicontinuous. For any x E Band t E [0, T] we have

sup IXt(8)1 -T~8~O

sup Ix(t + 8)1 -T~8~O

~ max{II4>II[_T,O]' b},

and hence iJ is uniformly bounded. Next, for all Xt E iJ and for 81, 82 E [-r, 0] ,

I (8)- (8)1 - 14>(t+81)-x(t+82 )1, ift+81 <0 andt+82 :2:0, (

Ix(t + 81) - x(t + 82 )1, if t + 81 :2: ° and t + 82 :2: 0,

Xt 1 Xt 2 - 14>(t+8t)-4>(t+82 )1, ift+81 <0 andt+82 <0, Ix(t + 81) - 4>(t + 82 )1, if t + 81 :2: ° and t + 82 < 0.

We now examine the cases. First, Ix(t+81)-x(t+82 )1 ~ b181-82 1 < c,ifI81 -82 1 < c/b. For case 3, 14>(t + 8J) - 4>(t + 82 )1 < c for all 181 - 821 < h(c) where h(c) is the uniformly continuous constant for 4>. For the second case of 14>(t + 81) - x(t + 82 )1, t + 81 < ° ~ t + 82, we can rewrite 14>(t + 8t) - x(t + 82)1 as

14>(t + 8J) - x(t + 82)1 ~ 14>(t + 8t) - 4>(0)1 + Ix(O) - x(t + 82)1 .

By the equicontinuity of B and uniform continuity of 4>, there exists hl(C) and h2(f) such that lSI - s21 < hI and SJ, S2 E [0, T] imply that Ix(sl) - x(s2)1 < c/2, and also that, 181 - 821 < h2 and 81, 82 E [-r,O] imply that 14>(81) - 4>(82)1 < c/2. Let h = min{hJ,h2}' So, if 81,82 E [-r,O] and if t + 81 < ° ~ t + 82 and 181 - 821 < h, then, It + 81 - 01 < h and It + 82 - 01 < h and so

IXt(8t) - Xt(82 )1 14>(t + 81) - x(t + 82)1

~ 14>(t + 8J) - 4>(0)1 + Ix(O) - x(t + 82)1

< c/2 + c/2

c.

Case 4, Ix(t + 8J) - 4>(t + 82)1, is similar. Thus iJ is uniformly equicontinuous with respect to CT and so there exists a compact DeC such that iJ ~ D C CT' Since B was an arbitrary bounded subset of C, we need to show F(B) is precompact. Choose a sequence {Fxv}:'1 in F(B), (so {xv} C B) . Thus {xv,} c iJ ~ D C CT' Define X = [O,T] X D X [-b,b] x .. · x [-b, b]. X is compact by Tychanoff's Theorem.

Let 8 = maxlf(t,uJ,u2, . .. ,un)l . By Jackson [14], there exist constants ",. ° < X 11 , _

i ~ n - 1, such that, for all t E [0, T],

fl ai

Jo laxiG(t,s)lds=,i(T-o)n-l, O~i~n-l.

273


For each 0:5 i :5 n - 1, let Hi = SUPO<t<T Ih(i)(t)l . Then, for all t E [0, T] and for all v EN, --

I(FXv)(i)(t) 1 $ loT I :~iG(t, s )f(s, Xv" x~(s), . .. , x~n-l)(s ))Ids + Hi

:5 9,iTn-i + Hi, 0:5 i :5 n - 1.

Let K = max{9,iTn-i + Hi : 0:5 i :5 n - I}. Then, IIFxvlln-1 :5 K for all v. Next, let v be given and choose any tI, t2 E [0, T]. Then,

I(Fxv)(i)(t l ) - (Fxv)(i)(t2 )1 = 11t2 (Fxv)(i+l)(s)dsl t,

:5 Kltl - t21, 0:5 i :5 n - 2.

Also,

I(Fxv)(n-I)(t l ) - (Fxv)(n-l)(t2)1 11t2 (Fxv)(n)(s )dsl t,

11t2 f(s , Xv" «s), . . . , xin-l)(s))dsl t,

:5 91t l - t21 ·

Thus, {Fxv} is uniformly equicontinuous with respect to II . lin-I . By the ArzelaAscoli Theorem, {Fxv} has a subsequence which converges with respect to II · lin-I. Thus F(B) is compact. Hence, F is completely continuous. By the hypothesis, £(F) = {xix = AFx, 0 < A < I} is bounded. And so, by the Leray-Schauder alternative, F has a fixed point x E C such that x = Fx. Then the function

z(t) = {x(t) , t E [0, T], ¢(t), t E [-r,O],

is a solution of (1), (2). Thus we have resolved the case where ¢(O) = o. Case II: ¢(O) -# O. Let y = x - ¢(O). Then (1), (2) is transformed to

x(n)(t)

f(t, Xt, X'(t), ... , X(n-l)(t))

f(t, Yt + ¢(O), y'(t), ... , y(n-l)(t))

J(t,Yt,y'(t), ... ,y(n-I)(t)), 0:5 t:5 T,

(5)

274

BVPs FOR FDEs

1

yo(s) = xo(s) - </>(0) = </>(s) - </>(0) == ~(s), -r S s so, y(r/j) = Aj,o - </>(0) == Aj,o, 2 S j S k - 1,

y(i)(7]j) = Aj,; == A j,; , 1 SiS mj - 1, 2 S j S k - 1, y(T) = Ak,o - </>(0) == Ak,o,

y(i)(T) = A k,; == Ak,;, 1 SiS mk - 1,

(6)

where L:~=2 ml = n - 1, 0 < T/j < T/j+1 < T, 2 S j S k - 3, and A j,; E JR. Also, note that ~(O) = O. By case I, (5) (6) has a solution y(t) . Then x(t) = y(t) + </>(0) is a solution of (1), (2). The proof is complete.

3. Applications

The applicability of the previous theorem depends upon the existence of an a prior bound for solutions of the BVP (3,\), (4,\) which is independent of A. The following theorem due to Eloe and Henderson [9) gives conditions under which such bounds exist .

Theorem 3.1 Let f : [0, T) x Cr x JRn-1 --+ JR be continuous. Let 8 E Cr be such that 8(0) = O. Also, assume there exists a positive real-valued function </>(t!, . . . , tn) , defined for t; 2 0, 1 SiS n, which is nondecreasing in each variable and such that

If(x, Y1, ·· · , Yn)1 S </>(iY11,·· ., IYnl)

for all (x, Y!' ... , Yn) E I x JR n. If

~ ti ~ L.J """"-~-7 --+ +00 as L.J t; ....... +00, ;=1 </>(t!, . .. , tn) ;=1

then there exists an M > 0 such that IIxlll-r,T] S M and IIx(i)llIo,T] S M , 1 SiS n-l , for all solutions x 0/ (3,\) with the boundary conditions

I xo(s) = </>(s), -r S s S 0, y(')(T/j) = 0, 0 SiS mj - 1, y(i)(T) = 0, 0 SiS mk - 1,

2 S j S k -1,

where L:~=2 mt = n - 1, and 0 < T/j < T/j+1 < T , 2 S j S k - 2.

(7)

The proof of the above theorem follows from Theorems 2.3 and 3.1 in Eloe and Henderson [9) .

As a consequence of Theorem 3.1 above, we have an immediate corollary.

Corollary 3.2 Assume the hypothesis of the Theorem. Then there exists a solution 0/(1) (7) .

275


References

1. R. P. Agarwal, Boundary value problems for differential equations with deviating arguments, J. Math. Phy Sci. 6 (1992), 425-438.

2. D. C. Angelova, Quickly, moderately, and slowly osciallatory solutions of a second order functional differential equation, Arch. Math. 21 (1985), 135-146.

3. D. C. Angelova and D. D. Bainov, On oscillation of solutions of forced functional differential equations of second order, Math. Nachr. 122 (1985), 289-300.

4. D. C. Angelova and D. D. Bainov, On the oscillation of solutions to second order functional differential equations, Bollettino U. M. I. (6) 1 -B (1982), 797-807.

5. D. C. Angelova and D. D. Bainov, On some oscillatory properties of the solutions of a class of functional differential equations, Tsukuba J. Math. 5 (1981), 39-46.

6. J. Ehme, P. W. Elce, and J . Henderson, Differentiability with respect to boundary conditions and deviating arguments for functional differential systems, Diff. Eqns. Dyn. Sys. 1 (1993), 59-71.

7. J. Ehme and J . Henderson, Functional boundary value problems and smoothness of solutions, Nonlin. Anal., in press.

8. P. W. Eloe and L. J. Grimm, Conjugate type boundary value problems for functional differential equations, Rocky Mtn. J. Math. 12 (1982), 627-633.

9. P. W. Eloe and J. Henderson, Nonlinear boundary value problems and a priori bounds on solutions, SIAM J. Math. Anal. 15 (1984), 642-647.

10. P. W. Eloe, J. Henderson, and D. Taunton, Multipoint boundary value problems for functional differential equations, Pan Am. Math. J., in press.

11. L. J . Grimm and K. Schmitt, Boundary value problems for delay differential equations, Bull. Amer. Math . Soc. 74 (1968), 997-1000.

12. L. J. Grimm and K. Schmitt, Boundary value problems for differential equations with deviating arguments, Arq. Math. 3 (1969), 24-38.

13. L. J. Grimm and K. Schmitt, Boundary value problems for differential equations with deviating arguments, Arq . . Math. 4 (1975) 176-190.

14. L. K. Jackson, Boundary value problems for ordinary differential equations, in "Studies in ordinary Differential Equations" (J. K. Hale, Ed.) pp. 92-127, MAA Studies in Mathematics, Vol. 14, Mathematical Association of America, Washington, DC, 1977.

276

BVPs FOR FDEs

15. G. A. Kamenskii and A. D. Myskis, Variational and boundary value problems for differential equations with deviating argument, Proc. Eqadiff IV, Lecture Notes in Mathematics 703 (1979), 179-188.

16. P. R. Krishnamoorthy and R. P. Agarwal, Higher order boundary value problems for differential equations with deviating arguments, Math . Seminar Notes 7 (1979), 253-260.

17. J. W. Lee and D. O'Regan, Existence results for differential delay equations - I, J. Diff. Eqns. 102 (1993), 342-359.

18. J. W. Lee and D. O'Regan, Existence results for differential delay equations - II, Nonlin. Anal. 17 (1991), 683-702.

19. S. K. Ntouyas, Y. G. Sficas, and P. Ch. Tsamatos, Existence principle for boundary value problems for second order functional differential equations, Nonlin. Anal. 20 (1993), p. 215-222.

20. D. O'Regan, Existence of solutions to some differential delay equations , Nonlin. Anal. 20 (1993) , 79-95.

21. P. Ch. Tsamatos and S. K. Ntouyas, Existence and uniqueness of solutions for boundary value problems for differential equations with deviating arguments, Nonlin . Anal. 22 (1994), 1131-1146.

22. P. Ch. Tsamatos and S. K. Ntouyas, Existence of solutions of boundary value problems for differential equations with deviating arguments via the topological transversality method, Proc. Royal Soc. Edinburgh USA (1991), 79-89.

23. H. Xia and T . Spanily, Global existence problems for singular functional differential equations, Nonlin. Anal. 20 (1993), 921-934.

277

THIRD ORDER BOUNDARY VALUE PROBLEMS FOR DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

P. CH. TSAMATOS Department of Mathematics, University of Ioannina

451 10 Ioannina, Greece

1. Introduction

In this paper we deal with the following differential equation with deviating arguments

x"'(t) = f(t, x(t), x(u(t)), x'(t), x'(g(t)), X"(t), X"( T(t))), a.e t E (0,1), (E)

where f : [0, 1) X ~ - R is a function satisfying Caratheodory's conditions and u, g, T

are continuous real valued functions defined on [0,1].

We suppose that

and

-00 < -r = min {u(t),g(t),T(t)} < ° tE[O,lj

1 < max{u(t),g(t),T(t)} = d < +00 . tE[O,lj

We also consider the following boundary conditions

m

x(t) = (/JI(t), t E [-r,O)

x(t) = rP2(t), t E [I,d)

L aiX(ei) = 'T/, i=l

m

x(t) = rP1(t), t E [-r, 0]

x(t) = rP1(t), t E [I,d)

L aix'(e;) = 'T/ 1:=1

(BCh

(BCh

278

and

m

BVPs FOR FDEs

x(t) = (/J!(t), x(t) = li>2(t),

tE[-r,O] tE[I,d]

L aix"((i) = 'T/, ;=1

(BCh

where ~l : [-r,O] -+ R, ~2 : [1, d] -+ R are differentiable functions such that ~~ and ~~ are absolutely continuous functions on [-r, O] and [1 , d] respectively. Also (i E (O,I),ai E R,i = 1, . .. ,m and 'T/ E R.

Recently, several papers have appeared which are concerned with the existence and uniqueness of solutions of boundary value problems (BVP for short) for third and higher order nonlinear differential equations (see [1], [3], [7], [12]). Also, during the last five years, boundary value problems for differential equations with deviating arguments were the subject of several articles (see [8-11] and [13-14]) .

The purpose of the present paper is to study some multi-point BVP for third order differential equations with deviating arguments . As usual, in BVP for differential equations with deviating arguments we search for solutions which satisfy the differential equation on a compact interval I and identify with an a priori given functions outside of I. Moreover, here we demand the solutions to satisfy a multi-point condition on I. In this way, we enlarge the class of boundary conditions considered in [3] and [7]. So, the results obtained here are new even for the usual case, that is for the case where u(t) = g(t) = T(t) = t, t E [0,1]. We notice that multi-point BVP for second order differential equations have been recently studied by Gupta, Ntouyas and Tsamatos in [4-6].

The existence results here are obtained by the help of the well known LerayShauder Alternative, which follows immediately from the Topological Transversality Theorem of Granas [2].The Topological Transversality Theorem of Granas and its corollaries are widely applied in the theory of boundary value problems. For applications of the Topological Transversality method on boundary value problems for differential equation with deviating arguments we refer the reader to [10], [13] and the references therein.

2. Preliminaries

If A is a compact interval ofthe real line, we denote by C(A), Ck(A), Lk(A) and L OO (A) the classical spaces of continuous, k-times continuously differentiable, measurable real valued functions whose k-th power of the absolute value is Lebesgue integrable on A, or measurable functions that are essentially bounded on A. We also use the Sobolev space W3 ,k ( A), k = 1, 2 defined by

W3.k(A) = {x : A -+ R x, x', x" abs. cont . on A with x E Lk[o, In

279

THIRD ORDER BOUNDARY VALUE PROBLEMS

We denote the norm in Lk(A) by 1I.lIk' and the norm in Loo(A) by 11.1100.

Definition 2.1. A function x : [-r, d] -+ R is called a solution of the BVP (E) - (BCh (resp. (E) - (BCh,(E) - (BCh) if x, x'I[-r, 0], x"I[O, 1], x'I[I, d] are absolutely continuous, xl[O, 1] satisfies the equation (E) and xl[-r, 0] = rPl, xl[l, d] = rP2, E~l aiX(ei) = "I (resp.E~l aix'(ei) = "I, E~l aix"(ei) = "I). Lemma 2.2.Let ai E R, i = 1, ... , m with all of the aj's having the same sign and ei E (0,1), i = 1, ... , m be given. Let also, x E W 3 ,1[0, 1] be such that x(O) = x(l) = 0 and E~l aiX(ei) = o. Then

Proof. Since x(O) = 0, we have x(t) = f~ x(s)ds, t E [0,1] and hence

Moreover, since x(O) = x(l) = 0, there exists 0 E (0,1) such that x'(O) = o. Therefore

IIx'lloo :::; IIx"lloo.

On the other hand the relation E~l ajX(ei) = 0 and the continuity of x imply that there exists e E [minl<i<m ei, maXl<i<m e;J such that (E~l ai)x(e) = 0 and hence x(e) = 0, since a/s, i = -( ... , n, have the same sign. Therefore x(O) = x(e) = x(l) = 0 which, by Rolle's Theorem, implies the existence of an f. E (0,1) such that x"(f.) = O. Consequently,

and the proof is complete.

When the real constants ai, i = 1, ... , m are not all of the same sign, we set

1+ = {i E {1, ... ,m}: ai > O},L = {i E {1, ... ,m}: ai < O}

and a+ = L aj,a_ = - L ai,

ieI+ ieI_

Then the following lemma holds.

Lemma 2.3. Let ai E R, i = 1, ... , m, a+, a_ be as defined above and e E (0,1) be given. Let also x E W 3,1[0, I] be such that x(O) = x(l) = 0 and E~l aix"(ei) = 0, with E~l ai =f O. Then

280

BVPs FOR FDEs

where

Proof. Since x(O) = 0, we have IIxlioo ~ IIx'lloo' Moreover, since x(O) = x(l) = 0 there exists 0 E (0,1) such that x'(O) = O. Therefore Ilx'lloo ~ IIx"lIoo.

On the other hand we have

L aix"(~;) = - L aix"(~i)' iEI+ iEI_

Hence, by the continuity of x" there exist ~+ E [miniEI+ ~i' maxiEI+ ~i) and ~- E [miD;El_ ~i,maXjEI_ ~d such that a+x"(~+) = a_x"(~_). Now, if a+ = 0 or a_ = 0 we have x"(~_) = 0 or x"(~+) = 0 respectively, since L~l ai f. O. Therefore

When a+ f. 0 f. a_, then for every t E [0,1) we have

Hence

and

provided that a+ > a_.

Similarly, we have

provided that a_ > a+.

For the existence results in the next paragraphs we shall use the following lemma which is referred to as the Leray-Schauder Alternative [2, p.61).

Lemma 2.4. Let C be a convex subset of a normed linear space E and assume 0 E C. Let F: C --+ C be a completely continuous operator and let £(F) = {x E C : x = >.Fx,for some 0 < >. < I}. Then either £(F) is unbounded or F has a fixed point.

281


3. Existence Results

Theorem 3.1.Let J : [0,1] x Ji!3 ---> R be a function satisfying Caratheodory's conditions. Assume that there exist functions Pi, i = 1, .. . ,6 and l in L1([0,1]) such that

6

IJ(t,x1,' " ,x6)1 :::; LPi(t)lx;j + l(t) i=1

for a.e t E [0.1] and all (Xl, "" X6) E Ji!3 Also, let ~i E (0,1), ai E R, i = 1, ... , m and TJ E R be given.

Then if ai, i = 1, ... , m are all of the same sign, the BVP (E) - (BCh has at least one solution provided that

(3.1)

Moreover, if ai E R, i = 1, .. . , m with E~l ai 1= 0, the BVP (E) - (BCh has at least one solution provided that

(3.2) i=1

where L is the constant defined in Lemma 2.3.

Proof. Consider first the BVP (E) - (BCh and the case ¢>1(0) = ¢>2(1) = TJ = O. Then the set

C = {x E C2([0, 1]) : x(O) = x(l) = O}

is obviously a subspace of C2([0, 1]).

Now, for any function X E C we define the function x : [-r, dJ ---> R by

and, for t E [0,1], we set

1 ¢>l(t), t E [-r,O]

x(t) = x(t), t E [0,1] ¢>2(t) , t E [1 , dJ

Tx(t) = III J(z,x(z),x(u(z)),x'(z),x'(g( z )),x"(z),x"(r(z )))drdBds

Next we define the operator F1 : C ---> C 2([0, 1]) by

where

Tx(l) E~l ai~i - E~l aiTx(~;} B E?:l aiTx(~i) - Tx(l) E?:l aiel Al = E~l ai~i(~i - 1) , 1 = E~l ai~i(~i - 1)

282

BVPs FOR FDEs

It is obvious that x is a solution of the BVP (E) - (BCh if and only if F1 x1[0, 1] = xl[O, 1] and xl[-r, 0] = 4>b xl[l,d) = 4>2.

Obviously F1(C) ~ C and Fl is completely continuous.

Now, it will be shown that the set £(Fl) = {x E C : x = AF1x,0 < A < I} is bounded.

Indeed, let x E £(Ft). Then we use the assumptions to get that

IIxllllh = Allf( t, x(t), x( u(t)), x'(t), x'(g( t)), x"(t) , x"( r( t)))II1

::; IIpl 111 II x 1100 + IIp2Ihmax{lIxIl 00 , 114>11100, 114>21100} + IIp31hllx'iloo + IIp4Ihmax{lIx'lIoo, 114>~1I00, 114>~1I00} + IIPslhllx"lloo + IIp6Ihmax{lIx"lIoo, 114>~1I00, 114>~1I00} + 1I1lh ::; IIp11hllxll00 + IIp211dllxll 00 + 114>11100 + 114>21100} + IIp311dl x'il00 + IIp411dllx'il00 + 114>~1I00 + 114>~1I00} + IIPslhllx"lIoo + IIp6l1dllx"1I00 + 114>~1I00 + 114>~1I00} + 1I1111 .

Using Lemma 2.2 we obtain

where

6

II XIII II 1 ::; (I: IIp;Jldll xlll lh + P, i=l

p = IIp2 lid 114>1 1100 + 114>2I1oo} + IIp4I1dll4>~1I00 + 114>~1I00} + IIp61h {1I4>~1I00 + 114>~1I00} + 1I1111'

Then (3.1) implies that

11 11111 < P -X 1 _ 1 _ ,,6 II II - C.

L..,.=1 P 1

Hence, by Lemma 2.2, we have

II xlioo ::; II x' 1100 ::; II x"lIoo ::; IIxllllh ::; c. Therefore the set £(F1) is bounded. According to the Lemma 2.4, the operator Fl has a fixed point x E C. Then it is obvious that the function

1 4>1 (t), t E [-r, 0]

z(t) = x(t), t E [0,1] 4>2(t), t E [1, d)

is a solution of the BVP (E) - (BCh.

For the proof of the theorem in the general case when 4>1(0)4>2(1)77 i= ° we note that the transformation x(t) = y(t) + p.(t), t E [0,1], where p.(t) = -A1t2 - Blt + 4>1(0), t E [0,1] and

Al = (4)1(0) - 4>2(1)) l:~1 ei + 77 - m4>1(0) l::'1(1 - ei)ei

reduces the BVP (E) - ( B C ) l to the following BVP

m

aiy (ti) = 0, i=l

where

Since &(0) = 0 = J2(1) we proceed as in the first step of the proof.

For the BVP (E) - (BC)3, and especially, for the case q51(0) = d 2 ( l ) = g = 0 we define the operator

F3x(t) = A3t2 + B3t + Tx( t ) , t E [O,l]

with x ( t ) = $l(t) , t E [-r,O],x(t) = dz(t) , t E [ 1 , 4 , and A3 = <?;i:(€i), B3 =

m .=I

xi=, aiTt(Ci)-2T=(1)CLl m d proceed as above, using Lemma 2.3. 2 xyLl ai

In the case where g51(0)$2(1)g # 0, we proceed similarly.

Theorem 3.2. Let f : [O,1] x R6 - R be as in Theorem 3.1. Then if t E ( 0 , l ) and ai E R , k = 1,. . . , m , with Cgl a;(l - 2t;) # 0, are given, the BVP (E) - (BC)2 has at least one solution provided that

Proof. As in the proof of Theorem 3.1, we consider first the case &(0) = $2(1) = 7 = 0, the space C and the operator F2 : C -+ C defined by

F z ~ ( t ) = A2t2 f Bzt + Tx( t ) , t E [O, 11,

with xi[-r,0] = $1,~1[1,d] = $2 and

284

BVPs FOR FDEs

It is obvious that x is a solution of the BVP (E) - (BCh if and only if (F2x)I[0, 1] = xj[O,l],xl[-r,O] = <PI and xj[l,d] = <P2. Moreover, as in the proof of Lemma 2.3, for any function x E C such that x = >.F2x, ° < >. < 1, we have

Ilxli oo ~ IIx'lloo ~ IIx"lloo.

Also, for every t E [0,1] we have

ITx(t)1 ~ llllf(z,x(z),x(u(z)),x'(z ),x'(g(z)),x"(z ),x"(r(z)))ldzd8ds

~ lll[lpI(z )llIxli oo + Ip2(z)I(lI x Ii00 + II <PI 11 00 + 1I<p21100) + IP3(z)llIx'1100

+lp4(z)I(llx'1l00 + 11<p~II°o + 1I<p;1I00) + Ips( z )llIx"lIoo

+IP6(z)I(lIx"lI°o + 1I<p~1I00 + 1I<p~1I 00 ) + Il(z)lldzd8ds

~ IIx"lloo ~ llllpi(Z)ldzd8ds ~ Ilx"lIoo ~ IIpilh + P,

where P is defined in the proof of Theorem 3.l.

Similarly, for every t E [0,1] we have

6

IT;(t)1 ~ Ilx"lloo L II Pi III + P 1::::1

and 6

T;(t)1 ~ Ilx"lloo L IIp;!II + P. 1'=1

Therefore for every t E [0,1] we have

6

~ (K + l)(llx"lIoo L II Pi III + P) . 1'=1

Hence 6

Ilx"lloo{1- (I< + 1) L IIp;!II) ~ (K + l)P 1=1

and, because of (3.3) and (3.4)

II II < II 'II < II "II < (K + l)P x 00 - x 00 - x 00 - 1 _ (K + 1) L:?=I IIpdh .

(3.4)

285


For the case '14>1(0)4>2(1) # 0 we proceed as in the proof of Theorem 3.1.

Following exactly the same arguments as above, we can prove an analogous result for the BVP (E) - (BCh, without assumptions on the sign of ai, i = 1, ... , m. From this point of view the next theorem completes Theorem 3.1.

Theorem 3.3. Let f : [0, IJ x It> -+ R be as in Theorem 3.1. Then if ~ E (0,1), ai E R, i = 1, .. . , m are given, the BVP (E) - (BCh has at least one solution provided that

6

(A + 1) L IIPilil < 1, ;=1

h A 41 ~ a,l were = I '=1 a,e,(l-{,lI"

4. Uniqueness Results

Theorem 4.1.Let f : [0,1] x It> -+ R be a function satisfying Caratheodory's conditions.Assume that there exist functions c;, i = 1, ... ,6 in £1([0,1]) such that

6

If(t, xl,"" X6) - f(t, Yl, · ··, ys)1 :::; L C;(t)IXi - Yil ( 4.1) ;=1

for a.e t E [0,1] and all (XI, ... , X6), (Yl," " Y6) in It> Let also ~i E (0,1), ai E R, i = 1, . .. , m and '1 E R be given.

Then if ai, i = 1, ... , m are all of the same sign, the BVP (E) - (BCh has exactly one solution, provided that

6

L IICilh < 1. (4.2) ;=1

Moreover, if ai E R, i = 1, .. . , m, with L:~l ai # 0, the BVP (E) - (BCh has exactly one solution, provided that

s L L IIcdll < 1. (4.3)

;=1

Proof. The existence of a solution of the BVP (E) - (BCh follows immediately from Theorem 3.1. by setting Ci = Pi,i = 1, ... ,6 and I(t) = If(t,O, ... ,O)I,t E [0,1]. Let now, X,Y be two solutions of the BVP (E) - (BCh· We then get

xlll(t) - ylII(t) = f(t,x(t),x(u(t)),x'(t),x'(g(t)),x"(t),X"(T(t))) - f(t, y(t), y(u(t)), y'(t), y'(g(t)), y"(t), y"( T(t))), t E [0,1]

and (x - y)I[-r,O] = 0 = (x - Y)I[I,dJ,L:~l ai(x - Y)(~i) = O. From (4.1) we get

IIXIll - ylllih :::; (lIcdh + IIC21h)lIx - Ylloo + (II C3Ih + II C4IDllx' - y'lioo + (lIcslh

+ lIeslldllx" - y"lIoo

286

BVPs FOR FDEs

Then by Lemma 2.3 we have

6

IIXIll - ylllill :S (L 11<; lid II XIII - ylllill' i=l

By (4.2), the last inequa.lityimpliesthat IIxlll - yllllll = O. Hence, x(t) = y(t) , t E [0,1) . Since x(t) = y(t), t E [-r,O) and x(t) = y(t), t E [d, 1], we have x = y.

The uniqueness for the BVP (E) - (BCh can be proved by a similar way.

Theorem 4.2. Let f : [0,1) x R!' -+ R be a function as in Theorem 4.1. Let also ei E (0,1), ai E R , i = 1, . .. , m and." E R be given .

Then if ai, i = 1, .. . , m are all of the same sign, the BVP (E) - (bch, (E) - (bch and (E) - (bC)4 have exactly one solution, provided that

6

L II Cili1 < 1. i=}

Moreover, if ai E R, i = 1, . . . , m with L:~1 ai i- 0, the BVPs (E) - (bch and (E) - (bc)s have exactly one solution, provided that

4

L L 11<;111 + IIcslh + 11<;116 < 1. i=l

The proof is omitted since it is similar to that of Theorem 5.1.

Remark 5.3 The uniqueness result in Theorem 5.1 is obtained via the Theorem 3.1. It is obvious that analogous results can be taken using Theorems 3.2 and 3.3. The same holds for the result of Theorem 4.2.

References

1. A. Boucherif and J . Henderson, Two point boundary value problems for fourth order ordinary differential equations, Pan. Math. J . 3 (1993) , 1-21.

2. J . Dugundji and A. Granas, "Fixed Point Theory", Vol. I, Monographie Matematyczne, PNW Warsawa, 1982.

3. C. Gupta and V. Lakshmikantham, Existence and uniqueness for a third order treee-point boundary value problem, Nonlinear Analysis TMA 16 (1991), 949-957.

4. C. Gupta , S. Ntouyas and P. Tsamatos, On an m-point boundary value problem for second order ordinary differential equations Nonlinear Analysis TMA 23 (1994), 1427-1436.

287


5. C. Gupta , S. Ntouyas and P. Tsamatos, Solvability of an m-point boundary value problem for second order ordinary differential equations, J. Math . Anal. Appl. 189 {1995}, 575-584.

6. C. Gupta , S. Ntouyas and P. Tsamatos,Existence results for m-point value problems, Diff. Equations and Dynamical Systems {to appear}

7. D. Rai and K. Schmitt, Boundary value problems for higher order differential equations, Nonlinear Analysis TMA 21 {1993}, 293-305.

8. J.Lee and D. O'Regan, Existence results for differential delay equations I, J . Differential Equations 102 {1993}, 342-359.

9. J.Lee and D. O'Regan, Existence results for differential delay equations II, Nonlinear Analysis 17 {1991}, 683-702.

10. S. Ntouyas and P. Tsamatos ,On well-posedness of boundary value problems for differential equations involving deviating arguments, Funcial. Ekvac. 35 {1992},137-147.

11. S. Ntouyas and P. Tsamatos ,Existence and uniqueness for secod order boundary value problems, Funcial. Ekvac. {to appear} .

12. D. J . O'Regan,Topological transversality:Applications to third order boundary value problemsSIAM J. Math. Anal. 18 {1987},630-641.

13. P. Tsamatos and S. Ntouyas, Existence of solutions of boundary value problems for differential equations with deviating arguments, via the Topological Transversality method, Prot. Royal Soc. Edinburgh. U8A {1991},79-89.

14. P. Tsamatos and S. Ntouyas, Existence and uniqueness of solutions for boundary value problems for differential equations with deviating arguments, Nonlinear Analysis TMA 22 {1994},1131-1146.

PERIODIC SOLUTIONS AND LIAPUNOV FUNCTIONALS

TINGXIU WANG Department of Mathematics and Computer Science Oakton Community College, 1600 East Golf Road

Des Plaines, fllinois 60016 USA

Abstract

It is well-known that for Equation (1), if X(t + T) is a solution of (1) when X(t) is a solution of (1), and ifthe solutions of (1) are uniformly bounded (U.B.) and ultimately uniformly bounded (U.U.B.), then (1) has a T-periodic solution. Based on the theorems the author obtained in his recent papers, in this paper we will examine some common functional differential equations and obtain not only U.B. and U.U.B ., but also periodic solutions. The examples we discussed here are nonlinear and nonautonomous. Since the functional differential equations are periodic systems, the results we get are quite clean and easy to use.

1. Introduction

We consider the functional differential equation with bounded delay

(1)

289

where Xt(s) = X(t + s) for -h ~ s ~ 0 and h is a positive constant . We will use the following notation.

Let (C, 11·11) be the Banach space of continuous functions r/>: [-h,O] -> RD with the norm 1Ir/>1I = maX_h<.<O 1r/>(s)1 and 1· 1 is any convenient norm in RD . To discuss the existence of periodi~ ~olutions, we generally assume that F : R x C -> RD be continuous and take bounded sets into bounded sets with F(t + T,r/» = F(t,r/» for some T > 0 and each (t, r/» E R x C. This assumption, in fact, implies that F takes bounded sets of r/> into bounded sets. Then it is known [1] that for each to E Rand each r/> E C there is at least one solution X(to, r/» of (1) satisfying Xta = r/> defined on an interval [to, to + a) and if there is an HI > 0 with IX(t, to, r/»I < HI, then a = 00.

Let R+ = [0,(0) . In this paper, we will work with wedges, denoted by Wi : R+ ->

R+, which are continuous and strictly increasing, which also satisfy Wi(O) = o. Let

290

BVPs FOR FDEs

V : R+ x C -+ R+ be continuous. The derivative of V with respect to (1) is defined by

V;(~)(t, 4» = lim sup[V(t + S, XtH(t, 4>)) - Vet, 4»)/S. 5_0+

Definition 1. Solutions of (1) are uniformly bounded (U.B.) iffor each BJ > 0 there exists a B2 > 0 such that [to ~ 0, 114>11 ~ B}, t ~ to) imply that IX(t , to , 4»1 0 there exists T > 0 such that [to ~ 0, 114>11 ~ B3, t ~ to + T) imply IX(t, to, 4»1 < B.

In the theory of existence of periodic solutions, the next theorem is quite wellknown. (see [1, p.249))

Theorem 1. Suppose that X(t + T) is a solution of (1) whenever X(t) is a 'solution of (1). If solutions of (1) are U.B. and U. U.B. for bound B, then (1) has aT-periodic solution.

Although there have been many results and methods on the existence of periodic solutions, these results and methods can only be applied with some restrictions. Grimmer [5) obtained the existence of periodic solutions with the comparison method. But setting up a comparison inequality for a Liapunov functional is often not simple. Makay [8) discussed the periodic solutions of dissipative functional differential equations. But there has not been a good theorem to obtain dissipativeness. Although in his paper [8), he gave a theorem of Liapunov functional type to obtain dissipativeness, the Liapunov functional of the theorem seems too difficult to construct . But dissipativeness is an interesting direction of research for periodic solutions. Some investigators [2, 4, 11] have also used homotopy to obtain the periodic solutions. But this method only works for a class of functional differential equations.

It is clear that Theorem 1 gives a very good criterion for the existence of periodic solutions if there is a good theorem on uniform boundedness and uniform ultimate boundedness. The author [9, 10] have obtained some quite general theorems on U.B. and U.U.B. In this paper, we are going to use the theorems in [9, 10) to obtain periodic solutions for some functional differential equations. Thus this paper is also a continuation of [9, 10). The following definitions and theorems are from [9 , 10) with some modifications. Although [12) also obtained a quite general theorem on U.B. and U.U.B., our Theorem 3 is not included by it. Our examples will show this .

Definition 2. A measurable function 'Y : R+ -+ R+ is said to be integrally positive with parameter a> 0 (IP(a» if whenever I = L~=l [am, .am] with am < .am < am+! and.am - am ~ a, m=l, 2,3, .. . , then JI'Y(s)ds = 00.

It is well-known (see [3)) that." is IP(a) for some a > 0 if and only if limt_co infJ/+"'.,,(s)ds > O. That is, ."dP(a) implies that there exist T > 0, and r > 0 such that for each t > T, Jtt+"'.,,(s)ds ~ r . Thus we also denote IP(a,r) = IP(a) . With this fact, we give a weaker definition than the last one.

Definition 3. A me&'urable function 'Y : R+ -+ R+ is said to be partially integrally

291

PERIODIC SOLUTIONS

positive with parameters a > 0, {3 > 0, and f > 0 (P IP( a, {3, f)) if there is a sequence {tnli with a ::; tn+! - tn ::; {3 such that It~+Ot .,,(s)ds ~ f.

Clearly if ." is T-periodic for some constant T > 0, then ."dP( cr, r} and ." E PIP(a,/3,r}, where f = It .,,(s)ds, and cr, {3 can be any constants with {3 ~ cr ~ T . Particularly, ." E IP( T, f) and ." E PIP( T, T, f).

We also need convex functions. For reference, we note that if W : R+ -+ R+ with W(~) ::; W(tl)~W(t2) for any t l , t2 E R+, then W is convex downward.

Definition 4. A functional D : R+ x C -+ R+ is said to be continuous along solutions of (1) if D(t,Xt ) is continuous on [to, co) for each solution X(t,to,¢» of (1) defined on [to, co).

Define m(¢» = min 1¢>(s)1 for each ¢> E C.

-"::;_::;0

Theorem 2. Let V : R+ x C --+ R+ be continuous and D : R+ x C --+ R+ be continuous along solutions of (1). Let II E P IP( cr, {3, f I) and 12 E IP( h, f 2)' Denote C = max{{3,h} . Let WI, W2, W3, W 4 , Ws be wedges with WI(r) --+ co, as r --+ co. Assume that

(i) WI(IX(t)1) ::; V(t,Xt ) ::; W 2{1X(t)1 + IL" D(u,Xu)du); (ii) V(t, ¢» ::; W3 (II</1I1); (iii) V(;)(t, Xt) ::; -/J(t)W4(m(Xt)) - 12(t)Ws(JL" D(u, Xu)du) + M; (iv) There is a e > 0 such that

Then solutions of (1) are U.B. and U. U.B.

Theorem 3. Let V : R+ x C --+ R+ be continuous and D : R+ x C --+ R+ be continuous along solutions of (1) . Let IE PIP(cr, {3,f) and denote C = max{{3,h}. Assume that WI(r) --+ co, as r --+ co, and that

to

(i) WJ{lX(t)l)::; V(t,Xd::; W 2(IX(t)1 + IL"D(u,Xu)du) ; (ii) D(t,</I)::; W3 (II</1I1); (iii) V(;)(t,Xt )::; -/(t)W4(m(Xt )) - Ws(D(t,Xt )) + M ; (iv) There are Uo > 0 and e > 0 such that W 4(e)f > 11MC and for each U ~ Uo,

h loU ( I( () C)) Ws(s)ds> llMC. W3 WI W2 2U + 11M 0

Then solutions of (1) are U.B. and U. U.B. Moreover, ifWs is convex downward, then Condition (ii) and (iv) can be weakened

(ii), V(t, </I) ::; W3 (11</I1I), and (iv) , W 4Wf > llMC for some e > o. The proofs of the above two theorems can be found in [9,10] .

292

BVPs FOR FDEs

Remark 1. It is known [10] that

(2)

is equivalent to

(3)

The difference is that (2) is convenient in proofs while (3) is often seen in applications.

2. Scalar Equations

Example 1. Consider the nonlinear equation

(4) x'(t) = -a(t)W(x(t)) + b(t)W(x(t - h)) + e(t)

where a: R --+ R+, b, e, and W: R --+ R are continuous.

Theorem 4. Let T > 0 be some constant. Assume that a, b, and e are T -periodic. Let M > 0 be the constant with le(t)1 ::; M. Assume the following conditions hold:

(a). There is a constant k > 1 such that a(t) - klb(t + h)1 := /(t) ~ O. Let r = Jl/(s)ds > 0, and C = max{T,h} . (b). W is odd, nondecreasing and there is a constant ~ > 0 with W(or > lIMC . Then the solutions of (4) are U.B. and U. U.B . and the equation also has a non-

trivial T -periodic solution.

Proof. Clearly x(t + T) is a solution of (4) when x(t) is a solution of (4). Because k > 1, take kl > 1 with k > kl > 1 and let 6 = k - kl > O. Then

-a(t) + k1lb(t + h)1 ::; -/(t) - 6lb(t + h)l.

Define

v(t, </» = 1</>(0)1 + kl lh Ib(t + s + h)IIW(</>(s))lds .

Let x(t) be a solution of (4). Then

v(t,Xt) = Ix(t)1 + kl Eh Ib(s + h)IIW(x(s))lds.

Obviously when W is restricted on [0,00), W is a wedge. Define D(t, </» := Ib(t + h)IIW(</>(O))I· Then D(t, </» ::; BI WUI</>II), where BI is an

upper bound of b(t). Clearly,

Ix(t)1 ::; v(t, Xt) ::; Ix(t)1 + kl Eh D(s, xs)ds.

293

PERIODIC SOLUTIONS

Also

V(4)(t,Xt) ~ -a(t)W(lx(t)l) + Ib(t)IIW(x(t - h))1 + le(t)1

+k1Ib(t + h)IIW(x(t))l- k1Ib(t)IIW(x(t - h))1

~ [-a(t) + k1lb(t + h)I]W(lx(t)1) + (1- k1)lb(t)IIW(x(t - h))1 + M

~ [-,(t) - hlb(t + h)IJW(lx(t)1) + M

~ -,(t)W(lx(t)1) - hD(t,xt) + M.

So all the conditions of Theorem 3 are satisfied and therefore solutions of (4) are U.B. and U.U.B. By Theorem 1, (4) has a T-periodic solution which obviously is nontrivial because e(t) =F 0 and x == 0 is not a solution.

Remark 2. It is not hard to see that none of theorems in [5, 8] can be applied here to obtain the periodic solution. [12] also gave a quite general theorem on U.B. and U.U.B. But the theorem in [12] can not be applied here either, because [12] requires b(t) satisfying (3 ~ IttH Ib(s)lds ~ 1 for some constants (3 > 0 and L > 0 and for each t E R+. For instance, b(t) = Isintl + sint will not satisfy the condition. In fact, if L < 11", then ftt;:+L Ib(s)lds = 0, where tn = (2n + 1)11", n = 0,1,2,··· . If L ~ 11", then I:;:+L Ib(s)lds ~ I:;:+7f Ib(s)lds = 4, where Sn = 2n1l", n = 0, 1,2,··· . This remark can also be applied to the later examples.

Example 2. (a special case of Example 1) The nonlinear scalar equation

, . 3 x(t) x(t - h) cos2t x (t) = -(I sm tl + "21 cos(t + h)1) 0.5 + Ix(t)1 - cos t 0.5 + Ix(t _ h)1 + ---U;'

where h = ~, has a nontrivial 211"-periodic solution.

Example 3. Consider the nonlinear integro-differential equation

(5) x'(t) = -a(t)W(x(t)) + l~h b(t - s)W(x(s))ds + e(t),

where a: R ~ R+, b: R+ ~ R, e and W: R ~ R are continuous.

Theorem 5. Let the functions a, and e be T-periodic, and r = I{ a(s)ds > O. Obviously a E PIP(T, T, r). Let C = max{T, h} and M > 0 be the upper bound of e(t). Assume all a, b, e, W satisfy the following conditions:

(a). There are constants k > 0 and h > 0 such that kh ~ 1 and

Ib(t - s)l- ka(s) ~ -ha(t), for - 00 < s ~ t < 00.

(b). W is odd and nondecreasing, and there is a constant ~ > 0 with W(~)hr > llMC.

Then solutions of (5) are U.B. and U. U.B. Therefore (5) has a nontrivial Tperiodic solution.

294

BVPs FOR FDEs

Proof. Obviously, if x(t) is a solution of (5), then x(t + T) is also a solution of (5) . Define

v(t,Xt) = Ix(t)1 + kjO t a(s)IW(x(s))lds, -h 1t+6

and define D(t,¢» = a(t)IW(¢>(O))I. Then

v(S)(t,Xt) ::; -a(t)W(lx(t)l) + lh Ib(t - s)IIW(x(s))lds + le(t)1

+kha(t)IW(x(t))I- k lh a(s)IW(x(s))lds

::; (-1 + kh)a(t)W(lx(t)l) + l)lb(t - s)l- ka(s))IW(x(s))lds + M

::; -Slh a(s)IW(x(s))lds + M

::; - S~ W(m(xt)) - ~ lh a(s)IW(x(s))lds + M

::; - Sf W(m(xt)) - ~2 t D(s, x.)ds + M. 2 1t-h

The other conditions of Theorem 2 can be verified easily. By Theorem 2, the solutions of (5) are V.B. and V.V .B. And by Theorem 1, (5) has a T-periodic solution, which is nontrivial.

Example 4. (a special case of Example 3) Consider the integro-differential equation

x'(t) = -(2 + 1 cos tl)x3(t) + t 1 x3(s )ds + M sin 3t, 1t-l t - s + 1

where M is any given constant, has a nontrivial 7r-periodic solution.

Example 5. Consider the integro-differential equation with infinite delay

(6) x'(t) = -a(t)W(x(t)) + Loo g(t - s)W(x(s))ds + e(t)

where a : R -+ R+, 9 : R+ -+ R, e : R -+ R are continuous.

Theorem 6. Let the functions a, and e be T-periodic, f = K{ a(s)ds > O. Obviously a E PIP(T, T,f). Let M > 0 be the upper bound of e(t). Assume all a, g, e, W satisfy the following conditions:

(aJ. 9 E Ll[O,OO) and 2:::"=og(t + nT) is uniformly convergent on [O,T). (b). There are constants k > 0 and S > 0 such that kh ::; 1 and

00

1 E g(t + nT)I- ka(s) ::; -Sa(t), for ~ 00 < s ::; t < 00.

n=O

295

PERIODIC SOLUTIONS

(c). W is odd and nondecreasing, and there is a constant e > 0 with W(e)6r > 22M.

Then (6) has a nontrivial T -periodic solution.

Proof. Consider the equation

(7) x'(t) = -a(t)W(x(t)) + l~T ~ g(t - s + nT)W(x(s))ds + e(t).

In Theorem 5, take b(t) = L~=o9(t + nT). If x(t) is a T-periodic solution and Assumption (a) holds, then

loo g(t - s)W(x(s))ds f rt

-

nT g(t - s)W(x(s))ds

n=O it-(n+l)T

r 00

it Lg(t-v+nT)W(x(v))dv. t-T n=O

So a T-periodic solution of (6) is a T-periodic solution of (7) and vice versa. By Theorem 5, Equation (7) has a T-periodic solution, and hence (6) has also a

T-periodic solution, which clearly is nontrivial. The equivalence between (6) and (7) with respect to periodic solutions was first

given by Grimmer [5].

3. Vector Equations

All the equations of the examples discussed in Section 2 are scalar. In this section, we are going to discuss vector equations. The difference is that all the scalar equations are nonlinear. But for vector equations, we will discuss linear equations. Throughout this section,

AT denotes the transpose of a matrix A and IXI = ~ ~ Xf, X E R".

Example 6. Consider the n-dimensional integro-differential equation

(8) X'(t) = A(t)X(t) + th B(t - s)X(s)ds + F(t),

where X E R", A is an n x n real matrix of continuous functions defined on R, B is an n X n real matrix of continuous functions defined on R+ = [0, 00), and F : R -+ R" is continuous.

Theorem 7. Let A and F be T -periodic for some constant T > O. Assume the following conditions hold:

296

BVPs FOR FDEs

(a). Let A(t) = ~(AT(t) + A(t)), and Al(t), A2(t), " ' , An(t) be the n eigenvalues of A(t) with

A(t) := max{Al(t), A2(t) , .. · ,An(t)} :::; 0, and r := loT IA(s)lds > O.

(b). Assume that there is a constant k > 0 such that kh < 1 and

IB(t - s)l- kIA(S)1 :::; 0 for - 00 < s:::; t < 00 .

Then the solutions of (8) are U.B. and U.U.B. (8) has a nontrivial T- periodic solution.

To prove the theorem we need the following lemma.

Lemma 1. Let A(t) = !(AT(t) + A(t)) where A(t) is a real n x n matrix. Let Al(t), A2(t), . .. , An(t) be the n eigenvalues of A(t) with

If X(t) is a solution of (8), then

DrIX(t)1 :::; A(i)IX(t)1 + th IB(t - s)IIX(s)lds + IF(t)l,

where Dr denotes the derivative from the right.

Proof. Because A(t) is a real symmetric matrix, there is an orthogonal matrix O(t) such that

[

Al(t) 0 .. · OAt .. .

OT(t)A(t)O(t) = : 2i) . .

o 0

o o

Let Y = OT(t)X, that is X = O(t)Y or XT = yTOT(t). Then

[

Al(t) 0

XT A(t)X yTOT(t)A(t)O(t)Y = yT ~ A2it)

o 0

:::; A(t)yTy = A(t)XT X = A(t)IX(tW,

o o

By Lemma 6.1 in [6], DrIX(t)1 :::; IX'(t)l. So clearly when X(t) = 0,

DrIX(t)1 :::; A(t)IX(t)1 + th IB(t - s)IIX(s)lds + IF(t)l·

y

297

PERIODIC SOLUTIONS

When X(t) =I 0,

DrIX(t)1 DrJXT(t)X(t)

1 [XT(t)X(t)l' 2JXT(t)X(t)

1 :::; X

2JXT(t)X(t)

[2XT(t)A(t)X(t) + 2IX(t)1 {h IB(t - s)IIX(s)lds + 2IX (t)IIF(t)l]

:::; A(t)IX(t)1 + l~h IB(t - s)IIX(s)lds + IF(t)l ·

Proof of Theorem 7. Because A is T-periodic, A(t) is T-periodic and obviously A(t) is a real symmetric matrix. Therefore all eigenvalues of A(t) are real and Tperiodic. Hence A(t) is T-periodic and IAI E PIP(T, T, r). Let M > 0 be the upper bound of F. Define

V(t, Xd = IX(t)1 + k lh 1:. IA(U)IIX(u)lduds.

Then

V(~)(t,Xt) :::; -IA(t)IIX(t)1 + Lh IB(t - s)IIX(s)lds + M

+khIA(t)IIX(t)l- k {h IA(S)IIX(s)lds

< (-1 + kh)IA(t)IIX(t)1 + {)IB(t - s)l- kIA(S)llIX(s)lds + M

:::; -1; kh IA(t)IIX(t)1 + -1; kh D(t, Xd + M

where D(t,Xt) = IA(t)IIX(t)l. Clearly

IX(t)1 :::; V(t,Xt) :::; IX(t)1 + kh {h IA(S)IIX(s)lds.

The other conditions of Theorem 3 can be verified easily and therefore the solutions of (8) are U.H. and U.U.B. and (8) has a nontrivial T-periodic solution.

Example 7. Consider the n-dimensional integro-differential Volterra equation with infinite delay

(9) X' = A(t)X(t) + loa H(t - s)X(s)ds + F(t),

298

BVPs FOR FDEs

where X E Rn, A is an n x n real matrix of continuous functions defined on R, H is an n x n real matrix of continuous functions defined on R+ = [0,00), and F : R --+ Rn is continuous.

Theorem 8. Let A and F be T-periodic for some constant T > o. Assume the following conditions hold:

(a). HE L1[0, 00) and "L':=o H(t + nT) is uniformly convergent on [0, Tj. (b). Let A(t) = !(AT(t) + A(t)) and A1(t), A2(t), . .. , An(t) be the n eigenvalues

of A(t) with

A(t) := max{A1(t), A2(t),.· · , An(t)} ~ 0, and loT IA(s)lds > O.

(c). There is a constant k > 0 such that kh < 1 and

1 L: H(t - s)l- kIA(S)1 ~ 0 for - 00 < s ~ t < 00 .

n=O

Then (9) has a nontrivial T -periodic solution. Theorem 8 is a direct corollary of Theorem 7 and the proof is similar to that of

Theorem 6 and it is omitted here. Many investigators [2, 4, 5) have discussed Equation (9) and obtained periodic

solutions. But they assumed A is constant. Our Theorem 8 is clearly different from theirs.

4. References

1. T.A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, Orlando, Florida, 1985.

2. T.A. Burton, P.W. Eloe, and M.N. Islam, Periodic solutions of linear integrodifferential equations, Math . Nachr. 147(1990), 175-184.

3. T.A. Burton and L. Hatvani, Stability theorems for nonautonomous functional differential equations by Liapunov functionals, Tohoku Math. J. 41(1989), 65-104.

4. T .A. Burton and B. Zhang, Periodic solutions of abstract functional differential equations with infinite delay, J. Differential Equations 90 (1991), 357-396.

5. R. Grimmer, Existence of periodic solutions of functional differential equations, J. Math. Anal. Appl. 72 (1979), 666-673.

6. J. Hale, Ordinary Differential Equations, Wiley, New York, 1969.

7. J. Hale and O. Lopes, Fixed point theorems and dissipative processes, J. Differential Equations 13 (1973), 391-402.

299

PERIODIC SOLUTIONS

8. G. Makay, Periodic solutions of dissipative functional differential equations, Tohoku Math. J. 46 (1994), 417-426.

9. T. Wang, Uniform boundedness with the condition, V'(t,Xt ) ::; -,(t)W4(m(Xt ))

-Ws(D(t,Xt )) +M, Nonlinear Analysis, to appear.

10. T. Wang, Some general theorems on uniform boundedness for functional differential equations, Dynamic Systems and Applications, to appear.

11. B. Zhang, Periodic solutions of nonlinear abstract functional differential equations with infinite delay, Funkcialaj Ekvacioj 36 (1993), 433-478.

12. B. Zhang, Boundedness in functional differential equations, Nonlinear Analysis 22 (1994) 1511-1527.

BOUNDARY VALUE PROBLEMS OF SECOND ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS

BO ZHANG Department of Mathematics and Computer Science

Fayetteville State University Fayetteville, NC 28301

1. Introduction

301

We consider the boundary value problem of second order functional differential equations

(A(t)x'(t)), = f(t, Xt, x'(t)), x(t) E W

Xo = "p, Ax(T) + Bx'(T) = v

(1.1)

where A(t) is an n x n continuous matrix defined on [O,T] . A and Bare n x n constant matrices, v E W, and "p E C = C([-h, 0], Ir') the space of all continuous functions r/> : [-h,O] -+ Ir' with the suppremum norm 11 · 11 where h is a positive real number or 00. If x is a continuous function of u defined on -h ~ u < A, A> D , then for each t with D ~ t < A ,Xt is an element of C defined by xM) = x(t + 0) for 0 E [-h,D].

Problem (1.1) was studied by many authors, for reference and history we refer to the work of Ladde and Pachpatte [1], DalE and Zhang [2], Ntouyas [3], Tsamatos and Ntouyas [4] . The purpose of the present paper is to generalize the results metioned above and to provide a systematic approach for the boundary value problems of functional differential equations.

Let R = (-00, +00), R+ = [0, +00), R- = (-00,0]' and 1·1 denote the Euclidean norm on Ir'. For an n x n matrix A, define the norm IAI of A by IAI = sup{ IAxl : x E Ir', Ixl ~ 1}. Let a : [0, T] -+ Rn be a bounded function, define lIall = sup{la(t) 1 : t E [0, T]}.

2. Existence and Uniqueness

The following lemma concerning a boundary value problem of the second order ordinary differential equation will play an important role for establishing the existence and uniqueness of problem (1.1).

302

BVPs FOR FDEs

Lemma 2.1. Suppose that A(t) is an n x n continuous invertible matrix defined on [O,T), A and B are n x n constant matrices, and 9 : [0, T) -+ Rn is continuous. If B + A Jt A-I (s )dsA(T) is invertible, then the problem

(A(t)x'(t))' = g(t), x(t) E Rn

x(O) = e, Ax(T) + Bx'(T) = v

has a unique solution on [0, T) for any e, vERn.

Proof. We define

A*(t) = lA-1(s)ds, D(T)=B+AA*(T)A(T).

Integrate (2.1) from t to T to obtain

A(T)x'(T) - A(t)x'(t) = iT g(u)du,

Thus,

A(t)x'(t) A(T)x'(T) -iT g(u)du,

x'(t) A-1(t)A(T)x'(T) - A-1(t) iT g(u)du,

x(t) x(O) + l A-1(s)dsA(T)x'(T) -l A-l(S) J.T g(u)duds,

e + l A-1(s)dsA(T)x'(T) -l A-l(S) J.T g(u)duds.

(2.1)

Ax(T) - AA *(T)A(T)x'(T) = Ae - A loT A-I (s) J.T g( u )duds (2.2)

Substituting the condition Ax(T) + Bx'(T) = v to (2.2) to obtain

(B + AA*(T)A(T))x'(T) = v - Ae + A loT A -1(S) J.T g(u)duds .

This yields

and

x(t) (1 - A*(t)A(T)D-l(T)A)e

+ A*(t)A(T)D-1(T)(v+A loT A-l(S) J.T g(u)duds)

- l A-l(S) J.T g(u)duds.

303

SECOND ORDER FDEs

It also follows that

Ix(t)1 :::; (1 + IIA*IIIA(T)IID-1(T)IIADI(1 + IWIIIA(T)IID-1(T)llvl (2.3)

and

+ IIA*IIIA(T)IID-1(T)IIAI loT IA-I(S)I J.T Ig(u)lduds

+ loT IA-I(S)I J.T Ig(u)lduds

< (1 + IIA*IIIA(T)IID-1(T)IIADI(1 + IIA*IIIA(T)IID-1(T)llvl

+ (1 + IIA*IIIA(T)IID-1(T)IIADT loT IA-1(s)ldsllgll,

Ix'(t)1 :::; IIA -IIlIA(T)IID-1(T)I(IAliel + IVD

+ (1 + IAITIIA-11IIA(T)IID-1(T)DIIA-1IITllgll.

This completes the proof of Lamma 2.l.

(2.4)

We are ready to state our main theorem in this section. For simplicity in notations we define KI = Jt IA-1(s)lds and K2 = IIA-11I

Theorem 2.1. Suppose that f : [0, T] x ex R" -t R" is continuous and satisfies on E = [0, T] x C x Rn the Lipschitz condition

(2.5)

for some positive constants (JI and (J2. If

(2.6)

then there exists a unique solution of the boundary value problem (1.1) .

Proof. Let t/J E C and define S to be the space of continuous functions x : [-h, T]-t R" which are continuously differentiable on [0, T] with Xo = 1/;. Define

d(x, y) = max{K2 max Ix(t) - y(t)l, KI max Ix'(t) - y'(t)I} . (2.7) OS'$T OS'$T

Then S is a complete metric space with the distance function d. Let xES. Consider the problem

(A(t)u'(t))' = f(t, Xl, x'(t)), (2.8) u(O) = 1/;(0), Au(T) + Bu'(T) = v.

304

BVPs FOR FDEs

By Lemma 2.1, problem (2.8) has a unique solution u defined on (0, T) . Now we define xES by x(t) = u(t) on (0, T) and Xo = 1/;. Then xES. Next we define a mapping P: S -t S by

P(x)=x, xES

For any x, yES we define z = x - ii. Then z satisfies

(A(t)z'(t)), = f(t, XI, x'(t)) - f(t, YI, y'(t)) , Zo = 0, Az(T) + BZ'(T) = O.

It follows from (2.3) and (2.4) that

and

Jz(t)J ::; (1 + JAJT I<2JA(T)JJD-!(T)I)T I<!

+ max {Jf(t, XI, x'(t)) - f(t, YI, y'(t))J} 09~T

Jz'(t)J ::; (1 + JAJTI<2JA(T)JJD-!(T)I)I<2T

+ max {Jf(t,XI ,X'(t)) - f(t,YI,y'(t))J} . 09~T

Let L = 1 + JAJTI<2JA(T)JJD-!(T)J. By the definition of d we have

d(x, ii) max{I<2 max Jx(t) - y(t)J , I<! max Jx'(t) - y'(t)J} 09~T O~I~T

::; LTI<!I<2[9! max Jx(t) - y(t)J + 92 max Jx'(t) - y'(t)JJ 09~T 09~T

(2.9)

(2.10)

(2.11)

(2 .12)

(2.13)

Since (1 + TI<2JAJJA(T)JJD-!(T)I)T(9!I<1 + 92I<2) < 1, we conclude that P is a contraction mapping. Therefore there exists a unique xES such that P(x) = x. This implies that X is the unique solution of (1.1). We complete the proof of Theorem 2.1.

SECOND ORDER FDEs

3. Continuous Dependence

The result in Section 2 remains valid if we consider

(A(t)x'(t))' = f(t,Xt,X'(t),CI!), x(t) E Rn

Xo = tP, Ax(T) + Bx'(T) = v

where CI! is a real parameter and

on [0, T] X C X ~ X R for some positive constants 81 and 82 with

305

(3.1)

(3.2)

(3.3)

We will show that the solution of (3.1) depends continuously on the parameter CI! if

(3.4)

for some positive constant N.

Theorem 3.1. Suppose that f : [0, T] X C X Rn x R -t R is continuous. If (3.2), (3.3), and (3.4) are satisfied, then the solution of (3.1) obtained by Theorem 2.1 depends continuously on CI!.

Proof. By Theorem 2.1, there exists a unique solution x(t) = x(t, CI!) of (3.1) for each CI!. Let x(t) = X(t,CI!l) and y(t) = y(t,CI!2) be solutions of (3.1) with CI! = Cl!l and CI! = Cl!2, respectively. By (2.13) with L = 1 + IAIT I<2IA(T)IID-l(T)I, we have

d(x,y) :::; LT I<1I<2 max {If(t, Xt, x'(t), al) - f(t, Yt, y'(t), Cl!2)1} 09~T

(3.5)

:::; LTI<1I<2 max {If(t,xt,x'(t),CI!l) - f(t,Yt,y'(t),CI!l)l} 09~T

+LT I<1I<2 max {If(t, Yt, y'(t), Cl!l) - f(t,Yt, y'(t), Cl!2)1} 09~T

:::; LTI<1I<2 [01 max Ix(t) - y(t)1 + O2 max Ix'(t) - y'(t)IJ O~t~T 09~T

+LTI<1I<2NICI!l - Cl!21

:::; LT(81I<1 + 82I(2)d(x,y) + LTI<1I<2 N ICI!l - Cl!21·

Define LTI<1I<2N L* -

- 1 - LT(81I<1 + 82I(2)·

306

BVPs FOR FDEs

Then

(3.6)

This implies that the solution of (3.1) depends continuously on a.

Remark 3.1. It is clear from the proof of Theorem 3.1 that the solution of (3.1) also depends continuously on the boundary values (t/J, v).

Remark 3.2. The method used in Section 2 and Section 3 is still valid if f satisfies a Lipschitz condition of the form

If(t, t/J, u, a) - f(t, J;, u, a)1 :::; 91(t)IIt/J - J;II + 92(t)lu - ul

for some positive continuous functions 91 and 92 .

References

1. G. S. LaddIe and B. G. Pachpatte, Existence theorems for a class of functional differential systems, J. Math. Anal. Appl. 90 (1982), 381-392.

2. B. S. Lalli and B. G. Zhang, Boundary value problems for second order functional differential equations, Ann. Differential Equations 8 (1992), 261-268.

3. S. K. Ntouyas, A boundary value problem for second order functional differential equations, Hiroshima Math . J. 12 (1982), 453-468.

4. P. C. Tsamatos and S. K. Ntouyas, Existence and uniquenes of solutions of a general type boundary value problem for second order functional differential equations, J. Math. Anal. Appl. 173 (1993), 3-17.

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Boundary Value Problems for Functional Differential Equations › 2018 › ... · With the importance of boundary value problems for functional differential equa tions in applications,