OF THE UNIVERSITY OF ZILINAˇ - uniza.skfpv.uniza.sk/cddea2008/files/studies23.pdf · OF THE UNIVERSITY OF ZILINAˇ ... Mu˝egyetem rkp. 3, ... [email protected] Studies

STUDIESOF THE UNIVERSITY OF ZILINA

Volume 23 Number 1 October 2009Special Issue: Proceedings of Conference on Differential and Difference

Equations and Applications (CDDEA), Strecno, Slovakia

June 23–27, 2008, Editors: J. Diblık and M. Ruzickova

Editors-in Chief Managing Editor

V. Balint, Zilina J. Diblık, Brno, Zilina R. Blasko, Zilina

Editorial Board

A. Dekret, B. Bystrica V. Dolnikov, Yaroslavl F. Fodor, Szeged

R. Fric, Kosice A. G. Horvath, Budapest T. Kusano, Fukuoka

P. Mihok, Kosice S. Paluch, Zilina M. Ruzickova, Zilina

J. Skokan, Urbana I. P. Stavroulakis, Ioannina M. Wozniak, Krakow

M A T H E M A T I C A L S E R I E S

Editorial Board

Vojtech Balint (Editor-in Chief), Department of Mathematics, Faculty of Operation and Eco-

nomics of Transport and Communications, University of Zilina, Univerzitna 1, 010 26 Zilina,

Slovakia, [email protected]

Anton Dekret, Department of Informatics, Faculty FPB, University of Matej Bell, Tajovskeho

40, 974 01 Banska Bystrica, Slovakia, [email protected]

Josef Diblık, Department of Mathematical Analysis and Applied Mathematics, Faculty of Science,

University of Zilina, Hurbanova 15, 010 26 Zilina, Slovakia, [email protected], University

of Technology, 628 00 Brno, Czech republic, [email protected]

Vladimir Dolnikov, Department of Mathematics, Yaroslavl State University, Sovetskaya 14, 150

000 Yaroslavl, Russia, [email protected]

Ferenc Fodor, Geometriai Tanszek, Bolyai Intezet, Aradi Vertanuk tere 1., H-6720 Szeged,

Hungary, [email protected]

Roman Fric, MU SAV, Gresakova 6, 041 01 Kosice, Slovakia, [email protected]

Akos G. Horvath, BME, Muegyetem rkp. 3, H-1111 Budapest, Hungary, [email protected]

Takasi Kusano, Department of Applied Mathematics, Faculty of Science, Fukuoka University,

Fukuoka, 814-01, Japan, [email protected]

Peter Mihok, Department of Applied Mathematics, Faculty of Economics, Technical University

Kosice, Bozeny Nemcovej 32, 040 01 Kosice, Slovakia, [email protected]

Stanislav Paluch, Department of Mathematical Methods, Faculty of Management Science and

Informatics, University of Zilina, Univerzitna 1, 010 26 Zilina, Slovakia, [email protected]

Miroslava Ruzickova, Department of Mathematical Analysis and Applied Mathematics, Faculty

of Science, University of Zilina, Hurbanova 15, 010 26 Zilina, Slovakia,

[email protected]

Jozef Skokan, Department of Mathematics, University of Illinois at Urbana Champaign, 1409

West Green Street, Urbana, IL 61801, USA, [email protected]

Ioannis P. Stavroulakis, Department of Mathematics, University of Ioannina, 451 10 Ioannina,

Greece, [email protected]

Mariusz Wozniak, Wydzial Matematyki Stosowanej, AGH 30 059 Krakow, al. Mickiewicza 30,

Poland, [email protected]

Rudolf Blasko (Managing editor), Department of Mathematical Methods, Faculty of Mana-

gement Science and Informatics, University of Zilina, Univerzitna 1, 010 26 Zilina, Slovakia,

[email protected]

Studies of the University of Zilina • Mathematical seriesFaculty of Operation and Economics of Transport and Communications, Department of

Mathematics, University of Zilina, Univerzitna 1, 010 26 Zilina, Slovakiaphone: +42141 5133250, fax: +42141 5651499, e-mail: [email protected],electronic edition: http://fpedas.uniza.sk or http://frcatel.fri.uniza.sk/studies

Typeset using LATEX2e, Printed by: EDIS – Zilina University publisherc©University of Zilina, Zilina, Slovakia

ISSN 1336–149X

Preface

This volume contains 14 selected papers induced by the Conference on Diffe-rential and Difference Equations and their Applications (CDDEA 2008) held inthe nice historical village Strecno, Slovak Republic, 23rd–27th June 2008. Thisinternational Conference was the 20th continuation of the previous fourteen Sum-mer Schools on Differential Equations, the first of which was organized in 1964,and five International Conferences. The founders of the tradition were universityprofessors Pavol Marusiak and Ladislav Berger.

The conference was a worthy continuation of the tradition and was organizedby the Faculty of Science, University of Zilina. In the work of the conference 80participants from 24 countries and 4 continents participated.

The conference was prepared by Organizing Committee (B. Bacova, J. Diblık,B. Dorociakova, M. Kudelcıkova (secretary), M. Ruzickova (chairman), M. Zabor-sky,) and Programme Committee (A. Boichuk, E. Braverman, V. Covachev, J. Dib-lık, A. Domoshnitsky, Z. Dosla, O. Dosly, J. Dzurina, J. R. Graef, I. Gyori, J. Jaros,D. Khusainov, W. Kratz, T. Krisztin, E. Liz, J. Mawhin, M. Medved, N. Partsva-nia, I. Rachunkova, V. Rasvan, Y. Rogovchenko, M. Ronto, M. Ruzickova, F. Sa-dyrbaev, S. Schwabik, S. Stanek, I. P. Stavroulakis, M. Tvrdy, J. Werbowski).

The programme contained 17 invited lectures, 42 contributed talks and 14posters, and covered a broad part of mathematics connected with differential anddifference equations and their applications and was divided into five sections (Ordi-nary differential equations, Functional differential equations, Difference equations,Partial differential equations and Numerical methods in differential and differenceequations).

The conference was a successful and fruitful meeting stimulating scientific con-tacts and collaborations during nice time in Strecno.

Josef Diblık and Miroslava RuzickovaGuest editors

Manuscript submission

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CONTENTS

I. Boychuk, O. Starkova, S. Tchujko:Weakly perturbed nonlinear boundary-value problem in critical case . . . . . . 1

A. A. Chubatov, V. N. Karmazin:The on-line monitoring over atmospheric pollution source on the base of func-

tion specification method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

O. Dosly, S. Fisnarova:Perturbation principle in discrete half-linear oscillation theory . . . . . . . . . . 19

O. Dosly, J. Reznıckova:Principal and nonprincipal solutions in the oscillation theory of half-linear

differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

I. Dzhalladova:Research stability of solutions of the system linear differential equations with

random coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

T. Garbuza:On estimate of the number of solutions for quasilinear 6-th order boundary

value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

P. Hasil:On positivity of the three term 2n-order difference operators . . . . . . . . . . . . 51

J. Jansky:Some numerical investigations of differential equations with several propor-

tional delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

I. Korol:Investigating boundary value problems for ordinary differential equations in a

critical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

J. Laitochova:An algorithm to solve some types of first order difference equations . . . . . 77

M. Lascsakova:The numerical model of forecasting aluminium prices by using two initial

values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Z. Patıkova:On asymptotics of conditionally oscillatory half-linear equations . . . . . . . . 93

N. Sergejeva:On nonlinear spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

A. Slavık:An introduction to product integration on time scales . . . . . . . . . . . . . . . . . 111

Studies of the University of Zilina • Mathematical seriesFaculty of Operation and Economics of Transport and Communications, Department of

Mathematics, University of Zilina, Univerzitna 1, 010 26 Zilina, Slovakiaphone: +42141 5133250, fax: +42141 5651499, e-mail: [email protected],electronic edition: http://fpedas.uniza.sk or http://frcatel.fri.uniza.sk/studies

Typeset using LATEX2e, Printed by: EDIS – Zilina University publisherc©University of Zilina, Zilina, Slovakia

ISSN 1336–149X

Studies of the University of ZilinaMathematical SeriesVol. 23/2009, 1–8

1

WEAKLY PERTURBED NONLINEAR BOUNDARY-VALUE

PROBLEM IN CRITICAL CASE

I. BOYCHUK, O. STARKOVA and S. TCHUJKO

Abstract. We construct necessary and sufficient conditions for the existence of

solution of weakly nonlinear boundary value problem for a system of ordinary dif-ferential equations in critical case.

1. Statement of the Problem

We construct necessary and sufficient conditions for the existence of solutionz(t, ε) : z(t, ·) ∈ C1[a, b], z(t, ·) ∈ C[0, ε0] of weakly nonlinear boundary-valueproblem for a system of ordinary differential equations [1, 2]

dz

dt= A(t)z + f(t) + εZ(z, t, ε), ℓz(·, ε) = α + εJ(z(·, ε), ε). (1)

We seek a solution of the problem (1) in a small neighborhood of the generatingproblem

dz0

dt= A(t)z0 + f(t), ℓz0(·) = α, α ∈ Rm. (2)

Here A(t) is an (n × n) matrix, f(t) is an n dimensional column vector whoseelements are real functions continuous on the segment [a, b] and ℓz(·) is a linearbounded vector functional of the form ℓz(·) : C[a, b] → Rm.

The nonlinearities Z(z, t, ε) and J(z(·, ε), ε) are twice continuously differentiablewith respect to the unknown z in a small neighborhood of the generating solutionand are continuous in the small parameter ε in a small positive neighborhood ofzero. In addition, we assume that the vector function Z(z, t, ε) is continuous inthe independent variable t on the segment [a, b]. We investigate the critical case(PQ∗ 6= 0) and the condition

PQ∗

d

α − ℓK[

f(s)]

(·)

= 0; (3)

is considered to be carried out, where the generating boundary-value problem (2)has an r-parameter family of solutions

z0(t, cr) = Xr(t)cr + G[

f(s); α]

(t), cr ∈ Rr, r = n − n1.

2000 Mathematics Subject Classification. Primary 34B05,34B15 .Key words and phrases. boundary value problem, critical case.

2 I. BOYCHUK, O. STARKOVA and S. TCHUJKO

Here X(t) is the normal (X(0) = In) fundamental matrix of the homogeneouspart of the system (2), Q = ℓX(·) is the (m × n) matrix, rank Q = n1, Xr(t) =X(t)PQr

, PQris a (n × r) matrix composed of r linearly independent columns of

the (n × n) matrix(orthoprojector) PQ : Rn → N(Q), PQ∗

dis an (d × m) matrix

composed of d linearly independent rows of the (m × m) matrix (orthoprojector)PQ∗ : Rn → N(Q∗).

G[

f(s); α]

(t) = K[

f(s)]

(t) − X(t)Q+ℓK[

f(s)]

(·)

is the generalized Green operator of the boundary-value problem (2) and

K[

f(s)]

(t) = X(t)

∫ t

a

X−1(s)f(s)ds

is the Green operator of the Cauchy problem (2). Q+ is the pseudoinverse Moore-Pen-rose matrix [1, 2, 3]. The lemmas below gives a condition for the existence ofa solution of the problem (1) in the critical case [1, 2].

Lemma 1.1. Suppose that the boundary-value problem (1) corresponds to the

critical case (PQ∗ 6= 0) and condition (3) of the solvability of the problem (2) is

satisfied. Also assume that problem (1) has a solution z(t, ε) = z0(t, cr) + x(t, ε)that turns into the generating solution z0(t, c

∗

r) for ε = 0. Then the vector c∗r ∈ Rr

satisfies the equation

F0(cr) = PQ∗

d

J(z0(·, cr), 0) − ℓK[

Z(z0(s, cr), s, 0)]

(·)

= 0. (4)

Assume that equation (4) has real roots. Fixing one of the solutions c∗r ∈ Rr ofequation (4), we seek a solution of the problem (1) z(t, ε) = z0(t, c

∗

r) + x(t, ε) inthe neighborhood of the generating solution z0(t, c

∗

r) = Xr(t)c∗

r + G [f(s); α] (t).Perturbation x(t, ε) defines a boundary-value problem

dx(t, ε)

dt= A(t)x + εZ(z0(t, c

∗

r) + x(t, ε), t, ε),

ℓx(·, ε) = εJ(z0(·, c∗

r) + x(·, ε), ε). (5)

In the neighborhood of the points x = 0, ε = 0 the following equality is true:

Z(z0(t, c∗

r) + x(t, ε), t, ε) =

Z(z0(t, c∗

r), t, 0) + A1(t)x(t, ε) + εA2(t) + R1(z0(t, c∗

r) + x(t, ε), t, ε),

where

A1(t) =∂Z(z, t, ε)

∂z

∣

∣

∣

∣

∣

z = z0(t, c∗r)ε = 0

, A2(t) =∂Z(z, t, ε)

∂ε

∣

∣

∣

∣

∣

z = z0(t, c∗r)

ε = 0

.

Similarly we separate the linear part ℓ1x(·, ε) with respect to x and the linearpart εℓ2(z0(·, c

∗

r)) with respect to ε of the functional J(z0(·, c∗

r) + x(·, ε), ε) and

WEAKLY PERTURBED NONLINEAR BVP IN CRITICAL CASE 3

also a term J(z0(·, c∗

r), 0) of zero order with respect to ε in the neighborhood ofthe points x = 0 and ε = 0

J(z0(·, c∗

r) + x(·, ε), ε) =

J(z0(·, c∗

r), 0) + ℓ1x(·, ε) + εℓ2(z0(·, c∗

r)) + J1(z0(·, c∗

r) + x(·, ε), ε).

2. Sufficient conditions.

Defining (d × r) matrix

B0 = PQ∗

d

ℓ1Xr(·) − ℓK [A1(s)Xr(s)] (·)

we come to the operator system, which is equivalent to the problem of the findingof the problems’ solution (5)

x(t, ε) = Xr(t)cr + x(1)(t, ε),

B0cr = −PQ∗

d

ℓ1x(1)(·, ε) + εℓ2(z0(·, c

∗

r), 0) + J1(z0(·, c∗

r) + x(·, ε), ε)

− ℓK[

A1(s)x(1)(s, ε) + εA2(s) + R1(z0(s, c

∗

r) + x(s, ε), s, ε)]

(·)

,

x(1)(t, ε) = εG[

Z(z0(s, c∗

r) + x(s, ε), s, ε); J(z0(·, c∗

r) + x(·, ε), ε)]

(t). (6)

For the construction of a solution of the boundary-value problem (5) in thecritical case traditionally [1, 4] the condition PB∗

0= 0 is used, which guarantees

decidability of the second equation of the operator system (6); here PB∗

0is an

(d × d) orthoprojector matrix, Rd → N(B∗

0).As it is shown in the [1] there are boundary value-problems for which the con-

dition PB∗

0= 0 is not satisfied. For the simplification of the computations in the

case PB∗

06= 0 let’s assume that

∂

∂z

[

∂Z(z, t, ε)

∂zx

]

∣

∣

∣

∣

∣

z = z0(t, c∗r)ε = 0

= 0,∂

∂z

[

∂J(z(·, ε), ε)

∂zx

]

∣

∣

∣

∣

∣

z = z0(t, c∗r)

ε = 0

= 0.

In this case in a small convex neighborhood of the points x = 0 and ε = 0 havea place the expansions [5]

Z(z0(t, c∗

r) + x(t, ε), t, ε) = Z(z0(t, c∗

r), t, 0) + A1(t)x(t, ε) + εA2(t)

+ ε2A3(t) + εA4(t)x(t, ε) + R2(z0(t, c∗

r) + x(t, ε), t, ε), (7)


J(z0(·, c∗

r) + x(·, ε), ε) = J(z0(·, c∗

r), 0) + ℓ1x(·, ε) + εℓ2(z0(·, c∗

r), 0)

+ ε2ℓ3(z0(·, c∗

r), 0) + εℓ4x(·, ε) + J2(z0(·, c∗

r) + x(·, ε), ε), (8)

where

A3(t) =∂2Z(z, t, ε)

2 · ∂ε2

∣

∣

∣

∣

∣

z = z0(t, c∗r)

ε = 0

, A4(t) =∂Z(z, t, ε)

∂ε ∂z

∣

∣

∣

∣

∣

z = z0(t, c∗r)

ε = 0

accordingly (n× 1) and (n× n) are the matrices and the linear vector functionals

ℓ3(z0(·, c∗

r), 0) =∂2J(z(·, ε), ε)

2 · ∂ε2

∣

∣

∣

∣

∣

z = z0

ε = 0

, ℓ4x(·, ε) =∂2J(z(·, ε), ε)

∂ε ∂z

∣

∣

∣

∣

∣

z = z0

ε = 0

.

Particular solution of the problem (5) is represented in the form

x(1)(t, ε) = εG[

Z(z0(s, c∗

r), s, 0); J(z0(·, c∗

r), 0)]

(t) + x(2)(t, ε);

here

x(2)(t, ε) = εG[

A1(s)x(s, ε) + ε2A3(s) + εA4(s)x(s, ε) + εA2(s)

+ εA4(s)x(s, ε) + R2(z0(s, c∗

r) + x(s, ε), s, ε); ℓ1x(·, ε) + εℓ2(z0(·, c∗

r), 0)

+ ε2ℓ3(z0(·, c∗

r), 0) + εℓ4x(·, ε) + J2(z0(·, c∗

r) + x(·, ε), ε)]

(t).

Let’s define

B1 = PB∗

0PQ∗

d

ℓ1G1(·) + ℓ4Xr(·) − ℓK[

A1(s)G1(s) + A4(s)Xr(s)]

(·)

is a (d × r) matrix. On conditions that

F1(c∗

r) := PQ∗

d

ℓ1G[

Z(z0(s, c∗

r), s, 0); J(z0(·, c∗

r), 0]

(·) + ℓ2(z0(·, c∗

r), 0)

− ℓK[

A1(s)G[

Z(z0(τ, c∗

r), τ, 0); J(z0(·, c∗

r), 0)]

(s) + A2(s)]

(·)

= 0

the solution cr = c(0)r + c

(1)r , c

(1)r ∈ N(B0) of the second equation of the operator

system (6) determines the equality

c(0)r = −B+

0 PQ∗

d

ℓ1x(2)(·, ε)+ε2ℓ3(z0(·, c

∗

r), 0)+εℓ4x(·, ε)+J2(z0(·, c∗

r)+x(·, ε), ε)

− ℓK[

A1(s)x(2)(s, ε)+ ε2A3(s)+ εA4(s)x(s, ε)+R2(z0(s, c

∗

r)+x(s, ε), s, ε)]

(·)


and the equation

εB1c(1)r = −PB∗

0PQ∗

d

ℓ1x(3)(·, ε) + εℓ4Xr(·)c

(0)r + ε2ℓ3(z0(·, c

∗

r), 0)

+ εℓ4G[

Z(z0(s, c∗

r), s, 0); J(z0(·, c∗

r), 0)]

(·) + εℓ4

[

εG1(·)c(1)r x(3)(·, ε)

]

+ J2(z0(·, c∗

r) + x(·, ε), ε) − ℓK

A1(s)x(3)(s, ε) + εA4(s)Xr(s)c

(0)r

+ ε2A3(s) + εA4(s)G[

Z(z0(τ, c∗

r), τ, 0); J(z0(·, c∗

r), 0)]

(s)

+ εA4(s)[

εG1(s)c(1)r + x(3)(s, ε)

]

+ R2(z0(s, c∗

r) + x(s, ε), s, ε)

(·)

, (9)

decidability of which guarantees the decidability of the operator system (6).The equation (9) has a solution according to the condition PB∗

1PB∗

0= 0; where

PB∗

1: Rd → N(B∗

1) is orthoprojector matrix. Thus, in the case

PB∗

06= 0, PB∗

1PB∗

0= 0, (10)

the boundary-value problem (5) has at least one solution, which is defined by theoperator system

x(t, ε) = Xr(t)(

c(0)r + c(1)

r

)

+ x(1)(t, ε),

x(1)(t, ε) = εG[

Z(z0(s, c∗

r), s, 0); J(z0(·, c∗

r), 0)]

(t) + x(2)(t, ε);

x(2)(t, ε) = εG1(t)c(1)r + x(3)(t, ε), G1(t) = G

[

A1(s)Xr(s); ℓ1Xr(·)]

(t),

x(3)(t, ε) = εG

A1(s)[

Xr(s)c(0)r + x(1)(s, ε)

]

+ εA2(s) + ε2A3(s)

+εA4(s)[

Xr(s)c(0)r + x(1)(s, ε)

]

+ R2(z0(s, c∗

r) + x(s, ε), s, ε);

ℓ1

[

Xr(·)c(0)r + x(1)(·, ε)

]

+ εℓ4

[

Xr(·)c(0)r + x(1)(·, ε)

]

+εℓ2(z0(·, c∗

r), 0) + ε2ℓ3(z0(·, c∗

r), 0) + J2(z0(·, c∗

r) + x(·, ε), ε)

(t),

c(0)r = −B+

0 PQ∗

d

ℓ1x(2)(·, ε) + ε2ℓ3(z0(·, c

∗

r), 0) + εℓ4

[

Xr(·)cr + x(1)(·, ε)]

+J2(z0(·, c∗

r) + x(·, ε), ε) − ℓK[

A1(s)x(2)(s, ε) + ε2A3(s)

+εA4(s)[

Xr(s)cr + x(1)(s, ε)]

+ R2(z0(s, c∗

r) + x(s, ε), s, ε)]

(·)

,

εc(1)r (ε) = −B+

1 PB∗

0PQ∗

d

ℓ1x(3)(·, ε) + εℓ4Xr(·)c

(0)r + ε2ℓ3(z0(·, c

∗

r), 0)

+εℓ4G[

Z(z0(s, c∗

r), s, 0); J(z0(·, c∗

r), 0)]

(·)

+εℓ4

[

εG1(·)c(1)r + x(3)(·, ε)

]

+ J2(z0(·, c∗

r) + x(·, ε), ε)

−ℓK

A1(s)x(3)(s, ε) + εA4(s)Xr(s)c

(0)r + ε2A3(s)

+εA4(s)G[

Z(z0(τ, c∗

r), τ, 0); J(z0(·, c∗

r), 0)]

(s)

+εA4(s)[

εG1(s)c(1)r + x(3)(s, ε)

]

+R2(z0(s, c∗

r) + x(s, ε), s, ε)

(·)

. (11)

Thus, we have proved the following theorem.


Theorem 2.1. Suppose that the boundary-value problem (1) corresponds to the

critical case PQ∗ 6= 0 and the solvability condition (3) is satisfied for the generating

problem (2). Then, under the condition (10) and F1(c∗

r) = 0 for every root c∗r ∈ Rr

of equation (4) the problem (5) has at least one solution, which is defined by the

operator system (11), and for ε = 0 that turns into the zero solution x(t, 0) ≡ 0.Moreover, the problem (1) has a at least one solution z(t, ε) : z(t, ·) ∈ C1[a, b],z(t, ·) ∈ C[0, ε0], for ε = 0 turns into the generativing solution z0(t, c

∗

r) of the

problem (2).

For the construction of an approximate solution of the operator system (11) inthe case 10 and F1(c

∗

r) = 0 one can use the method of simple iterations [1, 4]. Thelength of the segment [0, ε∗], on which the method of simple iterations is used,can be estimated both by the means of the majorizing Lyapunov equations [2, 4],and directly from the condition of operator constructing, which is defined by thesystem (10), analogous to [6]. In a particular case, when PB0

PB1= 0 the solution

of the problem (1) is unique. Here

PB0: Rr → N(B0), PB1

: Rr → N(B1)

(r × r) are orthoprojectors matrixes. The presence of derivatives

A2(s) 6= 0, A3(s) 6= 0, ℓ2(z0(·, c∗

r), 0) 6= 0, ℓ3(z0(·, c∗

r), 0) 6= 0,

and also the condition F1(c∗

r) = 0 distinguish the proved theorem from the appro-priate theorems [2, p. 193], [3, p. 42].

3. Periodic problem for the Mathieu equation.

An example of the conditions fulfillment of the proved theorem is a problem of theperiodic solutions findings of the Mathieu equation [2, 4]

y′′ +(

k2 + εh(ε) + ε cos 2t)

· y = 0, h(ε) = − 12 − ε

32 + ε2

512 + · · · ,

which leads to the form (1) for k2 = 1,

A =

[

0 1−1 0

]

, Z(z, ε) = col[

0, (−h(ε) − cos 2t)z(a)]

, J(z(·, ε), ε) ≡ 0.

So far as Q = 0, we get the critical case and PQ∗ = I2, r = 2,

B0 = π

[

0 −10 0

]

, B+0 = 1

π

[

0 0−1 0

]

, PB∗

0=

[

0 00 1

]

, PB0=

[

1 00 0

]

.

The equation (4) determines a generating solution y0 = cos t, which meets thederivatives


A1(t) =

[

0 012 − cos 2t 0

]

, A2(t) =

[

0 0132 cos t 0

]

, A4(t) =

[

0 0132 0

]

.

Matrixes

B1 = π32

[

0 0−1 0

]

, B+1 = 32

π

[

0 −10 0

]

, PB1=

[

0 00 1

]

, PB∗

1=

[

1 00 0

]

let us check the fulfillment of conditions F1(c∗

r) = 0, PB∗

1PB∗

0= 0 and PB0

PB1= 0,

which guarantee a simple solvability of the given periodic problem for the Mathieuequation.

We obtained the solution of the Mathieu equation

y(t, ε) = cos t + ε16 cos 3t + ε2

768

(

− 3 cos 3t + cos 5t)

+ ε3

73 728

(

6 cos 3t − 8 cos 5t + cos 7t)

+ ε4

11 796 480

(

220 cos3t + 30 cos 5t − 15 cos 7t + cos 9t)

+ ε5

2 831 155 200

(

− 7 350 cos3t + 1 575 cos5t + 90 cos 7t − 24 cos 9t + cos 11t)

+ ε6

951 268 147 200

(

86 625 cos3t− 75 495 cos5t + 6 426 cos7t

+210 cos9t − 35 cos 11t + cos 13t)

+ ε7

426 168 129 945 600

(

7 808 640 cos3t + 1 215 396 cos5t − 417 088 cos7t

+19600 cos9t + 420 cos 11t− 48 cos 13t + cos 15t)

+ ε8

627 676 214 335 475 936 385 988 060 074 598

(

−1 968 211 441 083 579 268 857 856 cos3t

+350 747 430 860 537 215 320 064 cos5t+22 420 507 574 425 725 435 904 cos7t

−4 227 042 353 854 022 156 288 cos9t + 127 032 196 174 184 464 384 cos11t

+1 933 098 637 433 241 856 cos13t− 161 091 553 119 436 832 cos15t

+2 557 008 779 673 600 cos17t)

+ε9(

23 124 636 551 157 987 147 776 cos 3t170 941 947 432 867 407 345 576 843 780 151 − 49 573 353 873 678 276 755 456 cos 5t

512 825 842 298 602 222 036 730 531 340 453

+ 3 526 980 670 450 162 991 104 cos 7t512 825 842 298 602 222 036 730 531 340 453 + 11 583 302 494 456 020 992 cos 9t

46 620 531 118 054 747 457 884 593 758 223

− 15 315 969 020 122 769 408 cos 11t512 825 842 298 602 222 036 730 531 340 453 + 320 812 752 620 985 920 cos 13t

512 825 842 298 602 222 036 730 531 340 453

+ 3 655 985 784 854 540 cos 15t512 825 842 298 602 222 036 730 531 340 453 − 3 714 017 305 249 057 cos 17t

8 205 213 476 777 635 552 587 688 501 447 248

+ 5 942 427 688 398 491 cos 19t1 050 267 325 027 537 350 731 224 128 185 247 744

)


+ε10(

619 cos 3t26 301 584 573 107 + 665 cos 5t

157 220 021 405 446 − 1 427 cos 7t1 190 919 240 161 326 + 93 cos 9t

1 968 260 270 353 267

+ 2 cos 11t1 800 732 080 213 723 − cos 13t

10 768 431 115 775 110 + cos 15t695 370 276 593 575 323

)

+ε11(

− 48 cos 3t10 631 726 903 339 + 95 cos 5t

131 674 077 597 326 + 17 cos 7t321 262 834 093 616 − 10 cos 9t

1 208 646 206 118 963

+ cos 11t4 778 357 936 802 680 + cos 13t

287 805 286 716 556 383 − cos 15t4 668 789 321 617 042 696

)

+ε12(

17 cos 3t70 774 151 047 074 − 72 cos 5t

516 105 297 821 315 + 9 cos 7t1 010 301 450 341 120 + cos 9t

2 719 118 682 937 842

− cos 11t27 185 810 534 608 284 + cos 13t

1 538 357 407 774 944 487 + cos 15t124 428 038 411 659 803 874

)

,

the appropriate functions

h(ε) = 1 − ε2 − ε2

32 + ε3

512 − 49 069 620 853 122 733 632 535 504 ε4

1 205 935 002 347 320 358 093 606 851 695

− 739 763 748 051 653 752 784 988 614 ε5

79 332 800 113 011 957 966 445 439 248 959 + 883 297 340 186 403 212 576 412 671 ε6

680 475 051 105 599 093 883 469 360 870 040

− 23 423 065 001 327 456 614 137 105 848 ε7

514 221 418 938 961 300 622 006 615 245 553 005−5 707 023 251 823 254 232 346 871 873 ε8

627 308 288 266 042 889 352 835 052 033 355 148

+ 883 649 951 419 426 862 471 625 495 ε9

605 344 828 462 237 393 159 057 236 667 067 828 .

We obtained also the solutions and the eigenfunctions of Mathieu equation,appropriate k = 2, 3.

References

[1] Boichuk A. A., Samoilenko A. M., Generalized inverse operators and Fredholm boundary-

value problems, Utrecht, Boston: VSP, 2004, XIV, p. 317.[2] Boichuk A. A., Zhuravlev V. F., Samoilenko A. M., Generalized inverse operators and Fred-

holm boundary-value problems, Inst. Math. NAS of Ukraine, 1995, p. 318.[3] Boichuk A. A., Constructive methods for the analysis of boundary-value problems, Naukovadumka, 1990, p. 96.

[4] Grebenikov E. A., Ryabov Yu. A., Constructive methods for the analysis of nonlinear sys-

tems, Nauka, 1979, p. 432.[5] Chujko S. M., Seminonlinear boundary-value problem in special critical case, Ukr. Math. Zh.,2009, 61, pp. 548—562.

[6] Chujko A. S., Domain of convergence of an iterative procedure for a weakly nonlinear bound-

ary value problem, Nonlinear Oscillations, 8, 2005, pp. 277–287.

I. Boychuk, University of Slavjansk, Slavjansk, Ukraine

O. Starkova , University of Slavjansk, Slavjansk, Ukraine

S. Tchujko, University of Slavjansk, Slavjansk, Ukraine,

E-mail address: [email protected]


9

THE ON-LINE MONITORING OVER ATMOSPHERIC

POLLUTION SOURCE ON THE BASE

OF FUNCTION SPECIFICATION METHOD

A. A. CHUBATOV and V. N. KARMAZIN

Abstract. The work presents the approach allows to sequential estimate of actionintensity of atmospheric pollution source on the base of concentration measurementsof pollution impurity in several stationary control points. The inverse problem wassolved by means of the step-by-step regularization and the function specificationmethod. Stable numerical approximation of unknown intensity are received. Thesolution is presented in the form of the digital filter.

Introduction

The most universal models for producing quantitative and qualitative fields ofpollution distribution in the atmosphere are semi-empirical models [1].

Absence of initial data about action intensity of emission sources, distortion ofboundary and initial conditions, the inadequate assessment of meteorological char-acteristics of the atmosphere leads to essential divergences between calculated andexperimental data. Therefore joint solution of direct and inverse problems of impu-rity distribution in the atmosphere on the basis of data of impurity concentrationmeasurements in stationary or mobile control points is represented expediently.This approach allows to expect an essential increase of accuracy of modelling cal-culations using mathematical models of acceptable complexity.

The two-dimensional linear turbulent diffusion equation [1] is employed for thedescription of processes of impurity distribution in the atmosphere (domain D)

∂q

∂t+

∂

∂x(vxq) +

∂

∂y(vyq) =

∂

∂x

(

Kx

∂q

∂x

)

+∂

∂y

(

Ky

∂q

∂y

)

+ f(x, y) · g(t), (1)

at following conditions q∣

∣

t=0= h(x, y), q

∣

∣

∂D= 0, where q = q(x, y, t) — integral

is over height of concentration of an impurity, (vx; vy) — vector of wind speeds,

2000 Mathematics Subject Classification. Primary 35R30, 35R25, 65R32; Secondary 35Q80,86A22.

Key words and phrases. Inverse problem, ill-posed problem, regularization, function specifi-cation method, on-line monitoring, atmospheric pollution, turbulent diffusion equation.

The authors gratefully acknowledge the financial support by the Russian Foundation for BasicResearch and administration of Krasnodar region (the project 09-01-96506).

10 A. A. CHUBATOV and V. N. KARMAZIN

(Kx; Ky) — vector of coefficients of turbulent diffusion, f(x, y) — function de-scribing spatial arrangement of a pollution source (in elementary case, inside of asource the function is equal 1, and outside — 0), g(t) — action intensity of source.

The direct problem consists in definition of the concentration field q(x, y, t) atthe known values (vx; vy), (Kx; Ky), f(x, y), g(t). Let’s examine the inverse prob-lem consisting in definition of function g(t) at the known values (vx; vy), (Kx; Ky),f(x, y) and q(x, y, t).

In practice in definition of the concentration field q(x, y, t) there is a number ofproblems:

• concentration measurements are not taken in all domain D;• concentration cannot be measured time-continuously;• there are errors of measurements.

Knowing that we shall consider the following:

• in the points (xj , yj), j = 1, . . . , J stationary control points are located;• measurements are taken in identical time intervals ∆t.• error of concentration measurements is additive and has the normal distri-

bution.

Then we can write down the formula

cji = q(xj , yj , ti) + δ · γ,

where cji — concentration measured by jth sensor at the time moment ti = i ·∆t,δ — root-mean-square error of sensor measurements, γ — standardized Gaussianvariate (Average(γ) = 0, V ariance(γ) = 1).

1. The solution of the inverse problem

The inverse problem for the source is characterized by solution instability withrespect to errors of concentration measurements and demands special methods ofthe solution [2, 3, 4].

Such methods can be direct methods (step-by-step regularization) [2], Tikhonovregularization [3] and function specification method [4]. In the function specifica-tion method the functional form of unknown intensity is supposed in advance andincludes a number of the unknown parameters estimated by means of the least-squares method. The unknown intensity can be estimated simultaneously for allinterval of time [0, T ] (whole domain approximation) and it is sequential on eachtime interval [tN−1; tN ] (sequential approximation).

Methods of step-by-step regularization and sequential function specificationwere used to solve the inverse problem. The functional form at which g(t) acceptseach time interval [tN−1; tN ] constant value gN (constant piecewise approximation)is considered.

THE ON-LINE MONITORING OVER ATMOSPHERIC POLLUTION SOURCE 11

1.1. Duhamel’s theorem and sensitivity coefficients

Linearity of the problem (1) allows to use the superposition principle and numericalanalogue of Duhamel’s theorem

q(xj , yj , ti) = Qinit(xj , yj , ti) +

i∑

n=1

gn ·(

Q(xj , yj , ti−n+1) −Q(xj , yj, ti−n))

, (2)

where Q(x, y, t) — solution of the direct problem (1) at g(t) = 1 and q∣

∣

t=0= 0,

Qinit(x, y, t) — solution of the direct problem (1) at g(t) = 0 and q∣

∣

t=0= h(x, y).

Let’s enter designations Q(xj , yj , ti) = φji, φj(i+1) − φji = ∆φji.The magnitude φji is called step sensitivity coefficient. It represents impurity

concentration in a point of an arrangement jth sensor at the moment of time ti,at condition g(t) = 1. The magnitude φji is possible to present as the responseto unit step of intensity of a source, an event at the moment of time t0 = 0.Then the magnitude φj(i−n) can be treated as impurity concentration in a point

of an arrangement jth sensor at the moment of time ti provided that unit step ofintensity of a source has happened at the moment of time tn.

The magnitude ∆φj(i−n) is called pulse sensitivity coefficient and represents

itself an increment of impurity concentration in a point of an arrangement jth

sensor at the moment of time ti, at unit impulse (duration ∆t) of intensity of asource which was proceeding during the time interval [tn−1; tn].

1.2. Sequential function specification method

We shall estimate gN , considering g1, g2, . . . , gN−1 are known values, calculatedon the previous steps. To give stability to the solution of the inverse problem weshall consider g(t) on several (r) time intervals at once. Let’s consider, that gN ,gN+1, . . . , gN+r−1 are connected by some functional dependence. In case of r = 1the method step-by-step regularization turns out.

Using (2) for the moments of time tN , tN+1, . . . , tN+r−1 let’s write down thematrix equation

Q = Qinit + Q|g=0 + Φ · G, (3)

where

Q =

Q(N)Q(N +1). . . . . . . . . .Q(N +r−1)

, Q(i) =

q(x1, y1, ti)q(x2, y2, ti). . . . . . . . . . .q(xJ , yJ , ti)

, G =

gN

gN+1

. . . . . .gN+r−1

,

Qinit =

Qinit(N)Qinit(N + 1). . . . . . . . . . . . . . .Qinit(N +r−1)

, Qinit(i) =

Qinit(x1, y1, ti)Qinit(x2, y2, ti). . . . . . . . . . . . . . .Qinit(xJ , yJ , ti)

,


Q|g=0 =

Q|g=0(0)Q|g=0(1). . . . . . . . . .Q|g=0(r−1)

, Q|g=0(k) =

N−1∑

n=1

gn · ∆φ1(N+k−n)

. . . . . . . . . . . . . . . . . . . .N−1∑

n=1

gn · ∆φJ(N+k−n)

,

Φ =

φ(0)φ(1) φ(0)...

.... . .

φ(r−1) φ(r−2) · · · φ(0)

, φ(k) =

∆φ1k

∆φ2k

. . . . .∆φJk

.

The equation (3) can be solved exactly only for the case r = 1 and J = 1 (Stolzsolution). In this case the solution of the inverse problem is frequently instable. Inthe case of using several time steps (r > 1) or several sensors (J > 1) the equationcan be solved only approximately by means of the least-squares method.

Let’s minimize the sum of squares of differences between measured C and cal-culated Q values of concentration

S = (C − Q)T · (C − Q) → min, (4)

where

C =

C(N)C(N +1). . . . . . . . . .C(N +r−1)

, C(i) =

c1i

c2i

. . .cJi

.

It is possible to approach the solution of the equation (3) in two ways:

• to solve equations set with r unknown values gN , gN+1, . . . , gN+r−1;• to reduce number of unknown values considering, that gN+1, gN+2, . . . ,

gN+r−1 is expressed by mean of some functional dependence from gN andfrom the previous values gN−1, gN−2, . . . , gN−p.

In the first case values gN , gN+1, . . . , gN+r−1 can turn out unrelated valuesthemselves, though in practice values of intensity g(t) cannot be vary arbitrarily.

In the second case the chosen functional dependence provides improvement ofsmoothness of the solution.

Let this functional dependence looks like

G = A · gN + B · G0, (5)

where

G0 =

gN−1

gN−2

. . . . .gN−p

, A, B are the r×1 and r×p matrixes.

Then the sequential estimation algorithm will look like:


1. for the chosen functional dependence gN+1, gN+2, . . . , gN+r−1 from gN andgN−1, gN−2, . . . , gN−p we shall estimate the unique unknown value gN ;

2. we shall pass to a following step, temporarily assuming dependence g(N+1)+1,g(N+1)+2, . . . , g(N+1)+r−1 from g(N+1) and g(N+1)−1, g(N+1)−2, . . . , g(N+1)−p.

We examine the elementary case of functional dependence — the assumptionof a constancy gN during r the sequential intervals of time (constant intensityapproximation)

gN = gN+1 = · · · = gN+r−1,

in this case

A =

11. . .1

, p = 1, B =

00

. . .0

.

Also we examine the case of linear dependence between gN , gN+1, . . . , gN+r−1

(linear intensity approximation)

gN+i−1 = gN + (i − 1) · (gN − gN−1) = i · gN + (1 − i) · gN−1,

in this case

A =

12. . .r

, p = 1, B =

01

. . .r−1

.

Using (5) in (3) we have

Q = Qinit + Q|g=0 + Φ · B · G0 + Φ · A · gN .

Employing the last expression in (4) and having calculating the matrix deriva-tive, we can calculate the intensity estimation gN

gN =(

XT · X)

−1

· XT ·(

C − Qinit − Q|g=0 − Φ · B · G0

)

, (6)

where X = Φ · A.

1.3. The solution in the digital filter form

The solution (6) is a linear function of the measured concentration cji, i = 1, 2, . . . ,N +r−1 and it is possible to present in the form of the digital filter [5]

gN =N+r−1∑

i=1

J∑

j=1

fj(N−i) ·(

cji −Qinit(xj , yj, ti))

,

where fj(i−r) — coefficients of the filter, fj(i−r) = Gji, i = 1, 2, . . . , N + r−1,Gji — solution of the inverse problem (6) at conditions cjr = 1; cji = 0, i 6= r;Qinit(x, y, t) = 0.

The solution in the form of the digital filter is more effective than other formsin the computing relation because coefficients fjk are calculated once.


2. Computing experiments

For the solution of the inverse problem it is necessary to know sensitivity coeffi-cients which are calculated from the solution of the direct problem at g(t) = 1. Forthe solution of the direct problem the program is written in programming languageFortran. In the program was used implicit difference scheme. For modelling con-centration measurements the direct problem is solved at known function of actionintensity of a source g(t). The solution of the inverse problem and visualization ofcalculations are realized in mathematical package MatLab.

Parameters of algorithm of the inverse problem solution are:

• ∆t — step of concentration measurements;• r — quantity of sequential time steps;• matrixes A, B define the form of functional approximation;• J — quantity of sensors;• (xj , yj) — position of sensors (influences of sensitivity coefficients);• δ — root-mean-square error of measurements of the sensor.

2.1. Graphs of sensitivity coefficients

The pulse sensitivity function (∆Q(x, y, t) = Q(x, y, t)−Q(x, y, t−∆t)) representsthe response on unit impulse (during time ∆t) changement of action intensity ofa source (see figure 1). The pulse sensitivity coefficients (PSC) represent discretesamples of pulse sensitivity function in the equidistant moments of time (see fig-ure 2) ∆φji = Q(xj , yj , ti+1) −Q(xj , yj , ti) = ∆Q(xj , yj , (i + 1)∆t).

0.050 0.100 0.150 0.2000

0.02

0.04

0.06

0.08

∆ t=0.001

3th sensor2th sensor1th sensor

Figure 1. The pulse sensitivity function∆Q(x, y, t).

0.050 0.100 0.150 0.2000

0.5

1

1.5

2∆ t=0.033

∆φ11

∆φ10 3th sensor

2th sensor1th sensor

Figure 2. The pulse sensitivity coefficients(PSC) ∆φji.

Graphs of sensitivity functions have one maximum. In figure 1 the phenomenaof lag and damping are clearly visible. The sensor number 1 is most sensitive: it isfaster (lag less, than at others sensors) and stronger (damping less, than at othersensors) reacts to intensity change of a source.

If we arrange the sensor in an operative range of a source then there is not lagand damping practically is not observed (in this case we have the pseudo-inverseproblem). Increasing distance from a source the effects of lag and damping aregetting more strongly: the maximum of sensitivity function moves to the right,the maximal value of function decreases, the graph becomes more flat.


2.2. The step-by-step regularization method

In case of step-by-step regularization (r = 1) the matrixes A, B degenerate to the1×1 matrixes: A = (1), B = (0).

The solution (6) looks like

gN =(

φ(0)T · φ(0))

−1

· φ(0)T ·(

C(N) − Qinit(N) − Q|g=0(0))

,

or in the unwrapped kind (considering, that ∆φj0 = φj1)

gN =

J∑

j=1

φj1 ·

(

cjN − Qinit(xj , yj , tN) −N−1∑

n=1

gn · ∆φj(N−n)

)

J∑

j=1

φ2

j1

.

If we use one sensor we shall receive the Stolz solution

gN =

c1N − Qinit(x1, y1, tN ) −N−1∑

n=1

gn · ∆φ1(N−n)

φ11

.

It is necessary for stability of the Stolz solution to have as big as possible thepulse sensitivity coefficient ∆φ10 = φ11. Let’s allow jump of intensity happenedat the moment of time t = 0. It is better to make measurements at the momentof time when the response to this jump is maximal. It is necessary to choose thestep of concentration measurements ∆t so, that ∆φ10 = ∆φ11 (see figure 2).

In figure 3 there are the examples of an estimation of intensity at an identicalstep ∆t for three cases:

1. function g(t) is smooth; there aren’t errors of measurements;2. function g(t) has jump; there aren’t errors of measurements;3. function g(t) has jump; there are errors of measurements (δ = 0.01 · qmax).

The magnitude qmax is the maximal concentration measured by sensor. Hereand there in figures it is shown the true function of intensity g(t) by dashed lineand the estimation of intensity by continuous line.

0.500 1.000 1.500 2.0000

2

4

6

t

g(t)

δ = 0

0.500 1.000 1.500 2.0000

1

2

3

t

g(t)

δ = 0

0.500 1.000 1.500 2.0000

1

2

3

t

g(t)

δ = 0.01 · qmax

Figure 3. The step-by-step regularization method..

Even in case of the pseudo-inverse problem (the sensor is located in an operativerange of a source or on border of a source) it is possible to have fluctuations of thesolution at presence the errors of measurements, but the solution remains stable.


For each sensor there is the step ∆tst[1] (the minimal step) at which solution ofthe inverse problem is stable.

2.3. The preliminary filtration and the post-filtration

To reduce fluctuations of the solution it is possible to execute the preliminaryfiltration of input data (measured concentration) and the post-filtration of outputdata (estimated values of intensity). Results of the preliminary filtration andthe post-filtration are shown on figure 4. The filtration was made by means ofnonrecursive filter [5]

yi =xi−1 + 2xi + xi+1

4.

0.500 1.000 1.500 2.0000

1

2

3

4

t

g(t)

0.500 1.000 1.500 2.0000

1

2

3

t

g(t)

Figure 4. r = 1, δ = 0.03 · qmax. The filtration was not applied (left); thepreliminary and the post-filtration was applied (right).

2.4. The function specification method

Stability of the solution of the inverse problem can achieve at the step of measure-ments ∆t < ∆tst[1], using several sequential time steps (r > 1). The value ∆tst[r]

(the minimal time step at which the solution of the inverse problem is stable)decreases with increasing r (see figure 2.4).

2.5. About accuracy of estimation of desired intensity

The root-mean-square error was used for the account of accuracy of intensityestimation g(t)

σG =

√

√

√

√

1

Nmax

Nmax∑

n=1

(

g(

(n − 1

2) · ∆t

)

− gn

)2

.

It is experimentally possible to choose the time step ∆topt at which σG is minimal.It is observed that at r = 1 with increase ∆t dependence of value σG from

parameter δ weakens (see figure 6). At ∆t = const with increase r influence ofparameter δ on value σG weakens and at certain r = rc the value σG practicallydoes not depend from δ = 0 ÷ 0.03 · qmax (see figure 7).


0.500 1.000 1.500 2.0000

1

2

3

t

g(t)

r = 1, ∆t = ∆tst[1]

0.500 1.000 1.500 2.0000

1

2

3

t

g(t)

r = 4, ∆t =∆t

st[1]

6

0.500 1.000 1.500 2.0000

1

2

3

t

g(t)

r = 2, ∆t =∆t

st[1]

4

0.500 1.000 1.500 2.0000

1

2

3

t

g(t)

r = 8, ∆t =∆t

st[1]

10

Figure 5. The function specification method, δ = 0.03 · qmax..

0.01 0.02 0.030

0.02

0.04

0.06

0.08

0.1

δ/qmax

σ G/g

max

∆tst[1]

=0.010 r=1

∆t=0.010∆t=0.012∆t=0.016

Figure 6. Dependence of value σG from∆t and δ at r = 1.

0.01 0.02 0.030

0.02

0.04

0.06

0.08

0.1

δ/qmax

σ G/g

max

∆t=0.025 ∆tst[1]

=0.050

r= 2r= 3r= 5r= 7

Figure 7. Dependence of value σG from rand δ at ∆t = const = ∆tst[1]/2.

3. Results of computing experiments

The numerous quasi-real experiments are lead using number of methodical prob-lems. Stable numerical approximation to desired value intensity for sources ofvarious types (point, linear, areal, distributed) are constructed, at presence of


measurement errors in sensors (parameter δ = 0÷ 0.03 · qmax). Sensors are settleddown outside of an operative range of a source (f(x, y) = 0) and in an operativerange of a source (f(x, y) 6= 0).

For each sensor there is the critical step ∆tst[1], such, that for each step ofthe solution of the inverse problem ∆t > ∆tst[1] the solution is stable, i.e. thestep-by-step regularization effect takes place. But its opportunities are limited,because for some sensor ∆tst[1] can be big enough therefore the estimated solutionbecomes worse.

The desire to increase the accuracy of intensity estimation, reducing time step∆t, leads to instability of the solution of inverse problem. Using several sensors(J > 1) the sensor with smaller ∆tst[1] has the prevailing influence. Using twosensors with identical ∆tst[1] improve result in comparison with one sensor. Touse functional specification method with several (r > 1) sequential time stepsmakes the solution stability more effective.

The analysis of results of numerical experiments allows to draw a conclusion,that for pair numbers (∆t/∆tst[1], δ), ∆t/∆tst[1] ∈ [0, 1; 1], δ ∈ [0; 0.03 · qmax] it ispossible to select r at which the error of intensity estimation g(t) is minimal.

The data of concentration measurements from sensors is understanding sequen-tially in the considered method, that allows to organize the on-line monitoringover emissions of pollution impurity in the atmosphere.

References

[1] Marchuk G. I., Mathematical modelling in the problem of environment, Moscow, Nauka,1982.

[2] Alifanov O. M., Inverse problems of heat exchange, Moscow, Mashinostroenie, 1988.[3] Tikhonov A. N., Arsenin V. Ya., Methods of the solution of ill-posed problems, Moscow,

Nauka, 1986.[4] Beck J. V., Blackwell B., St. Clair C. R., Jr., Inverse Heat Conduction: Ill-posed Problems,

New York, Wiley, 1985.[5] Hemming R. V., Digital filters, New York, Dover Publications, 1997.

A. A. Chubatov, Department of Applied Mathematics, Kuban State University, 149 Stavropol-skaya st., Krasnodar, Krasnodar region, Russian Federation,E-mail address: [email protected]

V. N. Karmazin, Department of Applied Mathematics, Kuban State University, 149 Stavropol-

skaya st., Krasnodar, Krasnodar region, Russian Federation,



19

PERTURBATION PRINCIPLE IN DISCRETE

HALF-LINEAR OSCILLATION THEORY

ONDREJ DOSLY and SIMONA FISNAROVA

Abstract. We investigate oscillatory properties of solutions of the second order

half-linear difference equation

∆(rkΦ(∆xk)) + ckΦ(xk+1) = 0, Φ(x) := |x|p−2x, p > 1,

where this equation is viewed as a perturbation of another (nonoscillatory) half-linear equation of the same form.

1. Introduction

We deal with the oscillatory properties of the second order half-linear differenceequation

∆(rkΦ(∆xk)) + ckΦ(xk+1) = 0, Φ(x) := |x|p−2x, p > 1. (1)

Qualitative theory of (1) is summarized in the book [3, Chap. 3]. It is shownthere that solutions of (1) behave in many aspects similarly as those of the linearequation

∆(rk∆xk) + ckxk+1 = 0, (2)

which is the special case p = 2 in (1) and whose oscillation theory is deeplydeveloped from various points of view, we refer to the books [1, 3, 16] and thereferences given therein.

Another motivation for the investigation of (1) is its continuous counterpart,the half-linear differential equation

(r(t)Φ(x′))′ + c(t)Φ(x) = 0, (3)

which attracted considerable attention in recent years, we refer to the books [2, 9].In the classical approach to the oscillation theory of (3), this equation is re-

garded as a perturbation of the (nonoscillatory) one term equation

(r(t)Φ(x′))′ = 0. (4)

In this setting, the methods used to study (3) are very similar to those used inlinear oscillation theory since the solution space of (4) is linear. Recently, another

2000 Mathematics Subject Classification. Primary 39A10.Key words and phrases. Half-linear difference equation, recessive solution, summation char-

acterization, linearization method, Riccati technique.Research supported by the Grant 201/07/0145 of the Czech Grant Agency of the Czech

Republic and the Research Project MSM0022162409 of the Czech Ministry of Education.

20 ONDREJ DOSLY and SIMONA FISNAROVA

approach has been introduced in [7], the so-called perturbation principle, where(3) is regarded as a perturbation of a (nonoscillatory) half-linear equation of thesame form (whose solution space is no longer additive). In this approach, a Riccatitype differential equation appears and this equation involves a certain nonlinearfunction. Many results obtained by perturbation principle are then based on thequadratization of this nonlinear function. We will recall basic ideas of this methodin the next section.

Concerning the discrete version of the perturbation principle, the “preliminary”step has been made in [8]. There the concept of the recessive solution of thenonoscillatory half-linear difference equation

∆(rkΦ(∆xk)) + ckΦ(xk+1) = 0 (5)

has been introduced and it was shown that (1), viewed as a perturbation of (5),is oscillatory provided

∑

∞

(ck − ck)hpk+1

= ∞ where h is the positive recessivesolution of (5). In the proof of this statement no Riccati type difference equationappears and this proof is based on the so-called variational principle.

In the following papers [5, 6], the Riccati technique in perturbation principle hasbeen introduced and it turned that to find a quadratic approximation of nonlinearfunction in this equation is considerably more complicated than in the continuouscase and several problems remained open. We formulate some of these problemsin last section of the paper.

The paper is organized as follows. In the next section we recall basic ideasof the perturbation principle both in the continuous and discrete case. Section3 deals with oscillation of perturbed Euler-type differential equation, where someopen problems formulated in [5] can be solved due to the special structure ofthe nonlinearity in modified Riccati equation. The last section is devoted to theformulation of some open problems related to our research.

2. Perturbation principle

We start with the continuous case. Together with (3) we consider the equation ofthe same form

(r(t)Φ(x′))′ + c(t)Φ(x) = 0. (6)

We suppose that this equation is nonoscillatory and that h is its eventually positivesolution. Put v = hp(w − wh), where w is a solution of the Riccati equationassociated with (3)

w′ + c(t) + (p − 1)r1−q|w|q = 0, q :=p

p − 1, (7)

and wh = rΦ(h′/h). Then v solves the so-called modified Riccati equation

v′ + (c(t) − c(t))hp(t) + (p − 1)r1−q(t)h−q(t)H(t, v) = 0, (8)

whereH(t, v) := |v + G(t)|q − qvΦ−1(G(t)) − |G(t)|q

with G(t) := r(t)h(t)Φ(h′(t)) and Φ−1 the inverse function of Φ. There are severalreasons why we call (8) the modified Riccati equation. One of them is that if

PERTURBATION PRINCIPLE 21

p = 2, then (8) reduces to the equation

v′ + (c(t) − c(t))h2(t) +v2

r(t)h2(t)= 0,

which is the Riccati equation corresponding to the linear equation

(r(t)h2(t)y′)′ + (c(t) − c(t))h2(t)y = 0,

which results from (3) with p = 2 upon the transformation x = h(t)y where his a solution of (6) with p = 2. Another reason is that (8) reduces to (7) whenh(t) ≡ 1.

If a solution h of (6) satisfies additionally h′(t) 6= 0, the function H can bewritten in the form

H(t, v) = |G(t)|qF( v

G(t)

)

, F (z) := |1 + z|q − qz − 1.

This formula shows that H(t, v) ≥ 0 for v ∈ R with equality if and only if v =0, Hv(t, v) = 0 if and only if v = 0 and that H is convex in v. Moreover, ifv(t)/G(t) → 0 as t → ∞ for (positive) solutions of (8), one can approximate thefunction F by its second degree Taylor polynomial

F (z) =q(q − 1)

2z2 + o(z2) as z → 0.

Then, substituting this formula to (8) and neglecting the term o(z2), one can write(8) in the “approximative” form

v′ + (c(t) − c(t))hp(t) +q

2R(t)v2 = 0, R(t) := r(t)h2(t)|h′(t)|p−2. (9)

The last equation is the “classical” Riccati equation associated with the linear

second order equation

(R(t)x′)′ +q

2C(t)x = 0, C(t) := (c(t) − c(t))hp(t). (10)

This fact enables to use many “linear” results in the half-linear oscillation theory,see, e.g., [10]. Moreover, (9) and (10) can be solved explicitly in some special cases.For example, if

C(t) =1

2qR(t)(∫ t R−1(s) ds)2or C(t) = 0,

then v(t) = 1

q ∫t R−1(s) ds

or v(t) = 2

q ∫t R−1(s) ds

, respectively. This fact has been

used in [11] and [12] to study conditionally oscillatory half-linear equations and itsso-called nonprincipal solutions. In addition to the approximation formula nearv = 0 we have the following global inequalities for the last term in the left-handside of (8) which hold for all v ∈ R

(p − 1)r1−q(t)h−q(t)H(t, v) ≤q

2R(t)v2, q ≥ 2,

(p − 1)r1−q(t)h−q(t)H(t, v) ≥q

2R(t)v2, q ∈ (1, 2]. (11)

These inequalities have been used in [4] to study integral characterization of theso-called principal solution of (3).


Now we turn our attention to half-linear difference equation (1). Oscillatoryproperties of (1) are defined using the concept of the generalized zero which isdefined in the same way as for (2), see, e.g., [3, Chap. 3] or [9, Chap. 7]. Asolution x of (1) has a generalized zero in an interval (m, m + 1] if xm 6= 0 andxmxm+1rm ≤ 0. Since we suppose that rk > 0 (oscillation theory of (1) generallyrequires only rk 6= 0), a generalized zero of x in (m, m + 1] is either a “real” zeroat k = m + 1 or the sign change between m and m + 1. Equation (1) is said to bedisconjugate in a discrete interval [m, n] if the solution x of (1) given by the initialcondition xm = 0, xm+1 6= 0 has no generalized zero in (m, n + 1]. Equation (1) issaid to be nonoscillatory if there exists m ∈ N such that it is disconjugate on [m, n]for every n > m and is said to be oscillatory in the opposite case. This terminologytypical for linear equations is correct since the discrete Sturmian theory extendsalmost verbatim to (1).

Similarly as in the continuous case we regard equation (1) as a perturbation ofthe nonoscillatory equation (5). Let h be an eventually positive solution of (5),w = rΦ(∆h/h), and let w be a solution of the Riccati type equation associatedwith (1)

wk+1 + ck −rkwk

Φ(Φ−1(rk) + Φ−1(wk))= 0. (12)

Then v = hp(w − w) is the solution of the difference equation

∆vk + (ck − ck)hpk+1

+ H(k, v) = 0,

where

H(k, v) := v + rkhk+1Φ(∆hk) −rkhp

k+1(v + Gk)

Φ(hqkΦ−1(rk) + Φ−1(v + Gk))

(13)

withGk := rkhkΦ(∆hk). (14)

Moreover, wk + rk > 0 (i.e., a solution of (1) which gives w via the formulaw = rΦ(∆x/x) has no generalized zero in (k, k + 1]) if and only if vk + v∗k > 0,where

v∗k := rkhk(Φ(hk) + Φ(∆hk)). (15)

Concerning the quadratic approximation of the function H in (13), we have indisposal no global estimates like (11) in the continuous case yet, and in the nextstatement we need to distinguish the cases Gk < 0 and Gk > 0, see [5, 6].

Theorem 2.1. Let G and v∗ be defined by (14) and (15), respectively, and

Rk := 2

qrkhkhk+1|∆hk|

p−2. (16)

(i) Let p ∈ (1, 2]. If Gk > 0 for k ∈ N, then

H(k, v) ≥v2

Rk + vfor v ≥ 0, (17)

if Gk < 0 for k ∈ N, then v∗k ≤ Rk and (17) holds for v ∈ (−v∗k, 0].

(ii) Let p ≥ 2. If Gk > 0, then

H(k, v) ≤v2

Rk + vfor v ≥ 0, (18)

if Gk < 0 for k ∈ N, then v∗k ≥ Rk and (18) holds for v ∈ (−Rk, 0].


The reason why the term R + v appears in the denominator in (17), (18) isthat this term appears also in the Riccati equation corresponding to linear Sturm-Liouville difference equation (substitute p = 2 in (1) and (12)).

As a consequence of the previous theorem we have the following statements,see [5].

Theorem 2.2. Let ck ≥ ck for large k.

(i) If p ∈ (1, 2], h is the recessive solution of (5), and the linear equation

∆(Rk∆xk) + Ckxk+1 = 0, (19)

where R is given by (16) and

Ck := (ck − ck)hpk+1

(20)

is oscillatory, then equation (1) is also oscillatory.

(ii) If p ≥ 2,∑

∞

R−1

k = ∞ and the linear equation (19) is nonoscillatory, then

equation (1) is also nonoscillatory.

We finish this section by a theorem which does not distinguish between thecases p ∈ (1, 2] and p ≥ 2 which is also proved in [5]. We use this statement toprove the main result of our paper which is given in the next section.

Theorem 2.3. Let ck ≥ ck for large k and hk > 0 be the recessive solution

of (5) such that∞∑

(ck − ck)hpk+1

< ∞. (21)

Further suppose that the condition∑

∞ R−1

k = ∞ holds and

limk→∞

Gk = ∞, Gk := rkhkΦ(∆hk). (22)

If there exists ε > 0 such that equation

∆(Rk∆yk) + (1 − ε)Ckyk+1 = 0 (23)

is oscillatory, then equation (1) is also oscillatory.

3. Perturbed Euler-type equation

In this section we consider the Euler type difference equation

∆(Φ(∆xk)) + ckΦ(xk+1) = 0, (24)

where

ck := −∆Φ(∆hk)

Φ(hk+1), hk = k

p−1

p . (25)

Our motivation comes from the continuous case where h(t) = tp−1

p is the principalsolution of the Euler-type equation

(Φ(x′))′

+γp

tpΦ(x) = 0, γp :=

(p − 1

p

)p

(26)

whose properties were studied in several papers, see e.g. [14, 18]. It was shownin [5] that

ck =γp

(k + 1)p

[

1 + O(k−1)]

, (27)


and that

Gk =(p − 1

p

)p−1[

1 −p − 1

2pk+ o(k−1)

]

, (28)

both as k → ∞.In [5] we conjectured that under some technical assumption the perturbed Euler

equation

∆(Φ(∆xk)) + [ck + dk]Φ(xk+1) = 0 (29)

is oscillatory provided

lim infk→∞

log k

(

∞∑

j=k

dj(j + 1)p−1

)

>1

2

(p − 1

p

)p−1

.

In this section we prove that this conjecture is true for p ≥ 2. It has been shown

in the recent paper [6] that if p ≥ 2, then the sequence hk = kp−1

p is the recessive

solution of (24). This fact enables to apply Theorem 2.3 (with hk = kp−1

p and(24) instead of (5)) to equation (29).

Note that Theorem 2.3 cannot be applied directly to (29) because of condition(22), which is not satisfied in this case. However, going through the proof ofTheorem 2.3, one can see that condition (22) can be replaced by the following twoconditions

∞∑

H(k, v) = ∞ for every v > 0 (30)

and

lim infk→∞

Gk > 0, Gk = rkhkΦ(∆hk), (31)

which are less restrictive, see [5].We will need the following Hille-Nehari oscillation criterion for the linear equa-

tion (2).

Lemma 3.1 ([15]). Suppose that ck≥0, rk >0,∑

∞

r−1

k =∞,∑

∞

ck <∞. If

lim infk→∞

(

k−1∑

r−1

j

)(

∞∑

j=k

cj

)

>1

4,

then (2) is oscillatory.

Theorem 3.2. Let p ≥ 2. Consider equation (29) with ck given in (25) and

suppose that dk ≥ 0 for large k and

∞∑

dk(k + 1)p−1 < ∞. (32)

Then equation (29) is oscillatory provided

lim infk→∞

log k

(

∞∑

j=k

dj(j + 1)p−1

)

>1

2

(p − 1

p

)p−1

. (33)


Proof. We apply Theorem 2.3 with ck = ck + dk. We have ck − ck ≥ 0 for large

k and, since p ≥ 2, the sequence hk = kp−1

p is the recessive solution of (24), see[6]. Condition (32) implies

∞∑

(ck − ck)hpk+1

=

∞∑

dk(k + 1)p−1 < ∞,

hence (21) is satisfied. By a direct computation (see [5]) we obtain

hkhk+1|∆hk|p−2 =

(p − 1

p

)p−2

k(1 + O(k−1)), as k → ∞, (34)

which means that∑

∞ R−1

k = ∞. Condition (22) is not satisfied, but accordingto the comment above (30) it is sufficient to show that conditions (30) and (31)hold instead of (22). Condition (31) follows from (28). To verify condition (30)

we substitute rk = 1 and hk = kp−1

p into (13) and using (28) we have

H(k, v) = v +hk+1

hk

Gk −hp

k+1(v + Gk)

Φ(hqk + Φ−1(v + Gk))

= v +hk+1

hk

Gk − (v + Gk)(hk+1

hk

)p(

1 +Φ−1(v + Gk)

k

)1−p

= v +(

1 +1

k

)

p−1

p

Gk −(

1 +1

k

)p−1

(v + Gk)(

1 +Φ−1(v + Gk)

k

)1−p

= v +(

1 +p − 1

pk+ o(k−1)

)

Gk

−(

1 +p − 1

k+ o(k−1)

)

(v + Gk)(

1 +(1 − p)Φ−1(v + Gk)

k+ o(k−1)

)

=p − 1

k

[Gk

p− (v + Gk) + |v + Gk|

q]

+ o(k−1),

as k → ∞. Denote by A the expression in brackets and let α :=(

p−1

p

)p−1. Then

G = α(

1 −p − 1

2pk+ o(k−1)

)

and

A =α

p

(

1 −p − 1

2pk+ o(k−1)

)

+[

v + α(

1 −p − 1

2pk+ o(k−1)

)][

Φ−1

(

v + α −α(p − 1)

2pk+ o(k−1)

)

− 1]

=α

p−

α

2pqk+ o(k−1)

+[

v + α −α

2qk+ o(k−1)

][

Φ−1(v + α)Φ−1

(

1 −α

(α + v)2qk+ o(k−1)

)

− 1]

=α

p−

α

2pqk+ o(k−1)

+[

v + α −α

2qk+ o(k−1)

][

Φ−1(v + α)(

1 −α

2(v + α)pk+ o(k−1)

)

− 1]


=α

p−

α

2pqk+ o(k−1)

+[

v + α −α

2qk+ o(k−1)

][

Φ−1(v + α) −αΦ−1(v + α)

2(v + α)pk− 1 + o(k−1)

]

=α

p+ |v + α|q − (v + α) +

β

k+ o(k−1),

where the constant β can be computed explicitly but its value is not important.Consequently,

H(k, v) =p − 1

k

[α

p+ |v + α|q − (v + α)

]

+ o(k−1) as k → ∞.

The term o(k−1) is such that∑

∞ o(k−1) is convergent and by a direct computationwe find that α

p+ |v +α|q − (v +α) ≥ 0 for v ∈ R with equality if and only if v = 0.

This means that∞∑

H(k, v) = ∞

whenever v 6= 0.It follows from (33) that there exists ε > 0 such that

lim infk→∞

log k

∞∑

j=k

dj(j + 1)p−1 >1

2(1 − ε)

(p − 1

p

)p−1

=1

2q(1 − ε)

(p − 1

p

)p−2

,

i.e.,

lim infk→∞

log kq

2

( p

p − 1

)p−2∞∑

j=k

(1 − ε)Cj >1

4,

where C is given by (20). Using the discrete l’Hospital rule (see, e.g., [1])

limk→∞

∑k−1

j=1(1/j)

log k= 1,

and hence, in view of (16) and (34),

lim infk→∞

(

k−1∑

R−1

j

)(

∞∑

j=k

(1 − ε)Cj

)

>1

4.

According to Lemma 3.1 it means that equation (23) is oscillatory, and we haveconsequently by Theorem 2.3 that also equation (29) is oscillatory.

4. Open problems and conjectures

In this concluding section we formulate some open problems related to the resultsof the previous section.

(i) As mentioned before, the half-linear Euler differential equation (26) has the

principal solution h(t) = tp−1

p and nonprincipal solutions behave asymptotically

as x(t) = tp−1

p log2/p t, see [13]. In the discrete case, it is not clear what is the“right” Euler equation even in the linear case. In [1], the equation

k(k + 1)∆2xk + ak∆xk + bxk = 0 (35)


is referred to as the discrete second order Euler difference equation.The reason for this terminology is that similarly to the continuous case, a so-

lution of this equation is in the form xk = Γ(k+λ)

Γ(k), where λ is a solution of the

quadratic equation λ(λ−1)+aλ+b = 0. However, equation (35) is not in the self-adjoint form, so the theory of Sturm-Liouville second order difference equationsdoes not apply to (35). In the context of self-adjoint equations, as Euler equationis usually regarded the equation

∆2xk +b

k(k + 1)xk+1 = 0, b ∈ R, (36)

but in contrast to (35), solutions of (36) cannot be expressed by an explicit formula.(ii) In our treatment of the half-linear Euler type difference equation we started

“from the end”. Following the continuous case, we took the sequence xk = kp−1

p

and then we computed the sequence c in (24) by formula (27). However, as men-tioned in the previous section, we know that xk is the recessive solution of (24)only for p ≥ 2. We conjecture that this is the case also for p ∈ (1, 2]. We alsoconjecture, again based on the continuous case, that dominant solutions of (24)behave asymptotically as

xk = kp−1

p

(

k∑

j=1

1

j

)2/p

.

(iii) Another open problem is related to the so-called half-linear Riemann-Weberdifferential equation

(Φ(x′))′ +[ γ

tp+

λ

tp log2 t

]

Φ(x) = 0 (37)

with γ given in (26).

It is known that (37) is nonoscillatory if and only if λ ≤ µ := 1

2

(

p−1

p

)p−1

. We

conjecture that the discrete version of (37) (with c given by (27))

∆(Φ(∆xk)) +[

ck +λ

(k + 1)p log2(k + 1)

]

Φ(xk+1) = 0

is nonoscillatory if and only if λ ≤ µ. This conjecture holds for λ < µ (see [5,Theorem 5.1] and if p ≥ 2 is also for λ > µ (as a consequence of Theorem 3.2).Moreover, it is supported by the linear case p = 2, see [17], where, among others,perturbations of linear Euler type difference equations are investigated.

(iv) The last open problem concerns the results of the paper [19], where thepair of perturbed Euler type differential equations with different powers by “half-linearity” is investigated. Oscillatory properties of the equation

(Φp(x′))′ +

γp

tp

[

1 + qδ(t)]

Φp(x) = 0, (38)

where δ is a positive continuous function and q is the conjugate number to p,are compared with the equation of the same form containing the power functionΦp′(x) := |x|p

′−2x. In particular, it is proved that if 1 < p < p′ and the equation

with the power p′ is oscillatory then (38) has the same property. The subject ofthe present investigation is to extend these results to the perturbed half-lineardifference equation (29).


References

[1] Agarwal R. P., Difference Equations and Inequalities, Second Edition, Marcel Dekker, New

York – Basel, 2000.[2] Agarwal R. P., Grace S. R., O’Regan D., Oscillation Theory of Linear, Half-Linear, Super-

linear and Sublinear Dynamic Equations, Kluwer Academic, Dordrecht, 2002.[3] Agarwal R. P., Bohner M., Grace S. R., O’Regan D., Discrete Oscillation Theory, Hindawi,

2005.[4] Dosly O., Elbert A., Integral characterization of principal solution of half-linear differential

equations, Studia Sci. Math. Hungar. 36 (2000), 455–469.[5] Dosly O., Fisnarova S., Linearized Riccati technique and (non)oscillation criteria for half-

linear difference equations, Adv. Difference Equ., 2008 (2008), Article ID 438130, 18 pages.[6] Dosly O., Fisnarova S., Summation characterization of the recessive solution for half-linear

difference equations, to appear in J. Difference Equ. Appl.[7] Dosly O., Lomtatidze A., Oscillation and nonoscillation criteria for half-linear second order

differential equations, Hiroshima Math. J. 36 (2006), 203–219.[8] Dosly O., Rehak P., Recessive solution of half-linear second order difference equations, J.

Difference Equ. Appl. 9 (2003), 49–61.

[9] Dosly O., Rehak P., Half-Linear Differential Equations, North-Holland Mathematics Studies202, Elsevier, 2005.

[10] Dosly O., Unal M., Half-linear differential equations: Linearization technique and its ap-

plications, J. Math. Anal. Appl. 335 (2007), 450–460.

[11] Dosly O., Unal M., Conditionally oscillatory half-linear differential equations, Acta Math.Hungar. 120 (2008), 147–163.

[12] Dosly O., Unal M., Nonprincipal solutions of half-linear second order differential equations,Nonlinear Anal. 71 (2009), 4026–4033.

[13] Elbert A., Asymptotic behaviour of autonomous half-linear differential systems on the plane,Studia Sci. Math. Hungar. 19 (1984), 447–464.

[14] Elbert A., Schneider A., Perturbations of the half-linear Euler differential equation, Result.Math. 37 (2000), 56-83.

[15] Erbe L., G.Zhang B., Oscillation of second order linear difference equations, Chinese J.Math. 16 (1988), 239–252.

[16] Kelley W. G., Peterson A., Difference Equations: An Introduction with Applications, Acad.Press, San Diego, 1991.

[17] Luef F., Teschl G., On the finiteness of the number of eigenvalues of Jacobi operators below

the essential spetrum, J. Difference Equ. Appl. 10 (2004), 299–307.[18] Patıkova Z., Hartman-Wintner type criteria for half-linear second order differential equa-

tions, Math. Bohem. 132 (2007), 243–256.[19] Sugie J., Yamaoka N., Comparison theorems for oscillation of second order half-linear dif-

ferential equations, Acta. Math. Hungar. 111 (2006), 165–179.

Ondrej Dosly, Department of Mathematics and Statistics, Masaryk University, Kotlarska 2, CZ-611 37 Brno, Czech Republic,E-mail address: [email protected]

Simona Fisnarova, Department of Mathematics, Mendel University of Agriculture and Forestry

in Brno, Zemedelska 1, CZ-613 00 Brno, Czech Republic,



29

PRINCIPAL AND NONPRINCIPAL SOLUTIONS

IN THE OSCILLATION THEORY

OF HALF-LINEAR DIFFERENTIAL EQUATIONS

ONDREJ DOSLY and JANA REZNICKOVA

Abstract. We discuss the role played by principal and nonprincipal solutions ofthe half-linear second order differential equation

`

r(t)Φ(x′)´

′

+ c(t)Φ(x) = 0, Φ(x) := |x|p−2x, p > 1,

in the half-linear oscillation theory.

1. Introduction

In this paper we will study oscillatory properties of solutions of the half-linearsecond order differential equation

(r(t)Φ(x′))′

+ c(t)Φ(x) = 0, Φ(x) := |x|p−2x, p > 1, (1)

where r, c are continuous functions, r(t) > 0. Our principal concern is to showwhat role play principal and nonprincipal solutions in the oscillation theory of (1).

The concept of the principal solution of the second order linear differentialequation

(r(t)x′)′ + c(t)x = 0 (2)

was introduced by Leighton and Morse [15] and basic properties of this solutionwere investigated by Hartman, see [12] for a basic survey. It was shown thatnonoscillatory equation (2) possesses a unique (up to a nonzero multiplicativefactor) solution h, called the principal solution, with the property that

limt→∞

h(t)

x(t)= 0 (3)

for any solution x linearly independent of h. An equivalent characterization of theprincipal solution is

∫

∞ dt

r(t)h2(t)= ∞ (4)

2000 Mathematics Subject Classification. Primary 34C10.Key words and phrases. Half-linear differential equation, oscillation criterion, principal and

nonprincipal solution, Riccati technique.Research supported by the grants 201/08/0469 and 201/07/P297 of the Grant Agency of the

Czech Republic.

30 ONDREJ DOSLY and JANA REZNICKOVA

since this integral is convergent for any solution linearly independent of h. Solu-tions linearly independent of the principal solution are called nonprincipal solu-tions.

To explain the role played by the principal and nonprincipal solutions in theoscillation theory, we treat the linear equation (2) as first. Together with thisequation we consider the nonoscillatory equation

(r(t)x′)′ + c(t)x = 0, (5)

where c is a continuous function. Denote by h and x principal and nonprincipalsolutions of (5), respectively, for which r(x′h − xh′) ≡ 1 (such solutions alwaysexist).

The proofs of the following statements can be found in [2] and (sometimesimplicitly) in [17].

Theorem 1.1. Equation (2) is oscillatory provided one of the following condi-tions holds:(i) (Leighton-Wintner type criterion)

∫

∞

(c(t) − c(t))h2(t) dt = ∞; (6)

(ii) (Hille-Nehari type criterion with the principal solution) The integral in (6) isconvergent and

lim inft→∞

(∫ t

r−1(s)h−2(s) ds

)∫

∞

t

(c(s) − c(s))h2(s) ds >1

4; (7)

(iii) (Hille-Nehari type criterion with the nonprincipal solution)

lim inft→∞

(∫

∞

t

r−1(s)x−2(s) ds

)∫ t

(c(s) − c(s))x2(s) ds >1

4. (8)

The nonoscillatory counterpart of Theorem 1.1 reads as follows.

Theorem 1.2. Equation (2) is nonoscillatory provided one of the followingconditions holds:

(i) lim supt→∞

(∫ t

r−1(s)h−2(s) ds

)∫

∞

t

(c(s) − c(s))h2(s) ds <1

4,

lim inft→∞

(∫ t

r−1(s)h−2(s) ds

)∫

∞

t

(c(s) − c(s))h2(s) ds > −3

4; (9)

(ii) lim supt→∞

(∫

∞

t

r−1(s)x−2(s) ds

)∫ t

(c(s) − c(s))x2(s) ds < −1

4,

lim inft→∞

(∫

∞

t

r−1(s)x−2(s) ds

)∫ t

(c(s) − c(s))x2(s) ds > −3

4. (10)

HALF-LINEAR DIFFERENTIAL EQUATIONS 31

In the previous theorems, equation (2) is regarded as a perturbation of thenonoscillatory equation (5) and the previous theorems state, roughly speaking,that (2) becomes oscillatory (remains nonoscillatory) if the difference c − c issufficiently large (not too large) with respect to nonoscillation of (5). Note that

the term∫ t

r−1h−2 in (7) (which tends to ∞ as t → ∞) can be also expressed

as x(t)/h(t) and the term∫

∞

tr−1x−2 (which tends to zero) as h(t)/x(t). Rate

of the convergence h(t)/x(t) → 0 “measures” how much (5) is nonoscillatory.Consequently, from this point of view, in Hille-Nehari criteria from Theorems 1.1,1.2, the first term in the product characterizes the “nonoscillation stability” of (5)and the second term characterizes the “size of perturbation”.

The idea of the proof of the criteria given in this section is the following. Thetransformation x = hy resp. x = xy transforms (2) into the equation

(R(t)y′)′ + C(t)y = 0, (11)

where R = rh2, C = (c − c)h2 resp. R = rx2, C = (c − c)x2.Now, (11) is regarded as a perturbation of the one-term equation (R(t)y′)′ = 0.

The application of the classical methods of linear oscillation theory (variationalprinciple, Riccati technique) then gives (non)oscillation criteria for (11). Theseresults “transformed back” to (2) then yield criteria given in Theorems 1.1, 1.2.We refer to [2] for details.

2. Half-linear equations

In this section we show how the previous linear results can be extended to (1). Westart with the situation when (1) is regarded as a perturbation of the one termequation

(r(t)Φ(x′))′ = 0. (12)

We refer to [1, 13, 14] for the proofs of the next statements.

Theorem 2.1. Equation (1) is oscillatory provided one of the following condi-tions is satisfied:

(i)

∫

∞

r1−q(t) dt = ∞ and

∫

∞

c(t) dt = ∞, q :=p

p − 1. (13)

(ii) The first condition in (13) holds and

lim inft→∞

(∫ t

r1−q(s) ds

)p−1 ∫

∞

t

c(s) ds >1

p

(p − 1

p

)p−1

.

(iii) The integral∫

∞

r1−q(t) dt < ∞ (14)

and

lim inft→∞

(∫

∞

t

r1−q(s) ds

)p−1 ∫

∞

t

c(s) ds >1

p

(p − 1

p

)p−1

.


(iv) Condition (14) holds, c(t) ≥ 0 for large t, and

lim inft→∞

1∫

∞

tr1−q(s) ds

∫

∞

t

c(s)

(∫

∞

s

r1−q(τ) dτ

)p

ds >(p − 1

p

)p

.

The nonoscillatory counterparts of some results of the previous theorem arepresented in the next statement.

Theorem 2.2. Suppose that∫

∞

r1−q(t) dt = ∞ and∫

∞

c(t) dt converges. If

lim supt→∞

(∫ t

r1−q(s) ds

)p−1 (∫

∞

t

c(s) ds

)

<1

p

(p − 1

p

)p−1

, (15)

lim inft→∞

(∫ t

r1−q(s) ds

)p−1 (∫

∞

t

c(s) ds

)

> −2p − 1

p

(p − 1

p

)p−1

, (16)

then (1) is nonoscillatory. If (14) holds, the statement of the theorem remains to

hold provided∫ t

r1−q and∫

∞

tc in (15), (16) are replaced by

∫

∞

tr1−q and

∫ tc.

Now we recall the concept of the principal solution of nonoscillatory half-linearequation. A solution h of (1) is said to be principal (at ∞), if its logarithmicderivative is less than logarithmic derivative of any linearly independent solutionx of (1) for large t, i.e.,

h′(t)

h(t)<

x′(t)

x(t)for large t.

This also means that the principal solution of (1) defines via the formula w =rΦ(h′/h) the minimal solution of the associated Riccati equation

w′ + c(t) + (p − 1)r1−q(t)|w|q = 0, q =p

p − 1.

Note that this definition, applied to linear equation (2), defines the same solutionas characterized by (3) and (4), see [11, 16]. These two “linear” characterizationsof the principal solution do not extend directly to half-linear equation since theyare based on the linearity of the solution space and on the Wronskian identitywhich are lost in the half-linear case, see [6, Sec. 1.3].

In the remaining part of this section we regard (1) as a perturbation of thenonoscillatory general half-linear equation

(r(t)Φ(x′))′ + c(t)Φ(x) = 0. (17)

In contrast to the linear case, this equation cannot be reduced to a one-term equa-tion of the form (12). Hence, similarly to the construction of the principal solution,linear methods have to be substantially modified. The statements presented belowcan be found (in a slightly modified form with respect to the presentation givenhere) in the papers [5, 7, 10] .

We start with the Leighton-Wintner type criterion ([5]) which is the half-linearversion of the part (i) of Theorem 1.1.


Theorem 2.3. Let h be the positive principal solution of (17). If∫

∞

(c(t) − c(t))hp(t) dt = ∞,


The next statement is the half-linear extension of Theorem 1.1 (ii) and Theo-rem 1.2, its proof can be also found in [5].

Theorem 2.4. Suppose that (17) is nonoscillatory and possesses the principalsolution h such that h′(t) 6= 0 for large t. Denote

G(t) := r(t)h(t)Φ(h′(t)), R(t) := r(t)h2(t)|h′(t)|p−2. (18)

Further suppose that

lim inft→∞

|G(t)| > 0,

∫

∞

R−1(t) dt = ∞. (19)

(i) If

lim inft→∞

(∫ t

R−1(s) ds

)∫

∞

t

(c(s) − c(s))hp(s) ds >1

2q,


(ii) If

lim supt→∞

(∫ t

R−1(s) ds

)∫

∞

t

(c(s) − c(s))hp(s) ds <1

2q,

lim inft→∞

(∫ t

R−1(s) ds

)∫

∞

t

(c(s) − c(s))hp(s) ds > −3

2q,

then (1) is nonoscillatory.

We finish this section with the next statement. Its first part is proved in [7],while the second statement is a result which will be submitted for publication inthe next future.

Theorem 2.5. (i) Let x be a positive solution of (17) such that x′(t) > 0 forlarge t, say t > T ,

∫

∞

r−1(t)x−2(t)(x′(t))2−p dt < ∞, and denote

R(t) :=

∫

∞

t

ds

r(s)x2(s)(x′(s))p−2ds.

Suppose thatlim

t→∞

R(t)G(t) = ∞,

where the function G is given by formula (18) with x instead of h. If

lim supt→∞

R(t)

∫ t

T

(c(s) − c(s)) xp(s) ds <1

2q,


and

lim inft→∞

R(t)

∫ t

T

(c(s) − c(s)) xp(s) ds > −3

2q

for some T ∈ R sufficiently large, then (1) is nonoscillatory.

(ii) Let h be the positive principal solution of (17) such that (19) holds.If c(t) ≥ c(t) for large t and

lim inft→∞

1∫ t

R−1(s) ds

∫ t

T

(c(s) − c(s))hp(s)

(∫ s

R−1(τ) dτ

)2

ds >1

2q

for T sufficiently large, then equation (1) is oscillatory.

3. An oscillation criterion

In this section we prove a new oscillation criterion for (1). If h ≡ 1 in this criterion,it reduces to the main result of [3].

Theorem 3.1. Let p ≥ 2. Suppose that c(t) ≤ c(t) for large t, (17) is nonoscil-latory and possesses the positive principal solution h such that

∫

∞

r1−q(t)h−q(t)H(t, v) dt = ∞ (20)

for every v > 0, where

H(t, v) := |v + G(t)|q − qvΦ−1(G(t)) − |G(t)|q (21)

with G given by (18).Finally, suppose that

∫

∞

c∗(t) dt < ∞, where c∗ := (c − c)hp. If

∫

∞

r1−q(t)

∣

∣

∣

∣

wh(t) + h−p(t)

∫

∞

t

c∗(s) ds

∣

∣

∣

∣

q−2

× exp

−p

∫ t

r1−q(s)Φ−1

(

wh(s) + h−p(s)

∫

∞

s

c∗(τ) dτ

)

ds

dt < ∞, (22)

where wh = rΦ(h′/h), then (1) is oscillatory.

Proof. Suppose, by contradiction, that (1) is nonoscillatory and let x be itsprincipal solution. Since p ≥ 2, by [4]

I(x) :=

∫

∞ dt

r(t)x2(t)|x′(t)|p−2= ∞. (23)

Put v = hp(w−wh), w = rΦ(x′/x. Then the inequality c ≤ c implies that v(t) ≥ 0for large t, see [11], and v is a nonincreasing solution of the equation

v′ + (c(t) − c(t))hp(t) + (p − 1)r1−q(t)h−q(t)H(t, v) = 0. (24)


Hence there exists the nonnegative limit v(∞) and by (20) v(∞) = 0. Indeed,from (24), for T < t,

v(t) +

∫ t

T

c∗(s) ds + (p − 1)

∫ t

T

r1−q(s)h−q(s)H(s, v(s)) ds = v(T ), (25)

hence

(p − 1)

∫ t

T

r1−q(s)h−q(s)H(s, v(s)) ds ≤ v(T ).

Letting t → ∞ in (25) and replacing T by t, we can write (24) in the integral form

v(t) =

∫

∞

t

c∗(s) ds + (p − 1)

∫

∞

t

r1−q(s)h−q(s)H(s, v(s)) ds ≥

∫

∞

t

c∗(s) ds.

Hence,

w(t) = h−p(t)v(t) + wh(t) ≥ wh(t) + h−p(t)

∫

∞

t

c∗(s) ds.

Then, since x′ = Φ−1

(w(t))

Φ−1(r(t))x, we have

x(t) = exp

∫ t

r1−q(s)Φ−1(w(s)) ds

≥ exp

∫ t

r1−q(s)Φ−1

(

wh(s) + h−p(s)

∫

∞

s

c∗(τ) dτ

)

ds

and (we skip the argument t)

|x′|p−2 =

∣

∣

∣

∣

w

r

∣

∣

∣

∣

(q−1)(p−2)

|x|p−2 =

∣

∣

∣

∣

w

r

∣

∣

∣

∣

2−q

|x|p−2

≥ rq−2

∣

∣

∣

∣

wh + h−p

∫

∞

t

c∗∣

∣

∣

∣

2−q

exp

(p−2)

∫ t

r1−qΦ−1

(

wh + h−p

∫

∞

s

c∗)

ds

.

Consequently,

1

rx2|x′|p−2

<1

rq−1∣

∣wh + h−p∫

∞

tc∗

∣

∣

2−qexp

−p

∫ t

r1−qΦ−1

(

wh + h−p

∫

∞

s

c∗)

ds

= r1−q

∣

∣

∣

∣

wh+h−p

∫

∞

t

c∗∣

∣

∣

∣

q−2

exp

−p

∫ t

r1−qΦ−1

(

wh + h−p

∫

∞

s

c∗)

ds

.

This implies, by (22) that∫

∞ dt

r(t)x2(t)|x′(t)|p−2< ∞

and this contradiction with (23) completes the proof.


References

[1] Dosly O., Methods of oscillation theory of half-linear second order differential equations,

Czech. Math. J. 50 (2000), 657–671.[2] Dosly O., Oscillation theory of linear differential and difference equations, In Proceedings of

Seminar “Differential Equations” (Pavlov, Palava, 2002), Plzen: West Bohemia UniversityPress7 (2002), 7–73.

[3] Dosly O., Half-linear second order differential equations: Oscillation theory and principal

solution, Stud. Univ Zilina Math. Ser. 17 (2003), 69–78.

[4] Dosly O., Elbert A., Integral characterization of the principal solution to half-linear second-

order differential equations, Stud. Sci. Math. Hung. 36 (2000), 455–469.[5] Dosly O., Lomtatidze A., Oscillation and nonoscillation criteria for half-linear second order

differential equations, Hiroshima Math. J. 36 (2006), 203–219.[6] Dosly O., Rehak P., Half-Linear Differential Equations, North Holland Mathematics Stu-

dies, Elsevier, 2005.[7] Dosly O., Reznıckova J., Oscillation and nonoscillation of perturbed half-linear Euler dif-

ferential equation, Publ. Math. Debrecen 71 (2007), 479–488.

[8] Dosly O., Reznıckova J., Nonprincipal solutions in oscillation criteria for half-linear differ-

ential equations, to appear in Studia Sci. Math. Hungar.[9] Dosly O., Reznıckova J., A remark on oscillation constant in a half-linear oscillation cri-

terion, in preparation.[10] Dosly O., Unal M., Half-linear differential equations: Linearization technique and its ap-

plications, J. Math. Anal. Appl. 335 (2007), 450–460.

[11] Elbert A., Kusano T., Principal solutions of nonoscillatory half-linear differential equations,Adv. Math. Sci. Appl. 18 (1998), 745–759.

[12] Hartman P., Ordinary Differential Equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964.

[13] Kusano T., Naito Y., Oscillation and nonoscillation criteria for second order quasilinear

differential equations, Acta. Math. Hungar. 76, (1997), 81–99.[14] Kusano T., Naito Y., Ogata A., Strong oscillation and nonoscillation of quasilinear diffe-

rential equations of second order, Diff. Equations Dyn. Syst. 2 (1994), 1–10.[15] Leighton W., Morse M., Singular quadratic functionals, Trans. Amer. Math. Soc. 40 (1936),

252–286.[16] Mirzov J. D., Principal and nonprincipal solutions of a nonoscillatory system, Tbiliss. Gos.

Univ. Inst. Prikl. Mat. Trudy 31 (1988), 100–117.[17] Swanson C. A., Comparison and Oscillation Theory of Linear Differential Equations, Acad.

Press, New York-London 1968.

Ondrej Dosly, Department of Mathematics and Statistics, Masaryk University, Kotlarska 2, CZ-

611 37 Brno, Czech Republic,E-mail address: [email protected]

Jana Reznıckova, Department of Mathematics, Tomas Bata University, Nad Stranemi 4511, 760

05 Zlın, Czech Republic,



37

RESEARCH STABILITY OF SOLUTIONS

OF THE SYSTEM LINEAR DIFFERENTIAL EQUATIONS

WITH RANDOM COEFFICIENTS

I. DZHALLADOVA

Abstract. We find necessary and enough conditions for L2-stability of solutions ofthe linear system differential equations with random (Semi-Markovian) coefficients.

Introduction

Let’s consider the system of the linear differential equations

dx(t)

dt= A(t, ξ(t))X(t) (1)

where ξ(t) – semi Markovian random process, which have value θ1, . . . , θn.

We assume:

1. Semi-Markovian random process determined with intensities qsk

(s, k = 1, 2, . . . , n).2. Always assume boundaries of initial values X(0).3. In time t-j semi-Markovian process and solutions of system equations (1)

have jump which defined the equations

X(tj + 0) = CksX(tj − 0), detCks 6= 0. (2)

1. First section

Definition 1.1. A zero solution of the system (1) is named asymptotic ofstability in middle, if for the arbitrary solution X(t) of the system (1) a limitcorrelation is executed

limt→∞

M(t) = limt→∞

〈X(t)〉 = 0

2000 Mathematics Subject Classification. Primary 35R35, 49M15, 49N50.Key words and phrases. stochastic system, stability, semi-Markovian process.

38 I. DZHALLADOVA

Moments equations for the system (1) have kind:

Mk(t) = ψk(t)Nk(t)Mk(0) +

∫ t

0

ψk(t− τ)Nk(t− τ)Vk(τ)dτ, (3)

Vk(t) =

n∑

s=1

qsk(t)CksNs(t)Ms(t)Ms(0)

+

∫ t

0

n∑

s=1

qsk(t− τ)CksNs(t− τ)Vk(τ)dτ, (4)

where Mk(t) - moments of first order.

Definition 1.2. A zero solution of the system (1) is named asymptotic ofstability in middle quadratic, if at the arbitrary initial conditions executed limitcorrelation

limt→∞

〈‖X(t)‖2〉 = 0, (5)

‖X(t)‖2 =

n∑

k=1

|xk|2. (6)

It is possible to prove, that asymptotic of stability in middle quadratic equiva-lent implementation of correlation

limt→∞

D(t) = 0, D(t) ≡ 〈X(t)X∗(t)〉. (7)

For research of stability in middle quadratic it is possible to use the system ofthe equations for the matrix r.m. second order

Dk(t) = ψk(t)Nk(t)Dk(0)N∗

k (t) +

∫ t

0

ψk(t−τ)Nk(t−τ)Wk(τ)N∗

k (t−τ)dτ, (8)

Wk(t) =

n∑

s=1

qks(t)CksNs(t)Ds(0)N∗

s (t)C∗

ks +

+

∫ t

0

n∑

s=1

qsk(t−τ)CksNs(t−τ)Ws(τ)N∗

k (t−τ)C∗

ksdτ. (9)

Definition 1.3. A zero solution of the system of the equations (1) is namedL2-stability, if the integral convergence

I =

∫

∞

0

‖X(t)‖2dt. (10)

Obviously, that zero solution of the system (1) of L2-stability then and onlyafter, when the matrix integral convergence

D =

∫

∞

0

D(t)dt. (11)

That to find the condition of L2-stability terms, will point some auxiliary re-sults.

RESEARCH STABILITY OF SOLUTIONS. . . 39

Lemma 1.4. For matrices of second partial moments

Dk(t) =

∫ t

0

XX∗fk(t,X)dX, (k = 1, . . . , n).

inequality executed Dk(t) ≥ 0.

Proof. So far as fk(t,X) ≥ 0, at the arbitrary vector have inequality

Y ∗Dk(t)Y =

∫ t

0

(Y ∗X)(X∗Y )fk(t,X)dX =

∫ t

0

|X∗Y |2fk(t,X)dX ≥ 0.

That is was required to prove.

Lemma 1.5. For the system of the integral equations (8) unevenness are exe-cuted Wk(t) ≥ 0.

Proof. Solution of the system of the equations for Wk(t) it is possible to findmethod of the successive approaching at, taking

W(j+1)

k (t) =

n∑

s=1

qks(t)CksNs(t)Ds(0)N∗

s (t)C∗

ks

+

∫ t

0

n∑

s=1

qks(t−τ)CksNs(t−τ)W(j)s (τ)N∗

s (t−τ)C∗

ksdτ, W(0)

k (t) ≡ 0. (12)

From the system of the equation (9) swims out, that at W(j)

k (t) ≥ 0 inequalityare executed, so far as. From the system of the equation (8) swims out, thatinequality Wk(t) ≥ 0 (k = 1, . . . , n) are executed and-symmetric matrices.


Theorem 1.6. Let’s assume that (1)–(3) conditions are executed. For that azero solution of the system (1) was L2-stability, it is necessary, that were conver-gence unusual integrals

Ik =

∫

∞

0

ψk(t)Nk(t)Dk(0)N∗

k (t)dt <∞.

Proof. From the system of the equation (8) swims out, that inequalities areexecuted

Dk(t) ≥ ψk(t)Nk(t)Dk(0)N∗

k (t),

integrating which, find inequalities

Dk =

∫

∞

0

Dk(t)dt ≥

∫

∞

0


k (t)dt.


It is possible to replace the conditions (9) by conditions of convergence of matrixunusual integrals

Ik0 =

∫

∞

0

ψk(t)Nk(t)N∗

k (t)dt <∞. (13)

40 I. DZHALLADOVA

Indeed, for any symmetric matrix Dkk(0) ≥ 0 the number exists, such that. Thusinequalities are executed

Ik =

∫

∞

0


k (t)dt ≥

∫

∞

0

ψk(t)Nk(t)ρkEN∗

k (t)dt ≥ ρkIk0. (14)

We are integrating system of the equations (8) on interval from 0 to infinity.Noting

Dk =

∫

∞

0

Dk(t)dt,Wk =

∫

∞

0

Wk(t)dt. (15)

We will obtain the system of the linear algebra equations for matrices Wk, Dk

Dk =

∫

∞

0

ψk(t)Dk(0)N∗

k (t)dt+

∫

∞

0

ψk(t)Nk(t)Wk(t)N∗

k (t)dt, (16)

Wk =

n∑

s=1

∫

∞

0

qks(t)CksNs(t)(Ds(0) +Ws)N∗

s (t)C∗

ksdt. (17)

Lemma 1.7. If inequalities Ik0 > 0 (10) are executed, from boundaries ofmatrices the boundaries of matrices swim out.

Proof. From condition Dk(0) ≥ 0 and equation (13) the inequalities swim out

Dk ≥

∫

∞

0

ψk(t)Nk(t)Wk(0)N∗

k (t)dt. (18)

Will choose a scalar multiplier, such that inequalities were executed

ρkIk0 ≥ Dk.

From the matrix inequality swims out∫

∞

0

ψk(t)Nk(t)(ρkE −Wk)N∗

kdt ≥ 0.


Using the lemmas we obtain the following result:

Theorem 1.8. Let unusual matrix integrals

Ik0 =

∫

∞

0

ψk(t)Nk(t)N∗

k (t)dt

convergence and are positive definite symmetric matrices, that are Ik0 > 0. Thennecessary and sufficient condition of L2-stability solutions of the system of theequation (1) there is a boundary of symmetric matrices that is implementation ofkind inequalities.

Will convert the system of the matrix equation (13). We used by denotation

Bk = Dk(0) +W(k).

It is possible to write down the system of the equation (13) in kind

Bk = D(0) +

n∑

s=1

∫

∞

0

qks(t)CksNs(t), (19)

RESEARCH STABILITY OF SOLUTIONS. . . 41

Will consider monotonous linear operators, set by formulas

LksBs =

∫

∞

0

qks(t)CksNs(t)BsN∗

s (t)C∗

ksdt. (20)

Thus it is possible to write down the system of the equation (8) in the operatorform:

B = D(0) + LB, (21)

where

B =

B1

B2

. . .Bn

. (22)

Consider notion of efficiency for matrices B if, where

L11 L12 . . . L1n

L21 L22 . . . L2n

......

. . ....

Ln1 Ln2 . . . Lnn

. (23)

Thus the operator will be monotonous, so far as from inequalities the B(1) ≥ B(2)unevenness LB(1) ≥ LB(2) swims out. From theorem 2 swims out, that thesystem of the equation (13) has a solution then and only after, when the methodof the successive approaching coincides

B(j+1)

k = Dk(0) +n

∑

s=1

∫

∞

0

qks(t)CksNs(t)B(j)s N∗

s (t)C∗

ksdt. (24)

As an existence of limited matricesWk ≥ 0, which satisfy the system of theequations (13), pursuant to theorem 2 equivalent the solutions, we obtain thefollowing result.

Theorem 1.9. Let for the system of the linear differential equation (1) withjump of solutions (3) conditions of theorem 1, 2 are executed.

For a zero solution of the system (1) was L2-stability it is necessary and enough,that one of equivalent conditions was executed:

1) The system of the equation (15) at any Dk(0) > 0 had a positive solution.2) System of the equation (15) at Dk(0) = E had a positive solution.3) The method of the successive approaching coincided (17).4) The operator had a spectral radius, less from unit.

For the asymptotic L2-stability solutions of the system (1) enough, that at somematrices Bk > 0 inequality were executed

Bk −

n∑

s=1

∫

∞

0

qks(t)CksNs(t)BsN∗

s (t)C∗

ksdt > 0.

42 I. DZHALLADOVA

Example 1.10. Will find the condition of theL2-stability solutions of the equa-tion

dx(t)

dt= a(ξ(t))x(t), a(θk) ≡ ak, (25)

where ξ(t) – Semi-Markovian process, that acquires two states with the set inten-sities

q12 = q21 = 0, t > T

2T−2(T − t), t ∈ [0, T ] .

We assume, that the solution of the equation (18) has jumps

x(tj + 0) = cx(tj − 0),

which take place simultaneously with jumps of solutions of the equation (18).The system of the equation (14) collects a kind

b1 = D1(0) + c2e2a2T − 1 − 2a2T

2a22T 2

b2, (26)

b2 = D2(0) + c2e2a1T − 1 − 2a1T

2a21T 2

b1. (27)

Condition of the L2-stability of solutions is given by such inequalities:

c4e2a1T − 1 − 2a1T

2a21T 2

e2a2T − 1 − 2a2T

2a22T 2

< 1.

References

[1] Valeev K. G., Dzhalladova I. A., Operacijne chislennja ta jogo zastosuvannja, (ukrainian)Kiev, 2004.

[2] Dzhalladova I. A., Optimizatsija stochastichnich ssstem, (ukrainian) Kiev, 2006.[3] Valeev K. G., Dzhalladova I. A., Optimizatsija vipadkovich procesiv, (ukrainian) Kiev, 2008.

I. Dzhalladova, B. Getman Kyiv National University of Economics, Current address: Rudenkoh

Str., 21, r.134, 02140, Kyiv, Ukraine,



43

ON ESTIMATE OF THE NUMBER

OF SOLUTIONS FOR QUASILINEAR 6-TH ORDER

BOUNDARY VALUE PROBLEM

TATJANA GARBUZA

Abstract. A special technique based on the analysis of oscillatory behavior oflinear equations is applied to investigation of nonlinear boundary value problemof sixth order. A quasi-linear sixth order equation x(6) = f(t, x) is studied. Weget the estimation of the number of solutions to the boundary value problems ofthe type x(6) = a(t)x + g(t, x), x(a) = A, x′(a) = A1, x′′(a) = A2, x′′′(a) = A3,x(b) = B, x′(b) = B1, where g is continuous together with the partial derivati-ve gx which is supposed to be positive, and a(t) is continuous positive valued int ∈ (a; +∞) function. We assume also that at least one solution to the problemunder consideration exists.

Introduction

We employ the idea by A. Perov [1, ch. 15] who studied the multiplicity of solutionsto two-point the second order nonlinear boundary value problems. His approachis based on comparison of the behavior of solutions of the equation at some givensolution of the BVP and at infinity. The first behavior is described in termsof the linear equation of variations and the second behavior is a consequence ofrequirements on a function f in the right side. Using this idea and the technique ofthe angular function (see [7]) the multiplicity results were obtained. To apply thisidea to the study of higher order equations one have to choose another methodssince the angular function technique hardly can be applied in this case. As analternative the theory of oscillation of linear equations of higher order can be used.Multiplicity results for the third order BVPs were obtained in [2] by combining theidea of A. Perov and some facts from the linear theory of conjugate points. Thenotion of a conjugate point is useful in our considerations. We use the definitionof conjugate point by Kiguradze [5].

Definition 0.1. By m-th conjugate point ηm of a is called the minimalvalue of t = b, (b > a) where b is a (m+n−1)-th zero (counting multiplicities in[a; +∞)) of solution of n-th order equation which has a zero at t = a.

2000 Mathematics Subject Classification. Primary 34B15.Key words and phrases. differential equations of 6-th order, nonlinear boundary value prob-

lem, quasilinear ordinary differential equation, multiplicity of solutions, oscillation, even orderordinary differential equation.

Supported by ESF project Nr. 2004/0003/VPD1/ESF/PIAA/04/NP/3.2.3.1./0003/0065.

44 TATJANA GARBUZA

As a by-product, a technique was elaborated for treatment of the higher dimen-sional case. In [3], n-th order BVPs were considered by straight-forward general-izations of the results of [2]. However, only the case of n−1 (of total number n)boundary conditions prescribed at one end of the interval was treated.

We refer to the book by Kiguradze and Chanturia [5] with respect to two termedequations y(n) = p(t)y, and to the article by Hunt [4] for sixth order equation.

In this paper we study the boundary value problem

x(6) = f(t, x), where f = a(t)x + g(t, x), (1)

x(a) = A, x′(a) = A1, x′′(a) = A2, x′′′(a) = A3, (2)

x(b) = B, x′(b) = B1, (3)

under the assumption that there exists a particular solution ξ(t) of the above prob-

lem and functions a(t), g(t, x), gx = ∂g∂x

are continuous, a(t) > 0 for t ∈ [a, +∞)and gx(t, x) > 0 for (t, x) ∈ [a, +∞) × R. A solution of (1) can be described interms of oscillatory behavior of the corresponding linear equation of variations

y(6) = a(t)y + gx(t, ξ(t))y, (4)

which will play a significant role in our considerations. Our results are based onthe theory of 6-th order differential equations of the form

y(6) = p(t)y, (5)

with boundary conditions (2)—(3). We assume that p(t) is continuous and p(t)>0.Our principal result consists of estimation from below of the number of solutions

to the boundary value problem (1)—(3) provided that it has at least one solutionξ(t). This estimate depends on the oscillatory behavior of the equation (4).

The paper is organized as follows. In the subsequent section 1 we investigatelinear equation of the form (5).

The main theorem of this section describes the two-parametric set of solutionsto the equation (5), subject to initial data

y(a) = y′(a) = y′′(a) = y′′′(a) = 0. (6)

In section 2 the multiplicity result is proved. The nonlinear problem (1)—(3)is considered, provided that g is bounded. Our method of proof differs from thatused by Perov [1] and is based on representation of the nonlinear equation (1) asa family of linear equations, the coefficients of which depend on solutions of (1)satisfying the initial value conditions (2).

1. Linear equation

In this section we consider the equation (5)

y(6) = p(t)y,

with continuous coefficient p > 0.

NUMBER OF SOLUTIONS OF QUASILINEAR 6-TH ORDER BVP 45

Definition 1.1. Let us call a solution x(t) of (5) (l, 6− l)-solution iff thereexist points t1, t2 (a ≤ t1 < t2) such that this solution satisfies conditions

x(i)(t1) = 0, (i = 0, . . . , l−1),

x(i)(t2) = 0, (i = 0, . . . , 5−l).

Lemmas similar to Lemma 2.1 and Lemma 2.2 in [6] are valid for equation (5).

Lemma 1.2. If y(t) is a solution of (5) and the values of y(i), i = 0, . . . , 5 are

non-negative (but not all zero) for t = a, then the functions y(i)(t), i = 0, . . . , 5are positive for t > a.

Proof. Consider the case of y′(a) > 0. First of all we show that y′(t) > 0, t > a.From statement of lemma we get that all other derivatives are nonnegative in

t = a, therefore there exists an interval [a, t0] where y′(t) > 0 for any t ∈ [a, t0),y(t) > 0 for any t ∈ (a, t0], and y′(t0) = 0.

The right hand side of

y′(t) = y′(a) +

5∑

k=2

y(k)(a) (t−a)k−1

(k−1)!+

∫ t

a

∫ z

a

∫ v

a

∫ u

a

∫ r

a

p(s)y(s) ds (7)

is positive and grow up together with t until y(t) remain positive.We have that y′(t) ≥ y′(a) > 0 in (a, t0] and therefore y′(t0) > 0, a contradic-

tion.In other cases, if some of the values y(i)(a) (i = 0, 2, 3, 4, 5) is positive, the proof

is similar.

Lemma 1.3. If u(t), v(t) are two different solutions of (5) and u(i)(a)≥v(i)(a),i = 0, . . . , 5 (at least one inequality is strict) then u(i)(t) > v(i)(t), i = 0, . . . , 5for t > a.

Proof. The result follows from Lemma 1.2, if we consider the function w(t) =u(t)−v(t). This function is a solution of (5) and w(i), i = 0, . . . , 5 are non-negative(but not all zero).

Let us state result by Kiguradze and Chanturia [5, Lemma 1.10, p. 26] adaptedto the case of the 6-th order equation.

Lemma 1.4. The k-th conjugate point ηk (k = 1, 2, 3, . . .) of equation (5),(p(t) > 0), is represented by either (4, 2)-solution or (2, 4)-solution which has

exactly k−1 zeros of odd order in interval (a, ηk).

Now we can prove two important results.

Lemma 1.5. The k-th conjugate point ηk of equation (5) is represented by

either (4, 2)-solution (or (2, 4)-solution), which has exactly k−1 simple zeros in

interval (a, ηk).

46 TATJANA GARBUZA

Proof. Define the function Φ+(t) as

Φ+(t) = x′′x′′′ − x′x(4) + xx(5). (8)

The function Φ+(t) for the (4, 2)-solution of equation (5) is nondecreasing po-sitive valued function for any t > 0, since Φ+(a) = 0,

Φ′

+(t)=

(

(x(3))2+x′′x(4))

−(

x′′x(4)+x′x(5))

+(

x′x(5) + xx(6))

=x′′′2+p(t)x2≥0

for any t > 0, and there exists an interval (a; t∗) where Φ+(t) 6= 0.It means that after quadruple zero there are zeros of order not greater than

two, but from Lemma 1.4 we know that all zeros of (4, 2)-solution which relatedto ηk in interval (a, ηk) are odd, therefore simple.

Lemma 1.6. The k-th conjugate point ηk continuously depends on the coeffi-

cient p(t) > 0 of the equation (5).

Proof. Consider equation (5) together with the initial conditions

y(a) = y′(a) = y′′(a) = y′′′(a) = 0, y(4)(a) = α, y(5)(a) = β. (9)

Due to linearity of the equation we can consider the initial angle

Θ = arctan βα∈ (−π

2, 0). (10)

Let Θk be the angle corresponding to a solution yk(t) with the conjugate pointηk which is a double zero, unstable under perturbations of the coefficient p(t).However, by Lemma 1.2 a solution with Θ < Θk satisfies y(t) < yk(t) for t > 0. IfΘ is close enough to Θk the respective y(t) must have two simple zeros on oppositesides of ηk and close to it. These simple zeros change continuously together withp(t). Since ηk lies between, it changes continuously too.

It can be shown (like in the work [7] for fourth order linear two-termed equa-tions) that both sequences ηi and Θi are ordered as

−π2

< Θ2 < . . . < Θ2i < . . . < Θ2i+1 < . . . < Θ1 < 0

and

0 < η1 < η2 < . . . .

Evidently both monotone sequences Θ2i−1 and Θ2i have limits, say, Φodd

and Φeven respectively. Computations show that in the case of p(t) = 1 one hasthat Θodd = Θeven = −π

4.

2. Nonlinear equation

In this section, results for the nonlinear boundary value problem (1)—(3) arestated and proved.

Suppose ξ(t) is a solution of the boundary value problem (1)—(3). Let x(t, α, β)be a solution of the equation (1), subject to the initial conditions (2) and

x(4)(a) − ξ(4)(a) = r cosΘ, x(5)(a) − ξ(5)(a) = r sin Θ. (11)


We denote z(t) = x(t) − ξ(t). Consider auxiliary linear equations

z(6) = ϕ(t, r, Θ)z, (12)

where coefficient ϕ depends on x(t, r, Θ) and ξ(t) as follows:

ϕ =f(t, x(t, r, Θ)) − f(t, ξ(t))

x(t, r, Θ) − ξ(t).

We assume that at the points where denominators vanish the right hand sidesare substituted by appropriate value of fx.

Our further considerations are based on the following observation.

Lemma 2.1. A solution x(t) of the initial value problem (1), (2), (11) satisfies

also the conditions (3) (i.e. it is a solution of the problem (1)—(3)) if and only if

there exists an extremal function Z(t) of the linear equation (12), which represents

a conjugate to t = a point at t = b, and such that

arctan z(5)(a)

z(4)(a)= Θ. (13)

Proof. Let x(t) be a solution (x(t) 6= ξ(t)) of the problem (1)—(3). Necessitythen follows readily from the observation that x− ξ is an extremal function, sinceit satisfies the equation (12) and has a quadruple zero at t = a and a double zeroat t = b.

Consider now linear equation (12) corresponding to some solution x(t) of (1),(2), (11). Suppose that Z(t) is an extremal function for (12) with a conjugatepoint at t = b.

Without loss of generality we may assume that

Z(4)(a) = x(4)(a) − ξ(4)(a), Z(5)(a) = x(5)(a) − ξ(5)(a). (14)

Otherwise Z(t) should be multiplied by an appropriate constant. Both func-tions Z(t) and x(t) − ξ(t) are solutions to the same Cauchy problem for linearequation (12). Thus they are identical, and x(t) = Z(t) + ξ(t) satisfies also theboundary condition (3) at the right end of interval (a, b). Hence the proof.

Lemma 2.2. Let g in (1) be bounded. Let ξ(t) be a solution of the problem

(1)—(3) and x(t) be a solution of the initial value problem (1)—(2). Then the

difference x(t) − ξ(t) cannot have more than m + 5 zeros (counting multiplicities)

in [a, b) for large enough r2 = (x(4)(a)− ξ(4)(a))2 +(x(5)(a)− ξ(5)(a))2, if ordinary

differential equation y(6) = a(t)y has m conjugate points in (a, b), and t = b is not

a conjugate point.

Proof. We have that

(x − ξ)(6) = a(t)(x − ξ) + g(t, x) − g(t, ξ),

(x − ξ)

r2

(6)

= a(t)x − ξ

r2+

g(t, x) − g(t, ξ)

r2,

48 TATJANA GARBUZA

therefore functions u = x−ξr2 satisfy

u(6) = a(t)u + δ,

u(a) = u′(a) = u′′(a) = u′′′(a) = 0,

u(4)(a) = α, u(5)(a) = β, α2 + β2 = r2, (15)

where δ → 0 as r2 → +∞.The function u(t) tends to a solution y(t) of the problem

y(6) = a(t)y,

y(a) = y′(a) = y′′(a) = y′′′(a) = 0,

y(4)(a) = α, y(5)(a) = β, α2 + β2 = r2.

We have assumed that equation y(6) = a(t)y has exactly m conjugate points in(a; b) and t = b is not a conjugate point. Then y(t, α, β) for any (α, β) has notmore than m + 5 zeros, counting multiplicities in [a; b). The some behavior havesolutions of (15).

The proof is complete.

Theorem 2.3. Let ξ(t) be a solution of the problem (1)—(3). Suppose that

function g in (1) is bounded and there exists a solution of the problem (1)—(3) such

that the interval (a, b) contains exactly k conjugate points η1, η2, . . . , ηk (to t = a)

with respect to the equation of variations (4). Then the boundary value problem

(1)- (3) has at least 2|k − m| + 1 solutions (ξ counted), if ordinary differential

equation y(6) = a(t)y has m conjugate points in (a, b).

Proof. Fix Θ ∈ (−π/2, 0) and consider a one parametric family of linear equa-tions (12). For small r, solutions of (12) behave like ones of the equation of varia-tions. Hence, there exist k conjugate points η1, η2, . . . , ηk in the interval (a, b) forsmall r. On the other hand for large r, this interval contains m points ην . The onlyway for conjugate point to leave the interval (a, b), as r varies from zero to infinity,is to pass over t = b. Let Sk(Θ) = max[r : ηk = b]. For any Θ ∈ [−π/2, 0) andany k such Sk exists and form continuous one-parametric (dependent on Θ) curve.Denote by wk(Θ) the angle defining the first extremal function of the equation

z(6) = ϕ(t, Sk(Θ), Θ)z,

and consider the difference wk(Θ) − Θ.Note that w1(0) − 0 is positive and w1(−

π2) − (−π

2) is negative. Hence, there

exist Θk ∈ [−π/2, 0) such that wk(Θk) = Θk. By Lemma 2.1, the k-th extremalfunction of the equation (12) where r = Sk(Θk), Θ = Θk, generates a solution tothe boundary value problem (1)—(3). It’s true for any k. Next |k−m| solutions tothe boundary value problem (1)—(3) we get after consideration Θ ∈ (π/2, π).

Remark 2.4. If we suppose that function f(t, x) = g(t, x) (that is, a(t) ≡ 0)in (1). Where g is increasing bounded function and there exists a solution of theproblem (1)—(3) such that the interval (a, b) contains exactly k conjugate points


η1, η2, . . . , ηk (to t = a) with respect to the equation of variation (4). Than theboundary value problem (1)- (3) has at least 2k+1 solutions (ξ counted), becauseordinary differential equation y(6) = 0 has not conjugate points in (a, b), (that is,m = 0.)

Example 2.5. Let us show the example of using the results of Theorem 2.3.We consider the quasilinear boundary value problem

x(6) = 86x + arctan(87x),

x(0) = x′(0) = x′′(0) = x′′′(0) = x(1) = x′(1) = 0. (16)

By linearization we have the linear equation in variation with constant coeffi-cient

y(6) = (86 + 87)y,

which has two extremal solutions with conjugate points (double zeros) in (0; 1),approximately, η1 ≈ 0.58 and η2 ≈ 0.89; but for large enough ρ we have equationwith constant coefficient 86 which has exactly one extremal solution with conjugatepoint in (0; 1) at the point t ≈ 0.838.

By Theorem 2.3 the problem (16) has at least (2 − 1)2 = 2 solution (ξ doesn’tcounted ).

We have computed solutions of nonlinear BVP (16). They are given on theFigure 1. (Second solution is given by the variable change x = −x. A third one isa trivial solution ξ ≡ 0.)

-0.25 0.25 0.5 0.75 1 1.25t

-7.5´10-6

-5´10-6

-2.5´10-6

2.5´10-6

5´10-6

7.5´10-6

xHtL

x1HtL

x2HtL

Figure 1. The three solutions of nonlinear boundary value problem (16): x1 sa-

tisfies x(4)

1(0) = 0.00335685, x

(5)

1(0) = −0.0393, x2 satisfies x

(4)

2(0) = −0.00335685,

x(5)

2(0) = 0.0393, x3 ≡ 0.

Example 2.6. Let us show the example of using the result of Remark 2.4 toTheorem 2.3. We consider the nonlinear boundary value problem

x(6) = arctan(87x), x(0) = x′(0) = x′′(0) = x′′′(0) = x(1) = x′(1) = 0. (17)

By linearization we have the linear equation of variations

y(6) = 87y,

50 TATJANA GARBUZA

which has exactly two extremal solutions with conjugate point (double zero) in(0; 1). By computation η1 ≈ 0.592876, η2 ≈ 0.907561. We have computed five(2k+1 = 2 · 2+1 = 5) solutions of nonlinear boundary value problem. They aregiven on the Figures 2 and 3. Fifth solution is a trivial solution ξ ≡ 0.

-0.25 0.25 0.5 0.75 1 1.25t

-0.0001

-0.00005

0.00005

0.0001xHtL

x1HtL

x2HtL

Figure 2. Solutions of nonlinear boun-dary value problem (17): x1 satisfies

x(4)

1(0)=0.040216, x

(5)

1(0)=−0.307, x2 sa-

tisfies x(4)

2(0)=−0.040216, x

(5)

2(0)=0.307.

-0.25 0.25 0.5 0.75 1 1.25t

-4´10-6

-2´10-6

2´10-6

4´10-6

xHtL

x3HtL

x4HtL

Figure 3. Solutions of nonlinear boun-dary value problem (17): x3 satisfies

x(4)

3(0)=0.0017779, x

(5)

3(0)=−0.0205,

x4 satisfies x(4)

4(0)=−0.0017779,

x(5)

4(0)=0.0205, x5≡0.

References

[1] Krasnoselskii M., Perov A., Povolokii A., Zabreiko P., Plane vector fields, Acad. Press, NewYork, 1966.

[2] Gera M., Sadyrbaev F., Multiple solutions of a third order boundary value problem, Math.Slovaca., 42 (1992), n. 2, pp. 173–180.

[3] Rovderova E., Number of solutions of boundary value problems, Nonlinear Anal.: TMA, 21

(1993), no 5, pp. 363–368.[4] Hunt R. W., The behavior of solution of ordinary, self-adjont differential equations of ar-

bitrary even order, Pacific Journal of Mathematics, No. 12, 1962, pp. 945–961.[5] Kiguradze I. T., Chanturia T. A., Asymptotic properties of solutions of nonautonomous

ordinary differential equations (Russian), Moscow, Nauka, 1990.[6] Leighton W., Nehari Z., On the oscillation of solutions of self-adjoint linear differential

equations of the fourth order, Trans. Amer. Math. Soc., 89, pp. 325–377, 1961.[7] Sadyrbaev F., Multiplicity of solutions for fourth order nonlinear boundary value problems,

Proc. Latvian Acad. Sciences, 7/8 (576/577) (1995), pp. 107–111.[8] Sherman T., Properties of solutions of N-th order linear differential equations, Pacific Jour-

nal of Mathematics, Vol. 15, No. 3, pp. 1045–1060, 1965.

Tatjana Garbuza, Daugavpils University, Parades str. 1, Daugavpils, Latvia, LV-5401,



51

ON POSITIVITY OF THE THREE TERM 2n-ORDER

DIFFERENCE OPERATORS

P. HASIL

Abstract. We consider the symmetric, three-term, 2n-order difference equation

aky

k+ b

k+ny

k+n+ a

k+ny

k+2n= 0

and we show that this equation possesses a positive solution for k ∈ Z if and only ifthe infinite symmetric matrix associated with this equation is nonnegative definite.

1. Introduction

Let ak < 0, bk > 0, k ∈ Z, be real-valued sequences and consider the symmetric2n-order recurrence relation

(τy)k+n := akyk + bk+nyk+n + ak+nyk+2n.

We associate with the difference equation

τy = 0 (1)

the operator T : ℓ2(Z) → ℓ2(Z) with the domain

ℓ2

0(Z) =

y = ykk∈Z ∈ ℓ2(Z), only finitely many yk 6= 0

given by the formula T f = τf, f ∈ ℓ20(Z), and we show that (1) has a positive

solution yk, k ∈ Z, if and only if the operator T is nonnegative, i.e.

〈T y, y〉 ≥ 0 ∀y ∈ ℓ2

0(Z),

where 〈·, ·〉 denotes the usual inner product in ℓ2(Z). The statements presented inthis paper extend some results given in [1] where the case n = 1 is considered.

2000 Mathematics Subject Classification. Primary 47B36, 39A70.Key words and phrases. Positive solutions, matrix operator, difference equations.Research supported by the grant 201/07/0145 of the Czech Grant Agency.

52 P. HASIL

2. Preliminary results

Let T be infinite symmetric matrix associated in the natural way with the operatorT and let us denote for µ ≤ ν, µ, ν ∈ Z the truncations of T by

tµ,ν :=

bµ 0 · · · 0 aµ · · · 0

0 bµ+1 0 · · · 0. . .

...... 0

. . .... 0 aν−n

0...

. . . 0... 0

aµ 0 · · · 0 bµ+n 0...

0. . . 0 · · · 0

. . . 0· · · 0 aν−n 0 · · · 0 bν

∈ Mat(ν−µ+1)×(ν−µ+1),

and their determinants by

dµ,ν := det(

tµ,ν

)

.

For r, s ∈ Z, r ≡ s (mod n), we introduce the diagonal minors Dr,s, whichare determinants of the submatrix of tµ,ν consisting of rows and columns whichcontain diagonal elements br, br+n, br+2n, . . . , bs−n, bs.

The following statement gives us the relation between determinant dµ,ν and itsminors Dr,s.

Lemma 2.1. It holds that

n−1∏

i=0

Dµ+i,si= dµ,ν =

n−1∏

i=0

Dri,ν−i,

where, for i = 0, . . . , n − 1,

si = maxx ∈ Z : x ≡ µ + i (mod n), x ≤ ν,

ri = minx ∈ Z : x ≡ ν − i (mod n), x ≥ µ.

Proof. Without the change of the determinant, the matrix tµ,ν can be trans-formed into the block diagonal matrix tµ,ν with blocks Dµ+i,si

or Dri,ν−i,i = 0, . . . , n − 1.

Corollary 2.1. It holds that

Dr,ν

Dr,ν−n

· dµ,ν−1 = dµ,ν =Dµ,s

Dµ+n,s

· dµ+1,ν ,

where r := minx ∈ Z : x ≡ ν (mod n), x ≥ µ and

s := maxx ∈ Z : x ≡ µ (mod n), x ≤ ν.

THREE TERM OPERATORS 53

Corollary 2.2. It holds that

Dµ,s0= bµ · Dµ+n,s0

− a2

µ · Dµ+2n,s0, Dr0,ν = bν · Dr0,ν−n − a2

ν−n · Dr0,ν−2n.

Proof. Computing dµ,ν by expanding it along its first row we obtain

dµ,ν = bµ · dµ+1,ν − a2

µ ·

( n−1∏

i=1

Dµ+i,si

)

· Dµ+2n,s0.

Next, using Lemma 2.1, we have

n−1∏

i=0

Dµ+i,si= bµ ·

( n−1∏

i=1

Dµ+i,si

)

· Dµ+n,s0− a2

µ ·

( n−1∏

i=1

Dµ+i,si

)

· Dµ+2n,s0,

which is the first equality. The second equality can be proved similarly.

Lemma 2.2. Let ν > µ + n and suppose that tµ,ν ≥ 0 (i.e., the matrix tµ,ν is

nonnegative definite), then dµ,ν−n > 0, . . . , dµ,ν−1 > 0.

Proof. We prove this statement for dµ,ν−n. The rest can be proved similarly.The assumption tµ,ν ≥ 0 implies

(i) dµ,k ≥ 0, ∀k ∈ [µ, ν] ∩ Z,(ii) Dµ+i,skii

≥ 0, ∀ki ∈ [0, ν − µ − i] ∩ Z, where

skii = maxx ∈ Z : x ≡ µ + i (mod n), x ≤ ν − ki.

Denoteℓ := minx ∈ [µ, ν] ∩ Z : x ≡ ν (mod n), dµ,x = 0.

Because dµ,µ = bµ > 0, . . . , dµ,µ+n−1 =∏n−1

j=0bµ+j > 0, we have ℓ ≥ µ + n.

Now suppose by contradiction that dµ,ν−n = 0, which implies that ℓ ≤ ν − n

and compute

dµ,ℓ+n = bℓ+n · dµ,ℓ+n−1 − a2

ℓ ·

( n−1∏

i=1

Dri,ℓ+1−i

)

· Dr0,ℓ−n,

Dr0,ℓ+n = bℓ+n · Dr0,ℓ − a2

ℓ · Dr0,ℓ−n.

Using Corollary 2.1 we have 0 = dµ,ℓ = Dr0,ℓ ·dµ,ℓ−1

Dr0,ℓ−nwhich implies that Dr0,ℓ = 0

and using Lemma 2.1 we have 0 < dµ,ℓ−n = K · Dr0,ℓ−n, where K > 0, whichimplies that Dr0,ℓ−n > 0. Altogether, we obtained that Dr0,ℓ+n < 0, whichcontradicts our assumption tµ,ν ≥ 0.

Following statement can be proved easily using Lemma 2.2 (see [1, p. 74–75]).

Lemma 2.3. Operator T is nonnegative definite if and only if dµ,ν > 0, for all

µ, ν ∈ Z, µ ≤ ν.

54 P. HASIL

Now, we present two auxiliary statements, which give us the possibility toconstruct a solution of the difference equation (1).

Lemma 2.4. Let T ≥ 0, and r, s be the same as in Corollary 2.1. Then

(i)1

bµ

<Dµ+n,s

Dµ,s

<bµ−n

a2µ−n

;

(ii)1

bν

<Dr,ν−n

Dr,ν

<bν+n

a2ν

;

(iii) The sequence ci :=

Dµ+n,s+in

Dµ,s+in

is increasing for i ∈ N0.

(iv) The sequence di :=

Dr−in,ν−n

Dr−in,ν

is increasing for i ∈ N0.

Proof. We prove parts (i) and (iii). Parts (ii) and (iv) can be proved similarly.To prove the first part, we expand dµ,ν along its first row and using Lemma 2.1we obtain

Dµ,s0= bµ · Dµ+n,s0

− a2

µ · Dµ+2n,s0.

Because a2µ · Dµ+2n,s0

> 0, the left inequality in (i) holds.Now, we expand the determinant dµ−n,ν and we obtain

0 < Dµ−n,s0= bµ−n · Dµ,s0

− a2

µ−n · Dµ+n,s0

which proves the right inequality in (i).The part (iii) we prove by induction. Directly one can verify that

Dµ+n,µ+n

Dµ,µ+n

<Dµ+n,µ+2n

Dµ,µ+2n

.

Now, we multiply equalities

Dµ,s = bµ · Dµ+n,s − a2

µ · Dµ+2n,s,

Dµ,s+n = bµ · Dµ+n,s+n − a2

µ · Dµ+2n,s+n

by 1

Dµ+n,sand 1

Dµ+n,s+n, respectively, and subtract them. We obtain

Dµ,s

Dµ+n,s

−Dµ,s+n

Dµ+n,s+n

= a2

µ ·

(

Dµ+2n,s+n

Dµ+n,s+n

−Dµ+2n,s

Dµ+n,s

)

,

which proves the statement.

Lemma 2.5. Let f, g be solutions of τy = 0. If fj = gj, . . . , fj+n−1 = gj+n−1,

j ∈ Z, then

fk0

>

(=)

gk0, . . . , fk0+n−1

>

(=)

gk0+n−1 for some k0 ≥ j + n (k0 ≤ j − n) (2)


implies

fk>

(=)

gk ∀k ≥ j + n (k ≤ j − n).

Proof. We follow the idea introduced in [1, 2]. Let f and g be solutions of thelinear system

tj+n,k0−1 ·

yj+n

yj+n+1

...yk0−1

=

v

0...w

,

where

v =

−ajyj

...−aj+n−1yj+n−1

, w =

−ak0−nyk0

...−ak0−1yk0+n−1

.

Then fk>

(=)

gk for k ∈ [j + n, k0 + n − 1] ∩ Z. Now, the existence of a K ≥ k0 + n

such that fk < gk contradicts (2).

If we suppose that T ≥ 0, we can (due to Lemma 2.4, j ∈ [0, n−1]∩Z) introducethe limits

C[µ,j]+ := lim

i→∞

Dµ+n,sj+in

Dµ,sj+in

, C[ν,j]−

:= limi→∞

Drj−in,ν−n

Drj−in,ν

and a positive map u : Z × Z → R defined by

u(µ, k) :=

ℓ∏

i=1

(

−ak−in · C[k−(i−1)n,k−µ−ℓn]

+

)

, k ≥ µ + n,

1, k ∈ [µ − n + 1, µ + n − 1],

ℓ−1∏

i=0

(

−ak+in · C[k+in,µ−k−ℓn]

−

)

, k ≤ µ − n,

where ℓ =∣

∣

∣

⌊

k−µn

⌋∣

∣

∣, and ⌊ · ⌋ denotes the floor function (greatest integer function)

of a real number.Now, let us recall the definition of a minimal solution of (1).

Definition 2.1. We say that a solution u of (1) is minimal on [µ + n,∞) ∩ Z

if any linearly independent solution v of (1) with vµ = uµ, . . . , vµ+n−1 = uµ+n−1

satisfies vk > uk for k ≥ µ + n. The minimal solution on (−∞, µ − n] is definedanalogously.

Lemma 2.6. Let T ≥ 0. Then u(µ, k), µ ∈ Z is fixed, is the minimal positive

solution of τy = 0 on [µ + n,∞) ∩ Z and (−∞, µ − n] ∩ Z separately.

56 P. HASIL

Proof. We introduce

u[µ,ν]

k :=

1, k∈ [µ, µ+n−1]∩ Z,

(−ak−n)·(−ak−2n)· · ·(−ak−ℓn)·Dk+n,si

Dk−(ℓ−1)n,si

, k∈ [µ+n, s0−1] ∩ Z,

(−ak−n)·(−ak−2n)· · ·(−ak−ℓn)· 1

Dk−(ℓ−1)n,si

, k∈ [s0, sn−1] ∩ Z,

0, k∈ [sn−1+1, sn−1+n] ∩ Z,

where i = k − µ − ℓn, ℓ =∣

∣

∣

⌊

k−µn

⌋∣

∣

∣.

The sequence u[µ,ν]

k satisfies τy = 0 on [µ + n, sn−1] ∩ Z and it holds that

limsn−1→∞

u[µ,ν]

k = u(µ, ν), k ≥ µ,

so we can see that τu = 0 on [µ + n,∞) ∩ Z.Let v be a positive solution of τy = 0 on [µ+n,∞) ∩ Z such that vµ = 1, . . . ,

vµ+n−1 = 1, which is linearly independent on u(µ, ·).

Because vsn−1+1 > 0, . . . , vsn−1+n > 0, we have vk > u[µ,sn−1]

k on [sn−1 +

1, sn−1 + n] ∩ Z. Thus, by Lemma 2.5, vk > u[µ,sn−1]

k for all k ≥ µ + n. Hencevk ≥ u(µ, k) for all k ≥ µ. Assume, by contradiction, that there exists k0 ≥ µ + n

such that vk0= u(µ, k0). Then, again by Lemma 2.5, vk = u(µ, k) for all k ≥ µ,

which is a contradiction.One can show analogously that u is the minimal positive solution of τy = 0 on

(−∞, µ − n] ∩ Z using

u[µ,ν]

k :=

1, k∈ [µ−n+1, µ]∩ Z,

(−ak)·(−ak+n)· · ·(−ak+(ℓ−1)n)·Drj,k−n

Drj,k+(ℓ−1)n, k∈ [r0+1, µ− n] ∩ Z,

(−ak)·(−ak+n)· · ·(−ak+(ℓ−1)n)· 1

Drj,k+(ℓ−1)n, k∈ [rn−1, r0] ∩ Z,

0, k∈ [rn−1−1, rn−1−n] ∩ Z,

where j = µ − k − ℓn, ℓ =∣

∣

∣

⌊

k−µn

⌋∣

∣

∣.

3. Positive solutions of τy = 0

If T ≥ 0, as a consequence of the previous section, the minimal positive solutionsof the equation τy = 0 on [n,∞) ∩ Z and (−∞, n] ∩ Z are

u+

k :=

u(0, k), k ∈ N0,

u(k − i, i)−1, k ≡ i (mod n), −k ∈ N, i ∈ [0, n − 1] ∩ Z,


u−

k :=

u(0, k), −k ∈ N0,

u(k + i,−i)−1, k ≡ i (mod n), k ∈ N, i ∈ [0, n − 1] ∩ Z,

respectively.

Lemma 3.1. Let T ≥ 0. Then u+

k , u−

k are positive solutions of τy = 0 on Z.

Proof. We show that u+

k is a positive solution of τy = 0 on Z, the statement

for u−

k can be proved similarly.The sequence u+ is a positive solution of τy = 0 on [n,∞). It follows from the

definition of u(µ, k) that for

∀k, ℓ ∈ Z, ∀m ∈ [k, ℓ] ∩ Z, m ≡ k (mod n)

it holds that

u(k, ℓ) = u(k, m) · u(m, ℓ).

Let m ∈ (−∞, n − 1] ∩ Z be arbitrary and M ∈ −N such that M ≤ m − n befixed. Then for some i ∈ [0, n − 1] ∩ Z we have M ≡ m − i (mod n) and

u(M, i) = u(M, m − i) · u(m − i, i).

Hence, by the definition of u+, we obtain

u+

m = u(m − i, i)−1 =u(M, m − i)

u(M, i),

which implies that u+ is a solution of τy = 0 on [M + i, i].

Theorem 3.1. T ≥ 0 if and only if there exists a positive solution of τy = 0.

Proof. The necessity follows from Lemma 3.1. To prove sufficiency we show atfirst positivity of all determinants Dµ,ξ, µ ≤ ξ ≤ ν, ξ ≡ µ (mod n). Assume thatthere exists a positive solution u of τy = 0. Then u solves the system

tµ,ν ·

uµ

uµ+1

......uν

=

v

0...0w

,

where

v =

−aµ−nuµ−n

...−aµ−1uµ−1

, w =

−aν−n+1uν+1

...−aνuν+n

.

58 P. HASIL

By Cramer’s rule we obtain

uµdµ,ν =[

− aµ−nuµ−nDµ+n,s0

+ (−aσuσ+n)(−aµ)(−aµ+n) · · · (−as0−n)]

(

n−1∏

i=1

Dµ+i,si

)

,

where σ ∈ ν − n + 1, . . . , ν, σ ≡ µ (mod n). Hence

Dµ,s0uµ = −aµ−nuµ−nDµ+n,s0

+ (−aµ)(−aµ+n) · · · (−as0−n)(−aσ)uσ+n.

If s0 = µ + n then Dµ+n,s0= Dµ+n,µ+n = bµ+n > 0 which implies that

Dµ,µ+n > 0. Hence all determinants Dµ,ξ are positive.Similarly we can show positivity of the determinants Dµ+i,ξi

, µ + i ≤ ξi ≤ ν,ξi ≡ µ + i (mod n), i = 1, . . . , n − 1, which gives positivity of the determinantsdµ,ν , µ ≤ ν and the statement follows from Lemma 2.3.

In an analogous way as before we can generalize a number of statements of [1].For example, the following result, which characterizes the minimal solutions of (1),can be proved in the same way as in [1] (see also [3]).

Theorem 3.2. If T ≥ 0, then for all µ ∈ Z

±∞

∑

i=µ

(

−aiu±

i u±

i+n

)

−1= ∞.

References

[1] Gesztesy F. and Zhao Z., Critical and Subcritical Jacobi Operators Defined as Friedrichs

Extensions, J. Differ. Equations 103 (1993), 68–93.[2] Olver F. W. J., Note on Backward Recurrence Algorithms, Math. Comp. 26 No. 120 (1972),

941–947.[3] Patula W. T., Growth and Oscillation Properties of Second Order Linear Difference Equa-

tions, SIAM J. Math. Anal. 10 (1979), 55–61.

P. Hasil, Department of Mathematics and Statistics, Masaryk University, Kotlarska 2, 611 37

Brno, Czech Republic,



59

SOME NUMERICAL INVESTIGATIONS OF DIFFERENTIAL

EQUATIONS WITH SEVERAL PROPORTIONAL DELAYS

J. JANSKY

Abstract. This paper describes the application and asymptotics of the trapezoidalrule for the pantograph differential equation with several proportional delays. Somecomparisons with known results are given as well.

Introduction

We shall study some numerical properties of the delay differential equation

y′(t) = ay(t) +

k∑

i=1

biy(λit), t ≥ 0 , (1)

where a < 0, bi 6= 0 and 0 < λi < 1 are real scalars, i ∈ 1, 2, . . . , k. This equationis the pantograph equation with several delays. The “classical” pantograph equa-tion was derived in 1970 in the frame of solving of problem on the railways. Later,many other applications in various areas led to this equation and its modifications.Therefore this equation has become the subject of many qualitative and numericalinvestigations. The pantograph equation and its modifications were studied in thecontinuous case [2, 3, 5, 6, 9] as well as in the discrete case [1, 4, 7, 12, 10].

We focus on difference equations arising from (1) by use of the numerical dis-cretization. The Euler method [8, 11] and the trapezoidal rule [1, 4] represent thestandard ways of such discretization.

In this paper we wish to describe some asymptotic properties of the trapezoidalrule discretization of (1).

1. Trapezoidal rule for the equation (1)

In this section, we outline a discretization of equation (1) by use of the trapezoidalrule. Note that an analogous problem was discussed in [1]. After integration of (1)

2000 Mathematics Subject Classification. Primary 39A11, 39A12.Key words and phrases. pantograph equation, trapezoidal rule, asymptotic behaviour.The research was supported by the grant # 201/08/0469 of the Czech Grant Agency and

by the research plan MSN 0021630518 “Simulation modeling of mechatronic systems” of theMinistry of Education, Youth and Sports of the Czech Republic.

60 J. JANSKY

we obtain

y(t) = y(0) + a

∫ t

0

y(τ) dτ +

k∑

i=1

bi

∫ t

0

y(λiτ) dτ.

Now we introduce the substitution ui = λiτ in the second integral and get

y(t) = y(0) + a

∫ t

0

y(τ) dτ +

k∑

i=1

bi

λi

∫ λit

0

y(ui) dui.

Considering the grid of points (nh), n = 0, 1, . . . with stepsize h > 0 we can write

y(

(n+1)h)

= y(nh) + a

∫ (n+1)h

nh

y(τ) dτ +

k∑

i=1

bi

λi

∫ λi(n+1)h

λinh

y(ui) dui.

Now we approximate both integrals using the trapezoidal rule. The approxi-mation of the first integral is simple:

∫ (n+1)h

nh

y(τ) dτ ≈ 12h(yn + yn+1),

where yn ≈ y(nh). From the second integral, we obtain

bi

λi

∫ λi(n+1)h

λinh

y(ui) dui ≈hbi

2

(

y(λinh) + y(λi(n + 1)h))

.

Since the points λinh are generally not in the given grid, we have to approximatethem using a linear interpolation. Hence

yλin ≈(

1 − (λin − ⌊λin⌋))

y⌊λin⌋ + (λin − ⌊λin⌋)y⌊λin⌋+1,

yλi(n+1) ≈(

1 − (λi(n+1)− ⌊λin⌋))

y⌊λin⌋ + (λi(n+1)− ⌊λin⌋)y⌊λin⌋+1.

Overall we have

yn+1 − yn = 12ah(yn + yn+1) + h

k∑

i=1

bi

(

βn,iy⌊λin⌋ + αn,iy⌊λin⌋+1

)

, (2)

where

αn,i := λin − ⌊λin⌋ + λi

2 > 0, βn,i := 1 − αn,i.

From formula (2) we express yn+1 and get the final form of the discretizedequation (1)

yn+1 = Ryn + Sh

k∑

i=1

bi

(

βn,iy⌊λin⌋ + αn,iy⌊λin⌋+1

)

, (3)

where

R := 2+ah2−ah

, S := 22−ah

.

NUMERICAL INVESTIGATIONS OF DIFFERENTIAL EQUATIONS 61

2. some auxiliary results

In this section we present the inequality which is useful in our further calculations.This inequality has the form:

|S|h

k∑

i=1

|bi|(

|βn,i|ρ⌊λin⌋ + |αn,i|ρ⌊λin⌋+1

)

≤ (1 − |R|)ρn, n = 0, 1, . . . (4)

Further, we have

ηi := supn∈Z+

(

|βn,i| + |αn,i|)

< ∞ and |R| < 1, (5)

because of the assumptions a < 0, h > 0.

Lemma 2.1. Let (5) holds. Then the sequence

ρn :=

(

n − 11−λ

)

− logλ γfor γ ≥ 1,

(

n + 11−λ

)

− logλ γfor 0 < γ < 1,

(6)

where

λ := max(λ1, λ2, . . . λk) for γ ≥ 1,

min(λ1, λ2, . . . λk) for 0 < γ < 1(7)

and

γ :=

h|S|

k∑

i=1

|bi|ηi

1 − |R|(8)

is a solution of inequality (4).

Proof. We only deal with the case γ < 1 because the case γ ≥ 1 is analogical.If γ < 1, then (ρn) is a decreasing sequence. Hence we can write

|S|h

k∑

i=1

|bi|(

|βn,i|ρ⌊λin⌋ + |αn,i|ρ⌊λin⌋+1

)

≤ |S|h

k∑

i=1

ηi|bi|ρ⌊λin⌋.

Further

|S|h

k∑

i=1

ηi|bi|ρ⌊λin⌋ = |S|h

k∑

i=1

ηi|bi|(

⌊λin⌋ + 11−λ

)

− logλ γ≤

≤ |S|h

k∑

i=1

ηi|bi|(

λin − 1 + 11−λ

)

− logλ γ= |S|h

k∑

i=1

ηi|bi|(

λin + λ1−λ

)

− logλ γ

≤ |S|h

k∑

i=1

ηi|bi|(

λn + λ1−λ

)

− logλ γ= |S|h

k∑

i=1

ηi|bi|λ− logλ γρn = (1−|R|)ρn.

62 J. JANSKY

3. Main result

The main theorem of this paper is the following.

Theorem 3.1. Let (yn) be a solution of (3), let a < 0, bi 6= 0for all i ∈ 1, 2, . . . , k and let (ρn), λ, γ be given by (6)—(8). Then

|yn| ≤ Kn− logλ γ for n = σ0, σ0+1, σ0+2, . . . , (9)

where σ0∈Z+ satisfies σ0 ≥ max

2(1−λ)λ , 2 logλ γ

and K := B0 exp(L 11−λ

) with

B0 := sup(

|yn|

ρn, n ∈ [λ(σ0 − 1), σ0] ∩ Z

+)

(10)

and

L :=2 logλ γ

γ(1 − |R|)(σ0 −1+λ1−λ

).

Proof. We use the substitution zn = yn

ρnin (3), where ρn is given by (6). Then

n+1zn+1 = Rρnzn + Sh

k∑

i=1

bi

(

βn,iρ⌊λin⌋z⌊λin⌋ + αn,iρ⌊λin⌋+1z⌊λin⌋+1

)

. (11)

Now we choose σ0 ≥ max

2(1−λ)λ , 2 logλ γ

, σ0 ∈ Z+ and define points

σm+1 := ⌊σm−1λ

⌋, where m = 0, 1, . . . . After some calculations, we obtain

λ−m(

σ0 −1+λ1−λ

)

≤ σm ≤ λ−1σm−1, m = 1, 2, . . . . (12)

Next we introduce intervals I0 := [λ(σ0 − 1), σ0] ∩ Z+, Im+1 := [σm, σm+1] ∩ Z

+

and denote Bm := sup(|zs|, s ∈ ∪mj=0Ij), m = 0, 1, 2 . . . .

Now we choose n⋆ ∈ Im+1, n⋆ > σm arbitrarily and we distinguish two caseswith respect to R.

(i) First, we deal with the case R = 0. In this case

zn⋆ = Shρn⋆

k∑

i=1

bi

(

βn⋆−1,iρ⌊λi(n⋆−1)⌋z⌊λi(n⋆−1)⌋+αn⋆−1,iρ⌊λi(n⋆−1)⌋+1z⌊λi(n⋆−1)⌋+1

)

,

hence

|zn⋆ | ≤ Bm|S|h

ρn⋆

k∑

i=1

|bi|(

|βn⋆−1,i|ρ⌊λi(n⋆

−1)⌋ + |αn⋆−1,i|ρ⌊λi(n⋆

−1)⌋+1

)

.

Using (4), we arrive at

|zn⋆ | ≤ρn⋆

−1

ρn⋆Bm.

Assuming γ ≥ 1, (ρn) is the nondecreasing sequence and we obtain |zn⋆ | ≤ Bm.

Assuming 0 < γ < 1 we derive with respect to (6), (12) and the binomial formulathe relation

ρn⋆−1

ρn⋆=

(

n⋆ + 11−λ

− 1

n⋆ + 11−λ

)

− logλ γ

≤1

(

1+ 1σm

)

− logλ γ≤

1

1 + − logλ γ

σm

≤ 1+2 logλ γ

σm.


This inequality implies the following relation

|zn⋆ | ≤ Bm

(

1 +2 logλ γ

σ0 −1+λ1−λ

λm)

. (13)

(ii) Let R 6= 0 . We can multiply the equation (11) by 1Rn+1 . We get

∆(

ρnzn

Rn

)

= ShRn+1

k∑

i=1

bi

(

βn,iρ⌊λin⌋z⌊λin⌋ + αn,iρ⌊λin⌋+1z⌊λin⌋+1

)

.

If we sum this relation from σm to n⋆ − 1 than we obtain

n⋆ zn⋆

Rn⋆ −σmzσm

Rσm=

n⋆−1∑

p=σm

ShRp+1

k∑

i=1

bi

(

βp,iρ⌊λip⌋z⌊λip⌋ + αp,iρ⌊λip⌋+1z⌊λip⌋+1

)

,

i.e.

zn⋆ =σm

n⋆Rn⋆

−σmzσm+ Rn⋆

n⋆

n⋆−1∑

p=σm

ShRp+1

k∑

i=1

bi

(

βp,iρ⌊λip⌋z⌊λip⌋+αp,iρ⌊λip⌋+1z⌊λip⌋+1

)

.

Thus

|zn⋆ | ≤ Bm

(

σm

n⋆|R|n

⋆−σm + |R|

n⋆

n⋆

n⋆−1∑

p=σm

|S|h

|R|p+1

k∑

i=1

|bi|(

|βp,i|ρ⌊λip⌋ + |αp,i|ρ⌊λip⌋+1

)

)

.

Using (4), we get

|zn⋆ | ≤ Bm

(

σm

n⋆|R|n

⋆−σm + |R|

n⋆

n⋆

n⋆−1∑

p=σm

1−|R|

|R|p+1 ρp

)

.

Now using the relation1−|R|

|R|p+1 = ∆

(

1|R|

)p(14)

and summing by parts we get

|zn⋆ | ≤ Bm

(

σm

n⋆|R|n

⋆−σm + |R|

n⋆

n⋆

n⋆−1∑

p=σm

∆(

1|R|

)pρp

)

= Bm

(

ρσm

ρn⋆|R|n

⋆−σm + 1 −

ρσm

ρn⋆|R|n

⋆−σm −

|R|n⋆

n⋆

n⋆−1∑

p=σm

1|R|

p+1 ∆ρp

)

= Bm

(

1 −|R|

n⋆

n⋆

n⋆−1∑

p=σm

1|R|p+1 ∆ρp

)

.

Now using (14), we get

|zn⋆ | ≤ Bm

(

1 −|R|

n⋆

n⋆

n⋆−1∑

p=σm

∆ρp

1−|R|∆(

1|R|

)p)

.

64 J. JANSKY

If γ ≥ 1 then ρp is nondecreasing, therefore ∆ρp ≥ 0 and |zn⋆ | ≤ Bm. In thecase 0 < γ < 1, same simple calculations are necessary to derive that ∆ρp isnegative and nondecreasing. Hence we can write

|zn⋆ | ≤ Bm

(

1− |R|n⋆

1−|R|

∆ρσm

n⋆

n⋆−1∑

p=σm

∆(

1|R|

)p)

=Bm

(

1− |R|n⋆

1−|R|

∆ρσm

n⋆

(

1|R|

n⋆ − 1|R|σm

)

)

≤ Bm

(

1 + 11−|R|

−∆ρσm

n⋆

)

≤ Bm

(

1 + 11−|R|

−∆ρσm

σm+1

)

.

Substituting the corresponding form of ρn (see (6)) and using the binomialformula, we can derive

−∆ρσm= ρσm

− ρσm+1 =(

σm + 11−λ

)

− logλ γ−(

σm + 1 + 11−λ

)

− logλ γ

=(

σm + 11−λ

)

− logλ γ(

1 − (1 + 1σm+ 1

1−λ

)

− logλ γ)

≤(

σm + 11−λ

)

− logλ γ(

1 −(

1 + − logλ γ

σm+ 1

1−λ

)

)

≤(

σm + 11−λ

)

− logλ γ logλ γ

σm.

and analogically

ρσm+1=(

σm+1 + 11−λ

)

− logλ γ≥(

1λσm + 1

1−λ

)

− logλ γ

≥(

1λσm + 1

λ1

1−λ

)

− logλ γ= γ

(

σm + 11−λ

)

− logλ γ.

Considering (12) we arrive at

−∆ρσm

ρσm+1(1−|R|) ≤

logλ γ

γ(1−|R|)1

σm≤

logλ γ

γ(1−|R|)1

(σ0−1+λ1−λ

)λm.

Hence|zn⋆ | ≤ Bm

(

1 + logλ γ

γ(1−|R|)(σ0−1+λ1−λ

)λm)

. (15)

Using the definition of L, summarizing cases (i)—(ii) and using estimates (13)and (15) we get

|zn⋆ | ≤ Bm(1 + Lλm) as m → ∞

for arbitrary n⋆ ∈ Im+1, n∗ > σm. Thus

Bm+1 ≤ Bm(1 + Lλm) as m → ∞.

Now we can estimate Bm in this way:

Bm+1 ≤ Bm

(

1+Lλm)

≤ B0

m∏

j=0

(

1+Lλj)

≤ B0

∞∏

j=0

(

1+Lλj)

≤ B0exp(L 11−λ

).

ThusBm ≤ B0 exp(L 1

1−λ) as m → ∞.

The estimate (9) is proved.

Remark 3.2. The constant σ0 has to be proposed with respect to a concreteequation. If we choose σ0 ≥ max

2(1−λ)λ , 2 logλ γ

too small, then the constant

L can be too large and the estimate (9) becomes worse. If we choose σ0 too large,then it will be necessary to calculate the constant B0 in (10) in too large interval.


Example 3.3. In this example we show the aplication of Theorem 3.1. Let usconsider the following initial value problem

y′(t) = −y(t) − 0, 25y( t4 ) − 0, 2y( t

3 ), t ≥ 0, y(0) = 1. (16)

After a discretization (3) with the stepsize h = 0, 05 we obtain

y0 = 1,

yn+1 = 3941yn − 1

82 (βn,1y⌊n/4⌋+ αn,1y⌊n/4⌋+1)−2

205 (βn,2y⌊n/3⌋+ αn,2y⌊n/3⌋+1),

where

αn,1 := n4 −⌊n

4 ⌋+18 > 0, βn,1 := 1−αn,1, αn,2 := n

3 −⌊n3 ⌋+

16 > 0, βn,2 := 1−αn,2.

Now if we set σ0 = 488, then using Theorem 3.1 we obtain the estimate

|yn| ≤ 2, 138n−0,576, for n = 488, 489, . . . (17)

The Fig. 1 displays the real numerical solution of the problem (16) and itsestimate given by (17).

Figure 1. The solution and its estimate.

4. some comparisons

In this section we compare our results with results given in [6] and [1]. For thesake of simplicity we restrict our comparisons to the case k = 1 and denote b :=b1. Under these conditions the equation (1) is reduced to the scalar pantographequation

y′(t) = ay(t) + by(λt), t ≥ 0, (18)

where λ = λ1 and assume a < 0, b 6= 0.First we present the estimate of the exact equation (18). It was proved in [6]

that the asymptotic estimate

y(t) = O(t− logλ |ba|) as t → ∞

66 J. JANSKY

holds for any solution of (18). Furthermore it is known that the constant − logλ | ba|

is the best one and cannot be improved.If we denote αn := αn,1 and βn := βn,1 then the difference equation (3) becomes

yn+1 = Ryn + Shb(

βny⌊λn⌋ + αny⌊λn⌋+1

)

. (19)

Now we denote η := η1. It was shown in [1] that any solution yn of (19) fulfilling

|R| + hη|S||b| ≤ 1, a < 0

is bounded, and the asymptotic estimate

yn = O(n− logλ γ), γ := |R| + hη|S||b|, as n → ∞ (20)

holds for any solution of (19).It is interesting to compare the asymptotic estimate (20) with (9). It can be

shown that if γ < 1 then

γ = hη|Sb|

1−|R|< |R| + hη|Sb| = γ.

Moreover, γ can be expressed as

γ = | ba|η for h|a| ≤ 2, γ = h|b|η

2 for h|a| > 2.

It is known (see [1, Theorem 6]) that if λ = 1l

where l ∈ 2, 3, . . . then η = 1.

Hence we get the value | ba| known from the asymptotic estimate of the solution of

the exact equation (18) provided h|a| ≤ 2.

References

[1] Buhmann M. D., Iserles A., Stability of the discretized pantograph differential equation,Math. Comp. 60 (1993), 575–589.

[2] Cermak J., On a linear differential equation with a proportional delay, Math. Nachr. (2007),495–504.

[3] Cermak J., The asymptotics of solutions for a class delay differential equations, RockyMountain J. math. 33 (2003), 775–786.

[4] Iserles A., Numerical analysis of delay differential equations with variable delays, Ann.Numer. Math. 1 (1994), 133–152.

[5] Iserles A., On the generalized pantograph functional-differential equation, European J. Appl.Math. 4 (1993), 1–38.

[6] Kato T. and McLeod J. B., The functional-differential equation y′(x) = ay(λx) + by(x),Bull. Amer. Math. Soc. 77 (1971), 891–937.

[7] Koto T., Stability of Runge-Kutta methods for the generalized pantograph equation, Numer.Math. 84 (1999), 870–884.

[8] Kundrat P., On the asymptotics of the difference equation with a proportional delay, Opus-cula Math. 26/3 (2006), 499–506.

[9] Lehninger H. and Liu Y., The functional-differential equation y′(t) = Ay(t) + By(λt) +Cy′(qt) + f(t), European J. Appl. Math. 9 (1998), 81–91.

[10] Liu Y., Numerical investigation of the pantograph equation, Appl. Numer. Math. 24 (1997),309–317.

[11] Liu Y., On the θ-method for delay differential equations with infinite lag, J. Comput. Appl.Math. 71 (1996), 177–190.

[12] Liu M. Z., Yang Z. W. and Xu Y., The stability of the modified Runge-Kutta methods for

the pantograph equation, Math. Comp. 75 (2006), 1201–1215.

J. Jansky, Technicka 2, 616 69, Brno, Czech Republik,



67

INVESTIGATING OF BOUNDARY VALUE PROBLEMS FOR

ORDINARY DIFFERENTIAL EQUATIONS IN A CRITICAL

CASE

IHOR KOROL

Abstract. The numerical-analytic method for investigating and approximate con-structing of the solutions of boundary value problems for nonlinear differential sys-

tems in a critical case is suggested.

1. Introduction

The theory of boundary value problems (BVP) is an important branch of thegeneral theory of differential equations due to their strong connection with thepractical applications. Various types of methods (numerical, analytic, numerical-analytic, functional etc) [1–7] for investigating the problems of existing and ap-proximate constructing of the solutions are studied.

The modification of the numerical-analytic method, which is developed in thispaper allow us to study BVP

dx

dt= A(t)x + f (t, x) , B1x(a) + B2x(b) +

b∫

a

[

dB(t)]

x(t) = d

in a critical [2] case – when there exist a nontrivial solutions of the correspondencelinear homogeneous problem. We can use this modification when the boundarycondition is ”degenerate” [7]. Also we can note that due to using this modifica-tion the restriction for Lipschitz matrix concerns not the whole right part of thedifferential equation, but only the nonlinearity f(t, x), and this is less difficult todeal with.

2. Linear BVP

In the beginning let us consider a linear inhomogeneous system of differentialequations

dx

dt= A(t)x + h(t), (1)

Key words and phrases. ordinary differential equations, boundary value problems, numerical-analytic method, critical case, successive approximations.

68 IHOR KOROL

with the additional linear condition, which can be represent by means of Riemann-Stieltjes integral

B1x(a) + B2x(b) +

b∫

a

[

dB(t)]

x(t) = d. (2)

Let us assume that matrix A(t) : [a, b] → Rn×n and function h(t) : [a, b] → R

n

are continuous, B1, B2 ∈ Rn×n are the constant matrices, matrix B(t) : [a, b] →

Rn×n is of bounded variation, d ∈ R

n.A continuously differentiable n-dimensional function x ∈ C1([a, b], Rn), t ∈ [a, b]

is said to be a solution of the problem (1), (2) if it satisfies the equation (1) andverifies also the boundary condition (2).

It is known that the solution x(t, x0) of the differential system (1) with theinitial value x(a) = x0 is of the form

x(t, x0) = Ωtax0 +

t∫

a

Ωtsh(s)ds, (3)

where Ωta, Ωa

a = In is a matriciant of the corresponding to (1) homogeneous systemof differential equations

dx

dt= A(t)x, (4)

In is identity (n×n)-matrix.While substituting (3) into the boundary condition (2) one can see that x(t, x0)

satisfies the boundary condition (2) if and only if x0 is a solution of the algebraicsystem

Gx0 = d −

b∫

a

W (s)h(s)ds, (5)

where

W (s)=B2Ωbs+

b∫

s

[dB(t)]Ωts, G=B1+W (a)=B1+B2Ω

ba+

b∫

a

[

dB(t)]

Ωta. (6)

Obviously that in a noncritical case [2] – when a correspondent to (1), (2)homogeneous BVP

dx

dt= A (t)x, B1x(a) + B2x(b) +

b∫

a

[

dB(t)]

x(t) = 0 (7)

does not have the nontrivial solutions, the algebraic system (5) has a unique solu-tion

x0 = G−1

(

d −

b∫

a

W (s)h(s)ds

)

,

INVESTIGATING OF BOUNDARY VALUE PROBLEMS. . . 69

which is the initial value of a unique solution of the BVP (1), (2)

x(t) = ΩtaG−1d − Ωt

aG−1

b∫

a

W (s)h(s)ds +

t∫

a

Ωtsh(s)ds.

Let us consider the problem of existence of the solutions of the BVP (1), (2) ina critical case [2] i.e. when

A) the corresponding homogeneous BVP (7) has k nontrivial linearly indepen-dent solutions, k = n − rank(G), 1 ≤ k ≤ n.

Lemma 2.1. Assume that the linear homogeneous BVP (2), (4) has k linearly

independent solutions. Then for an arbitrary function h(t) there exist a function

H(t) such that inhomogeneous differential system

x′ = A(t)x + h(t) + H(t), (8)

possesses a k-parametric family of solutions, which satisfies the boundary condi-

tion (2).

Proof. From (5) we have [2, 4] that the BVP (8), (2) has a solution if and onlyif the condition

PG∗

(

d −

b∫

a

W (s)

(

h(s) + H(s)

)

ds

)

= 0 (9)

is fulfilled, where PG∗ is an orthoprojector from the space Rn to the null spa-

ce kerG∗. Respectively PG is an orthoprojector from the space Rn to the null

space kerG:

PG : Rn → kerG, kerG = y : y ∈ R

n, Gy = 0,

PG∗ : Rn → kerG∗, kerG∗ = z : z ∈ R

n, zG∗ = 0,

rankPG = rankPGk= rankPG∗ = rankPG∗

k= k.

We will denote by PGkthe (n × k)-matrix, which columns constituting a basis

of the kernel ker(G) and they are the linearly independent columns of PG. Re-spectively by PG∗

kwe denote (k×n)-matrix, which rows constituting a basis of the

ker(G∗) and are the linearly independent rows of matrix PG∗ .Let us denote

H(t) = W ∗(t)(PG∗

k)∗R−1

1PG∗

k

(

d −

b∫

a

W (s)h(s)ds

)

, (10)

where

R1 = PG∗

kR2(PG∗

k)∗,

R2 =

b∫

a

W (s)W ∗(s)ds = B2

b∫

a

ΩbsW

∗(s)ds +

b∫

a

[dB(t)]

t∫

a

ΩtsW

∗(s)ds. (11)

Substituting (10) into (9) we obtain

70 IHOR KOROL

PG∗

k

(

d −

b∫

a

W (s)

(

h(s) + H(s)

)

ds

)

= PG∗

k

(

d −

b∫

a

W (s)h(s)ds

−

b∫

a

W (s)W ∗(s)ds(PG∗

k)∗R−1

1PG∗

k

(

d −

b∫

a

W (s)h(s)ds

))

=

(

Ik − PG∗

k

b∫

a

W (s)W ∗(s)ds(PG∗

k)∗R−1

1

)

PG∗

k

(

d −

b∫

a

W (s)h(s)ds

)

= 0.

Thus condition (9) is fulfilled if H(t) is of the form (10). The solution x(t, x0)of the system (8), (10) with the initial value x(a, x0) = x0 has the form

x(t, x0)=Ωtax0+

t∫

a

Ωts

h(s)+W ∗(s)(PG∗

k)∗R−1

1PG∗

k

(

d−

b∫

a

W (s)h(s)ds

)

ds. (12)

Substituting x(t, x0) of the form (12) into the boundary condition (2) we cansee that x(t, x0) satisfies the boundary condition (2) if and only if the initial valuex0 is a solution of the algebraic system

Gx0 = d −

b∫

a

W (s)

h(s) + W ∗(s)(PG∗

k)∗R−1

1PG∗

k

(

d −

b∫

a

W (s)h(s)ds

)

,

which we can rewrite in the form

Gx0 =(

In − R2(PG∗

k)∗R−1

1PG∗

k

)

(

d −

b∫

a

W (s)h(s)ds

)

. (13)

Multiplying the right side of this equation from the left by matrix PG∗

kwe

will obtain the zero vector. It means that system (13) is solvable and (becauserankG = n − k) it has k-parametric solution of the form [2, 4]

x0 = PGkξ + G+

(

In − R2(PG∗

k)∗R−1

1PG∗

k

)

(

d −

b∫

a

W (s)h(s)ds

)

, (14)

where G+ is a unique Moore-Penrose generalized inverse (n×n)-matrix [2, 4, 8, 9],ξ ∈ Rk is an arbitrary vector. Substituting x0 of the form (14) into (12) we obtainthe general k-parametric solution of the BVP (2), (8):

x(t, x0) = x(t, ξ)

= ΩtaPGk

ξ + ΩtaG+

(

In − R2(PG∗

k)∗R−1

1PG∗

k

)

(

d −

b∫

a

W (s)h(s)ds

)

+

t∫

a

Ωts

h(s) + W ∗(s)(PG∗

k)∗R−1

1PG∗

k

(

d −

b∫

a

W (s)h(s)ds)

ds.


Finally we can rewrite it in the form

x(t, ξ) = ΩtaPGk

ξ + Ωta

(

G++(R(t) − G+R2)(PG∗

k)∗R−1

1PG∗

k

)

d +

b∫

a

L(t, s)h(s)ds,

where

R(t) =

t∫

a

ΩasW ∗(s)ds,

and

L(t, s) =

Ωts − Ωt

a

(

G++(

R(t)−G+R2

)

(PG∗

k)∗R−1

1PG∗

k

)

W (s), 0 ≤ s ≤ t ≤ b,

−Ωta

(

G++(

R(t)−G+R2

)

(PG∗

k)∗R−1

1PG∗

k

)

W (s), 0 ≤ t < s ≤ b.

The lemma is proved.

3. The numerical-analytic method for investigating nonlinear

differential systems with linear boundary condition

Now let us discuss the problem of existence and approximate constructing of thesolutions of nonlinear differential systems

dx

dt= A (t)x + f (t, x) , x, f ∈ Rn, (15)

which fulfills the additional condition (2). We will consider a critical case i.e. thecase when the condition A is fulfilled.

We suppose that on Ω = [a, b] × D, where D ⊂ Rn is a closed and bounded

domain, the following conditions are hold:

B) matrix A(t) : [a, b] → Rn×n and function f : Ω → R

n are continuous for t

and the following inequalities are hold:

|f(t, x)| ≤ m(t), |f(t, x′) − f(t, x′′)| ≤ K(t)|x′ − x′′|,

where vector m(t) and matrix K(t) are continuous and their componentsare nonnegative;

C) the domain D0 ≡

ξ ∈ Rk |B(x0(t, ξ), β) ⊆ D

is not empty:

D0 6= ∅,

where

β = maxt∈[a,b]

(

∣

∣

∣Ωt

a

(

G++(R(t)−G+R2)(PG∗

k)∗R−1

1PG∗

k

)

d∣

∣

∣+

b∫

a

∣

∣L(t, s)∣

∣m(s)ds

)

and B(y, ) = x ∈ Rn : |x − y| ≤ for fixed y, ∈ R

n;D) maximum eigenvalue of the following matrix Q is less then one:

Q = maxt∈[a,b]

b∫

a

∣

∣L(t, s)∣

∣K(s)ds.

72 IHOR KOROL

The notation | · | means the absolute value: |x| = col(|x1|, . . . , |xn|), wherex ∈ Rn and |A| = (|Aij |)

ni,j=1

. The inequalities are meant componentwise.Next lemmas give us necessary and sufficient conditions for existing solutions

of the BVP (15), (2).

Lemma 3.1 ([10]). Let the linear homogeneous BVP (7) has k, 1 ≤ k ≤ n

linearly independent solutions and the vector-function ϕ(t) ∈ C1([a, b], Rn) is a

solution of the BVP (15), (2) with the initial value

ϕ(a) = ϕ0 := PGkξ + G+

(

d −

b∫

a

W (s)f(s, ϕ(s))ds

)

. (16)

Then ϕ is a solution of the system of equations

x(t) = ΩtaPGk

ξ

+Ωta

(

G++(R(t)−G+R2)(PG∗

k)∗R−1

1PG∗

k

)

d +

b∫

a

L(t, s)f(s, x(s))ds, (17)

PG∗

k

(

d −

b∫

a

W (s)f(s, x(s))ds

)

= 0. (18)

Lemma 3.2. Let the linear homogeneous BVP (7) has k, 1 ≤ k ≤ n linearly

independent solutions. Then:

1) if the vector-function ϕ(t) satisfies the equation (17) then

ϕ(t) ∈ C1([a, b], Rn) and ϕ(t) satisfies the boundary condition (2);2) if furthermore function ϕ(t) = ϕ(t, ξ∗) satisfies the equation (18), then

ϕ(t) = ϕ(t, ξ∗) is a solution of the BVP (15), (2) with the initial value (16).

Proof. Let us assume that ϕ(t) satisfies the equation (17).Then ϕ(t) ∈ C1([a, b], Rn) and the identity

ϕ(t) ≡ ΩtaPGk

ξ + ΩtaG+

(

In − R2(PG∗

k)∗R−1

1PG∗

k

)

(

d −

b∫

a

W (s)f(s, ϕ(s))ds

)

+

t∫

a

Ωtsf(s, ϕ(s))ds

+

t∫

a

ΩtsW

∗(s)ds(PG∗

k)∗R−1

1PG∗

k

(

d −

b∫

a

W (s)f(s, ϕ(s))ds

)

(19)


is fulfilled. Substituting ϕ(t) of the form (19) into the boundary condition (2) andtaking into the consideration (6), (11) we obtain

d − B1ϕ(0) − B2ϕ(b) −

b∫

a

[dB(t)]ϕ(t)

= d −

(

B1 + B2Ωba −

b∫

a

[dB(t)]Ωta

)

×(

PGkξ + G+

(

In − R2(PG∗

k)∗R−1

1PG∗

k

))

(

d−

b∫

a

W (s)f(s, ϕ(s))ds

)

−

(

B2

b∫

a

Ωbsf(s, ϕ(s))ds+

b∫

a

[dB(t)]

t∫

a

Ωtsf(s, ϕ(s))ds

)

−

(

B2

b∫

a

ΩbsW

∗(s)ds +

b∫

a

[dB(t)]

t∫

a

ΩtsW

∗(s)ds

)

×(PG∗

k)∗R−1

1PG∗

k

(

d −

b∫

a

W (s)f(s, ϕ(s))ds

)

= d − GG+

(

In − R2(PG∗

k)∗R−1

1PG∗

k

)

(

d −

b∫

a

W (s)f(s, ϕ(s))ds

)

−

b∫

a

W (s)f(s, ϕ(s))ds − R2(PG∗

k)∗R−1

1PG∗

k

(

d −

b∫

a

W (s)f(s, ϕ(s))ds

)

= (In − GG+)(

In − R2(PG∗

k)∗R−1

1PG∗

k

)

(

d −

b∫

a

W (s)f(s, ϕ(s))ds

)

= PG∗

k

(

In − R2(PG∗

k)∗R−1

1PG∗

k

)

(

d −

b∫

a

W (s)f(s, ϕ(s))ds

)

=(

PG∗

k− PG∗

kR2(PG∗

k)∗R−1

1PG∗

k

)

(

d −

b∫

a

W (s)f(s, ϕ(s))ds

)

= 0.

Therefore ϕ(t) satisfies the boundary condition (2). If furthermore (18) holds,then from (19) it follows that ϕ(t) satisfies the identity

ϕ(t) ≡ ΩtaPGk

ξ + ΩtaG+

(

d −

b∫

a

W (s)f(s, ϕ(s))ds

)

+

t∫

a

Ωtsf(s, ϕ(s))ds, (20)

i.e. ϕ(t) is a solution of the system (15) with the initial value (16).Thus the lemma is proved.

74 IHOR KOROL

For investigating the problem of existence and approximate constructing of thesolutions of the BVP (15), (2) we consider the k-parametric sequence of functionsgiven by the formula

xm(t, ξ)=x0(t, ξ)+Ωta

(

G++(

R(t)−G+R2

)

(PG∗

k)∗R−1

1PG∗

k

)

d

+

b∫

a

L(t, s)f(s, xm−1(s, ξ))ds, x0(t, ξ)=ΩtaPGk

ξ, ξ∈Rk, m=1, 2, . . . . (21)

It can be easily verified that each of this functions satisfies the boundary con-dition (2). Let us now establish the main result of this paper.

Theorem 3.3. Assume that for the BVP (2), (15) the conditions A—D are

hold. Then:

1) the sequence of functions (21) converges as m → ∞,

uniformly in (t, ξ) ∈ [a, b]×D0 to the limit function x∗(t, ξ) and the following

error estimate holds

|x∗(t, ξ) − xm(t, ξ)| ≤ (E − Q)−1Qmβ; (22)

2) the limit function x∗ (t) = x∗ (t, ξ∗) is a solution of BVP (2), (15) if and

only if ξ∗ is a solution of the determining equation ∆(ξ) = 0, where

∆(ξ)def= PG∗

k

(

d −

b∫

a

W (s)f(s, x∗(s, ξ))ds

)

(23)

and its initial value is

x∗(a) = PGkξ∗ + G+

(

d −

b∫

a

W (s)f(s, x∗(s))ds

)

. (24)

Proof. Considering the difference

|x1(t, ξ) − x0(t, ξ)| ≤∣

∣

∣Ωt

a

(

G+ +(

R(t) − G+R2

)

(PG∗

k)∗R−1

1PG∗

k

)

d∣

∣

∣

+

b∫

a

|L(t, s)f(s, x0(s, ξ))| ds ≤ β,

we see that x1(t, ξ) ∈ D. It can be shown by induction that xm(t, ξ) ∈ D forall ξ ∈ D0, m ∈ N. Furthermore, from the Lipschitz condition we obtain theestimates:

|xm+1(t, ξ) − xm(t, ξ)| ≤

b∫

a

|L(t, s)| · |f(s, xm(s, ξ)) − f(s, xm−1(s, ξ))| ds

≤ Q|xm(t, ξ) − xm−1(t, ξ)| ≤ Q2|xm−1(t, ξ) − xm−2(t, ξ)| ≤ · · · ≤ Qmβ,


and thus

|xm+j(t, ξ) − xm(t, ξ)| ≤

j−1∑

i=0

|xm+i+1(t, ξ) − xm+i(t, ξ)|

≤

j−1∑

i=0

Qm+i |x1(t, ξ) − x0(t, ξ)| ≤

j−1∑

i=0

Qm+iβ. (25)

From (25) and the condition D it follows that xm(t, ξ) is a Cauchy sequence.Therefore it uniformly converges to a continuous function x∗(t, ξ). Passing to thelimit as j → ∞ in (25), we prove the error estimate (22). Since all functions satisfythe boundary condition (2), we conclude that so does the limit function x∗(t, ξ)for arbitrary ξ ∈ R

k. Taking the limit as m → ∞ we get that x∗(t, ξ) satisfies theequation (17). According to the lemmas 3.1, 3.2, the limit function x∗(t, ξ∗) is asolution of the BVP (15), (2) if and only if the condition ∆(ξ∗) = 0 is fulfilled andin this case its initial value is of the form (24).

Thus the theorem is proved.

The next statement gives us the satisfactory conditions for the existence of thesolution of the boundary value problem (2), (15). This conditions are based uponthe properties of the approximations xm(t, ξ) and not upon those of the limitfunction x∗(t, ξ).

Theorem 3.4. Assume that for the BVP (2), (15) the conditions A—D are

hold and furthermore:

1) there exists closed convex subset D1 ⊂ D0 ⊂ Rk such that for some fixed

m ∈ N the approximate equation

∆m(ξ)def= PG∗

k

(

d −

b∫

a

W (s)f (s, xm(s, ξ)) ds

)

= 0 (26)

possesses only one isolated solution ξ = ξ0m of non-zero index;

2) on the boundary ∂D1 of the set D1 the condition

infξ∈∂D1

|∆m(ξ)| > Q1(In − Q)−1Qmβ, (27)

is satisfied, where Q1 =b∫

a

∣

∣PG∗

kW (s)

∣

∣K(s)ds.

Then there exists a solution x = x∗(t) = x∗(t, ξ∗) of the boundary value problem

(2), (15), where ξ∗ ∈ D1.

Proof. We introduce the continuous for ξ ∈ ∂D and θ ∈ [a, b] vector field family

∆(θ, ξ) = ∆m(ξ) + θ(∆(ξ) − ∆m(ξ)), 0 ≤ θ ≤ 1,

connecting the vector fields ∆(0, ξ) = ∆m(ξ) and ∆(1, ξ) = ∆(ξ). Let us assumethat there exists θ0 ∈ [0, 1] such that ∆(θ0, ξ) = 0. Then

∆m(ξ) = −θ0

(

∆(ξ) − ∆m(ξ))

. (28)

76 IHOR KOROL

From (22), (23), (26) and the Lipschitz condition we have

|∆(ξ) − ∆m(ξ)| ≤

b∫

a

∣

∣PG∗

kW (s)

∣

∣ · |f (s, x∗(s, ξ)) − f (s, xm(s, ξ)) ds| ≤

≤

b∫

a

∣

∣PG∗

kW (s)

∣

∣K(s)|x∗(s, ξ) − xm(s, ξ)|ds ≤ Q1 (In − Q)−1Qmβ.

But in this case from (28) we obtain the inequality

|∆m(ξ)| ≤ |∆(ξ) − ∆m(ξ)| ≤ Q1 (In − Q)−1

Qmβ,

which contradict the condition (27). It means that the field family ∆(θ, ξ) doesnot assume the value zero on ∂D1, therefore vector fields ∆(ξ) and ∆m(ξ) arehomotopic. It means that the rotation of vector field ∆(ξ) on the boundary ∂D1

is also non-zero and consequently ∆(ξ) assumes the value zero at least in one pointξ = ξ∗ ∈ D1. Thus the theorem is proved.

References

[1] Ronto M., Samoilenko A. M., Numerical-Analytic Methods in the Theory of Boundary Value

Problems, Singapore: World Scientific, 455 p., 2000.[2] Boichuk A. A., Samoilenko A. M., Generalized Inverse Operators and Fredholm Boundary

Value Problems, VSP Utrecht Boston, 320 p., 2004.[3] Samoilenko A. M., Laptinskij V. N., Kenjebaev K. K., Constructive Methods for Investi-

gating of Periodoc and Multipoint Boundary Value Problems (in Russian), The Papers ofthe Institute of Mathematics of the Academy of Sciences of Ukraine, v. 29, K.: IM NAS ofUkraine, 220 p., 1999.

[4] Schwabik S., Tvrdy M., Vejvoda O., Differential and Integral Equations, Boundary Value

Problems and Adjoints, Praha, Academia, 248 p., 1979.[5] Luchka A. Ju., Projector-Iterative Methods (in Russian), K.: Naukova Dumka, 288 p., 1993.[6] Grebenikov E. A., Ryabov Ju. A., Constructive Methods for Analysing of the Nonlinear

Systems (in Russian), M.: Nauka, 432 p., 1979.[7] Ronto A., A Note on Boundary Value Problems with “Degenerate” Conditions, Publ. Univ.

of Miskolc, Series D. Natural Sciences. Vol 38, Mathematics (1998), pp. 83–96.[8] Moore E. H., On the Reciprocal of the General Algebraic Matrix, Bull. Amer. Math. Soc.,

pp. 394–395, 1920.[9] Penrose R., A Generalized Inverse for Matrices, Proc. Cambridge Philos. Soc., 1955, 55, 3,

pp. 406–413.[10] Boichuk A. A., Nonlinear Boundary Value Problems for Systems of Ordinary Diferential

Equations, Ukrainian Math. Journ., 1998, 50, N2., pp. 162–171.

Ihor Korol, Uzhhorod National University, Ukraine,

E-mail address: korol [email protected]


77

AN ALGORITHM TO SOLVE SOME TYPES

OF FIRST ORDER DIFFERENCE EQUATIONS

J. LAITOCHOVA

Abstract. The classical theory of first-order linear functional and difference equa-tions is obtained as a special case of the theory developed here for the functionalequation model

f Φ − Af = B.

The setting is a special space S of continuous strictly monotonic functions, where

is a group multiplication defined on S. Iterative solution formulas are given in thetwo cases when A, B are constant and when A, B depend on x.

Introduction

The difference equation

f(x + 1) − 2f(x) = 5 (1)

and the functional equation

f(

1

1+x2 − 1

2+ x

)

− xf(x) = 0 (2)

are instances of equations represented in this article by a unifying abstract model

f Φ − Af = B (3)

where is a group multiplication defined on monotonic functions, defined in alater section.

Within the abstract framework, solution formulas for (3) can be developed(see [5]) and the structure of solutions can be documented (see [6, 7]). The formulasapply equally to difference equations like (1) and functional equations like (2).

Examples are given in the last section of the paper.Some basic references for the topic of the paper are [1, 2, 8].

1. First order difference equations in the space S

Denote by C0(J) the set of continuous functions on the interval J = (−∞,∞),Z the set of integers and R the set of real numbers.

2000 Mathematics Subject Classification. Primary 39B05, 39B12.Key words and phrases. First order linear difference equation, first order linear functional

equation, space of continuous strictly monotonic functions, group multiplication.The author gratefully acknowledge the financial support from the project Narodnı program

vyzkumu II MSMT, c. 2E06029 – STM Morava (2006–2008).

78 J. LAITOCHOVA

The setting for the model of functional and difference equations is in the space S:

Definition 1.1. Let a, b be real or extended numbers, a < b. The set of allfunctions f defined on the interval J = (−∞,∞) which satisfy

(1) f ∈ C0(J),(2) f maps one-to-one the interval J on the interval (a, b),

will be denoted by the symbol S and called a space of strictly monotonic

functions.

Definition 1.2. An arbitrarily chosen increasing function X = X(x), X ∈ S,will be called a canonical function in S. The inverse to the canonical functionX will be denoted by X∗.

Remark 1.3. In the classical theory of difference equations, set S is the spaceof monotonic functions defined on J and the canonical function is the identityfunction X(x) = x.

Definition 1.4. Let α, β ∈ S. Let X∗ be the inverse to the canonical functionX ∈ S. The composite function

γ(x) = α(

X∗(β(x)))

is defined on J and it will be called a product of functions α, β ∈ S in the classS, denoted as

γ = α β.

Remark 1.5. In the classical theory of difference equations we use compositionof functions for the operation .

Definition 1.6. Let X ∈ S be a canonical function. Let Φ ∈ S. The iterates

of a function Φ in S are given by

Φ0(x) = X(x), (4)

Φn+1(x) = (Φ Φn) (x), x ∈ J, n = 0, 1, 2, . . . , (5)

Φn−1(x) =(

Φ−1 Φn)

(x), x ∈ J, n = 0,−1,−2, . . . , (6)

where Φ−1 = Φ is the inverse element to the element Φ in S according to themultiplication .

Remark 1.7. In the classical theory of difference equations we have X(x) = xand

Φ1(x) = Φ(x), Φn+1(x) = Φ(Φn(x)), n = 1, 2, . . . .

Definition 1.8. Let Φ ∈ S. A function p = p(x), p ∈ C0(J), is called auto-

morphic over Φ ifp(x) = (p Φ) (x) for x ∈ J. (7)

Assume hereafter Φ ∈ S, X ∈ S is a canonical function and Φ(x) > X(x) forx ∈ J . Let xµ = X∗(Φµ(x0)), where µ ∈ Z, symbol x0 denotes an arbitrary pointof J and X∗ is the inverse of X . The set of points xµ = X∗(Φµ(x0)), µ ∈ Z, iscalled an orbit of the point x0.

AN ALGORITHM FOR DIFFERENCE EQUATIONS 79

Definition 1.9. Let f ∈ C0(J), Φ, X ∈ S. Let X be a canonical function.The difference operator of a function f relative to Φ is denoted by ∆Φf(x) anddefined by the equation

∆Φf(x) = (f Φ) (x) − f(x).

Remark 1.10. In the classical theory of difference equations we have

∆f(x) = f(x + h) − f(x), h > 0.

It is easy to prove that the n-th difference of a function f(x) relative to Φ atx ∈ J is

∆nΦf(x) =

n∑

i=0

(−1)n−i

(

n

i

)

(

f Φi)

(x).

The new model equation, that is the linear functional (difference) equation

of k-th order in S, is given by the following equation

pk(x)∆kΦf(x) + pk−1(x)∆k−1

Φf(x) + · · · + p0(x)f(x) = Q(x). (8)

Equation (8) can be re-written in the form

ak(x)(

f Φk)

(x) + · · · + a0(x)(

f Φ0)

(x) = Q(x), (9)

where ai, Q ∈ C0(J), i = 0, 1, . . . , k, x ∈ J . We look for a solution f ∈ C0(J)which satisfies equation (9) identically on J in the case of the functional equationor x ∈ xµ

∞

µ=0, where xµ = X∗(Φµ(x0)), x0 ∈ J in the case of the difference

equation.The general theory of linear kth order functional and difference equations in S

was developed in [6, 7].The linear homogeneous functional (difference) equation of kth order with con-

stant coefficients is of the form

ak

(

f Φk)

(x) + ak−1

(

f Φk−1)

(x) + · · · + a0

(

f Φ0)

(x) = 0, (10)

where ak = 1 and a0, . . . , ak−1 are constants.

1.1. Abel functional equations in S

In the space S we define the generalized equation of Abel type in S as

(α Φ) (x) = (Φ α) (x),

where X , Φ are given and α is the unknown function. If X(x) = x, then itspecializes to the Abel functional equation in S,

(α Φ) (x) = X(x + 1) α(x).

Remark 1.11. In the classical theory of difference equations, the Abel func-tional equation reduces to

α(Φ(x)) = α(x) + 1.

Definition 1.12. A fixed point x of Φ is defined by the equation

Φ(x) = X(x),

where X(x) is a canonical function in S.

80 J. LAITOCHOVA

Given an increasing function Φ, possibly having fixed points in its domain (a, b),a group-theoretic iterative explicit construction is given in reference [4] for in-finitely many solutions α of the Abel functional equation in S. Each solution α isinfinite at fixed points of Φ and otherwise monotonic.

Theorem 1.13. Let equation (10) be given and denote by X∗ the inverse func-

tion of canonical function X. Let λ1, λ2, . . . , λk be simple positive roots of the

characteristic equation

akλk + ak−1λk−1 + · · · + a1λ + a0 = 0. (11)

Let α be a solution of the associated Abel functional equation

(α Φ) (x) = X(x + 1) α(x). (12)

Then the functions

f1(x) = λX∗

(α(x))

1, f2(x) = λ

X∗

(α(x))

2, . . . , fk(x) = λ

X∗

(α(x))

k (13)

are linearly independent solutions of equation (10).

For a proof see [3].In the following subsections, we cite some special results from [5], where the

construction appears.

1.2. First order linear functional and difference equations with constant

coefficients and a constant right side

Consider the first order linear functional equation

f Φ(x) = Af(x) + B,

where A, B are constants, A 6= 0. The following solution formulas are proved inreference [5].

Theorem 1.14. Assume A = 1 and B is a constant. The general solution of the

nonhomogeneous equation (f Φ) (x) = Af(x)+B is a sum of the general solution

of the homogeneous equation (F Φ) (x) = AF (x) and a particular solution of the

nonhomogeneous equation (f Φ) (x) = Af(x) + B. That is,

f(x) = C(x)a(x) + ∆−1

ΦB,

where a(x), C(x) are automorphic over Φ and B ∈ R.

Theorem 1.15. Assume A 6= 1, A > 0, B are constants. Then each solution

of the nonhomogeneous equation (f Φ) (x) = Af(x) + B is a sum of the general

solution of homogeneous equation (F Φ) (x) = AF (x) and a particular solution

of the nonhomogeneous equation (f Φ) (x) = Af(x) + B. Therefore,

f(x) = C(x)AX∗

(α(x)) + B1−A

,

where C(x) is automorphic over Φ and α(x) is a solution of the Abel equation

(α Φ) (x) = X(x + 1) α(x).


1.3. First order linear functional and difference equations with noncon-

stant coefficients

Consider for continuous functions A, B on 〈x0,∞), A(x) never zero, the equation

(f Φ) (x) − A(x)(

f Φ0)

(x) = B(x).

Theorem 1.16. The general solution f of the first order nonhomogeneous equa-

tion (f Φ) (x)−A(x)f = B(x) with continuous nonconstant coefficients A, B on

jn+1 is a continuous function

f(x) =(

A Φ−1)

(x)(

A Φ−2)

(x) · · ·(

A Φ−n−1)

(x)G(x),

where

G(x) = ∆−1

Φ

[

B(x)

(A Φ0) (x) (A Φ−1) (x) · · · (A Φ−n−1) (x)

]

+ k(x),

and k(x) is automorphic over Φ.

A proof can be found in [5].

x

y14

−8

x = −7.6 x = −1 x = 1 x = 5.4

Figure 1. An iterative solution y = α(x) for the functional

equation α(Φ(x)) = α(x) + 1, where Φ(x) = 1

1+x2−

1

2+ x.

1.4. Examples

Example 1.17. Define X(x) = x and consider the Abel functional equation

α(Φ(x)) = α(x) + 1, where Φ(x) = 1

1+x2 − 1

2+ x.

Graph possible solutions of the equation.

Solution. The natural domain of Φ is the real line (−∞,∞), but we can considerother possible domains (−∞,−1), (−1, 1) and (1,∞) equally natural. This isbecause the fixed points x = ±1 of Φ, which satisfy Φ(−1) = −1, Φ(1) = 1,prevent x = ±1 from being used in the given equation. We get a solution α on J

82 J. LAITOCHOVA

which is defined piecewise on three intervals. A graphic showing one possible globaliterative solution α(x) appears in Figure 1. The fixed points separate regions ofincrease and decrease of α. The black and grey segments of the solution α representthe iterative steps used to produce the graphic. Generally, infinitely many solutionsare required to fully represent α. The tick marks on the x-axis show the intervalson which the iterations are made.

The Abel functional equation α(Φ(x)) = α(x) + 1, where Φ(x) = 1

1+x2 − 1

2+ x,

can be considered a difference equation, in which case the orbit of a point x0

generates the iterative solution. This solution y = α(x) is a discrete sequence ofpoints in the plane, depicted as the open circles in Figure 2.

Figure 2. A discrete iterative solution y = α(x) for the difference

equation α(Φ(x)) = α(x) + 1, where Φ(x) = 1

1+x2 −

1

2+ x.

Example 1.18. Consider the space S, with J replaced by [0,∞). The canonicalfunction X is defined by X(x) = ex. We find a solution f of the nonhomogeneousequation

(f Φ) (x) − f(x) = 1,

where

Φ(x) =

1

1+x2 , x < −1,

x + 1.5, x ≥ −1.

Solution. The problem can be converted to

f(F (x)) = G(f(x)),

where G(u) = 1 + u and F (x) = X−1(Φ(x)) =

ln(x + 1.5), x > −1,

ln 1

1+x2 , x ≤ −1.Figure 3 shows a possible solution f .

Example 1.19. For the space S and canonical function X(x) = x, considerthe nonhomogeneous equation with nonconstant coefficients

f Φ(x) = (2 + sin(6πx))f(x) + 1.


0

6

−4

−10 0 15

Figure 3. A solution f(x) for the functional equation (f Φ) (x)−f(x) = 1.

We solve for f whenΦ(x) = 1

1+x2 − 1

2+ x.


f(F (x)) = A(x)f(x) + B(x),

where A(x) = 2 + sin(6πx), B(x) = 1 and F (x) = 1

1+x2 − 1

2+ x.

Because F has two fixed points x = 1 and x = −1, iterative methods can beapplied on the three intervals between fixed points; see Figure 4.

Figure 4. An iterative solution y = f(x) for the difference equation

f(Φ(x)) = (2 + sin(6πx))f(x) + 1, where Φ(x) = 1

1+x2−

1

2+ x.

Example 1.20. Consider the problem

(f Φ) (x) = f(x) + (0.1 + sin(6πx)).

The canonical function X is X(x) = x. We solve for f when

Φ(x) = 1

1+x2 − 1

2+ x.

84 J. LAITOCHOVA


f(F (x)) = A(x)f(x) + B(x),

where A(x) = 1, B(x) = 0.1 + sin(6πx) and F (x) = 1

1+x2 − 1

2+ x.

The function F is continuously differentiable and F ′(x) > 0. Because F hastwo fixed points x = 1 and x = −1, iterative methods can be applied on the threeintervals between fixed points. A solution f(x) is displayed in Figure 5.

Figure 5. A solution for f(Φ(x)) = f(x) + (0.1 + sin(6πx)).

References

[1] Elaydi, S., N., An Introduction to Difference Equations, Springer (New York, Berlin, Hei-delberg), 1999.

[2] Kuczma, M., Choczewski, B. and Ger, R., Iterative Functional Equations, Cambridge Uni-versity Press (Cambridge), Warszawa, 1990.

[3] Laitochova, J., A remark on k-th order linear functional equations with constant coeffi-

cients, Hindawi Publishing Corporation (Cairo, New York, 2006), Advances in DifferenceEquations, vol. 2006, Article ID 72615, 8 pages.

[4] , Group iteration for Abel’s functional equation, Nonlinear Analysis: Hybrid systems1 (2007), 95–102.

[5] , Linear difference operators in the space of strictly monotonic functions, Tatra Mt.Math. Publ. 38 (2007), 111–121.

[6] , General Theory of Linear Functional Equations in the Space of Strictly Monotonic

Functions, to appear in conference proceedings ICDEA 2007, Lisboa.[7] , Solution structure for homogeneous linear functional equations in S, Folia FSN

Universitatis Masarykianae Brunensis, Mathematica 16, CDDE, Brno 2006, 123–129.[8] Pragerova, A., Diferencnı rovnice, SNTL Prace (Praha, 1971).

J. Laitochova, Department of Mathematics, Faculty of Education, Palacky University, Zizkovo

nam. 5, 771 40 Olomouc, Czech republic,



85

THE NUMERICAL MODEL OF FORECASTING ALUMINIUM

PRICES BY USING TWO INITIAL VALUES

MARCELA LASCSAKOVA

Abstract. In mathematical models forecasting the prices on the commodity ex-changes statistical methods are usually used. In the paper we derive the numericalmodel based on the numerical solution of the Cauchy initial problem for the 1storder ordinary differential equations to prognose the prices of Aluminium on theLondon Metal Exchange. When forecasting monthly average prices, we comparethe accuracy of the prognoses acquired either in the direct way or as the arithmeticmean of daily prognoses. The advantages of the studied types of forecasting duringdifferent movements of Aluminium prices are analyzed.

Introduction

One of the most important factors determining the prices of the non-ferrous metalsis the impact of the London Metal Exchange (LME). It is the world’s premier non-ferrous metals market. The origins of LME can be traced back as far as the openingof the Royal Exchange in London in 1571. Although there are other commodityexchanges where metals are traded (for example NYMEX in the USA or SIMEXin Singapure) the majority of producers and businessmen respect just the officialprices daily closed on LME.

Observing trends and forecasting the movements of metal prices is still a currentproblem. There are a lot of approaches to forecasting the price movements. Someof them are based on mathematical models. Forecasting the prices on the commo-dity exchanges often uses statistical methods that need to process a large numberof historical market data. The amount of needed market data can sometimes bea disadvantage. In such cases other mathematical methods are required.

We have decided to use numerical methods. Their advantage is that much lessmarket data is needed in comparison with the statistical models. Our numericalmodel for forecasting prices is based on the numerical solution of the Cauchy initialproblem for the 1st order ordinary differential equations.

Let us consider the Cauchy initial problem in the form

y′ = f(x, y), y(x0) = y0. (1)

2000 Mathematics Subject Classification. Primary 65L05, 65L06.Key words and phrases. forecasting, numerical modelling, ordinary differential equation.Support of Slovak VEGA Grant 1/4005/07 is acknowledged.

86 MARCELA LASCSAKOVA

We assume that there exists just one solution y(x) of the problem (1) in theinterval 〈α, β〉, which has an exponential character. Based on this assumption weshall consider the Cauchy initial problem in the form

y′ = a1y, y(x0) = y0. (2)

The particular solution of the problem (2) is y=a0ekx, where a0 =y0e

−a1x0 , k=a1.In our prognostic model we came out of the Aluminium prices presented on

LME. We dealt with the monthly averages of the daily closing Aluminium prices“Cash Seller&Settlement price” in the period from December 2002 to June 2006.We obtained the market data from the official web page of the London MetalExchange [3]. The course of the Aluminium prices on LME (in US $ per tonne)in the observing period is presented in Figure 1.

Figure 1. The course of the Aluminium prices on LME in years 2003–2006.

As we can see in Figure 1 the course of the Aluminium prices in the consideredperiod changes dramatically.

1. Mathematical model

We shall consider the Cauchy initial problem in the form (2)

y′ = a1y

THE NUMERICAL MODEL OF FORECASTING ALUMINIUM PRICES 87

with the initial conditiony(x0) = Y0,

where [x0, Y0] are the known values (x0 is the order of month, let x0 = 0 and Y0

is the Aluminium price (stock exchange) on LME in the month x0).To determine the value of unknown coefficient a1, the second known point

[x1, Y1] is used, where x1 = 1 and Y1 is the Aluminium price on LME in themonth x1. That means [x1, Y1] are the values corresponding to the next month incomparison with those of [x0, Y0].

Substituting the point [x1, Y1] to the particular solution of the problem (2) wehave

Y1 = Y0ea1(x1−x0).

After some manipulations we obtain the formula of unknown coefficient a1

a1 =1

x1 − x0

ln

(

Y1

Y0

)

.

Now we can substitute a1 to the Cauchy initial problem (2) and we acquire

y′ =1

x1 − x0

ln

(

Y1

Y0

)

· y, y(x1) = Y1.

Generalizing the previous principle we can get the Cauchy initial problem in thepoint xi in the following form

y′ =1

xi − xi−1

ln

(

Yi

Yi−1

)

· y, y(xi) = Yi, i = 1, 2, 3, . . . . (3)

The unknown values of Aluminium prices are forecasted by the numerical solu-tion of the Cauchy initial problem (3) using the method of exponential forecasting.This numerical method is based on the exponential approximation of the solutionof the Cauchy initial problem for the 1st order ordinary differential equations [1].

The method is using the following numerical formulae

xi+1 = xi + h, yi+1 = yi + b h + Qevxi

(

evh − 1)

,

for i = 1, 2, . . . , where h = xi+1 − xi is the step of constant size.The unknown coefficients are calculated by means of these formulae

v =f ′′(xi, yi)

f ′(xi, yi), Q =

f ′(xi, yi) − f ′′(xi, yi)

(1 − v)v2evxi

, b = f(xi, yi) −f ′(xi, yi)

v.


If we consider the Cauchy initial problem (3), the function f(xi, yi) has theform f(xi, yi) = a1yi, and then

f ′(xi, yi) = a1y′(xi) = a2

1yi, f ′′(xi, yi) = a2

1y′(xi) = a3

1yi,

where a1 = 1

xi−xi−1

ln(Yi/Yi−1).

Using two known initial values [xi−1, Yi−1] and [xi, Yi] we calculate the prognosisyi+1 in the month xi+1, i = 1, 2, 3, . . . . In this way, using the known Aluminiumstock exchanges in December 2002 and January 2003, we gradually determineprice prognoses in the next months in the period from February 2003 to June2006. Thus, we obtain the prognoses for 41 months. Each Aluminium price iscalculated by using a current form of the Cauchy initial problem (3).

Two types of forecasting are created:

1. Monthly forecasting.

Using the values [xi−1, Yi−1] and [xi, Yi] we directly obtain [xi+1, yi+1] fori = 1, 2, 3, . . . , 40.

2. Daily forecasting.

In this case the prognosis yi+1 in the month xi+1 is obtained from the va-lues [xi−1, Yi−1] and [xi, Yi] using more partial computations. The interval〈xi, xi+1〉 of the length h = 1 month is divided into n parts, where n is thenumber of trading days on LME in the month xi+1. We gain the sequenceof points xi0 = xi, xij = xi + h

n· j, for j = 1, 2, . . . , n, where xin = xi+1.

For each point of the subdivision of interval a current form of the Cauchyinitial problem (3), which is solved by the chosen method of exponentialforecasting, is created. In this way we obtain the prognoses of Aluminiumprice on trading days yij . By computing the arithmetic mean of the dailyprognoses we obtain the monthly prognosis of Aluminium price in the monthxi+1. Thus, yi+1 = (

∑nj=1

yij)/n.This type of forecasting responds more to creating the real monthly avera-ges of the daily closing Aluminium prices on the London Metal Exchange.Consequently, we assume that this type of forecasting gives us more accurateprognoses than monthly forecasting.

The calculated prognoses are compared with the real stock exchanges. Weevaluate the difference between them, which is denoted by δs = ys −Ys (prognosisdeviation) and the ratio of prognosis deviation from the real price, i.e.

ps =δs

Ys

· 100%, s = 2, 3, . . . , 42.

The numerical computations are made by using programs written in Turbo Pascal.


2. Results

2.1. Comparing the accuracy of both monthly forecasting and daily

forecasting

We deal with the comparison of the accuracy of the chosen types of forecastingduring solving the Cauchy initial problem (3) by the numerical method of exponen-tial forecasting. The forecasting types were compared by means of the followingcriterions:

• the number of more accurate price prognoses when comparing the two cho-sen types of forecasting in the observing months,

• the number of critical values of forecasting (it means the prognosis with theabsolute percentage error exceeded 10 % of the real stock exchange),

• the arithmetic mean of all absolute percentage monthly deviations,∑

42

s=2|ps|/41.

The obtained results are presented in Table 1.

CriterionMonthly Daily

forecasting forecasting

The number of more accurate13 28

price prognoses

The number of critical values 3 1

The arithmetic mean of all absolute4.65% 3.99%

percentage monthly deviations

Table 1. The results of price forecasting by means of the method of expo-nential forecasting.

As we can see in Table 1 the arithmetic means of all absolute percentage monthlydeviations are not different markedly. But we can say that daily forecasting is moreaccurate. The lower arithmetic mean of all absolute percentage monthly devia-tions, fewer critical values and more than twice higher number of more accurateprognoses than monthly forecasting point at this fact. Better results of daily fore-casting are also confirmed by the distribution of the number of prognoses accordingto their absolute percentage monthly deviation from the real price, see Table 2.

In Table 2 we can see that by daily forecasting we obtained more prognoses withthe lower absolute percentage monthly deviation than by monthly forecasting.

2.2. The analyze of the higher accuracy of daily forecasting

Comparing the values of both types of forecasting and the Aluminium stock ex-change in the observing months we have found out that the daily forecasted prog-noses are lower when the price increases and they are higher when the price


The absolute percentage Monthly Dailymonthly deviation forecasting forecasting

< 5% 27 30

〈5%, 7.5%) 8 7

〈7.5%, 10%) 3 3

≥ 10% 3 1

Table 2. The distribution of the number of the prognoses according to theirabsolute percentage monthly deviation from the real price.

decreases in comparison with the prognoses obtained by monthly forecasting.Thereby neither the increase nor the decrease of the daily forecasted prognosisis large. Therefore daily forecasting can be considered slighter than monthly fore-casting. The advantages of daily forecasting are especially visible when the pricetrend is changed. It can be seen in the following situations:

• the trend of the prices is increasing (Yi−1 < Yi), but the following prognosedprice Yi+1 falls,

• the trend of the prices is decreasing (Yi−1 > Yi), but the following prognosedprice Yi+1 increases.

The computed prognosis keeps the trend of two previous initial values (it meansthese relations hold Yi−1 < Yi < yi+1, Yi−1 > Yi > yi+1, respectively). In the in-creasing trend the prognosis increases, too. The increase of the daily forecastedprognosis yi+1 is smaller than the increase of the prognosis obtained by monthlyforecasting. Thus, the daily forecasted prognosis is closer to the real stock ex-change, which is fallen. In the decreasing trend the prognosis falls, too. Thedecline of the daily forecasted prognosis yi+1 is also smaller than the decline ofthe monthly forecasted prognosis. Therefore the prognosis obtained by daily fore-casting is closer to the real stock exchange in the increase. In the described casesof the variable trends daily forecasting was always more successful than monthlyforecasting.

There are also stable trends in the course of the Aluminium prices:

• the trend of the prices is increasing and the next prognosed price increases,too (Yi−1 < Yi < Yi+1),

• the trend of the prices is decreasing and the next prognosed price falls, too(Yi−1 > Yi > Yi+1).

In these cases both chosen types of forecasting are more accurate under the dif-ferent circumstances. The success of the forecasting types depends on the intensityof either the increase or the decrease in the three observed months [xi−1, Yi−1],[xi, Yi] and [xi+1, Yi+1]. If the increase or the decrease in the price is rapid, it


means |Yi+1 − Yi| > |Yi − Yi−1|, then monthly forecasting is more accurate. Inthe moderate increase or decrease, it means when |Yi+1 − Yi| < |Yi − Yi−1|, dailyforecasting is more accurate. It holds for all 20 prognoses computed in the pe-riods of the stable trend. Table 3 shows the number of more accurate prognosesby comparing the mentioned types of forecasting according to the trends of theAluminium prices.

The trend of the pricesMonthly Daily

Totalforecasting forecasting

the trend of the prices is increasing11 6 17

Stable and the prognosed price increases

trend the trend of the prices is decreasing2 1 3

and the prognosed price falls

the trend of the prices is increasing0 11 11

Variable and the prognosed price falls

trend the trend of the prices is decreasing0 10 10

and the prognosed price increases

Total 13 28 41

Table 3. The number of more accurate prognoses by comparing the mentioned types offorecasting according to the trends of the Aluminium prices on LME.

Having the lower number of critical values daily forecasting is more accurate.The lower number of critical values forecasting gains, the more successful it is.From the set of 41 calculated prognoses, only 3 monthly forecasted prognoses (inMay 2004 (11.31%), April 2005 (10.18%) and June 2006 (26.08%)) and 1 dailyforecasted prognosis (in June 2006 (20.95%)) acquired the absolute percentageerror higher than 10% of the real price. (The number in the brackets presentspercentage deviation of the prognosis from the real price.)

The cause of the forecasting fail in the above mentioned months is the rapid fallin the prices after the increase. The more rapid change is, the bigger error of theprognosis is. Considering the previous analyze of the accuracy of the forecastingtypes depending on the trends of the prices, we can assume that daily forecastingis more accurate during the trend of the prices changes. In May 2004 and April2005, where the change was not so rapid (the decrease in the price in May 2004presented 6.16% of the price in April 2004 and the decrease in the price in April2005 was 4.44% of the price in March 2005), by daily forecasting we obtained theprice prognoses with the percentage errors less than 10% of the real price (in May2004 it was 9.03% and in April 2005 the percentage error was 7.52%). But thebreak in June 2006 was so large (13.42% of the price in May 2006) that dailyforecasting was not able to catch the break after the rapid increase, either.


3. Conclusion

To forecast the prices on the commodity exchanges, numerical models need muchless market data than statistical models. The presented prognostic numericalmodel has no problem to catch the rapid increase in the increasing trend, or therapid fall in the decreasing trend. The changes in the stable trends cause theproblems in forecasting. If the change is too large, forecasting does not catch thechange and the absolute price error rises over 10% of the stock exchange. Dailyforecasting can make forecasting more accurate. It can achieve the trend changesbetter than monthly forecasting.

References

[1] Penjak V., Lascsakova M., Solution of the Cauchy problem for the ordinary differential

equation y′ = f(x, y) by means of the exponential approximation, Studies of University inZilina, Mathematical-Physical Series 13 (2001), 163–166.

[2] Varga M., Prognozovanie na komoditnej burze, Diplomova praca, EkF TU, Kosice, 2005.[3] http://www.lme.co.uk.

Marcela Lascsakova, Department of Applied Mathematics, Faculty of Mechanical Engineering,

Technical University, Letna 9, 043 87 Kosice, Slovak Republic,



93

ON ASYMPTOTICS OF CONDITIONALLY OSCILLATORY

HALF-LINEAR EQUATIONS

ZUZANA PATIKOVA

Abstract. We use a combination of Riccati technique together with the conceptof perturbations of a general nonoscillatory half-linear differential equation

(r(t)Φ(x′))′ + c(t)Φ(x) = 0, Φ(x) = |x|p−1 sgn x, p > 1,

where r, c are continuous functions, r(t) > 0, to derive an asymptotic formula fora nonoscillatory solution of the conditionally oscillatory half-linear second orderdifferential equation.

1. Introduction

Qualitative properties of solutions of a general half-linear second order differentialequation

(r(t)Φ(x′))′ + c(t)Φ(x) = 0, Φ(x) = |x|p−1 sgn x, p > 1, (1)

where t ∈ [t0,∞), r, c are continuous functions and r(t) > 0, have recently beenstudied by many authors (see e.g. [1, 7]). It is well known that this equation canbe classified in the same way as in the theory of linear second order differentialequations as oscillatory if its every nontrivial solution has infinitely many zerostending to infinity and as nonoscillatory otherwise. In the next let us supposethat equation (1) is nonoscillatory and let d(t) be a positive continuous function.If there exists a positive constant µ0 such that equation

(r(t)Φ(x′))′ + [c(t) + µd(t)]Φ(x) = 0 (2)

is oscillatory for µ > µ0 and nonoscillatory for µ < µ0, we say that (2) is condi-

tionally osillatory with a critical constant µ0.Having a nonoscillatory equation (1), authors of [3] constructed a conditionally

oscillatory equation by adding a suitable term to the left hand side of (1) in theform

(r(t)Φ(x′))′ +

[

c(t) +µ

hp(t)R(t)(∫ t

R−1(s) ds)2

]

Φ(x) = 0, (3)

2000 Mathematics Subject Classification. Primary 34C10.Key words and phrases. Half-linear differential equation, conditionally oscillatory half-linear

equation, modified Riccati equation.

94 ZUZANA PATIKOVA

where h(t) is a positive solution of nonoscillatory equation (1) such that h′(t) 6= 0on some interval [T0,∞), R and G are defined by

R(t) := r(t)h2(t)|h′(t)|p−2, G(t) := r(t)h(t)Φ(h′(t)) (4)

and the conditions∫

∞ dt

R(t)= ∞, lim inf

t→∞

|G(t)| > 0

hold. The critical constant of equation (3) is µ0 = 1/(2q), where q is the conjugatenumber to p, i.e., p−1+q−1 = 1. In [3] it is also shown that (3) has for this constantµ = µ0 a solution of the asymptotic formula

x(t) = h(t)

(∫ t

R−1(s) ds

)

1

p

(

1 + O

(

(∫ t

R−1(s) ds

)−1))

as t → ∞.

This result was qualitatively improved in [8], where a more exact formula for the

term 1+O((∫ t

R−1(s) ds)−1) was given. More precisely, according to [8], equation(3) has a solution of the form

x(t) = h(t)

(∫ t

R−1(s) ds

)

1

p

L(t), (5)

where L(t) is a generalized normalized slowly varying function of the form

L(t) = exp

∫ t ε(s)

R(s)∫ s

R−1(τ) dτds

and ε(t) → 0 for t → ∞.

Denote γp =(

p−1p

)p. For the special case r(t)≡1, c(t)=γpt

−p and h(t) = tp−1

p

the conditionally oscillatory equation (3) with µ = 12q

, becomes the Euler-Weber

(or alternatively Riemann-Weber) half-linear differential equation

(Φ(x′))′ +

[

γp

tp+

µp

tp log2 t

]

Φ(x) = 0 (6)

with the critical coefficient µp = 12

(

p−1p

)p−1. Asymptotics of solutions of (6) were

firstly studied by Elbert and Schneider in [4], where equation (6) was consideredas as a perturbation of the half-linear Euler equation

(Φ(x′))′ +γp

tpΦ(x) = 0.

In [4] asymptotic formulas of two linearly independent solutions of (6) weregiven and these results were later improved and discussed in terms of varyingfunctions in [9] and [10]. According to these papers, equation (6) has a pair oflinearly independent solutions of the forms

x1(t) = tp−1

p (log t)1

p L1(t), x2(t) = tp−1

p log t1

p (log(log t))2

p L2(t), (7)

where L1(t) is a normalized slowly varying function in the form

L1(t) = exp

∫ t ε1(s)

s log s

ASYMPTOTICS OF CONDITIONALLY OSCILLATORY EQUATIONS 95

with ε1(t) → 0 as t → ∞ and L2(t) is a function in the form

L2(t) = exp

∫ t ε2(s)

s log s log(log s)ds

with ε2(t) = o(log(log t)).

Getting motivated by the results for the Euler-Weber equation, the aim of thispaper is to give the asymptotic formula of the second solution of equation (3),linearly independent to (5).

2. Preliminaries

One of the basic concepts of the half-linear theory is the Riccati technique, whichrelates a solution of half-linear equation (1) to a solution of a linear first orderdifferential equation. More precisely, if x is an eventually positive or negativesolution of the nonoscillatory equation (1) on some interval of the form [T0,∞),

then w(t) = r(t)Φ(

x′

x

)

solves the Riccati type equation

w′ + c(t) + (p − 1)r1−q(t)|w|q = 0. (8)

Conversely, having a solution w(t) of (8) for t ∈ [T0,∞), the correspondingsolution of (1) can be expressed as

x(t) = C exp

∫ t

r1−q(s)Φ−1(w) ds

,

where Φ−1 is the inverse function of Φ and C a constant. This means that similarlyas in the linear oscillation theory, the nonoscillation of equation (1) is equivalentto the solvability of a Riccati type equation (8) (for details see [1]).

Using the concept of perturbations it is convenient to deal with the so calledmodified (or generalized) Riccati equation (for details see e.g. [2] and [3]). Let h

be a positive solution of (1) and wh(t) = r(t)Φ(

h′

h

)

be the corresponding solution

of the Riccati equation (8). Let us consider another nonoscillatory equation

(r(t)Φ(x′))′ + C(t)Φ(x) = 0 (9)

and let w(t) be a solution of the Riccati equation associated with (9). Let us denotev(t) = (w(t) − wh(t))hp(t). Then the nonoscillation of equation (1) is equivalentalso to the solvability the modified Riccati equation

v′ + (C(t) − c(t))hp + (p − 1)r1−qh−q|G|qF( v

G

)

= 0, (10)

where G(t) is defined by (4) and

F (u) = |u + 1|q − qu − 1. (11)

Now let us recall the terminology of regularly and slowly varying functions inthe sense of Karamata (see [5, 6, 7] and the references therein).

Let a continuously differentiable function Q(t) : [T0,∞) → (0,∞) be such that

Q′(t) > 0 for t ≥ T0, limt→∞

Q(t) = ∞

and let g(t), ε(t) be some measurable functions satisfying

limt→∞

g(t) = g ∈ (0,∞) and limt→∞

ε(t) = ∈ R.

96 ZUZANA PATIKOVA

A positive measurable function f(t), such that f Q−1 is defined for all larget, and which can be expressed in the form

f(t) = g(t) exp

∫ t

t0

Q′(s)ε(s)

Q(s)ds

, t ≥ T0

for some T0 > t0, is called generalized regularly varying function of index with

respect to Q (the notation f ∈ RVQ() is then used). If g(t) ≡ g (a constant),f(t) is called normalized regularly varying function. For = 0 the terminology(normalized) slowly varying function is used.

3. Main result

As a main result we introduce the asymptotic formula for the second solution ofequation (3) with the critical coefficient µ = 1

2q, which is linearly independent

to the one given by (5). The following statement is also the answer to the openproblem conjecturing such result presented in [8].

Theorem 1. Let µ = 12q

. Suppose that (1) is nonoscillatory and posseses a

positive solution h(t) such that h′(t) 6= 0 for large t and let conditions∫

∞ dt

R(t)= ∞, (12)

and

lim inft→∞

|G(t)| > 0. (13)

hold. Then equation (3) has a solution of the asymptotic formula

x(t) = h(t)

(∫ t

R−1(s) ds

)

1

p(

log

(∫ t

R−1(s) ds

))

2

p

L(t),

where L(t) is a function in the form

L(t) = exp

∫ t ε(s)

R(s)∫ s

R−1(τ) dτ log(∫ s

R−1(τ) dτ) ds

with ε(s) = o(

log(∫ t

R−1(s) ds))

.

Proof. The proof proceeds in a similar way as the proof of the similar statementin [8]. First we formulate the modified Riccati equation associated with (3), whichreads as

v′(t) +µ

R(t)(∫ t

R−1(s) ds)2+ (p−1)r1−q(t)h−q(t)|G(t)|qF

(

v(t)

G(t)

)

= 0, (14)

where G and R are defined by (4) and F by (11).Assumptions (12) and (13) imply the convergence of the integral

∫

∞

r1−q(t)h−q(t)|G(t)|qF

(

v(t)

G(t)

)

dt,

from which follows that v(t) → 0 and v(t)G(t) → 0 for t → ∞ (see [3]).


Let C0[T,∞) be the set of all continuous functions on the interval [T,∞) (con-crete T will be specified later) which converge to zero for t → ∞ and let us considera set of functions

V = ω ∈ C0[T,∞) : |ω(t)| < ε, t ≥ T ,

where ε > 0 is such that(q + 1) ε < 1

2 . (15)

It is obvious that the relation(

q2 + 1

)

ε ≤ 1 (16)

is implied.We assume that a solution of the modified Riccati equation (14) is in the form

v(z, t) =

1q

log(

∫ tR−1(s) ds

)

+ 2q

+ z(t)

∫ tR−1(s) ds log

(

∫ tR−1(s) ds

) ,

where z ∈ V . For brevity let us denote J(t) =∫ t

R−1(s) ds.Substituting the derivative v′(t) into the modified Riccati equation (14) and

considering r1−q(t)h−q(t)|G(t)|qR(t) = G2(t), we get the equation

z′(t) +−z(t) − 2

q− 2

qlog J(t)

R(t)J(t) log J(t)

+− 1

2qlog2 J(t) − z(t) log J(t) + (p − 1)J2(t)G2(t)F

(

v(z,t)G(t)

)

log2 J(t)

R(t)J(t) log J(t)= 0,

which can be rewritten as

z′(t) +z(t)

R(t)J(t) log J(t)+

E(z, t)

R(t)J(t) log J(t)= 0 (17)

for

E(z, t) = − 2q− 2

qlog J(t) − 1

2qlog2 J(t) − 2z(t) − z(t) log J(t)

+ (p − 1)J2(t)G2(t) log2 J(t)F

(

v(z, t)

G(t)

)

.

Now, let us turn our attention to the behavior of the function F (u), which playsan important role in estimating of certain needed terms (for details see [3, 8]).

Studying the behavior of F (u) and F ′(u) for u in a neighbourhood of 0, we have

F (u) = F ′′(0)2 u2 + F ′′′(ζ)

6 u3 = q(q−1)2 u2 + q(q−1)(q−2)

6 |1 + ζ|q−3 sgn(1 + ζ)u3,

where ζ is between 0 and u. For |u| small enough there exists a positive constantMq such that

∣

∣

∣F (u) − q(q−1)

2 u2∣

∣

∣≤ (q − 1)Mq|u|

3. (18)

Similarly,

F ′(u) = F ′′(0)u + F ′′′(ζ′)2 u2 = q(q−1)u + q(q−1)(q−2)

2

∣

∣1+ζ′∣

∣

q−3sgn(1+ζ′)u2,

98 ZUZANA PATIKOVA

where ζ′ is between 0 and u. Again, for |u| small enough we have

|F ′(u) − q(q − 1)u| ≤ 3(q − 1)Mq|u|2. (19)

The estimate for the function E(z, t) for z ∈ V reads as

|E(z, t)|

=

∣

∣

∣

∣

− 2q− 2

qlog J(t)− 1

2qlog2J(t)−2z(t)−z(t) logJ(t)+ q

2v2(t)J2(t) log2J(t)

+(p−1)J2(t)G2(t) log2J(t)F( v

G

)

− q2v2(t)J2(t) log2J(t)

∣

∣

∣

∣

≤

∣

∣

∣

∣

− 2q− 2

qlogJ(t)− 1

2qlog2J(t)−2z(t)−z(t)logJ(t)+ q

2

(

1qlogJ(t)+ 2

q+z(t)

)2∣

∣

∣

∣

+

∣

∣

∣

∣

∣

(p−1)G2(t)J2(t) log2J(t)

(

F

(

v(z, t)

G(t)

)

−q(q−1)

2

v2(t)

G2(t)

)∣

∣

∣

∣

∣

≤∣

∣

q2z2(t)

∣

∣+

∣

∣

∣

∣

∣

∣

∣

Mq

(

1q

log J(t) + 2q

+ z(t))3

G(t)J(t) log J(t)

∣

∣

∣

∣

∣

∣

∣

≤ q2 |z

2(t)| +

∣

∣

∣

∣

∣

∣

∣

MqK

(

1q

log J(t) + 2q

+ z(t))3

J(t) log J(t)

∣

∣

∣

∣

∣

∣

∣

,

where (18) was used for u = vG

and K := supt≥T |G(t)|−1 is a finite constant forT sufficiently large because of (13). According to (12) and using the L’Hospitalrule, there exists T1 such that the last term in the previous inequality convergesand is less than ε2 and therefore

|E(z, t)| ≤ q2ε2 + ε2 ≤ ε2

(

q2 + 1

)

for t ≥ T1. (20)

Futhermore, for z1, z2 ∈ V we have

|E(z1, t) − E(z2, t)| =

∣

∣

∣

∣

(−2 − log J(t))(z1 − z2)

+ (p − 1)G2(t)J2(t) log2 J(t)

[

F

(

v(z1, t)

G(t)

)

− F

(

v(z2, t)

G(t)

)] ∣

∣

∣

∣

,

which, by the mean value theorem with a suitable z(t) ∈ V , becomes

≤ ‖z1 − z2‖

[

∣

∣(−2 − log J(t)) + qv(z, t)J(t) log J(t)∣

∣

+∣

∣

∣(p − 1)J(t) log J(t)G2(t)F ′

(v(z, t)

G(t)

)

− qv(z, t)J(t) log J(t)∣

∣

∣

]

≤ ‖z1−z2‖

[

q|z(t)| +∣

∣

∣(p−1)J(t) logJ(t)G(t)

(

F ′

(v(z, t)

G(t)

)

−q(q − 1)v(z, t)

G(t)

)∣

∣

∣

]

≤ ‖z1 − z2‖

[

q|z(t)| +∣

∣

∣3Mq

(1q

log J(t) + 2q

+ z(t))2

G(t)J(t) log J(t)

∣

∣

∣

]

,


where (19) was used. Similarly as in the previous estimate, there exists T2 suchthat the last term in the last row of the inequality is less than ε and hence

|E(z1, t) − E(z2, t)| ≤ ε(q + 1) · ‖z1 − z2‖ for t ∈ [T2,∞). (21)

Now, let us consider a function

r(t) = exp

∫ t 1

R(t)J(t) log J(t)ds

.

The equation (17) is then equivalent to

(r(t)z(t))′ +r(t)

R(t)J(t) log J(t)E(z, t) = 0. (22)

Let us define an integral operator F on the set of functions V by

(Fz)(t) = −1

r(t)

∫ t r(s)

R(s)J(s) log J(s)E(z, s) ds.

Since J(t) =∫ t

R−1(s) ds, it is easy to see that∫ t r(s)

R(s)J(s) log J(s)ds = r(t) − c

for some suitable positive constant c and the inequality

1

r(t)

∫ t r(s)

R(s)J(s) log J(s)ds ≤ 1

holds.For T = maxT1, T2, by (20) and (15) we have

|(Fz)(t)| ≤1

r(t)

∫ t r(s)

R(s)J(s) log J(s)|E(z, s)| ds ≤

(

q2 + 1

)

ε2 ≤ ε,

which means that F maps the set V into itself, and by (21) and (16) we see that

|(Fz1)(t) − (Fz2)(t)| ≤1

r(t)

∫ t r(s)

R(s)J(s) log J(s)|E(z1, s) − E(z2, s)| ds

≤ ‖z1 − z2‖ε(q + 1) < 12‖z1 − z2‖,

which implies that F is a contraction. Using the Banach fixed-point theorem wecan find a function z(t), that satisfies z = F z. That means that z(t) is a solution

of (22) and also of (17) and v(t) =1

qlog J(t)+ 2

q+z(t)

J(t) log J(t) is a solution of (14).

Expressing the solution of the standard Riccati equation for (3) correspondingto the solution v(z, t) of the modified Riccati equation, we have

w(t) = h−p(t)v(z, t) + wh(t) = wh(t)

(

1+v(z, t)

hp(t)wh(t)

)

= wh(t)

(

1+

1q

log J(t) + 2q

+ z(t)

hp(t)wh(t)J(t) log J(t)

)

= wh(t)

(

1+

1q

log J(t) + 2q

+ z(t)

G(t)J(t) log J(t)

)

.

100 ZUZANA PATIKOVA

Since a solution of (3) is given by the formula x(t) = exp

∫ tr1−q(s)Φ−1(w) ds

,

we need to express

r1−q(t)Φ−1(w) =h′(t)

h(t)

(

1+

1q

log J(t) + 2q

+ z(t)

G(t)J(t) log J(t)

)q−1

=h′(t)

h(t)

(

1+(q−1)

1q

log J(t) + 2q

+ z(t)

G(t)J(t) log J(t)+ o

( 1q

log J(t) + 2q

+ z(t)

G(t)J(t) log J(t)

))

=h′(t)

h(t)+

1p

R(t)J(t)+

2p

R(t)J(t) log J(t)+

(q − 1)z(t)

R(t)J(t) log J(t)

+o

( 1q

log J(t) + 2q

+ z(t)

R(t)J(t) log J(t)

)

.

The solution of (3) is in the form

x(t) = exp

log h(t) + log J1

p (t) + log(log2

p J(t))

+

∫ t (q − 1)z(s) + o(1q

log J(t) + 2q

+ z(t))

R(s)J(s) log J(s)

.

As J(t) =∫ t

R−1(s) ds, z ∈ V and hence z(t) → 0 for t → ∞, the statement of

the theorem holds for ε(t) = (q − 1)z(t) + o(1q

log J(t) + 2q

+ z(t)).

References

[1] Dosly O., Rehak P., Half-Linear Differential Equations, North Holland Mathematics Studies202, Elsevier, Amsterdam, 2005.

[2] Dosly O., Patıkova Z., Hille-Wintner type comparison criteria for half-linear second order

differential equations, Arch. Math. 42 (2006), 185–194.

[3] Dosly O., Unal M., Conditionally oscillatory half-linear differential equations, to appear inActa Math. Hungar.

[4] Elbert A., Schneider A., Perturbations of the half-linear Euler differential equation, Result.Math. 37 (2000), 56–83.

[5] Howard H. C., Maric V., Regularity and nonoscillation of solutions of second order linear

differential equations, Bull. T. CXIV de Acad. Serbe Sci. et Arts, Classe Sci. mat. nat. Sci.math. 20 (1990), 85–98.

[6] Jaros J., Kusano T., Tanigawa T., Nonoscillation theory for second order half-linear differ-

ential equations in the framework of regular variation, Result. Math. 43 (2003), 129–149.[7] Jaros J., Kusano T., Tanigawa T., Nonoscillatory half-linear differential equations and

generalized Karamata functions, Nonlin. Anal. 64 (2006), 762–787.[8] Patıkova Z., Asymptotic formulas for non-oscillatory solutions of conditionally oscillatory

half-linear equations, to appear in Math. Slov.[9] Patıkova Z., Asymptotic formulas for non-oscillatory solutions of perturbed Euler equation,

Nonlinear Analysis (2007), doi:10.1016/j.na.2007.09.017.[10] Patıkova Z., Asymptotic formulas for solutions of half-linear Euler-Weber equation, E. J.

Qualit. Theory of Diff. Equ., Proc. 8’th Coll. Qualit. Theory of Diff. Equ., 15 (2008), 1–11.

Zuzana Patıkova, Department of Mathematics, Tomas Bata University in Zlın, Nad Stranemi

4511, 760 05 Zlın, Czech Republic,



101

ON NONLINEAR SPECTRA

N. SERGEJEVA

Dedicated to the memory of prof. RNDr. Pavol Marusiak, DrSc.

Abstract. We construct the Fucık spectrum for some second order boundaryvalue problem with nonzero integral condition. The spectrum contains only finitenumber of branches.

Introduction

The Fucık spectrum is a set of points (µ, λ) such that the problem −x′′=µx+−λx−,x(0) = 0, x(1) = 0 has a nontrivial solution ([1]). This spectrum was studied firstin the works of S. Fucık as an object related to “slightly” nonlinear problems. Theinterest to problems of this type grew up also in connection with the theory ofsuspension bridges [3]. From the mathematical point of view the Fucık equationbecame a source of numerous investigations generalizing and refining the resultsby Fucık .

Let us mention some directions in which the study of Fucık spectra continued.The Fucık spectra have been investigated for the second order equation with differ-ent two-point boundary conditions (the Neumann, Sturm - Liouville and periodicones). Equations with a nonzero right sides were considered in [5]. There are fewerworks on the problems of higher order. The third order and fourth order equationsof Fucık type were studied by Pope [6], P. Krejci [2], P. Habets and M. Gaudenzi[4] and others. Some authors (Arias, Campos) have studied non-autonomous Fucıktype equations. Of the recent works let us mention also [7].

This paper is devoted to Fucık type problems with the integral condition. Therespective spectrum is essentially different from the classical one. It is not yeta countable set of hyperbolas but rather a two-sided wave spreading along thebisectrix of the first quadrant in the parameter (µ, λ) plane. The respective solu-tions still may have arbitrary number of zeroes. This spectrum was studied anddescribed in the author’s work [8] (see also [9]).

2000 Mathematics Subject Classification. Primary 35R35, 49M15, 49N50.Key words and phrases. Fucık problem, Fucık spectrum, integral boundary value problem.Supported by ESF project Nr. 2004/0003/VPD1/ESF/PIAA/04/NP/3.2.3.1./0003/0065.

102 N. SERGEJEVA

In the present work we consider the boundary conditions x(0) = 0, |x′(0)|= 1,∫ 1

0 x(t)dt=c, where c is a given constant. The normalization condition is introdu-ced in order to avoid continuous spectra. The analysis of these new spectra is themain goal of this paper.

The paper is organized as follows.Section 1 is devoted to the Fucık type problem with the zero integral condition.

Basic definitions and basic notations are given also.In Section 2 we present the results on the Fucık spectrum for the problem with

nonzero integral conditions. The spectrum is obtained under the normalizationcondition |x′(0)| = 1, because otherwise problems may have continuous spectra.This spectrum differs essentially from the known Fucık spectra, it contains onlyfinite number of branches, where the number of branches depends on the value ofc. To the best of our knowledge Fucık spectra for such problems have not beenconsidered previously. These are the main results of the work.

In Section 3 we consider a modified problem. The normalization condition|x′(0)| = 1 is replaced by another one in which x′(0) depends on a value c of theintegral. We describe properties of the spectrum.

Let us remark that usually two-index description of the branches of the classicalFucık spectrum is employed (and this is enough for λ and µ positive). The lowerindex shows how many zeroes has the respective solution in the interval (0; 1),and the upper index refers to a sign of the derivative of a solution at the initialpoint. In our notation we use three indices in description of some branches. Thiscomplication of notation may be justified by the following reasons. It makes sensein the case of integral boundary conditions to consider solutions which are asso-ciated with (µ, λ) also from the second and the fourth quadrants of the plane ofparameters. Usually only positive (non-negative) λ and µ are employed. Thus theadditional “-” sign at the lower index in a description of a branch shows that therespective λ or µ is negative. For instance, the notation F+

0− refers to solutions ofthe equation x′′ = −µx, which do not have zeros in the interval (0, 1], which arepositive, and µ < 0 (these solutions are exponents, not sines and cosines!)

1. About some nonlinear problem with zero integral condition

Consider the problem

−x′′ = µx+ − λx−, µ, λ ∈ R, (1)

x+ = maxx, 0, x− = max−x, 0, with the boundary conditions

x(0) = 0,

1∫

0

x(s)ds = 0. (2)

Definition 1.1. The spectrum is a set of points (µ, λ) such that the problem (1),(2) has nontrivial solutions.

ON NONLINEAR SPECTRA 103

The first result describes decomposition of the spectrum into branches F+i and

F−

i (i = 0, 1, 2, . . . ) according to the number of zeroes of a solution to the prob-lem (1), (2) in the interval (0, 1).

Proposition 1.2. The Fucık spectrum consists of the set of curves

F+i = (µ, λ)| x′(0) > 0, the nontrivial solution of the problem (1), (2)

x(t) has exactly i zeroes in (0, 1),F−

i = (µ, λ)| x′(0) < 0, the nontrivial solution of the problem (1), (2)x(t) has exactly i zeroes in (0, 1).

Theorem 1.3. The Fucık spectrum∑

=+∞⋃

i=1

F±

i for the problem (1), (2) con-

sists of the branches given by

F+1 = F+

1+

⋃

F+1−

⋃

(µ, λ)∣

∣

∣µ = (2 + π)2, λ = 0)

,

F+1+ =

(µ, λ)∣

∣

∣

2µ− 1

λ+

cos(

√

λ−q

λµ

π

)

λ= 0, π

√µ

< 1, π√

µ+ π

√

λ≥ 1

,

F+1− =

(µ, λ)∣

∣

∣

2µ− 1

λ+

cosh(

√

−λ−q

−λµ

π

)

λ= 0, µ > 0, λ < 0

,

F+2i =

(µ, λ)∣

∣

∣

2i+1µ

− 2iλ−

cos(

√µ−

√µλ

πi+πi

)

µ= 0,

i π√

µ+ i π

√

λ< 1, (i + 1) π

√µ

+ i π√

λ≥ 1

,

F+2i+1 =

(µ, λ)∣

∣

∣

2i+2µ

− 2i+1λ

−cos

(

√

λ−q

λµ

π(i+1)+π(i+1))

λ= 0,

(i + 1) π√

µ+ i π

√

λ< 1, (i + 1) π

√µ

+ (i + 1) π√

λ≥ 1

,

F−

i =

(µ, λ)∣

∣(λ, µ) ∈ F+i

, where i = 1, 2, . . . .

Proof. The proof of this theorem for the branches from the first quadrant isgiven in the present work [9]. The proof for other branches is similar.

First five branches of the spectrum for the problem (1), (2) are depicted inFigure 1.

2. Spectrum for the Fucık type problem with nonzero integral

condition

Consider the equation (1) with the boundary conditions

x(0) = 0, |x′(0)| = 1,

1∫

0

x(s)ds = c, c ∈ R. (3)

The meaning of the notation of the spectrum is the same as earlier.First we give some lemmas.

104 N. SERGEJEVA

F1

-

F1

+

F2

+

F2

-F

3

+

F3

- F4

+

F4

-

F5

+

F5

-

-100 100 200 300 400 500Μ

-100

100

200

300

400

500

Λ

Figure 1. The spectrum for the problem (1), (2).

Lemma 2.1. The branch F+0− of the spectrum for the problem (1), (3) exists

only for c > 12 .

Proof. Let µ < 0, λ ∈ R, x′(0) = 1. Consider the solutions of the problem (1),(3) without zeroes in the interval (0, 1). Such (µ, λ) values belong to the F+

0−. Weobtain that the problem (1), (3) reduces to the eigenvalue problem x′′ = −µx,x(0) = 0, x′(0) = 1, µ < 0. The solution of this problem is x(t) = 1

√

−µsinh

√−µt.

In view of this we obtain

1∫

0

x(s)ds = 1µ(1 − cosh

√−µ). (4)

It follows that the integral value tends to +∞ as µ tends to −∞ and it tends to12 as µ tends to 0 from the left.

The function (4) is the monotone function for negative µ. That is why thebranch F+

0− exists only for c ∈ (12 , +∞).

Lemma 2.2. The branch F+0+ of the spectrum for the problem (1), (3) exists

only for 2π2 ≤ c < 1

2 .

Proof. Consider the solutions of the problem (1), (3) without zeroes in theinterval (0, 1) in the case of µ > 0, λ > 0. Such (µ, λ) values correspond tothe branch F+

0+. We obtain that the problem reduces to eigenvalue problem in


this case. It follows that µ ∈ (0, π2]. In view of the solution of the problem isx(t) = 1

√µ

sin√

µt, we obtain

1∫

0

x(s)ds = 1µ(1 − cos

√µ). (5)

In view of it we obtain that the maximal integral value tends to 12 as µ tends

to 0, but the minimal integral value is 2π2 for µ = π2.

The function (5) is the monotone function in the considered interval (0, π2].That is why the branch F+

0+ exists only for c ∈ [ 2π2 , 1

2 ).

Lemma 2.3. For any − 12 < c < 1

2 there exists an integer i(c) value such that

the branches F+j of the spectrum for j > i do not exist.

Proof. Now we consider the solutions of the problem (1), (3) with one zero inthe interval (0, 1).

First we consider the solutions for µ > 0, λ > 0. Such (µ, λ) values correspondto the branch F+

1+.The solutions of the problem (1), (3) consist of two parts. The first one (in the

interval t ∈ [0, τ ]) is the solution of the eigenvalue problem

x′′ = −µx, x(0) = x(τ) = 0, x′(0) = 1, (6)

where τ is a zero of the respective solution of the problem (1), (3). We considerthe solutions of the problem (6) which have not zeroes in (0, τ).

But the second one (in the interval t ∈ [τ, 1]) is the solution of the eigenvalueproblem

x′′ = −λx, x(τ) = 0, x′(0) = −1. (7)

The solutions of this problem have not zeroes in (τ, 1).The calculations show that τ = π

√µ, the solutions of the problem (6) are the

functions x(t) = 1√

µsin

√µt, but the solutions of the problem (7) are the functions

x(t) = − 1√

λsin(

√λt − τ).

It is clear that the maximal value of integral will be reached if τ tends to 1, butthe minimal one will be reached if τ tends to 0. That is the maximal value will beobtained if µ → π2, λ > 0, but the minimal one if µ → +∞, λ → 0.

For this reason we obtain that c ∈(

− 12 , 2

π2

)

. That is why the branch F+1+

exists only for such c values.Now we consider the solutions of the problem (1), (3) with one zero in the

interval (0, 1) for µ > 0, λ < 0. Such (µ, λ) values correspond to the branch F+1−.

Similarly as for the previous branch we obtain that the maximal value tends to2

π2 if µ → π2, λ < 0, but the minimal one tends to − 12 if µ → +∞, λ → 0 from

the left. It follows that the branch F+1− exists only for c ∈

(

− 12 , 2

π2

)

.Now we consider the solutions of the problem (1), (3) with two zeroes in the

interval (0, 1). Such (µ, λ) values belong to the branch F+2 .

106 N. SERGEJEVA

The solutions of the problem (1), (3) consist of three parts. The first one (inthe interval t ∈ [0, τ1]) is the solution of the eigenvalue problem

x′′ = −µx, x(0) = x(τ1) = 0, x′(0) = 1, (8)

where τ1 is the first zero of the respective solution of the problem (1), (3) and thesolutions of the problem (8) which do not have zeroes in (0, τ1).

The second one (in the interval t ∈ [τ1, τ2]) is the solution of the eigenvalueproblem

x′′ = −λx, x(τ1) = x(τ2) = 0, x′(0) = −1, (9)

where τ2 is the second zero of the respective solution of the problem (1), (3) andthe solutions of the problem (9) do not have zeroes in (τ1, τ2).

The third one (in the interval t ∈ [τ2, 1]) is the solution of the eigenvalue problem

x′′ = −µx, x(τ2) = 0, x′(0) = 1. (10)

The solutions of the problem (10) do not have zeroes in (τ2, 1) also.The calculations show τ1 = π

√µ, τ2 = π

√µ+ π

√

λ, the solutions of the problems (8),

(9) and (10) are the functions x(t) = 1√

µsin

√µt, x(t) = − 1

√

λsin(

√λt − τ1),

x(t) = 1√

µsin(

√µt − τ2) correspondingly.

Therefore, similar to above, we obtain that the maximal value of integral willbe obtained if µ → π2, λ → +∞, but the minimal one if µ → +∞, λ → π2.

In view of this we obtain that c ∈(

− 2π2 , 2

π2

)

. That is why the branch F+2

exists only for such c values.Analogously for another branches of the spectrum we obtain that the branches

F+2i exist only for c ∈

(

− i 2(iπ)2 , i 2

(iπ)2

)

and the branches F+2i+1 exist only for

c ∈(

− i 2(iπ)2 , (i + 1) 2

((i+1)π)2

)

, where i = 1, 2, . . . .

The similar result is valid for the negative branches also.

Lemma 2.4. The branch F−

0− of the spectrum for the problem (1), (3) exists

only for c < − 12 .

Lemma 2.5. The branch F−

0+ of the spectrum for the problem (1), (3) exists

only for − 12 ≤ c < 2

π2 .

Lemma 2.6. For any − 12 < c < 1

2 there exists the integer value i(c) such that

the branches F−

j of the spectrum for j > i do not exist.

Remark 2.7. Analogously as in previous lemmas we obtain that the branch F−

0+

exists only for c ∈ (− 12 ,− 2

π2 ], but the branch F−

0− exists only for c ∈ (−∞,− 12 ),

the branches F−

1+ and F−

1− exist only for c ∈(

− 2π2 , 1

2

)

.

The branches F−

2i exist only for c ∈(

− i 2(iπ)2 , i 2

(iπ)2

)

and the branches F−

2i+1 exist

only for c ∈(

− (i + 1) 2((i+1)π)2 , i 2

(iπ)2

)

, where i = 1, 2, . . . .


0.2 0.4 0.6 0.8 1.0

0.5

1.0

1.5

2.0

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

-0.4

-0.3

-0.2

-0.1

0.1

0.2

(µ, λ) ∈ F+

0−(µ, λ) ∈ F

+

0+(µ, λ) ∈ F

+

0+(µ, λ) ∈ F

+

1+

0.2 0.4 0.6 0.8 1

-0.4

-0.3

-0.2

-0.1

0.1

0.2

0.2 0.4 0.6 0.8 1

-1

-0.8

-0.6

-0.4

-0.2

0.2

0.2 0.4 0.6 0.8 1

-0.2

-0.1

0.1

0.2

0.2 0.4 0.6 0.8 1

-0.2

-0.1

0.1

0.2

(µ, λ) ∈ F+

1+(µ, λ) ∈ F

+

1−(µ, λ) ∈ F

+

2(µ, λ) ∈ F

+

2

0.2 0.4 0.6 0.8 1

-0.2

-0.1

0.1

0.2

0.2 0.4 0.6 0.8 1

-0.2

-0.1

0.1

0.2

0.2 0.4 0.6 0.8 1

-0.2

-0.1

0.1

0.2

0.2 0.4 0.6 0.8 1

-0.2

-0.1

0.1

0.2

(µ, λ) ∈ F+

3(µ, λ) ∈ F

+

3(µ, λ) ∈ F

+

4(µ, λ) ∈ F

+

4

Figure 2. Some solutions of the problem (1), (3).

Some solutions of the problem (1), (3) with different number of zeroes in theinterval (0, 1) are depicted in Figure 2. This Figure is an illustration for Lem-mas 2.1 — 2.3.

Theorem 2.8. The Fucık spectrum for the problem (1), (3) for any c 6= 0,c 6= ± 1

2 consists of the finite number of branches.

Proof. It is a direct consequence of Lemmas 2.1 — 2.6.

Remark 2.9. Theorem results are summarized in the Table 1.

c values The spectrum consists of

`

−∞;− 1

2

´

F−

0−

`

−1

2;− 2

π2

˜

F−

0+, F

+

1

`

− i 2

(iπ)2;−(i + 1) 2

((i+1)π)2

˜

F±

1, F

±

2, . . . , F

±

2i, F

+

2i+1

. . . . . .ˆ

(i + 1) 2

((i+1)π)2; i 2

(iπ)2

´

F±

1, F

±

2, . . . , F

±

2i, F

−

2i+1

ˆ

2

π2; 1

2

´

F+

0+, F

−

1

`

1

2; +∞

´

F+

0−

Table 1. Theorem 2.8 results.

108 N. SERGEJEVA

Theorem 2.10. The branches of the Fucık spectrum for the problem (1), (3)are given by such equations

F+0+ =

(µ, λ)∣

∣

∣

1−cos√

µ

µ= c

x′(0) , 0 < µ ≤ π2, λ 6= 0

,

F+0− =

(µ, λ)∣

∣

∣

1−cosh√

−µ

µ= c

x′(0) , µ < 0, λ 6= 0

,

F+1 = F+

1+

⋃

F+1−,

F+1+ =

(µ, λ)∣

∣

∣

2µ− 1

λ+

cos(

√

λ−q

λµ

π

)

λ= c

x′(0) ,π√

µ< 1, π

√µ

+ π√

λ≥ 1

,

F+1− =

(µ, λ)∣

∣

∣

2µ− 1

λ+

cosh(

√

−λ−q

−λµ

π

)

λ= c

x′(0) , µ > 0, λ < 0

,

F+2i =

(µ, λ)∣

∣

∣

2i+1µ

− 2iλ−

cos(

√µ−

√µλ

πi+πi

)

µ= c

x′(0) ,

πi√

µ+ πi

√

λ< 1, π(i+1)

√µ

+ πi√

λ≥ 1

,

F+2i+1 =

(µ, λ)∣

∣

∣

2i+2µ

− 2i+1λ

−cos

(

√

λ−q

λµ

π(i+1)+π(i+1))

λ= c

x′(0) ,

π(i+1)√

µ+ πi

√

λ< 1, π(i+1)

√µ

+ π(i+1)√

λ≥ 1

,

F−

i =

(µ, λ)∣

∣(λ, µ) ∈ F+i

.

Proof. The proof of this theorem is similar to the proof given in the work [9]for the (µ, λ) from the first quadrant for c = 0.

Some first branches of the spectrum to the problem (1), (3) are depicted inFigure 3.

3. Spectrum for the Fucık problem with normalization condition,

which depends on integral value

Consider the equation (1) with the boundary conditions

x(0) = 0, |x′(0)| = 30πc,

1∫

0

x(s)ds = c, c ∈ R. (11)

The meaning of the notation of the spectrum is the same as earlier.

Theorem 3.1. The Fucık spectrum for the problem (1), (11) consists of the

branches F±

1 , . . . , F±

38, F−

39 given by equations from Theorem 2.10,

where |x′(0)|=30πc.

Proof. Let us change variables in the problem (1), (11) as follows

X(τ) = 130πc

x(t).


F1

-

F1

+F

2

+

F2

-

F3

+

F3

-F

4

+

F4

-

F5

+

F5

-

200 400 600 800Μ

200

400

600

800

Λ

F1

-

F1

+

F2

+

F2

-

F3

+

F3

-

F4

+

F4

-

F5

+

F5

-

200 400 600 800Μ

200

400

600

800

Λ

F1

-

F1

+

F2

+

F2

-

F3

+

F3

-

F4

+

F4

-

F5

+

F5

-

200 400 600 800Μ

200

400

600

800

Λ

c = 0.01 c = 0.02 c = 0.03

F1

-

F0+

+

200 400 600 800Μ

200

400

600

800

Λ

F0-

+

200 400 600 800Μ

200

400

600

800

Λ

F0-

+

200 400 600 800Μ

200

400

600

800

Λ

c = 0.3 c = 10 c = 100

Figure 3. The Fucık spectrum of the problem (1), (3) for |x′(0)| = 1 anddifferent c values.

It follows than X ′(τ) = 130πc

x′(t), X ′′(τ) = 130πc

x′′(t).In view of this we obtain the problem

−X ′′ = µX+ − λX−; (12)

with boundary conditions

X(0) = 0, |X ′(0)| = 1,

1∫

0

X(s)ds = 130π

. (13)

The problem (12), (13) is a problem considered in Section 2, where c = 130π

.That’s why we can use all results about the spectrum from the previous Section.In view of this we obtain that

(i + 1) 2((i+1)π)2 ≤ 1

30π< i 2

(iπ)2 or 60−ππ

≤ i < 60π

.

It follows that i = 19. The proof of Theorem is complete.

Some first branches of the spectrum to the problem (1), (11) are depicted inFigure 4.

Acknowledgement. The author wishes to thank F. Sadyrbaev for supervisingthis work.

110 N. SERGEJEVA

F1

-

F1

+F

2

+

F2

-

F3

+

F3

-F

4

+

F4

-

F5

+

F5

-

200 400 600 800Μ

200

400

600

800

Λ

Figure 4. The Fucık spectrum for the problem (1), (11) in the case for|x′(0)| = 30πc.

References

[1] Kufner A., Fucık S., Nonlinear Differential Equations (in Russian), Nauka, Moscow (1988).[2] Krejci P., On solvability of equations of the 4th order with jumping nonlinearities, Cas. pest.

mat., 108, 1983, 29–39.[3] Lazer A. C., McKenna P. J., Large-Amplitude Periodic Oscillations in Suspension Bridges:

Some New Connections with Nonlinear Analysis, SIAM Review, 32, 1990, No. 4.[4] Habets P., Gaudenzi M., On the number of solutions to the second order boundary value

problems with nonlinear asymptotics, J. Differential Equations, 34, 1998, No. 4, 471–479.[5] Necesal P., Nonlinear boundary value problems with asymetric nonlinearities – periodic

solutions and the Fucık spectrum, PhD thesis, University of West Bohemia, Czech Republic,2003.

[6] Pope P. J., Solvability of non self-adjoint and higher order differential equations with jump-

ing nonlinearities, PhD Thesis, University of New England, Australia, 1984.[7] Rynne B. P., The Fucik Spectrum of General Sturm-Liouville Problems, J. Differential

Equations, 161, 2000, 87–109.[8] Sergejeva N., Fucık Spectrum for the Second Order BVP with Nonlocal Boundary Condition,

Nonlinear Analysis: Modelling and Control, 12, 2007, No. 3, 419–429.[9] Sergejeva N., On the unusual Fucık spectrum, Discrete and Continuous Dynamical Systems,

Supplement, 2007, 920–926.

N. Sergejeva, Daugavpils University , Current address: Parades 1, LV-5400 Daugavpils, Latvia,



111

AN INTRODUCTION TO PRODUCT INTEGRATION

ON TIME SCALES

ANTONIN SLAVIK

Abstract. This paper represents a short survey of product integration on time

scales. We introduce the notion of product ∆-integral of a matrix function definedon an arbitrary time scale, and thus generalize the classical definition of productintegral. We summarize its properties and describe its relation to linear systemsof dynamic equations. Finally, we generalize the notion of the matrix exponentialfunction.

Introduction

The notion of product integral goes back to V. Volterra (see e.g. [3, 5, 7]). Givena matrix function A : [a, b] → R

n×n (where Rn×n denotes the set of all real n× n

matrices), a partition a = t0 < t1 < · · · < tm = b, and a collection of tagsξi ∈ [ti−1, ti], i ∈ 1, . . . , m, we form the product

(

I + A(ξm)(tm − tm−1))

· · ·(

I + A(ξ1)(t1 − t0))

(where I stands for the identity matrix). The product integral of A over [a, b],

which is usually denoted by the symbol∏b

a

(

I + A(t) dt)

, is defined as the limitof these products as the norm of the partition approaches zero (provided the limitexists).

The interest in product integration stems mainly from the fact that if the matrixfunction A is a Riemann integrable, then the indefinite product integral

Y (t) =t

∏

a

(

I + A(s) ds)

, t ∈ [a, b],

is continuous and satisfies the integral equation

Y (t) = I +

∫ t

a

A(s)Y (s) ds.

2000 Mathematics Subject Classification. Primary 26A42, 28B10, 34A30, 39A12.Key words and phrases. Product integral, matrix exponential, linear dynamic system, regres-

sive functions.Supported by grant KJB101120802 of the Grant Agency of the Academy of Sciences of the

Czech Republic, and by grant MSM 0021620839 of the Czech Ministery of Education.

112 ANTONIN SLAVIK

Thus, if A is continuous, then Y ′(t) = A(t)Y (t), Y (a) = I, and Y is a fundamentalmatrix of the linear system of equations y′(t) = A(t)y(t), where y : [a, b] → R

n.In [2] and [4], M. Bohner and G. Guseinov have developed the basic theory of

Riemann integration on time scales. Inspired by their methods, we introduce thenotion of product integral on time scales. We assume that the reader is familiarwith the basic concepts of the calculus on time scales (see e.g. [1] or [2]). The aimof this paper is to provide an introduction to the theory of product integration ontime scales; the proofs will be given elsewhere (see [6]).

1. Basic definitions

Let T be a time scale. We use the symbol [a, b] to denote a compact interval in T,i.e. if a, b ∈ T, then [a, b] = t ∈ T; a ≤ t ≤ b. The open and half-open intervalsare defined in a similar way.

A partition of [a, b] is a finite sequence of points

t0, t1, . . . , tm ⊂ [a, b], a = t0 < t1 < . . . < tm = b.

Given such a partition, we put ∆ti = ti − ti−1. A tagged partition consistsof a partition and a sequence of tags ξ1, . . . , ξm such that ξi ∈ [ti−1, ti) forevery i ∈ 1, . . . , m. The set of all tagged partitions of [a, b] will be denotedby the symbol D(a, b). We will always assume that the division points are calledt0, t1, . . . , tm and the tags ξ1, . . . , ξm.

If δ > 0, then Dδ(a, b) denotes the set of all tagged partitions of [a, b] such thatfor every i ∈ 1, . . . , m, either ∆ti ≤ δ, or ∆ti > δ and σ(ti−1) = ti. Note thatin the latter case, the only way to choose a tag in [ti−1, ti) is to take ξi = ti−1.

The following concept of Riemann ∆-integral represents a time scale general-ization of the classical Riemann integral (see [2, 4]):

Definition 1.1. Let T be a time scale and a, b ∈ T. A bounded functionf : [a, b]→R is called Riemann ∆-integrable if there exists a number S∈R with theproperty that for every ε>0, there exists a δ>0 such that

∣

∣

∑m

i=1f(ξi)∆ti−S

∣

∣<εfor every D ∈ Dδ(a, b). The number S is called the Riemann ∆-integral of f over[a, b] and we write

∫ b

a

f(t)∆t = S.

The Riemann ∆-integral of a matrix function A : [a, b] → Rn×n,

where A = aijni,j=1

, is defined in a straightforward way as the matrix

∫ b

a

A(t)∆t :=

∫ b

a

aij(t)∆t

n

i,j=1

(provided all the integrals exist).

AN INTRODUCTION TO PRODUCT INTEGRATION ON TIME SCALES 113

We now proceed to the notion of product ∆-integral of a matrix function, aconcept which represents a time-scale generalization of the classical product inte-gral.

Let T be a time scale and a, b∈T, a<b. Given a matrix function A : [a, b]→Rn×n

and a tagged partition D ∈ D(a, b), we denote

P (A, D) =

1∏

i=m

(

I + A(ξi)∆ti)

=(

I + A(ξm)∆tm)

· · ·(

I + A(ξ1)∆t1)

.

(The order is important because matrix multiplication is not commutative.)

Definition 1.2. A bounded matrix function A : [a, b] → Rn×n is called product

∆-integrable if there exists a matrix P ∈ Rn×n with the property that for every

ε > 0, there exists a δ > 0 such that ‖P (A, D) − P‖ < ε for every D ∈ Dδ(a, b).The matrix P is called the product ∆-integral of A over [a, b] and we write

b∏

a

(

I + A(t)∆t)

= P.

We make the agreement that if a = b, then∏b

a

(

I + A(t)∆t)

= I for everyfunction A.

Remark 1.3. The previous definition assumes that the space Rn×n is equipped

with a certain norm. Since all norms on a finite-dimensional space are equivalent,the definition does not depend on the choice of a particular norm.

The following theorem says that if T = R, then the product ∆-integral coincideswith the classical notion of product integral. On the other hand, if T = Z (or moregenerally T = hZ), then the product ∆-integral reduces to a product of certainmatrices.

Theorem 1.4. Let a, b ∈ T, a < b.

(i) If T = R, then A : [a, b] → Rn×n is product ∆-integrable if and only if it is

product integrable in the classical sense, and then

b∏

a

(

I + A(t)∆t)

=

b∏

a

(

I + A(t) dt)

.

(ii) If T = Z, then every function A : [a, b] → Rn×n is product ∆-integrable and

b∏

a

(

I + A(t)∆t)

=

a∏

t=b−1

(

I + A(t))

.

114 ANTONIN SLAVIK

(iii) If T=hZ, then every function A : [a, b]→Rn×n is product ∆-integrable and

b∏

a

(

I + A(t)∆t)

=

a/h∏

t=b/h−1

(

I + A(th)h)

.

2. Product integral properties

It is not immediately obvious whether a given matrix function A : T → Rn×n is

product ∆-integrable. The following theorem provides a useful sufficient condition;its proof is rather technical and long, but is similar to its classical counterpart.

Theorem 2.1. Every Riemann ∆-integrable function is product ∆-integrable.

Example 2.2. Every constant function A is Riemann ∆-integrable, and there-fore also product ∆-integrable. If T = Z, then (see Theorem 1.4)

b∏

a

(

I + A∆t)

=(

I + A)(b−a)

.

Now, let T = R. For every m ∈ N, let Dm be a tagged partition that divides [a, b]into m subintervals of length (b − a)/m (the tags can be arbitrary). Then

b∏

a

(

I + A∆t)

= limm→∞

P (A, Dm) = limm→∞

(

I + A(b−a)

m

)m= eA(b−a).

The next statement is an easy consequence of the previous theorem and thedefinition of product ∆-integral.

Theorem 2.3. If A : [a, b]→Rn×n is Riemann ∆-integrable and c∈ [a, b], then

b∏

a

(

I + A(t)∆t)

=

b∏

c

(

I + A(t)∆t)

·

c∏

a

(

I + A(t)∆t)

.

We now turn our attention to the properties of the indefinite product integral.Again, the proof of the following theorem is rather long and technical, but followsits classical version.

Theorem 2.4. If A : [a, b] → Rn×n is Riemann ∆-integrable, then the indefi-

nite product ∆-integral

Y (t) =

t∏

a

(

I + A(s)∆s)

, t ∈ [a, b],

AN INTRODUCTION TO PRODUCT INTEGRATION ON TIME SCALES 115

is a continuous function, which satisfies

Y (t) = I +

∫ t

a

A(s)Y (s)∆s, t ∈ [a, b].

Corollary 2.5. If A : [a, b] → Rn×n is rd-continuous and y0 ∈ R

n, then thevector function

y(t) =

t∏

a

(

I + A(s)∆s)

y0

is a solution of the dynamic equation y∆(t) = A(t)y(t) such that y(a) = y0.

3. Regressive functions

We now demonstrate that the theory of product integration on time scales leadsto a simple proof of the existence-uniqueness theorem for the linear initial-valueproblem y∆(t) = A(t)y(t), y(t0) = y0, where A : T → R

n×n is a regressive rd-con-tinuous function. (A different proof without product integration is given in [1].)

Definition 3.1. A function A : T → Rn×n is called regressive if the matrix

I + A(t)µ(t) is regular for every t ∈ T.

In the classical case, the product integral of an arbitrary matrix function is aregular matrix. This is no longer true on a general time scale; to guarantee theregularity, we have to assume that our matrix function is regressive:

Theorem 3.2. If A : [a, b] → Rn×n is a regressive Riemann ∆-integrable

function, then∏b

a

(

I + A(t)∆t)

is a regular matrix.

Definition 3.3. If a < b, we define∏a

b

(

I + A(t)∆t)

=(

∏b

a

(

I + A(t)∆t)

)

−1

provided the right-hand side exists.

Theorem 3.4. If A : T → Rn×n is a regressive rd-continuous function, t0 ∈ T

and y0 ∈ Rn, then the vector function

y(t) =

t∏

t0

(

I + A(s)∆s)

y0

represents a unique solution of the dynamic equation y∆(t) = A(t)y(t) such thaty(t0) = y0.

The monograph [1] contains the following definition: If A : T → Rn×n is a

regressive rd-continuous matrix function and t0 ∈ T, then the initial value problemY ∆(t) = A(t)Y (t), Y (t0) = I has a unique solution, which is denoted by eA(·, t0)and called the matrix exponential.

116 ANTONIN SLAVIK

Using product integration, the notion of matrix exponential can be extendedto a wider class of all regressive Riemann ∆-integrable functions. The followingtheorem shows that the generalized matrix exponential inherits some importantproperties of the matrix exponential as given in [1].

Theorem 3.5. Let T be a time scale and A : T → Rn×n a regressive Riemann

∆-integrable function. Then the function

eA(t, t0) =

t∏

t0

(

I + A(s)∆s)

, t0, t ∈ T,

has the following properties:

(i) e0(t, s) ≡ I and eA(t, t) ≡ I,(ii) eA(σ(t), s) =

(

I + A(t)µ(t))

eA(t, s),

(iii) eA(t, s)−1 = eA(s, t),(iv) eA(t, s)eA(s, r) = eA(t, r).

Let us finally remark that the whole theory of product ∆-integral presented inthis paper can be easily modified to obtain the corresponding notion of product∇-integral, which is related to the linear dynamic system y∇(t) = A(t)y(t).

References

[1] Bohner M. and Peterson A., Dynamic Equations on Time Scales: An Introduction with

Applications, Birkhauser, Boston, 2001.[2] Bohner M. and Peterson A., Advances in Dynamic Equations on Time Scales, Birkhauser,

Boston, 2003.[3] Dollard J. D. and Friedman C. N., Product Integration with Applications to Differential

Equations, Addison-Wesley Publishing Company, Reading, Massachusetts, 1979.[4] Guseinov G. Sh., Integration on time scales, J. Math. Anal. Appl. 285, No. 1, 107–127

(2003).[5] Slavık A., Product Integration, its History and Applications, Matfyzpress, Prague, 2007.[6] Slavık A., Product integration on time scales, Submitted.[7] Volterra V. and Hostinsky B., Operations infinitesimales lineaires, Gauthier-Villars, Paris,

1938.

Antonın Slavık, Charles University, Faculty of Mathematics and Physics, Sokolovska 83, 186 75

Praha, Czech Republic,


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