Oscillation of functional differential equations

RLgEVIER

Available online at www.sciencedirect.com MATHEMATICAL AND

OClIENOI C ~ . ' RRO'lr° COMPUTER MODELLING

Mathematical and Computer Modelling 41 (2005) 417-461 www.elsevier.com/locate/mcm

Oscillation of Functional Differential Equations

R. P. AGARWAL Department of Mathematical Sciences

Florida Institute of Technology Melbourne, FL 32901, U.S.A.

agarwal@f it. edu

S. R. GRACE Department of Engineering Mathematics Faculty of Engineering, Cairo University

Orman, Giza 12221, Egypt srgrace©eng, cu. edu. eg

I. KIGURADZE A. Razmadze Mathematical Institute of the Georgian Academy of Sciences

Tbilisi, Georgia kig©rmi, acnet, ge

D. O'REGAN Department of Mathematics

National University of Ireland Galway, Ireland

donal, oregan@nuigalway, ie

(Received May 2004; accepted June 2004)

Abs t rac t - -Some new criteria for the oscillation of functional differential equations of the form,

d [ 1 d i d a d ]~ an--l(t) -~ an-2(t) d-t"'al(t) dt x(t) +Sq(t) f(x[g(t)])=O,

are presented in this paper. © 2005 Elsevier Ltd. All rights reserved.

Keywords--Osci l la t ion, Nonoscillation, Functional, Nonlinear, Comparison.

1. I N T R O D U C T I O N

In this paper, we are concerned with the oscillatory behavior of all solutions of the functional differential equation,

Lnx (t) + 5q (t) f (x [g (t)]) = 0, (1.1; 5)

0895-7177/05/$ - see front matter (~) 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm. 2004.06.018

Typeset by A~ts-~X

418 R.P. AGARWAL et al.

where n >_ 2, ~ = -t-1, and

n o x ( t ) = x( t ) ,

1 d L k x (t) -- L k - l X (t) k = 1, 2, ., n - 1,

ak ( t ) d t ' " "

d L n x (t) = ~ [L~- lX (t)] ~ .

In what follows, we shall assume that

(i) ~ e c([t0, ~ ) , ~t+ = (0, ~ ) ) , to > 0, and

f ~ a i ( s ) = ~ , = 1 , 2 , . . . , n - 1, ds i

(ii) q e C([to, c~), R+),

(1.2)

(1.z)

(iii) g ~ C([t0, co), R -- ( -oc , co)) is nondecreasing and limt__.= g(t) = ec, (iv) f E C(R, ]R) is nondecreasing and x f ( z ) > O, for x ~ O, (v) a is a quotient of positive odd integers.

The domain :D(Ln) of Ln is defined to be the set of all functions x : [tz, c~) ~ R, t~ > to, such that Ljx(t) , j = O, 1 , . . . , n exist and are continuous on [t~, c~). Our attention is restricted to those solutions x E 7)(Ln) of equation (1.1; 5) which satisfy sup{lx(t)l : t > T} > o, for every T > t~. We make the standing hypothesis that equation (1.1; 5) does possess such solutions. A solution of equation (1.1; 5) is called oscillatory if it has arbitrarily large zeros; otherwise, it is called nonoscillatory. Equation (1.1; 5) is called oscillatory if all its solutions are oscillatory.

The problem of obtaining sufficient conditions to ensure that all solutions of certain higher-order nonlinear functional differential equations of type (1.1; 5) are oscillatory when a = 1 and/or a > 0 has been studied by a number of authors, see [1-29] and the references cited therein.

Our main objective is to present a systematic study on the oscillation of equation (1.1; 5) and establish some new oscillation criteria.

In Section 2, we shall give the proofs of some important lemmas which are useful throughout this paper. Section 3 is devoted to the study of equation (1.1; 5) when f satisfies the condition f±o~ d u / f l / ~ ( u ) < cc and in Section 4, we present some results for equation (1.1; 5) when f satisfies the condition f(x)sgn x > Ix[ ~, for x ¢ 0. In Section 5, we give results for equation (1.1; 5) when f satisfies a condition of the form f+o d u / f ( u l / ~ ) < ~ " Section 6 is devoted to the linearization of nonlinear oscillation results and via comparison with the oscillatory behavior of second-order linear equation, we present results for equation (1.1; 5) when f satisfies the condition f(x)sgn x >_ lxt ~, for x ¢ 0, where/3 is the quotient of positive odd integers, fl > a,

= ~, and 13 < ~. In Section 7, we present some oscillation criteria for equation (1.1; 5) and its special case when ai = 1, i = 1, 2 , . . . , n - 1.

The obtained results extend, improve and correlate a number of existing results.

2. P R E L I M I N A R I E S

To formulate our results, we shall use the following notation from [15]: for a~ e C([t0, oc), R), i = 1 , 2 , . . . , we define I0 = 1,

// / ~ ( t , ~ ; ~ , ~ - ~ , . . . , ~ 0 = a ~ ( ~ ) I ~ _ ~ ( ~ , ~ ; ~ _ ~ , . . . , ~ ) d~, i = 1 , 2 , . . . .

It is easy to verify from the definition of Ii that

Ii ( t , s ; a l , . . . , a i ) = ( -1 ) ' £ ( s , t ; a ~ , . . . , a l )

and

f I i ( t , s ; a l , . . . , a ~ ) = a i ( u ) I i - l ( t , u ; a l , . . . , a i - 1 ) du.

We shall need the following lemmas.

Functional Differential Equations 419

LEMMA 2.1. I f X e D(Ln), where Ln is Ln defined by (1.2) with a = 1, then, the following formu/as hold, for 0 < i < k < n - 1 and t, s E [to, oo)

and

k - 1

Lix (t) = E Ij_~ (t, s; a,+l, . . . , ak-1) L~x (s) j= i

/st + Ik-~-~ (t,~; a~+~,..., a~_~) a~ (u) Lkz (u) d~

(2.1)

k--1

L~x (t) = E ( - 1 ) J - ~ / j - i (s, t; a j , . . . , ai+l) Ljx (s) J=~ (2.2) f' + ( - 1 ) -~ - ~rk_~_l (u, t; ak_~,..., a~+~) ak (~) Ck~ (~) a~.

This lemma is a generalization of Taylor's formula with remainder encountered in calculus. The proof is immediate.

LEMMA 2.2. Suppose condition (1.3) holds. I f x E ~)(L~) where L~ is defined as in Lemrna 2.1 is eventually of one sign, then, there exist a tz >_ to >_ 0 and an integer ~, 0 < e < n with n + ~, even for x( t)[mx(t) nonnegative eventually, or n + g odd, t'or x(t)L~x(t) nonpositive eventually and, such that, for every t >__ t~,

e > 0 implies x ( t )Lkx (t) > O, k = O, 1 , . . . , l (2.3)

e < n - 1 implies ( -1) e-k x (t) Lkx (t) > 0, k = e, l + 1 , . . . , n.

This lemma generalizes the Lemma 1.1 from [15] and can be proved similarly. We note that if x(t) is a solution of equation (1.1; 5) which is eventually of one sign, then,

d - - a L , _ l x (t) L , x (t), dt ( L ~ - l x ( t ) ) = ~-1

and since a satisfies (v), we see that Lnx and Lnx have the same sign. Moreover, one can easily see that Ljx(t) , 0 <_ j <_ n are eventually of one sign.

It will be convenient to make use of the following notation in the remainder of this paper. For any T > to and all t _> T, we let

wt[t,T] = I i - l ( t , s ; a l , . . . , a l - 1 ) a e ( s ) I n - ~ - l ( t , s ; a n - l , . . . , a t + l ) ds,

Wl[t,T] = a l (t) h _ ~ ( t , s ; a 2 , . . . , a l _ l ) a t ( s ) I , ~ _ l - l ( t , s ;a ,~- l , . . . ,ae+l) ds,

N [ t , T ] = rain N~[t,T] and 2<t<n-1

For t > s > T, we let

~1 [t, s] = al (s)I,~-2 (t, s; a , , - 1 , . . . , as) ,

~ [t,T] = min{~l It, At], ~ [At, T], 2 < ~ < n - 1, and 0 < A < 1},

w , [t, s] ---- In-1 (t, s; a l , . . . , an - l ) ,

~n [t,S] : al (t) In-2 (t, s ; a2 , . . . , an_ l ) ,

w__ 0 It, T] : max I, ( t , T ; a l , . . . , a ~ ) , l<_i<n--1

~ o It, s] : In -1 (t, s; a , - 1 , . . . , a l ) .

w [ t , T ] = min we[t ,T]. l < t < n - - i

1 < g < n - i ,

2 < g < n - i ,

t >_ T/A,

Now, we present the following lemmas.

420 R . P . AGARWAL et al.

L E M M A 2 . 3 .

(il) Then, for 1 < g < n - 1 and a11 t > T > to,

Let x E l)(Ln) be eventually positive, (-1)'~-~6 = - 1 and condition (1.3) hold.

x (t) > w~ [t, T] ( 6 L , _ l x (t)). (2.4)

(i2) Then, for 2 < l < n - 1 and a11 t >_ T >_ to,

x' (t) >_ ~[t , T] ( 6 n , _ l x (t)). (2.5)

(i3) For 1 < l < n - 1, let ), be a constant with 0 < ~ < 1. Then, there exists a T* >_ T/A, such that

x' (At) _> ~ a [t, T] (65 ,_1x (t)), for t > T*. (2.6)

PROOF. Let x E 7)(Ln) be eventually positive and ~, 1 < / < n - 2 be the integer assigned to the function x as in Lemma 2.2. From (2.2) with i, k, t, and s replaced b y / , n - 1, s, and t, respectively, we obtain in view of (2.3),

n- -2

Lex (s) = ~ ( -1 ) ~-~ Ij_~ (t, s; aj, . . . , a~+l) L j x (t)

i t (2.7) + ( 1 ) n- t -1 In_g_2(u,S;an_2,...,a~+l)an_l(COLn_lX(~Z) du - - J , g

>_ xn_~_s (~, s; an-s , . . . , a~+l) a~-i (~) (~L~_lx (~)) a~.

Using the fact that 6L~_lX is noninereasing on [tl, oe), for some tl _> to _> O, we obtain

Lex (s) >_ I,~-t-1 (t, s; a ~ - l , . . . , at+l) (6L,~_lx (t)), for t > s > tl. (2.8)

On the other hand, using (2.1) with i, k, and s replaced by 0,~ and t~, respectively, and (2.3), we get

~-1 t

Ij (t, tl; al,..., aj) Ljx (tl) +/~ I~-i (t, s; at,..., ai-1) a~ (s) L~x (s) ds (2.9) (t) X

j=O

> h - 1 ( t , s ; a l , . . . , a e _ l ) a ~ (s) L~x(s) ds, for t > tl . (2.10)

Substituting (2.8) in (2.10), we obtain

[/i ] x (t) _> /~-1 (t, s; a l , . . . , a~-l) a~ (s) I , -~ -1 (t, s; a n - l , . . . , a~+l) ds ( S L y - i x (t)),

for t >_ tl.

The case when g = n - 1 is contained in (2.10). This completes the proof of (il). To prove (is), we let 2 < Z < n - 1. As before, we obtain (2.8). On the other hand, using (2.1)

with i, k, and s replaced by 1 , / a n d tl , respectively, and (2.3), we find

F £--1

x' (t) = al (t) [~-~"//(t, t l ; a 2 , . . . , a~) L~x (tl)

] + h-~ (t, ~; as, . . . ,a~-~) a~ (s) L~x (s) as (2.11) 1

> al (t) h - s (t, s; a s , . . . , a~-l) at (s) L~x (s) ds, for t > t l . 1

Combining (2.11) with (2.8), we obtain the desired inequality.

Functional Differential Equations

To prove (i3), we only need to prove the case when g = 1. replaced by 1, n - 1, s, and t, respectively, we have

F

X' (S)----al (s) / ~ (--1) j '-I I j -1 (t ,s;aj , . . . ,a2) Ljx(t)

Li=1 z' ] + ( - I F -~ &-a (~, s; a ._~. , . . . ,a2) a . _ l (~) L._~x (~) d~

Using the fact that 5Ln-tx is nonincreasing on [tl, e~), we have

x' (s) >__ at (s) I,~-2 (t, s; a ~ - l , . . . , a2) ( ~ L , _ l z (t)) = ~ [t, s] (~L~_~z (t)), For a constant A with 0 < )~ < 1, there exists a T1 >__ tl/A, such that

x' (at) > ~1 [t, At] (SL~_~x (t)), for all t > T1,

and for 2 < g < n - 1, inequality (2.5) becomes

z' (At) > ~ [At, tl] (SL~_~x (t)), for t > T1.

Combining (2.12) and (2.13), we obtain (2.6). This completes the proof.

LEMMA 2.4. Let x E Z)(L~) t > s > T > t o ,

and

421

From (2.2) with i, k, t, and s

for t > s > tl.

for t > s > tl.

(2.12)

(2.13)

|

be eventually positive, condition (1.3) hold and ~ = 0. Then, for

(2.14)

--X / (8) > Wl [t,$] ( ( - -1 ) n -1 nn_lX(t)). (2.15)

Let x c T)(L~) be eventually positive and g = 0. From Lemma 2.1, there exists a tl > to, PROOF. such that

( -1) y L j x ( t ) > 0 , O < _ j < n - 1 , t > t l . (2.16)

From (2.2) with i, k, t, and s replaced by 0, n - l , s, and t respectively, we obtain in view of (2.16) n--2

x (s) = ~ ( -1 ) j I~ (t, s; a j , . . . , al) L~x (t) j=o

+ ( -1 ) "-1 &-2 (~, s; a~ -2 , . . . , al) a ._ l (~) L~_lx (~) d~

>_ ( -1 ) "-1 In-2 (u, s; a , -2 , . . . , al) an-1 (u) L,~-lX (u) du, t > s > tl.

Using the fact that ( -1 ) '~ - lLn_ lx is nonincreasing on [tl, c¢), we have

x(slzI,_l(t,s;a~_~,...,~l)((-1F-1L~_l~(t)), for t> s > t l .

Next, from (2.2) with i, k, t, and s replaced by 1, n - 1, s, and t, respectively, we have P

L J=l

+ (--1) n-1 In-3 (u, s; a n - 2 , . . . , a2) an-1 (u) Ln-lX (u) d

> a l ( s ) I , - 2 ( t , s ; a , _ l , . . . , a 2 ) ( ( - 1 ) " - l L n _ l x ( t ) ) , for t > s > t l .

This completes the proof. |

422 1~. P. AGARWAL et al.

LEMMA 2.5. Let x E ~(L~) be eventually positive, condition (1.3) hold and £ = n. Then, for t > s > T ,

x (t) > w. It, s] L ._ lx (~) (2.17)

and

x' (t) _> @,~ [t, s] Ln- lx (s).

PROOF. Let x E T)(Ln) be eventually positive and let ~ = n. By Lemma 2.2, there exists a t l _> to, such that

Ljx (t) > O, 0 <_ j <_ n and t _> tl . (2.18)

From (2.1) with i and k replaced by 0 and n - 1, respectively, and in view of (2.18), we obtain

n- -2 t

x (t) = E / J (t, s; a l , . . . , a~-2) Ljx (s) + f In-2 (t, u; a l , . . . , a,_2) a,~-i (u) Ln_lx (u) du j=o J 8

>_ In-2(t ,u;al , . . . ,an-2)ar~-l(u)Ln-lX(U) du

>In - l ( t , s ;a l , . . . , a~_ l )Ln - lX ( s ) , f o r t > s > t l . (2.19)

JV" = .,V' l u J r '3 u • • • u H , , _ I ,

J r " = .A/'o u J r '2 u • • • u H , _ I ,

J r " = .,V'o u N ' 2 u • • • u . ,V ' ,~ ,

N = J V l u . A / ' 3 u • • • u . ~ , ~ ,

if 6 = 1 and n is even,

i f 6 = l a n d n i s o d d ,

if 6 = - 1 and n is even,

if 6 = - 1 and n is odd,

where Af~ denotes the subset of Af consisting of all x satisfying (2.3). Next, for a function g E C([t0, c~), R), we put

A~ = (t e [to, oo): g(t) > t}, ~ = {t e [to, oo): g(t) < t}.

Also, we shall introduce the notation,

v( t ) = max {min {s,g (s)} : 0 <_ to <_ s <_ t } ,

Note that the following inequalities hold,

g(s) <_T(t), f o r T ( t ) < s < t , and

p(t) = min{max {s,g(s)} : s >_ t}.

g (s) > p ( t ) , for t < s < p ( t ) .

Once again, from (2.1) with i and k replaced by 1 and n - 1, respectively, we have

F n--2

x' (t) = al (t) [ E Ij (t, s; a2,.. . , an-2) Ljx (s) L~=I

; ] + I ,_3( t ,u;a2 , . . . ,a ,_2)an_l (u)L , - lX(U) du (2.20)

/' ___ax(t) I , , _ ~ ( t , ~ ; ~ , . . . , ~ , _ ~ ) a ~ _ ~ ( ~ ) L ~ _ ~ ( ~ ) e~

>_ al (t) I~-2 (t, s; a2,.. . , a,-1) L ,_ l x (s), for t > s > tl.

This completes the proof. |

Finally, we shall denote the sets of all proper solutions, all oscillatory solutions and all nonoscillatory solutions of equation (1.1; 6) by 3, (9, and Af, respectively. It is clear that 8 = O tAAf. It is easy to see that Af has a decomposition, such that


3. O S C I L L A T I O N C R I T E R I A I

In this section, we shall present some criteria for the oscillation of equation (1.1; 5) when the function f satisfies

f :t:c~ du f l /a (u-----~ < oo. (3.1)

According to the subsets of A: for equation (1.1; 6), we shall consider the two separate cases,

(il) g E {1,2, . . - , n - 1} and (i2) g = n.

3.1. Af~ = 0, g E { 1 , 2 , . . . , n - 1}

In this section, we shall provide some sufficient conditions which ensure that Aft = 0.

THEOREM 3.1. Let 1 < g < n - 1, ( - 1 )n - t5 = -1 , a > 1, and Conditions (i)-(v), (1.3), (3.1) hold. Moreover, assume that there exist two functions ~ and ~ E Cl([to, c~),]~+), such that

a (t) _< inf {t,g (t)}, l i m a (t) = 0% and a ' (t) > 0, for t > to, (3.2) t -"'+ ~

and, for all large T >_ to,

~' (t) >_ o ( ) (~' (t))x/~ < O, (3.3) and w* [a (t) , T] a ' (t)

where

/ f

{ ~ [ a ( t ) , T ] ,

w* [a (t), T] = ~ [a (t), T],

when g > 2 and a ( t ) > T,

when g > 1 and, for some constant A,

O<A<landa(t)>T/A.

(3.4)

j oo ~ (s) q (s) : 0% (3.5) ds

then, N~ = O.

PROOF. Let x E ~r, and assume that x(t) > 0, for t > to _> 0. By Lemma 2.2, there exists a tl > to, such that (2.3) holds, for all t _> tl. Now, we consider the following two cases,

(I1) g > 2 a n d (I2) g _> 1.

CASE (I1). Assume g > 2. Define

5L~_lx (t) w (t) = ¢ (t) f (x [o (t)]) ' for t > t2, for some t2 >__ tl. (3.6)

Then,

5L~ ix (t) W ' ( t )=~ ( t ) 5(L~_lx(t)) ' +~'(t) _ f (x [a (t)]) f (x [o" (t)])

-- ~ (t) 5 (L~_lX (t)) (x [a (t)]) x' [a (t)] a' (t) f2 (X [O" (t)])

f (x [g (t)]) + ~, (t) ~ L ~ _ : (t) _< -~ (t) q (t) : (x [~ (t)]) f (x [~ (t)])

SLY_ix (t) ___ - ~ (t) q (t) + ~' (t) f (~ [~ (t)]) ' for t ___ t2.

(3.7)

(3.8)


By Lemma 2.3(i2), there exists a t3 _> t2, such that t >_ a(t) > t3 and

x' [a (t)] _> ~[c~(t) ,t3] (SLn_:(t)), for t > ta. (3.9)

Using (3.9) in (3.S), we have

~' (t) { ~' [o (t)] ~' (t) ] ~ w ' (t) < -~ (t)q (t) + : (~ [~ (t)]) \ ~ - 5 ~ ~ t ) /

=-((t)q(t)+ ~[gNi t3 - -~ ' (t) p/~(x[~(t)])]' for t >_ t4, for some t4 > ta.

Integrating (3.10) from t4 to t, we obtain

W (t) < W (t4) - ~ (s) q (s) ds + \ ~ [a (s), t3] a' (s) fl/(~ (x [a (s)]) ds (3.11)

f: { x, lo-(,,)l,,,(s) _< W (t4) - t, ( (s) q (s) ds + ~ [a (s), tz] a' (s) f l /~ (x [(7 (s)]) "

However, by the Bonnet second mean value theorem, for a fixed t >_ t4 and, for some y E [t4,t], we have

~ [O" ($), t3] O" (8) <fl/a (X [O" (8)]),]

= ~ [O" (t4), t3] O" (t4) f ' / " (X [a (S)]) (3.12)

{ ((, (_t4))1%) fx[a<o,] dtt

<( : / - 7 - - -- M, - ~N[a(t4),ta]a'(t4) [~(t4)] (u)

where M is a positive constant. Using (3.12) in (3.11), we have

f i~(~)q (s) _< (t) + ds - W W (t ,) + M%

Letting t ---+ cc in (3.13), we arrive at a contradiction to condition (3.5).

CASE (I2). Assume e > 1. Define

5L~_ix (t) w (t) = ~ (t)

f (~ [~o (t)]) '

(3.13)

(3.14)

for some constant A,0 < A < 1 and, for t > t2, for some t2 _> tl//~. Then,

_ 5L~ ix ( t ) (L~ : (t))' + ~, (t) - W' (t) = ~ (t) f (x [ ~ (t)]) f (x [ ~ (t)])

(3.1s) 5L~_lx (t) ' (x [ia (t)]) x' [A(7 it)] A~r' (t) -~( t ) f i F[~v(T)]) f

Lg_lx (t) (3.16) _< -~ (t) q (t) - ~ (t) f (x b o (t)])"

By applying Lemma 2.303 ) and the procedure of Case ( I J we arrive at the desired contradiction. This completes the proof. |

The following result is immediate and so the proof is omitted.


THEOREM 3.2. Let condition (3.3) in Theorem 3.1 be replaced by

f ~ as < oo, (3.17) ( ' (t) _> 0 and w* [a (s), T] a ' (s)

where w*[~(t),T] is as in (3.4). Then, the conclusion of Theorem 3.1 holds.

Next, we present the following results when

Q (t) = q (s) ds < o0. (3.18)

THEOREM 3.3. Let I < ~ < n - 1, ( -1 )n-e5 = -1 , Conditions (i)-(v), (1.3), and (3.1) hold, and suppose that there exists a function a 6 Cl([t0, oo),R+), such that (3.2) holds with a'(t) >_ O, for t >_ to. if, for a11 large T _> to,

f oo [~r (s) ,T] (s) (s) = co, (3.19) w* o .! QUa ds

where w*[a(t),T] is as in (3.4), then, Af~ -- 0.

PROOF. Let x E Aft and assume that x(t) > 0, for t _> to _> 0. As in the proof of Theorem 3.1, we consider the cases,

(I1) g _ > 2 a n d (I2) e > 1.

CASE (I1). Suppose l _> 2. Define W(t) as in (3.6) with ~(t) -- 1. Then, we obtain

and hence, for any t > t2,

or

f*i 6LT~_: (t~) q (s) ds < f (x [a (t2)])'

5L~_lX (t) Q (t) < : (~ [~ (t)])

5Ln-lx (t) Q~/~ (t) <_ f~/~ (~ [~ (t)])"

Using (3.9) in (3.20), we have

~[a(t),t3]cr'(t)Q1/~(s) ds <_

Integrating (3.21) from t3 to t, we obtain

x'[a(t)]cr'(t)

fl/~ (~ [~ (t)]) ' t _> t3 ~ t2.

(3.20)

(3.21)

f,~ f,i x'[~(s)]o'(s) dE= [~t~c~)l dv ~3 ~ [° (s), t3] ~' (s) Q1/~ (s) ds < ix/ . (x [~ (s)]) Jxt~c~3)l 11/~ (~) f,~ ~ dv

< fl/"--""-Z~ < (x), x[~(t3)] (v)

which contradicts condition (3.19).

CASE (I2). Suppose £ > 1. Define W(t) as in (3.14) with ~(t) = 1. Then, as in the above proof one can easily see that, for all t > t2

5L,~_lx (t) QI/~ (t) < f~/~ (~ [:~ (t)])"

By applying Lemma 2.3(i3) and proceeding as in the proof of Case (I1) above, we arrive at the desired contradiction. This completes the proof. |

426 R . P . AGARWAL st al.

THEOREM 3.4. Let 1 < ~ < n - 1, (--1)n--t~ = --1, Conditions (i)-(v), (1.3), and (3.1) and (3.18) hold. Moreover, assume that there is a function ~ E Cl([ t0,oe) ,R+), such that condition (3.2) holds with #( t ) >_ O, for t >_ to and let

f' (~) f(~-~)/" (z) > k > o, ~o~ ~ ¢ o. (3.22)

If, for all large T >_ to,

f oo (w* [a (s), T] (7 ! (s)) (3.23) ] 1/or

× Q (s) + (~,* [o (~), TI ~' (~)) Q(I+~)/~ (u) d~j as = ~ ,

where w*[o'(t),T] is as in (3.4), and

~ = { k, wheng_>2,

c, c is any positive constant, when ~ _> 1,

then, A~ = 9.

PROOF. Let x E Aft and assume that x(t) > O, for t >_ to >_ 0. As in the proof of Theorem 3.1, we consider the two cases,

(I1) g _ > 2 a n d (I2) e _> 1.

CASE (I1). Assume e >__ 2. Define W(t) as in (3.6) with ~(t) = 1 and obtain (3.7) which takes the form,

5L~_lx (t) f , (x [a (t)]) x' [a (t)] a ' (t) t > t2. (3.24) w ' (t) < - q (t) /~ (~ [~ (t)]) ' -

Again, as in the proof of Theorem 3.1, case (I1), we obtain (3.9), for t _> t3 >__ t2. Using (3.9) in (3.24), we obtain

W' (t) < - q (t) - k~ [a (t), t3] a ' (t) W (l+a)/a (t), for t > t4 _> t3. (3.25)

Integrating (3.25) from t to v > t and letting v -~ c~, we have

5 ( L , - l x (t)) a >_ f (x [a (t)])

× Q ( t ) + k ~ [ a ( s ) , t a ] a ' ( s ) W cl+~)/~(s) ds , for t>_t4 ,

and hence, W (t) ___ Q (t), for t >_ t4. (3.27)

Using (3.9) and (3.27) in (3.26), we obtain

[ x' [a (t)] a ' (t) > ~ [~ (t) ta] or' (t) Q (t) + k N [a (s), t3] ~' (s) Q(I+~)/~ (s) ds /1/~ (x [~ (t)]) -

Integrating this inequality from t4 to t and using condition (3.1), we obtain a contradiction to condition (3.23).

CASE (I2). Assume g > 1. Define W(t) as in (3.14) with ¢(t) = 1. Then,

5g~_lx (t) f , (x [Aa (t)]) x' [A~ (t)] ;ka' (t) t > t2. w ' (t) < - q (t) / 2 (x [~o (t)]) ' -

By applying Lemma 2.3(i3) and proceeding as in the above case, we arrive at the desired conclusion. This completes the proof. |

We note that condition (3.22) can be replaced by

f ' (x) fO-~)/~ (x) = "r (x), for x ¢ 0, (3.28)

where V(x) is a positive nondeereasing function, for x # 0. Now, we have the following result.


THEOREM 3.5. Let condition (3.22) in Theorem 3.4 be replaced by (3.28). If, for all large T > to and all positive constant -k, condition (3.23) holds, then, Af~ = 0.

PROOF. Let x C Af~ and let x(t) > 0, for t _> to _> 0. As in the proof of Theorem 3.4, Case (I1), we see that inequality (3.24) takes the form,

W'(t) <_ -q ( t ) -~[(x[o'(t)])~[cr(t),t3]o-I(t)w(lq-c~)/a(t), for t _>~ t 4 __> t 3. (3.29)

Since x(t) is increasing on It1, co) and ~/(z) is a nondecreasing function, there exist a constant m > 0 and ts _> t4, such that

x [cr (t)] _ m, for t >_ tb. (3.30)


W ' (t) _<~ - q (t) - ~/(m) w [o" ( t ) , t3] o -! (t) W (1-['°~)/Cz ( t ) , for t _> tb.

The rest of the proof is similar to that of Theorem 3.4, and hence, is omitted. |

To prove our next linearization result, we need the following lemma which is given in [4].

LEMMA 3.1. The ha/f-//near equation,

(~ (t) (~' (t))~) ' + q (t) ~ (t) = 0,

where ~ is as in (v), ~(t), q(t) e C([t0, co), R+) is nonoscillato~y if and only if there e~ist a

number T >_ to and a function v(t) E Cl([t0, co), ]~) which satisfies the inequality

~' (t) + ~ - 1 / ~ (t)Iv (t)[ (~+")/" + q (t) _< o, on [T, co) .

THEOREM 3.6. Let 1 < ~ < n - 1, ( - 1 ) n - t 6 = - 1 , Conditions (i)-(v), (1.3), (3.1), and (3.22) hold and suppose that there exists a function a 6 Cl([t0, oo),R+), such that condition (3.2) holds. If, for all large T > to, the second-order half-Iinear equation,

(w . [~ ( t ) :~r ]~ , ( t ) )~ (W(t)) ~ + q ( t ) y ~ ( t ) = o , (3.31)

is oscillatory, where w*[o(t),T] is as in (3.4) and ~ is as in Theorem 3.4, then, ~ = O.

PROOF. Let x 6 Af~ and assume that x(t) > 0, for t _> to _> 0. Proceeding as in the proof of Theorem 3.4 and considering the two cases,

(I1) ~ _ > 2 a n d (I2) ~ _> I .

CASE (I1). Suppose ~ _> 2. Define the function W(t) as in (3.6) with ~(t) = 1 and proceed as in the proof of Theorem 3.4, Case (I1) to obtain (3.25). Now, applying Lemma 3.1 to inequality (3.25), we conclude that the equation (3.31) is nonoscillatory, which is a contradiction.

CASE (I2). Suppose ~ > 1. Define the function W(t) as in (3.14) with ~(t) = 1. The rest of the proof is similar to that of Theorem 3.4, Case (I2) and that given above. We omit the details.

Similarly, we can obtain the following result.

THEOREM 3.7. Let condition (3.22) in Theorem 3.6 be replaced by condition (3.28). If, for all large T _> to and all positive constant k, the equation (3.31) is oscillatory, then, A;~ = 0.

REMARK 3.1. According to the above results, if the hypotheses of any of these hold, then, we can easily see the following.

(1) For even-order equation (1.1; 1), every solution x 6 0 . (2) For odd-order equation (1.1; 1), any solution x 6 0 tA Af0. (3) For even-order equation (1 .1; -1) , any solution x 6 0 tAAf0 UAf~. (4) For odd-order equation (1 .1; -1) , any solution x 6 0 U A/',~.

428 R.V. AGARWAL et al.

3 . 2 . A f n = O

We note that this case occurred for equation (1.1; 5) when 5 = -1 . We shall employ either one of the following conditions on the function f .

(il) f satisfies -- f (--xy) >_ f (xy) > f (x) f (y),

or, we assume that 02) inf { f (~x)/f (~) : f # 0} > 0, for any x > 0.

For this purpose, we need the function defined by

f*(x) = { sgn x inf{f (~ lx l )0 , f (5) :"z

for zy > 0, (3.32)

>0~, if x#0, (3.33)

i f x = 0.

It is easy to see that i f (x) has the following properties.

f* (x) is nondecreasing on R and x f* (x) > 0, for x # 0, (3.34)

We set

If(~ [xl)l >--If(~)llf*(x)l, for ~x > 0.

f (x), if condition (3.32) holds,

F ( x ) = f* (x), if condition (3.35) holds,

and suppose that +oo du

F ( u l / " )

For more details see [25]. Now, we present the following results.

THEOREM 3.8. and

(3.35)

(3.36)

< oo. (3.37)

Let 5 = -1, Conditions (i)-(v), (1.3), and either condition (3.32) or (3.35) hold

- - -+ 0, as x --* oo, (3.38)

where F as in (3.36). If

X a

F(x)

f p(t)

lim sup q(s)f(w~[g(s) ,p(t)]) ds >O, t - ~ o 0 J t

(3.39)

then, A/'. = O.

PROOF. Assume that x C Afn and let x(t) > 0, for t _> to > 0. Then, there exists a tl > to, such that

L j x (t) > O, 0 _wn[g(s),p(t)]L~_lx[p(t)], fortl <_t < s q (s) f (w,~ [g (s) , p (t)] L,~-xx [p (t)])

> q (s) f (wn [g (s), p (t)]) F (L~_lx [p (t)]), (3.42)

fort1 ~ t < s _ q(s) f (w~[g(s ) ,p ( t ) ] )F(Ln_ lx[p ( t ) ] ) ds , I t

fp( t ) L~_lx [p (t)] > q (s) f (Wn [g (S), p (t)]) ds. (3.43) F (L~_lx [p(t)]) - j,

Taking lim sup of both sides of inequality (3.43) as t --* co and by (3.38), we obtain a contradiction

to condition (3.39). This completes the proof. |

Let 5 = -1 , Conditions (i)-(v), (1.3), end either condition (3.32) or (3.35) and THEOREM 3.9. (3.37) hold.

A q (s) f (wn [g (s), s]) ds = (3.44) OC, g

then, N n = 0.

PROOF. Let x e Af,, and assume that x(t) > 0, for t > to _> 0. From (2.17) w i t h t and s be replaced by g(t) and t, respectively, we have

x [g (t)] >_ w, [g (t), t] n~_ix (t), for t • Ag n [tl, co), (3.45)

where tl is as in the proof of Theorem 3.8. Set u(t) = LX_~x(t). Then, u(t) satisfies

du d---[ = L~x (t) = q (t) f (x [g (t)])

_> q (t) I (w~ [g (t), t] L~_lZ (t)) > q (t) I [9 (t), t]) F (L _lx (t))

=q( t ) f ( w n [ g ( t ) , t ] ) F ( u l / ~ ( t ) ) , f o r t E A g N [ t l , c o ) .

Dividing the above inequality by F(u~/a(t)) and integrating on Ag n [t~,Tl], where TI(_> tl) is arbitrary, we obtain

du q (s) f (w~ [g (s), s]) ds < ds = - -

gn[~;1)T1 ] - - F (,~I/G~ (S)) J . ( t l ) F (V/~3t) "

Letting T1 ---* ec, we find

q (s) f (w~ [g (s), s]) ds < du gfl[tl,CO) -- (tl) F (U l / a ) < CO.

This contradicts condition (3.44). This completes the proof. I

THEOREM 3.10. Let 5 = -1 , g(t) > t, for t >_ to, Conditions (i)-(v), and (1.3) hold and

X c~

- * 0, as x --* co. ( 3 . 4 6 ) f (x)

If

lira sup In-2 (g ( t ) , u ;a l , . . . ,an-2)an-1 (u) q(s) ds du > O, t ......+ Oo J t

then, Afn = 0. PROOF. Let z E Af~ and assume that x(t) > 0, for t _> to _> 0. Then, there exists a tt >_ to, such that (3.4o) holds, for t _> tl. Let T1 > tx be, such that inf{g(t) : t _> T1} _> tl. From (2.19) (see the proof of Lemma 2.5) with t and s replaced by g(t) and t, respectively, we have

f g(t)

z [g (t)] >_ I , - 2 (g (t), u; a l , . . . , an-2) an-1 (u) L ,_ lX (u) du, for t _> T1. (3.47) J t

430 R . P . AGAI~WAL et al.

Integrating equation (1.1;-1) from t to u, we obtain

L ~ _ ~ (u) - i~_~x (t) = q (~) f (x [g (~)l) d~

o r

(I L,~-lX (u) k q (s) f (x [g (s)]) ds , for u > t > T1.

Using (3.48) in (3.47), one can easily see that

(3.48)

x [g (t)] > In-2 (g (t), U; al , . . . , am-z) an-1 (u) q (s) ds ds. I ~ l <" (z [g (t)]) - .,~

The rest of the proof is similar to that of Theorem 3.8, and hence, is omitted. I

In the following result, condition (3.32) or (3.35) is disregarded.

THEOREM 3.11. Let 5 = -1 , Conditions (i)-(v), (1.3), and (3.1) hold. If

/A f f (~ ) g u l/<~ (3.49)

×an_l(u)(/s q(r) dr) duds=-cc,

then, dkfn -~ O. PROOF. Let x EAf, and suppose that x(t) > 0, for t > to _> 0. Then, there exists a tl _> to, such that (3.40) holds, for t _> tl. From (2.20), (see the proof of Lemma 2.5) with t and s replaced by g(t) and t, respectively, we get

f g(t)

x' [g (t)] > al [g (t)] ~t 1~_3 (g (t), u; a2 , . . . , a~-2) (3.50)

×a~-l(u)n~_lx(u)du, t eAgA[t l ,oc) .

As in the proof of Theorem 3.10, we obtain (3.48). Using (3.48) in (3.50), one can easily see that

x' [g (t)] g' (t) > ~ [~ (t)] g' (t) fl/~ (x [g (t)]) - 1/~ (3.51)

f ~,"(') I~-s (g (t), u; 32,... ,a~-2)a~-1 (u) q (r) dr) du.

Integrating (3.51) on A 9 N [tl,T1], where T1 ~ (tl) is arbitrary, we obtain

,n%,T~lal[g(s)lg'(s)S"(~)I,~-a(g(s),u;a2,...,a~-2)a,~-l(u) q ( r ld r ) l / "duds

/ ~[a(T~)l dx J~[g%)l (x)

Letting T1 ~ co, we find

(/; ,nc<,,~) 31 [g (s)] g' ( s ) /g( ' ) s~_3 (g (~),~; a~, . . . , ~_~)a~_1 (~) q(~) dr) 1/, ~

L ~ dx <- f l / ~ < oo. [g(t~)] (x)

This contradicts condition (3.49) and completes the proof. I


REMARK 3.2. By combining the hypotheses of any of the results of Section 3.1 for the equation (1.1;-1) with the appropriate ones of Section 3.2 for equation (1.1;-1) , one can easily see the following.

(1) For even-order equation (1.1;-1), any solution x E (_9 UAf0. (2) For odd-order equation (1.1;-1), every solution x E O.

We can also conclude the following.

(3) For odd-order equation (1.1;1) every unbounded solution x E O. (4) For even-order equation (1.1;-1) every unbounded solution x E O.

4. O S C I L L A T I O N C R I T E R I A I I

In this section, we shall give some oscillation criteria for equation (1.1; 5) when the function f satisfies

f (x) sgn x _> Ix[ s , for x # 0. (4.1)

Regarding the subsets of AY for equation (1.1; 5), we shall distinguish the following three separate cases,

(il) £ E { 1 , 2 , . . . , n - - 1}, (i2) / = 0, and (ia) f = n.

4.1. Ale = 0, £ E { 1 , 2 , . . . , n - 1}

In this section, we shall present some sufficient conditions which ensure that AYe = ~. For this, we shall need the following lemma due to [19].

LEMMA 4.1. I f X and Y are nonnegative numbers, then,

X x - A X Y ~-1 + ( A - 1)Y~' k O, A > 1,

and

X ~ - A X Y ~-1 - (1 - A) Y~ _< 0, 0 < A < 1.

In the above inequafities, the equality holds i f and only i f X = Y .

Now, we prove the following result.

THEOREM 4.1. Let 1 _< e _< n - 1, ( -1)n-e5 = - 1 , Conditions (i)-(v), (1.3), and (4.1)

hold. I f there exist two functions a( t ) , [ ( t ) E Cl([t0, oo),~[+), such that (3 .2)holds and, for

all large T >_ to,

lim sup ~ (s) q (s) - 0 t---~O0

(~., (~))~+1 (e (s) ~, (s) ~ . [~ (s), T])"

ds = co, (4.2)

where w*[a(t),T] is as in (3.4) and

(a + 1) -(a+l) , 0 = A -~ ( a + 1) -(~+I),

if ~ _> 2,

O < A I,

then, AY~ = 0.

PROOF. Let x E AYe and assume that x(t) > 0, for t > to _> 0. By Lemma 2.2, there exists a t l k to, such that (2.3) holds, for all t >_ tl . Now, we consider the following two cases,

(I1) t > z and (I2) t _> 1.

432 R . P . AGARWAL et at.


5Lg_lx (t) w (t) = ~ (t) ~ [~ (t)] ' for t ___ t~ _> t~. (4.3)

Then, for t > t2,

W' (t) : - ~ (t) q (t) f (x [g (t)]) + ~, (t) 5Lg-lx (t) ~- [o (t)] ~ [~ (t)]

(4.4)

~L~_~x ( t )~o_~ [~ (t)] x' [~ (t)] o' (t) - ~ (t) ~ [~ (t)]

< -~ (t) q (t) + ~' (t) W ('" - aa' (t) ~ (t) 5L~-lx (t) x' [a (t)]. (4.5) - ~ (t) '~; ~+~ [~ (t)]

By Lemma 2.3 (i2), there exists a t3 _> t2, such that t 2 a(t) >_ t2 and (3.9) holds. Using (3.9) in (4.5), we have

W' (t) < - ~ (t) q (t) q- ~' (t) W (t) -- ~ (t) (4.6)

--SO d (t) W [0" ( t ) , t2] ~--l/a (t) W (a+l)/a (t), for t >_ t3.

Fix t > t3 and set

x = ( ~ [o ( t) , t21 ~' (t)) "/(~+~) ~-~/(~+~) (t) w (t) ~ ~ + 1 , =-->I,

and

y = ( a .~)a [~' (t) /:W(~+l) (t) (a~[a(t) , t2]a, ~ (t) ~

Now by Lemma 4.1, for t _> t3, we obtain

~' (t) w (t) - ~ [~ (t) , t~] o' (t) w (~+~)/" (t) (t)

( 1 ) a + l [ ( ~ , ( t ) ~ a + l ] - a _< ~ ~ (t) \ ~ (t) ] ~ [a (t) , t21 a' (t)

Thus, (4.6) reduces to

w ' (t) < - ~ (t) q (t) + 1 (~' (t))~+l for t _> t3. (Or q- 1) a+ l [~ ( t) w [ff ( t ) , t2 ] 0 4 it)] a '

Integrating the above inequality from t3 to t, we have

0 < w it) _< w (t2) - ( i s ) q is) - 0 [ ( (s) ~' (s) ~ [o (s ) , t2]] ~

Taking lim sup of both sides of this inequality as t - , c~, we obtain a contradiction to condi- t ion (4.2).

CASE (I2). Assume g _> 1. Define

6L~_lx (t) (4.7) w (t) = ~ (t) x - [ ~ (t)]'

for some constant )~, 0 < A < 1 and, for t _> t2, for some t2 _> t l /A. Then,

5L~ ix (t) ~' (t) w (t~ - ~ ' (t) ~ (t) - z ' [ ~ (t)]. (4.s) W'( t ) _< - ~ ( t ) q(t) + ~(t) ' J x ~+1 [)~a (t)]

Next, by applying Lemma 2.303 ) and the procedure of Case (I1), we arrive at the desired contradiction. This completes the proof. |

For each t > to, we let gi t) < t and define

#( t ) = sup{s >_ to : g(s) < t}.

Clearly, #(t) >__ t and g o/z(t) = t. Now, we give the following result.


THEOREM 4.2. Let 1 < ~ < n - 1, ( -1)n-~5 = -1 , Conditions (i)-(v), (1.3), and (4.1) hold. If, for all large T > to,

l imsupw~ [t,T] q(s) ds > 1, (4.9) t - ~ (t)

then, Af~ = O.

PROOF. Let x E All and suppose that x(t) > 0, for t > to >_ 0. By Lemma 2.2, there exists a tl _> to, such that (2.3) holds, for t > tl. Integrating equation (1.1; 5) from t _> tl to u > t and letting u --* o% we obtain

5 (Ln - l x (t)) a ~ q (s) x a [g (s)] ds, t > tl. (4.10)

By Lemma 2.3 (il), there exists a t2 _> tl , such that

Thus,

x (t) >_ wt It, t2] (hL~_lx (t)), for t ~_ t2. (4.11)

(S ) x ~ (t) > w 7 [t, t2] (hLn- lx (t)) ~ >_ w~ [t, t2] q (s) x ~ [g (s)] ds , t > t2.

Now, by #(t) > t and the fact that x'(t) > 0 and g(s) > t, for s > #(t), it follows that

x ~ (t) > w~ [t, t2] q (s) x ~ [g (s)] ds > w~ [t, t2] x ~ (t) q (s) ds. (4.12) (t) (t)

Dividing both sides of (4.12) by x~(t), we have

w~ [t, T] q (s) ds < 1, for t > t2. (4.13) (t)

Taking l imsup of both sides of (4.13) as t ~ ~ , we o b t a i n a contradiction to condition (4.9). This completes the proof. |

We note that inequality (2.4) may take the following form,

x (t) >_ w It, T] (hL~_lz (t)), for all large T _> to and t _> T. (4.14)

Clearly, condition (4.9) becomes

lff l imsupw" [t,T q(s) ds > 1. (4.15) t--,oo (t)

In the case of advanced equation (1.1; 5), i.e., g(t) _> t, for t _> to, Theorem 4.2 takes the following form.

THEOREM 4.3. Let 1 < g < n - 1, (-1)'~-~5 ---- - 1 , g(t) > t, for t > to, Conditions (i)-(v), (1.3), and (4.1) hold. If, for all large T > to and t > T,

lira sup w~ It, T] q (s) ds > 1, (4.16) t---4OO

then, ~ = 0.

Next, we present the following theorem when condition (3.18) holds.


THEOREM 4.4. Let 1 < hold. Moreover, assume holds with a'(t) >__ O, for

e < n - 1, ( -1 )~- t5 = -1 , Conditions (i)-(v), (1.3), (3.18), and (4.1) that there is a function a C Cl([to, oo), JR+), such that condition (3.2) t >_ to. If, for all large T >_ to,

lim sup w [a (t), T] Q (s) + k t---+OO

w* [~ (s), T] ~' (s) Q(~+I)/~ (s) ds > 1, (4.17)

where w*[a(t),T] is as in (3.4) and

k = ~ a, when g >_ 2, [ aA, AE(O, 1), when~>_l ,

then , = O.

PROOF. Let x E Aft and assume that x(t) > 0, for t > to >_ 0. As in the proof of Theorem 4.1, there exists a t l _> to, such that (2.3) holds, for t >_ t l and the two cases,

(I1) ~>_2and

(h) e _ 1, are considered.

CASE (I1). Assume ~ _> 2. Define W(t) as in (4.3) with ~(t) = 1 and as in the proof of Theorem 4.1, we obtain (4.6) which takes the form,

W' (t) ~ - q (t) - c~#' (t) ~ [a (t), t2] W (a+1)/a (t), for t > ta. (4.18)

Integrating (4.18) from t to u k t and letting u --~ co, we find

5(L---n:-~x(--t)'~ > Q ( t ) + a ~ [ a ( s ) , t 2 ] a ' ( s ) W ( ~ + l ) / ~ ( s ) ds t > t 2 . (4.19) \ [ o ( t ) ] ) - ' -

Now, one can easily see that W (t) >__ Q (t), for t >_ t3. (4.20)

Using (4.14) with T = t2, (4.20) in (4.19), we obtain

[ ],- 1 >w[cr( t ) , t2] Q ( t ) + a ~[~(s ) , t2]a ' ( s )Q(~+l) / '~ (s ) ds

Taking lira sup of both sides of the above inequality as t --* co, we obtain a contradiction to condition (4.17).

CASE (I2). Assume g _> 1. Define W(t) as in (4.7). Then, it is easy to see that

W' (t) <_ - q (t) - Aa~a [~ (t), t2] W (~+1)/~ (t), for t > ta. (4.21)

The rest of the proof is similar to that of Case (11) above, and hence, is omitted. |

Next, we have the following comparison results.

THEOREM 4.5. Let 1 < g < n - 1, (-1)n-e(~ = -1 , Conditions (i)-(v), (1.3), and (4.1) hold and assume that there exists a function a E Cl([t0, co),]R+), such that condition (3.2) holds with a'(t) >_ O, for t >_ to. If, for all large T >_ to with a(t) > T, every solution of the retarded first-order equation,

y' (t) + q (t) wg [a (t), T l y [a (t)] = 0, (4.22)

is oscillatory, then, Af~ = O.


PROOF. Let x e Afe and suppose that x(t) > 0, for t > to _> 0. There e x i s t s a t l _> to, such that (2.3) holds, for t > tl. By Lemma 2.3(il), there exist t3,t2 >_ tl, such that a(t) > t2, for all

t > t3, and x [a (t)] > we [a (t), t2l (SLy_ix [a (t)]), for t > t3. (4.23)

Using condition (4.1) and (4.23) in equation (1.1; 6), we obtain

_ s d (L~_ lx ( t ) )~= d dt dt (SLn- lx (t) ) = q (t) f (x [g (t)]) > q (t) x ~ [a (t)]

> q (t) (t) , t:] ( t ) ] F , for t _> t3.

Set y(t) = (SLn_lX(t)) ~ > O, t > t3, we get

y' (t) + q (t) w~ [a (t), t2] y [a (t)] _< 0, for t _> t3. (4.24)

Integrating (4.24) from t > t3 to u and letting u --* c~, we have

y (t) _> q (s) w~ [a (s), ta] y [a (s)] ds, t >>_ta.

As in [23], it is easy to conclude that there exists a positive solution y(t) of the equation (4.22) with l i m t - ~ y(t) = 0, which contradicts the fact that equation (4.22) is oscillatory. This completes

the proof. 1

THEOREM 4.6. Let 1 < £ < n - 1, ( -1 )n-e5 = -1 , Conditions (i)-(v), (1.3), and (4.1) hold and assume that there exists a function a 6 Cl([t0, cc),N+), such that condition (3.2) holms. If, for 411 large T >_ to, the second-order equation,

(w*[a(t) i'T]cr'(t))" (y'(t))" + q ( t ) y" ( t ) = 0 , (4.25)

is oscillatory, where w*[a(t),T] is as in (3.4) and k is as in Theorem 4.4, then, Aft = O.

PROOF. Let x 6 Are and assume that x(t) > 0, for t > to _> 0. As in the proof of Theorem 4.4,

we consider the two cases,

(I1) g _ > 2 a n d (I2) e > 1.

CASE (I1). Let g >_ 2. Proceeding as in the proof of Theorem 4.4, Case (I1), we obtain inequality (4.18). By applying Lemma 3.1 to inequality (4.18), we conclude that the equation (4.25) is

nonoscillatory which is a contradiction.

CASE (I2). Let ~ >_ 1. Proceeding as in Theorem 4.4, Case (I2), to obtain inequality (4.21) and applying Lemma 3.1 to (4.21) to obtain the desired conclusion. This completes the proof. 1

4.2. X 0 = 0

This case occurs for equation (1.1; 5) when

(1) 5 : 1 and n is odd, and (2) 5 : - 1 and n is even.

We shall present some sufficient conditions which ensure that the subset Af0 for equation (1.1; 5)

is empty.

436

THEOREM 4.7.

then, Ho = O.

R. P. AGARWAL et aL

Let ( -1)n5 = -1 , Conditions (i)-(v), (1.3), and (4.1) hold. If

lim sup f t q (s) w~ [T (t), g (S)] ds > 1, t-.-*oo J r ( t )

(4.26)

PROOF. Let x E No and assume that x(t) > 0, for t > to _> 0. By Lemma 2.2, there exists a t l _> to, such that

( -1 ) i L~x (t) > 0 (0 _Wo[T(t),g(s)l((--1)~-lL~_lx[T(t)]), f o r t > s > t l . (4.28)

Using condition (4.1) and (4.28) in equation (1.1; 5), we find

d ((-1)'~-l L,~_lx(s)) ~ ds -- q (s) f (x [g (s)]) > q (s) x ~ [g (s)] (4.29)

( >_ q (s) w~' [r (t), g (s)l (-1) ~-1 g n - l z [r (t)l , for t > s > tl.

Integrating both sides of (4.29) from r(t) to t, we obtain

> ( ( -1) ~-1Ln_lX[r( t ) ] ) q(slw~[,(t),g(s)] ds, (t)

o r o[; ] ((-1)'~-~L,~_~x[r(t)]) q(s)w~[r(t),g(s)lds-1 <_0, for t_> tl .

(t)

However, this is inconsistent with (4.26). This completes the proof. |

REMARK 4.1. If g(t) < t, for t _> to, condition (4.26) becomes

lim sup f t q (s) w~ [g (t) , g (s)] ds > 1. (4.30) t-~oo Jg(t)

THEOREM 4.8. Let ( -1 )n5 = -1 , g(t) < t, for t >_ to, Conditions (i)-(v), (1.3), and (4.1) hold. If

(L I,~-2(u,g(t);a,,_2,. . . ,al)an_l (u) q(~) d~- du > 1, (4.31) t~oo Jg(t)

then, JV'o = O.

PROOF. Let x E JV'o and assume that x(t) > 0, for t :> to _> 0. There exists a t l >_ to, such that (4.27) holds, for all t > t~. Choose a t2 _> tl , such that inf{g(t) : t > t2} > t~. Replacing i, k, t, and s by 0, n - 1, g(t), and t, respectively, in (2.1), we can easily see that

x [g ( t ) l __ I n _ 2 (u , g ( t ) ; a ,~ -2 , • • • , a~) a ,~_~ (~) (t) (4.32)

((-1) (u)) du, for t > g(t) > t2.


Integrating equation (1.1; 5) from u to t, we get

((--1) n - 1 L , ~ _ l x ( u ) ) a > ( ( - 1 ) n - 1 L n _ l x ( t ) ) a + ~utq(~-)f (x [g (7-)])d'r, for t > u > t2,

o r

( ( - 1 ) n - i L k - i x ( u ) ) > ( ~ t q ( z ) f (x[g('r)]) dT) ~/~

Using (4.1) and (4.33) in (4.32), we obtain

(4.33)

~gt (~uu t )l/c~ x [g (t)] > x [g (t)] In-2 (u, g (t) ; a ~ - 2 , . . . , al) a~-i (u) q (T) dT du,

(t)

o r t t 1 / a

1>_ ~ In_2(u,g(t);an_2, . . . ,al)an_l (U) ( ~ q(T) d~') du. (t)

Taking l imsup of both sides of the above inequality as t ~ ~ , we obtain a contradiction to condition (4.30). This completes the proof. 1

Next, we present the following comparison result.

THEOREM 4.9. Let ( -1)n5 = -1 , g(t) < t, for t > to, Conditions (i)-(v), (1.3), and (4.1) hold and assume that there exists a function ~ E C([t0, c¢), R), such that ~'(t) > 0 and g(t) <<_ ~(t) <_ t, for all t >> to. If the first-order delay equation,

y ' ( t )+q( t )w~ [y(t),g(t)]y[~(t)] = 0 , (4.34)

is oscillatory, then, No = 0. PROOF. Let x e No and assume that x(t) > 0, for t > to > 0. There exists a t 1 > to, such that (4.27) holds, for all t > tl. Choose a t2 > tl, such tha t inf{g(t) : t > t2} > tl. Next, replacing t and s by ~(t) and g(t), respectively, in (2.14), we have

x[g(t)]>_wo[~l(t),g(t)]((-1)n-lL,~-lx[~l(t)]), f o r t > 7 1 ( t ) > g ( t ) > t 2 .

Using (4.1) and (4.35) in equation (1.1; 5), we have

-d- t ( -1 ) L~_lx(t) = q ( t ) f ( x [ g ( t ) ] ) > q ( t ) x a[g(t)]

> q(t)w~[~l(t),g(t)] ( ( - 1 ) n - 1 L n _ l x [ ~ ( t ) ] ) ~ ,

for t > ~/(t) > g (t) > t2.

(4.35)

Set u(t) =- ((-1)n-lL~_lx(t)) ~, t > t2, we have

du(t) dt

- ~ > q(t)w~[~(t),g(t)]u[~(t)], for t > v (t) > g (t) > t2.

The rest of the proof is similar to that of Theorem 4.5 and hence, is omitted.

4.3. A f t = 0

This is the case for equation (1.1; 5) when 5 -- - 1 . In this section, we shall present some sufficient conditions which ensure tha t the subset flfn for

equation (1, 1; - 1 ) is empty.

438

THEOREM 4.10.

then, Afn = ~.

R. P. AGARWAL e t al.

Let (i = -1 , Conditions (i)-(v), (1.3), and (4.1) hold. I f

[ p(t) limsup q(s)w~ [g(s) ,p(s)] ds > 1,

t --~ oo d t (4.36)

PROOF. Let x E Af,, and let x(t) > 0, for t >_ to _> 0. There exists a tl >_ to, such that (3.40) holds, for t > tl . As in the proof of Theorem 3.8, we obtain (3.41). Using (3.41) and (4.1) in equation (1.1;-1) , we have

, 4 -~- (L~ 1 x (8)) = q ($) f (X [g (8)]) > q ( S ) X a [g (S)] ds -

> q (s) wn a [g (s), p (t)] L~_lX [p (t)], for t~ < t < s q (s) w~ [g (s), p (s)] ds.

dt

Taking limsup of both sides of this inequality as t --* c~, we have a contradiction to condition (4.36). This completes the proof. |

THEOREM 4.11. Let 5 -- -1 , g(t) >__ t, for t >_ to, Conditions (i)-(v), (1.3), and (4.1) hold. I f

l imsup fg( t ) ( j u ) 1 / a In-2 (g (t), u; a l , . . . , an-z) an- t (u) q (T) dr du > 1, t - ~ oo J t

(4.38)

then, Afn = ~.

PROOF. Let x cAfn and suppose that x(t) > O, for t > to ~ 0. There exists a t1 >__ to, such that (3.40) holds, for t _> tl. As in the proof of Theorem 3.10, we obtain the inequalities (3.47) and (3.48), for u > t > T1 >_ tl. Now, using (4.1) and (3.48) in (3.47), we can easily see that

1 _> limsup [ g ( O I n _ 2 ( g ( t ) , u ; a l , . . . , a n _ 2 ) a n - 1 (u) q ( T ) tiT) 1/a d B ,

t--* c~ J t • .

which contradicts condition (4.38) and completes the proof. |

Finally, we present the following comparison result.

THEOREM 4.12. Let 5 = -1 , g(t) ~ t, fort > to, Conditions (i)-(v), (1.3), and (4.1) and assume that there exists a function ~ e C([to, c~), •), such that ~'(t) >_ 0 and g(t) >_ ~(t) >_ t, for t >_ to. If the first-order advanced equation,

y'(t) =0, (4.39)

is oscillatory, then, flfn = ~.

PROOF. Let x C Af~ and assume that x(t) > 0, for t > to >_ 0. There exists a tl _> to, such that (3.40) holds, for t >__ tl. Choose a t2 _> tl , such that inf{g(t) : t ~ t2} > tl. From (2.14) with t and s be replaced by g(t) and ~(t) respectively, we have

x [g (t)] _> w,~ [g (t), ~ (t)] L n - l x [~ (t)], for g (t) _> ~ (t) >__ t >__ t2. (4.40)


Using (4.1) and (4.40) in equation (1.1;-1), we get

d ot ~ L ~ _ l x (t) = q (t) f (x [g (t)]) >_ q (t) x ~ [g (t)]

:> q (t)wn a [g ( t ) ,~ (t)] Lan_l x [~ (t)], for g (t) >_ ~ (t) >_ t __ t2. (4.41)

Set u(t) = L~_ lx ( t ) , t >_ t2, we have

(t) - - > q (t) w~ [g (t) ~ (t)] u [~ (t)], for t > t2.

dt - ' -

However, in view of a similar result to Corollary 3.2.3 in [31], equation (4.39) has an eventually positive solution, a contradiction. This completes the proof. |

REMARK 4.1. It is known that equation (4.34) is oscillatory if

fv t 1 (4.42) lim inf q (s) w~ [7 (s), g (s)] ds > - , t-*co (t) e

while equation (4.39) is oscillatory if

/ ((t) 1 l iminf q(s)wa~ [g(s) ,~(s)] ds > - . (4.43)

t .--.* o o , I t e

REMARK 4.2. It is easy to see that appropriate combinations of the results of Sections 4.1-4.3 will imply the oscillation of equation (1.1; 5). In fact, one can easily see the following.

(1) Any of the results of Section 4.1 with l • {1, 3 , . . . , n - 1} implies the oscillation of even- order equation (1.1;1).

(2) Any of the results of Section 4.1 with g • {2, 4 , . . . , n - 1} together with appropriate results of Section 4.2 implies the oscillation of odd-order equation (1.1;1).

(3) Any of the results of Section 4.1 with g • {2, 4 , . . . , n - 2} together with suitable results from Sections 4.2 and 4.3 implies the oscillation of even-order equation (1.1;-1) .

(4) Any of the results of Section 4.1 with ~ • {1, 3 , . . . , n - 2} together with a suitable result of Section 4.3 implies the oscillation of odd-order equation (1.1;-1).

5. O S C I L L A T I O N C R I T E R I A III

In this section, we shall present some criteria for the oscillation of equation (1.1; 5) when the function f satisfies either condition (3.32), or (3.35) and

f ~ du o F < (s.1)

where F is as in (3.36).

According to the subsets of Af, for equation (1.1; 5), we shall consider the two cases,

(il) g • { 1 , 2 , . . . , n - 1} and 02) g -= 0.

5.1. ~ = ~,~ C { 1 , 2 , . . . , n - 1}

Here, we shall provide some sufficient conditions which ensure that Af~ = 0.

440 R. P. AGARWAL eg al.

Let 1 < ~ _< n - 1, ( -1)n-~6 = -1 , Conditions (i)-(v), (1.3), either (3.32) THEOREM 5.1.

or (3.35) and (5.1) hold. If, for a11 large T >_ to with g(t) >_ T,

i ¢~ q (s) f (wt [g (s), T]) = c~, (5.2) ds

then, Hi = O.

PROOF. Let x 6 Aft and assume that x(t) > 0, for t >_ to _> 0. There exists a t l > to, such that (2.3) holds, for t >_ tl . By Lemma 2.3 01), there exists a t2 _ tl , such that

x [g (t)] >_ we [g (t), tl] (6L,_tx (t)) , for t >_ t2. (5.3)

Using (3.32) or (3.35) and (5.3) in equation (1.1; 5), we have

d dt (6L~_lx (t) ) ~ = q (t) f (x [g (t)]) >_ q (t) f (w~ [g (t), tl] (6L~_lx (t)))

> q (t) f (w~ [g (t) , tl]) E ( ~ n . _ l x (t)) , for t > t~.

Substituting u(t), for (6L=_lx(t)) ~, t > t2, we have

i du ~ ~ > q ( t ) f ( w l [ g ( t ) , t l l ) F ( u l / " ( t ) ) for t > t2. (5.4) dt - ' -

Dividing both sides of (5.4) by F(ul/a(t)) and integrating from t2 to t, we have

q (s) f (w~ [g (s), tl]) ds <_ F (uX/~ (s)) J~(t) f (-~T/~)"

Letting t --~ oc, we conclude that

/ ; 1 ~(~2) du q(s) f (w~ [g(s),tl]) ds < -- J0 F ( u l / ° 0 < (x),

which contradicts condition (5.1). This completes the proof. |

REMARK 5.1. By (3.35) with ~ = 1, f (x) > f(1)F(x), for x > 0, it follows that

/0 s (x~io~ -< F (~, i< , ) ' for m > o.

Thus, (5.1) yields

S - t - ~ du

f (~1~) < o~. (5.5)

THEOREM 5.2. Let 1 < ~ < n - 1, ( -1)n-~6 = -1 , Conditions (i)-(v), (1.3), and (3.18) hold. Moreover, assume that there is a function a E Cl([to, oo), ]~+), such that condition (3.2) holds with a ' ( t ) >_ O, for t >_ to and assume that

f (x) sgn x _> Ixl ~ , for ~ # o, (5.6)

where/~ < a is a quotient of positive odd integers. If, for every constant c > O, and all large T >_ to,

c w* a' (s) Q(l+~)ln 11~ lim~_~sup Q1/~ (t) w [a (t), T] 1 + ~ [a (s), T] (s) ds = oc, (5.7)

where Q(t) > 0, fo~ t > to, ~* [Ht), T] is as i . (3A), then, ~ = ~. PROOF. Let x EAft and assume that x(t) > 0, for t > to >_ 0. There exists a t 1 _> to, such that (2.3) holds, for t _> tl. Now, there are two cases to consider,

(I1) ~ _ > 2 a n d (I~) ~ > 1.


CASE (I~). Assume ~ k 2. Define

w (t) =

Then, for t > t2, we have

5L~_~x (t) • , [~ (t)] ' t>_tl.

5L~_lx (t) , W' (t) <_ - q (t) - fla' (t) - ~ - ~ (-~)] .x [a (t)],

As in the proof of Theorem 3.1, we obtain (3.9), for t >__ t3 _> t2. Thus,

w ' (t) <_ - q (t) - ~-~ [~ (t) , t~] ,~' ( t) w ( ~+~)/" (t) ~(~-")/~ [~ (t)],

Integrating (5.8) from t k t4 to u and letting u -~ oo, we find

L'~_lX (t) ~ x ~ [(7 (t)]

x Q (t) + ~ ~ [c~ (s) , ta] ~' (s) w (~+~)/~ (s) x (~-~)/~ [a (s)] ds ,

and hence, W(t) >_ Q(t), for t _> t4. There exist a constant el > 0 and a ts _> t4, such tha t

5L,_lx (t) < el, for t > ts.

For t _> ts, it follows from (5.9) and (5.10) tha t

for t k to.

for t k t4 k t3. (5.8)

for t >__ t4,

x B/c~ [a (t)] < elQ -1/~ (t) or x [Cr (t)] _< Cl/~Q -1/~ (t) ,

(5.9)

(5.10)

and hence, x (~-~)/~ [a (t)] __k c~-~)/~Q (~-~) /~ (t) , for t >__ ts. (5.11)

Using (5.11) in (5.9) yields

aLn_lX (a-~>/a [a (t)] ~ x [a (t)]

[ / too ] 1 / , (5.12) × Q (t) + ~c~ ~-'~)/f~ ~ [a (s), t3] a' (s) Q(~+I)/~ (s) ds

Using (4.14) with T = t3 in (5.12), we have

• ( - -~) / - [~ (t)] _> ~ [o (t), t3] [ f ~ ]~/,~ (5.13) × Q (t) + ~c~ ~-~)/~ ~ [~ (s) , t~] o' (s) Q(~+I)/~ (s) ds

Again, using (5.11) in (5.13) yields

[ ]~C~fl_,:x)/fl/o ° ]l/a Taking l imsup of both sides of this inequality as t ~ c~, we obtain a contradiction to condi- t ion (5.7).


5L~_lx (t) for t > t2 w (t) = ~ [ ~ (t)]'

and proceed as in Case (11) we arrive at the desired conclusion. |


THEOREM 5.3. Let i < g < n - 1, ( - 1 ) " - t 5 = -1 , Conditions (i)-(v), (1.3), and (5.6) hold. In addition, suppose that there exists a function (7 E Cl([t0, oo),]~+), such that condition (3.2) holds. If, for every constant c > 0 and all large T :> to, the second-order half-linear equation,

(w* [(7 (t) ,T] a' (t)) ~ (y' (t))" + cq (t) y" (t) = O, (5.14)

is oscillatory, where w*[a(t), T] is as in (3.4) and

I, if t _> 2,

P = A E (0, i), i f i > i,

then, Af~ = 0.

PROOF. Let x 6 Aft and assume that x(t) > 0, for t >_ to >_ 0. As in the proof of Theorem 5.2, we consider the two cases,

(I1) ~_>2 and t > 1.

CASE (I~). Assume ~ _> 2. Proceeding as in the proof of Theorem 5.2, Case (I~), and obtain (5.8), for t > t 4. It is easy to see that there exist a t5 _> t4 and a constant cl > 0, such that

X [(7 (t)] _< ClW___0 [(7 ( t ) , t4], for all t _> t5. (5.15)


w ' (t) _< - q (t)

(~-~)1~ (~-~)1~ [a ~ [a , t4] (7' W (1+~)1~ - p c l (t) , t4] (t) (t) ( t ) , for t _> ts. (5.16)

By applying Lemma 3.1 to inequality (5.16), we conclude that the equation (5.14) is nonoscillatory, which is a contradiction.

CASE (I2) . Assume t >_ 1. Define W(t) as in the proof of Theorem 5.2, Case (I2) and proceed as in the proof of Case (I1), we arrive at the desired conclusion. This completes the proof. |

5.2. 2 5 0 = 0

This case occurs for equation (1.1; 5) when

(1) 5 = 1 and n is odd, and (2) 5 = - 1 and n is even.

We shall provide some sufficient conditions which ensure that the subset 25o for equation (1.1; 5)

is empty.

THEOREM 5.4. Let (-1)~5 -- -1 , Conditions (i)-(v), (1.3), and either (3.32) or (3.35) hold and let ~c~

- - - -+ 0 , a s u ~ o o . ( 5 . 1 7 ) F

I f

then, Afo = O.

~ t l imsup q(s) f ( w o [T(t) ,g (S)]) ds > O, t-~¢¢ (t)

(5.18)

PROOF. Let x E 25o and assume that x(t) > 0, for t >_ to >_ 0. There exists a t 1 >_ to, such that (4.27) holds, for all t _> tl. Replacing t and s by ~'(t) and g(s), respectively, in (2.14), we obtain (4.28), for t > s > tl .


Using condition (3.32) or (3.35) and (4.2s) in equation (1.1; 5), we find

d ((_1),~_i L~_lx(s)) a =q(s) f (x[g(s )] ) ds

>__ q (s) f (w0 [r (t), g (s)] ((--1) n Ln-lX [7- (t)]))

_> q (s) f (w0 [T (t), g (s)]) F ( ( -1 ) ~ Ln- lx [7- (t)]).

Integrating (5.19) from 7-(t) to t and setting u(t) = ((-1)'~L,~_lx(t)), we obtain

u~[r(t)] > q(s)f(wo[7-(t),g(s)]) ds. F (~ [7- (t)]) - (t)

Taking limsup of both sides of this inequality as t -~ e~, we obtain

0 > lira sup f t _ q (s) f (w0 [7- ( t ) ,g (s)]) ds, t-*oo Jr(t)

which contradicts condition (5.18). This completes the proof.

THEOREM 5.5. hold. If

443

(5.19)

|

Let ( -1)n6 = -1 , Conditions (i)-(v), (1.3), (5.1), and either (3.32) or (3.35)

Te q (s) f (Wo [s,g (s)]) ds = oc, (5.20 / g

then, No = @.

PROOF. Let x 6 Af0 and assume that x(t) > 0, for t > to _> 0. There exists atl > to, such

that (4.27) holds, for t >_ tl. Choose a t2 >_ tl, so large that t2 >_ tl and inf{g(t) : t >_ t2} > tl. Replacing t and s by t and g(t), respectively, in (2.14), we have

x[g(t) l>_wo[t ,g( t )]((-1)n- lL,~_lx( t ) ) , for t e 7~g N [t2, c~). (5.21)

Using (5.21), condition (3.32) or (3.35) in equation (1.1; (f) and letting u(t) = ((-1)n-lL~_lx(t)) ~ > 0, for t > t2, we have

du(t) at = q (t) S (x[g (t)]) _> q (t) f (~o [t,g (t)] u 1/" (t) )

>q(t) f (wo[ t ,g ( t ) ] )F(u l / " ( t ) ) , f o r t e U g n [ t 2 , oc ).

Choose a t3(> t2) arbitrarily. Dividing both sides of the above inequality by F(ul/~(t)) and integrating over T~g n [tz, ta], we find

f n f t l ~ u' (s) f~ ( " ) du q (s) f (wo [s,g (s)]) ds < ds = --

Letting t3 --* c~, we conclude that

fT¢. fu(tZ) du q(s) f (wo [s,g(s)]) ds < < c~, ~nIt2,o~) - . ,o F (ul/~)

which contradicts condition (5.20/. This completes the proof. |


THEOREM 5.6. Let (-1)n(~ = -1 , Conditions (i)-(v), (1.3), and (5.17) with F be replaced by f hold. H g(t) <_ t, for t >_ to and

t t i/c~

lira sup f~ In_2(u,g(t);an_2, . . . ,al)an_l (U) (~u q(T) dT) du > O, (5.22)

then, Afo = 0.

PROOF. Let x EN'o and assume that x(t) > 0, for t_> to >_ 0. There exists a t l _> to, such that (4.27) holds, for all t >_ tl. Choose a t2 _> tl so large that t2 _ tl and inf{g(t) : t >_ t2} > tl. As in the proof of Theorem 4.8, we obtain (4.32) and (4.33). Now, by using (4.33) in (4.32) yields

/: (// x[g(t)] >_ fl/~(x[g(t)]) I~-2(u ,g( t ) ;an-2 , . . . ,a l )an_l (u) q(T) dT/1/~ du. (t)

The rest of the proof is similar to that of Theorem 5.4 and hence, is omitted. |

THEOREM 5.7. Let (-1)~5 = -1 , Conditions (i)-(v), and (1.3) hold and let g(t) <_ t, for t > to, and

0 fl/a (%~-------~ < 00. (5.23)

If

f ~ al [g(s)] is) g'

(5.24) (~s ( / u S ) l / a ) I~-3 (u,g (s);an-2,. ,a2) a=-i (u) q (T) dT du ds = c~,

(s)

then, No = 0.

PROOF. Let x E N0 and let x(t) > 0, for t > to _> 0. There exists a tl _> to, such that (4.27) holds, for t > tl . Choose a t 2 _> tl so large that t2 _> tl and inf{g(t) : t _> t2} > tl. Next, replacing i, k, t, and s by 1, n - 1, g(t), and t, respectively in (2.2), one can easily find

xt[g(t)] ~ax[g(t)] In-3(u,g( t ) ;an-2, . . . ,a2)an- l (U) (-1)n-lLn_lX(U) du. (t)

As in the proof of Theorem 4.8, we obtain (4.33). Thus,

t t 1/a x' [g (t)] g' ( t )> ax [g (t)] g' (t)fg In-3 (u,g(t);an-2,..., a2)an-1 ( u ) ( f q ( r ) d ' r ) du. :i/~ (~ [g (t)]) - (~)

The rest of the proof is similar to that of Theorem 5.5 and hence, is omitted. |

Finally, we present the following comparison result.

THEOREM 5.8. Let (-1)'~6 ---- -1, g(t) < t, for t >_ to, Conditions (i)-(v), (1.3), and either (3.32) or (3.35) hold. Moreover, assume that there exists a function r/ E C([t0, oo),N), such that r/(t) >_ 0 and g(t) <_ ~(t) <_ t, for all t >_ to. If the first-order delay equation,

y' (t) + q ( t ) : (wo [,7 (t), g (t)]) F (y ' / ° [,7 (t)]) = o, (5.25)

is oscillatory, then, A[o = O.

PROOF. Let x E A/o and let x(t) > 0, for t _> to _> 0. As in the proof of Theorem 4.9, we obtain (4.35). Next, by using condition (3.32) or (3.35) and (4.35) in equation (1.1; 5), we have

d ( ( _ l ) n _ 1Ln_ lx ( t ) ) ~ = q (t) f (x [g (t)]) dt (5.26)

> q (t) f (wo [7 (t), g (t)]) F ( ( - 1 ) ~-1 nn- l x [~? (t)]) ,


for t > y(t) > g(t) > t2. Set u(t) = ( ( -1 )n - lLn_ lx ( t ) )% t >_ t2. Then, (5.26) takes the form,

_du(t) >q(t)f(wo[rl(t),g(t)])F(ul/=[rl(t)] ) for t > t2. dt - ' -

The rest of the proof is similar to that of Theorem 4.5, and hence, is omitted. This completes the proof. I I

REMARK 5.1. As before, appropriate combinations of the results of Sections 5.1 and 5.2 will imply the oscillation of equation (1.1; 6).

Now, we have the following.

(1) Any of the results of Section 5.1 with £ E {1, 3 , . . . , n - 1} implies the oscillation of even- order equation (1.1; 1).

(2) Any of the results of Section 5.1 with £ E {2, 4 , . . . , n - 1} together with appropriate results of Section 5.2 implies the oscillation of odd-order equation (1.1; 1).

(3) Any of the results of Section 5.1 with £ E {1, 3 , . . . , n - 2} implies that any solution x(t) of odd-order equation (1.1;-1) belongs to the subset O UAfn.

(4) Any of the results of Section 5.1 with ~ E {2, 4 , . . . , n - 2} together with appropriate ones of Section 5.2 implies that any solution x(t) of even-order equation (1.1;-1) belongs to the subset O U Afn.

6. L I N E A R I Z A T I O N OF N O N L I N E A R O S C I L L A T I O N R E S U L T S

By combining Theorems 3.6, 4.6, and 5.3, we obtain the following result.

THEOREM 6.1. Let 1 < £ < n - 1, (-1)'~-~6 = -1 , Conditions (i)-(v), and (1.3) hold and

f (x) sgn x > [ x f , for x ~ 0, (6.1)

where/3 is the quotient of positive odd integers. Moreover, suppose that there exists a function a E Cl([t0, oo),R+), such that condition (3.2) holds. If, for all large T >_ to >_ O, the half-linear differential equation,

(t), T] (t)) @' + q (t) it) ; o, (6.2)

is oscillatory, where

cl, cl > 0 is any constant, when/3 > a,

r (t) = (a/k) ~ , k is as in Theorem 4.4, when fl = a,

c2_~ -~ [va (t), T], c2 > 0 is any constant, v is as in Theorem 5.3, when/3 < a,

then, Aft = 0.

Next, we shall proceed further and obtain similar results as above, but instead of comparing with oscillation of half-linear equation (6.2), we shall compare with the oscillation of certain second-order linear ordinary differential equations.

To obtain our next results, we shall need the following lemma given in [34].

LEMMA 6.1. Let r 6 C([T,c~),R+), T >_ to. I f there exhsts a function w E CI([T,c~),R), such that

q (t) _< - w ' (t) - r (t) w 2 (t), for every t _> T,

then, the second-order linear ordinary differential equation,

(x ' + q (t) x (t) = o, (t) r ( t ) )

is nonoscillatory.

Now, we present the following theorem.

446 R .P . AGARWAL e t al.

THEOREM 6.2. Let a > 1, 1 < £ < n - 1, ( -1 )~- t5 = -1 , Conditions (i)-(v), (1.3), and (6.1) hold. Moreover, suppose there exists a function a E Cl([to, ce) ,R+), such that condition (3.2) holds. Then, Aft = 0, if there exists a constant A E (0, 1) when g >_ 1 and, for a// large T > to _> 0, the second-order linear differential equation,


i !

(63)

~t(t) =Aa ' ( t )w*[or ( t ) ,T]w~- l [a ( t ) ,T] , a(t) > T,

w*[a(t),T] is as in Theorem 3.1 and A E (0,1), i[£ >_ 1, A = 1, iY£ ~ 2 and

el,

(t) : 1,

c2_~ -~ [Aa (t), T],

cl > 0 is any constant, when fl > a,

when fl = a,

c2 > 0 is any constant, when/3 < a.

PROOF. that (2.3) holds. Now, we consider the two cases,

(il) l _ > l a n d (i~) t ___ 2.


5Lg_lx (t) w (t) - x~ [ ~ ( t ) ] '

Then, for t _> t l , we have

Let x E Af~ and assume that x(t) > 0, for t > to >_ 0.

for t ~ tl and some A E (0, 1).

~' [~or (t)] xB-1 [~or (t)]. w ' (t) ~_ - q (t) - A~or' (t) w 2 (t) - 5 L ~ _ l x (t)

There exists a t l ~ to, such

x [Aor (t)] > w [or ( t ) , All ( S L n - l X (t)).

Using (6.5) and (6.6) in (6.4), we have

W ! (t) __~ - q (t) - ~flort (t) wA [ °r ( t ) , All w (~-1 [(7 ( t ) , t l ] W 2 (t) x f~-a [~or (t)] ,

or w ' (t) _< - q (t) - ~ a (t) w 2 (t) x ~ - " [~or (t)], for t >_ t2.

As in the proof of Theorem 3.6 when 13 > a and Theorem 5.3 when ~ < a, one can easily see

that

W' (t) _< - q (t) - f l ~ W 2 (t), for t _> t2. (6.7)

Applying Lemma 6.1 to inequality we conclude that equation (6.3) is nonoscillatory, which is a

contradiction.

CASE (I2). Assume g _> 2. Here, we define W(t) as in Case (iz) with A = 1. The proof will be the same, and hence, we omit the details. |

When a E (0, 1], we shall present the following criterion.

and (6.6)

for t > t2,

(6.a)

From Lemma 2.3, it follows that there exist a constant A E (0, 1) and a t2 > t l /A, such that

x' [Ao" (t)] ~ WX [o" (t), All (SLn-lx (t)) (6.5)


THEOREM 6.3. Le t0 < a _< 1, 1 < ~ < n - l , ( - 1 ) ~ - t 5 ---- - 1 , Conditions (i)-(v), (1.3), and (6.1) hold, Q(t) > 0, for t >_ to. Moreover, assume that there exists a function a(t) E Cl([t0,(:x)), ~+), such that (3.2) holds. Then, Af~ = 0 is there exists a constant A 6 (0, 1) when £ > 1 and, for all large T > to > O, the second-order linear equation,

(t) (t) + (t) y (t) = o, (6.8)


~x (t) ---- Aw* [a (t), T] Q(1-a)/~ (t) a ' (t), a (t) > T,

w*[a(t),T] is defined as in Theorem 3.1 and A C (0,1) if£ _> 1, A = 1 ifg > 2 and

Cl, cl > 0 is any constant, when fl > a,

r l (t) = 1, when j3 = a,

c2w_~ (~-~)/~ [Aa (t), T], c2 > 0 is any constant, when ~ < a.

PROOF. Let x E Af~ and assume that x(t) > 0, for t > to > 0. As in the proof of Theorem 6.2, we obtain (6.4) which takes the form,

W' (t) < - q (t) - A~a' (t) @:~ [a (t), tl] W 2 (t) W (1-a)/a (t) x (~-'~)1~ [Aa (t)], t > tl. (6.9)

As in the proof of Theorem 3.3, we have

W (t) _> Q (t) > 0, for t __ tl. (6.10)


W' (t) _~ - q (t) - After' (t) @x [~r (t), tl] Q(1 - , ) / , (t) W 2 (t) x (~-a)/`~ [Aa (t)], t > tl.


Next, we shall provide proof of Theorem 4.1 when a > 1 and a C (0, 1] without employing Lemma 4.1.

THEOREM 6.4. Let a > 1 and the hypotheses of Theorem 4.1 hold and condition (4.2) is replaced by

l i m s u p f t [ ~ ( 1 ) (~'(s))2 1 ] (s) q (s) - . = (6.11) J r a ( s ) d s

where ~t(t) is defined as in Theorem 6.2. Then, Aft = 0.

PROOF. Let x E Af~ and assume that x(t) > 0, for t _ to _> 0. Define

5Lan_l x (t) W (t) = ~ (t) x [Aa (t)] ' for t >_ t~ > to.

Proceeding as in the proof of Theorems 4.1 and 6.2, one can easily find

. . W'(t) <_ -~(t)q(t) + - ~ W ( t ) - aA ~[cr( t ) , t l ]w"- l [a( t ) , t l ]W2( t ) ,

for t > t2 >_ T/A, for some A E (0, 1), or

W' (t) _< - ~ (t) q (t) + ~ - - ~ W (~) - E - ~ t (t) (t)

2

[ 1 ] f°rt>-t . < - ~(t) q(t) 4a ( ( t ) a ( t ) '

The rest of the proof is similar to that of previous results, and hence, is omitted. |

448

THEOREM 6.5.

tion (4.2) is replaced by

R. P. AGAR,WAL et al.

Let 0 < a _< 1 and the hypotheses of Theorem 4.1 hold, Q(t) > O, and condi-

*[ 1 (~'(s))2 1 limsupt_~oo JT f ~(s) q ( s ) - 4a ~-(~) ~;l'(s) ds = oo, (6.12)

where f~l(t) is defined as in Theorem 6.3. Then, Aft = 0. PROOF. Let x e Ale and assume that x(t) > 0, for t > to >_ 0. Define the function W(t) as in the proof of Theorem 6.4 and proceed as in the proof of Theorems 4.1 and 6.3, to obtain

~' (t) w' (t) < -~ (t) q (t) + T s ~ w (t)

- ~A~' (t) ~ [~ (t), tl] ¢-1/~ (t) w ~ (t) w (1-~)/" (t), for t > t~

and w (t) > ~ (t) Q (t), for t > t2.


7. O S C I L L A T I O N C R I T E R I A

From the previous results, we shall state as a sample some oscillation criteria for equation (1.1; 5). The formulations and statements of other results are left to the reader.

THEOREM 7.1. Let n be even and assume that (i)-(v) and (1.3) hold. Moreover, assume that there exists a function ~ E Cl([ t0 ,oo) ,R+), such that (3.2) holds.

(Sl). Suppose that

f 4-~ du f~/~ (~--------~ < o0

and there exists a function ~ E Cl([t0, co), R +) and a constant A E (0, 1), such that for all large T _> to >_ 0,

( )' (~' (t))l/a _ O, for t > cr (t) > T/A. ~' (t) > 0 and ~;~ [a (t) , T] a ' (t)

Then, the condition,

/ ~ ~ (s) q (s) = co, ds

is sufficient for equation (1.1;1) to be oscillatory.

(s2). Suppose that f(x)sgn x >__ Ixl", for x # O. If there exist a function ~(t) E Ct([t0, co), R +) and a constant A E (0, 1), such that for all large T > to with a(t) > T/A, either

r ~-~ (," (s)F +: f lim sup ] [~ (s) q (s) ds = co, t--*c~ JT (a-~- 1) a+l (~(S) O't(S)~X [a(s),T])"

or, a > 1 and

/:I 1 limsuPt__.~ ~(s)q(s) 4aA ~(s) a ' (s)~x[a(s) ,Tlw~-l[a(s) ,T] ds=co,

or, a < 1 and

limsupt~ Jr [ ( ( s ) q ( s ) - ~ ~,. -~-~ ] ~'(s)~ , [~(s) ,T]Q(l-~) /~(s)

where Q(t) > O, then equation (1.1;1) is oscillatory.


(S3). Suppose that either (3.32) or (3.35) holds and

f± du o F (~1/~) < oo,

where F is as in (3.36). Then, for all large T _> to, the condition,

f oo q (s) (w[g (s), T]) 0% f ds


(s4). Suppose that a > 1 and f(x)sgn x > [xlZ, for x # 0, where/~ is the quotient of positive odd integers. Equation (1.1;1) is oscillatory if there exists a constant A E (0, 1), such that for all large T _> to _> 0, the second-order linear equation,

r(t ) ha ! (t)W~ [a (t), T] w ~-1 [a (t), T]

is oscillatory, where r(t) is as in Theorem 6.2.

!

y' (t) + Zq (t) y (t) = 0,

(sh). Suppose that 0 < c~ < 1 and f(x)sgn x > Ix[ ~, for x ¢ 0, where fl is the quotient of positive odd integers. Equation (1.1;1) is oscillatory if there exists a constant A 6 (0, 1), such that for all large T _> to _> 0, the second-order linear equation,

( r l( t) ( t )y , ( t ) ) ' ~ , (t) : ~ [~ (t), T] Q(:-~) /~ + ~q (t) y (t) = 0,

is oscillatory, where r l ( t ) is as in Theorem 6.3 and Q(t) = f~o q(s)ds > O.

THEOREM 7.2. Let n be odd and assume that (i)-(v) and (1.3) hold. Moreover, assume that there exists a function a 6 Cl([t0, oo), R+), such that (3.2) holds.

(ol) . Suppose that g(t) <_ t, for t > to, and

/: o~ du o f l / ~ (~----~ < oo.

Then, for all large T >_ to with a(t) >_ T, the conditions,

/ ~ [o(s ) ,T] (s) (s) = o o , G ! Q1/~ ds

where

io o (~. (~s >l/a ) al [g (s)] g' (s) fn-3 (u, g (s) ;an-2, . . . , a2) an-1 (u) q (T) dT du ds = c~

(~)

are sufficient for equation (1.1;1) to be oscillatory.

(02). Suppose that f (x)sgn x _> Ix[ ~, for x ¢ 0. Then, for all large T > to with a(t) > T, the conditions,

[ ].° lim sup w [a (t), T] Q (t) + a ~ [a (s), T] a' (s) Q(l+a)/~ (s) ds > 1,

$---~OO

where // lim sup q (s) w~ [~- (t), g (~)1 as > 1 t-~oo (t)

axe sufficient for equation (1.1;1) to be oscillatory.


(03). Suppose that either (3.32) or (3.35) holds and

du o F (~/") < o0,

where F is as in (3.36). Then, for all large T > to with g(t) > T, the conditions,

oo q (s) f (w[g (s), T]) ds = oo

and

limsup fl ~ q(s) f (wo [7( t ) ,g(s)] ) ds > 0, t ---*oo J r ( t )


(04). Suppose that f (x)sgn x >_ Ix[ ~, for x ¢ 0, where fl is the quotient of positive odd integers. Equation (1.1;1) is oscillatory if, for all large T >_ to with a(t) > T, the half-linear second-order equation, ( )' (t) (y' (t))"

[~ (t), T] ~' (t) + q (t) y" (t) = 0,


r ( t ) = / 1, w h e n f l = a , [ c*--/3

c2w__ 0 [a (t) , T], c2 > 0 is any constant, when B < a,

and

~ t lira sup q (s) w~ [T (t), g (S)] ds > 1, when fl = a, t-~oo (0

// lira sup q (s) wg [r (t), g (s)] ds > 0, when fl < a. t -~oo (t)

(05). Suppose that a >__ 1 and f(x)sgn x > Ix] z, for x # O, where ~ is the quotient of positive odd integers. Equation (1.1;1) is oscillatory if, for all large T > to with a(t) > T, the linear second-order equation,

( )' r (t) y, ~[o( t ) ,T]w-- l [o( t ) ,T]o , ( t ) (t) + Z q ( t ) y ( t ) =o ,

is oscillatory, where r(t) is as in (04) and

t

lim sup q (s) w~ [r (t), g (s)] ds > 1, t--*oo J r ( t )

limsup f t q(s)wg[r(t) ,g(s)J ds > O, t -~oo J r ( t )

when ~3 = a,

when ~ < a.

THEOREM 7.3. Let n be odd and assume that (i)-(v) and (1.3) hold. Moreover, assume that there exists a function a e Cl([t0, cx~), R+), such that (3.2) holds.

(31). Suppose that

f ±oo du

f ~ l . (~---~ < ~ '


and there exist a function ~ E Cl([t0, oo), R +) and a constant A E (0, 1), such that, for all large T _> to with ~(t) > T,

~' (t) _> O, for t _> to, and N~ [a (s), T] a' (s)

Then, the conditions,

/ o~ ~ (s) q (s) = cx~ ds

and

L (/ al[g(s)]g'(s)~g(')In_a(g(s),u;a2,.. . ,a~_2)an_l(U) q('r) d ' r )V~duds=oo,

are sufficient for equation (1.1;-1) to be oscillatory.

(J2). Suppose that f (x)sgn x > Ixl a, for x ¢ O. Equation (1.1;-1) is oscillatory if, for all large T > to with or(t) > T, the first-order delay equation,

y' (t) + q (t) w" [o (t) , T] y [0 (t)] = 0,

is oscillatory and p(O

lim sup q (s) w n [g (s), p (t)] ds > 1. t - -+ o o , I t

(J3). Suppose that either (3.32) or (3.35) holds and

/ +~ du ~o F (~1/.) < o~,

where F is as in (3.36). Then, for all large T >_ to with g(t) > T, the conditions,

/ ~ q ( s ) (w [g(s), oo, f T]) ds

/.4 q(s) f (w,~ [g(s),s]) ds = oo, 9


(J4). Suppose that f (x)sgn x > ]x] ~, for x ~ 0, where fl is the quotient of positive odd integers. Equation (1.1;-1) is oscillatory if, for all large T >_ to with a(t) > T, the half-linear equation,

r( t ) ( ~ [a (t), T] o' (t)


I CI, r ( t ) = I,

and

I

(y' (t))" + q (t) y" (t) = 0, for some A E (0, 1),

cl > 0 is any constant, when ~ > a,

when fl = a,

f p(t)

l imsup q (s) w~ [g (s), s] ds > O, t'---~ O0 J t

p(t) lira sup q (s) w x [g (s), 0 (t)] as > 1,

t - .* o o , I t

when ~ > a,

when ~ = a.


(is). Suppose that 0 < a < 1 and f(x)sgn x >_ Ix[ ~, for x ¢ 0, where f~ is the quotient of positive odd integers. Equation (1.1;-1) is oscillatory if, for all large T _> to with a(t) > T, the second-order equation,

~ (t) ) ' A ~ [cr (t), T] o' (t) Q(1-a)/a (t) y' (t) if- ~q (t) y (t) = O, for some )~ E (0, 1),

is oscillatory, where r(t) is as in 04), Q(t) = f t q(s) ds > 0 and

limsup fP(t) q(s)w~[g(s),s] ds > O, t - * c ~ J t

p(t) lim sup q (s) w~ [g (s), p (t)] ds > 1,

t --* o o , I t

when/3 > ~,

when j3 = a.

THEOREM 7.4. Let n be even, assume that (i)-(v) and (1.3) hold. Moreover, assume that there exists a function 0 E Cl([t0, oo), R+), such that (3.2) holds.

(M1). Suppose that either (3.32) or (3.35) holds and

~oo du

o F (~1/ . ) < o~,

where F is as in (3.36). Then, for all large T > to with a(t) > T, the conditions,

ff ff [o (s), T] 0' (s) QI/~ (s) ds = c¢, Q (t) = q (s) ds,

f q (s) I ( ~ . [g(s) ,s]) d8 = ~ , g

and

limsup - [ t q(s ) f (wo [~'(t),g(s)]) ds > 0, t--+oo JT(t)


(M2). Suppose that f (x)sgn x > [xl", for x ¢ 0. Then, for all large T > to with a(t) > T, the conditions,

l imsupw[o(t) ,T] Q ( t ) + a ~[o(s) ,T]o'(s)Q(l+a)/~(s) ds t --+ O o

> 1,

where t

lira sup q i s) w~ [~ (t), g (s)] ds > 1 , -~oo (t)

and

f p(t)

~m sup q (s) w~ [g (s), p (t)] ds > 1 t - ~ o o J t

are sufficient for equation (1.1;--1) to be oscillatory.


~ oo du o P (~1/~) < oo,


where F is as in (3.36). Then, for all large T > toe the conditions,

/ oo q (s) f (w [g (s), T]) = c~, ds

lim sup f t q (s) f (Wo IT (t), g (S)]) ds = oo, t--,oo d~'(t)

and

453

is oscillatory,

and


and

(Ms). if, for all large T > to with (7(t) > T, the second-order equation,

(t)y ( ) ) ' 1 ' t +aq( t ) y ( t ) = o ,

[(7 (t) , T] (7' (t) Q(I--)/ .

f t lim sup q (s) w~ [7- i t ) , g (s)] ds > 1 t--*oo (t)

f p(t)

lira sup q (s) w~ [g (s), p (t)] ds > 1. t - - ~ J t

These theorems seem to be new even when specialized to the equation,

x ("-1) (t) + ~q (t) f (x [g (t)]) = 0, dt

i t

lim sup q (s) w~ IT (t), g (S)] ds > 1 t---~oo J r ( t )

f p(t)

lim sup q (s) w,~ [g (s), p (t)] ds > 1. t--coo .It

Suppose tha t 0 < a _< 1 and f (x)sgn x > Ixl a, for x # O. Equat ion (1 .1 ; -1) is oscillatory

(7.1; 5)

f q(s) f ( w n [g(s),s]) ds = 00 , g

are sufficient for equation (1 .1; -1) to be oscillatory.

(Ma). Suppose tha t c~ _> 1 and f (x)sgn x ~ Ixl ~, for x # O. Equat ion (1 .1 ; -1) is oscillatory if, for all large T > to ~ O, the second-order equation,

( )' 1 y~ [(7 (t) , T] W a - 1 [(7 it), T] (7' (t) (t) + aq (t) y (t) = O,

454 R .P . AGARWAL et al.

for which Conditions (ii)-(v) are satisfied. noting that in this case, for T >_ to and all t > T,

(t - T) '~-1 wt [t,T] = ( n - 1 ) ( t - 1)! ( n - t - 1)!'

(t - T) n-2 w--t [t,T] = (n - 2) (e - 2)! (n - e - 1)!'

~ [t, T] = [A (1 - A)] '~-2 t '~-2 (~ - 2 ) ! '

Atn -2 [t, s] = (,~ _ ~)!,

Atn-1 w[a( t ) ,T] = ( n - 1)!'

WO [ t ,S] -~ Wn [t,s] ---- ( t - - s ) n -1 (~ - 1)! '

~' ( t ) > o

Then, the condition,

So, we state some of them below as corollaries, by

l < l < n - 1 ,

2 < l < n - 1 ,

A C (0, 1), for all large t > T/A,

for some A E (0, 1), and all large t _> T,

for some A E (0, 1), and all large t >_ T,

t > s > T .

S OO ~ ( s ) q ( s ) = ~¢, ds


(s2). Suppose that f(x)sgn x >_ Ixl", for x 7~ 0. If there exists a function ~(t) E Cl([to, e~),R +) and a constant A E (0, 1), such that, for all large T _> to and t _> T either

((n - 2)!)" A-" (~' (s)) ~+1 1 ] (a + 1) ~+1 (A (1 - A)) ('~-2)~ (~ (s) (a '~-2 (s) a' (s)) ~] ds = ec

l i m sup f t [ ~ ( s ) q ( s ) - ( n - 2), ( ( n - 1)[) ~ - ' ( (~ ' (s))2~l t--+~ .ST 4aA<~ (A ( i - A)) n -2(~( '~ - ' i y~-~- - - -~(s )~ ' ' (s ) ~t ~ ' ~ , ] J d s = o o

or, 0 < a < _ l and

~T~'[ (n-- 2)! O(a-Za__ !S__) ] l ip_+sup ~(s)q(s) -4aA(~Tf -_- -~) )n_2a.~_2(s)a , (s ) j d s = o ¢ ,

Q(t) > 0, then, equation (7.1;1) is oscillatory.

(s3). Suppose that either (3.32), or (3.35) holds and

i+ du o F (ull ") < e~,

lim sup ~ (s) q (s) - t---*oo T

or, a > 1 and

COROLLARY 7.1. Let n be even and assume that (ii)-(v) hold. Moreover, assume that there exists a function cr e Cl([to, oo),R+), such that (3.2) holds.

(Sl). Suppose that

S +~ du

S ~1~ @------7 < ~

and there exists a function ~(t) E Cl([t0, ec), R+), such that

and ~ a , _ 2 ( t ) a , ( t ) ) _<0, f o r t > T , for some T _> t0.

Functional Differential Equat ions 455

where F is as in (3.36). Then, the condition,

f =q(~) f (g"-~ (8)) ~s= ~¢,


(s4). Suppose that ~ > 1 and f(x)sgn x > Ixl ~, for x ¢ 0, where/3 is the quotient of positive odd integers. Equation (7.1;1) is oscillatory if there exists a constant A ~ (0, 1), such that for all large t > to, the second-order linear equation,

(~.-_?),_o ~(t) ~.-~ (1 - ~) -~ ~(~-~).+~-2 (t) ~' (t)

y' (t))' + / 3 q (t) u (t) = 0,


el '

r (t) = 1,

c2a(n-1)(~-~) (t) ,

cI > 0 is any constant, when/3 > a,

when/3 = a,

c2 > 0 is any constant, when/3 < a.

(ss). Suppose that 0 < a < 1 and f (x)sgn x > Ixl~, for x # 0, where/3 is the quotient of positive odd integers. Equation (7.1;1) is oscillatory if there exists a constant A ~ (0, 1), such that for all large t > to, the linear equation,

y' +/3q (t) y (t) ---- O, (~ 2)! Q(a-1)/a (t) r l (t)

(t]~' A ~-1 (1 - A) '~-2 a n-2 (t) c ' (t) " " ]


Cl '

r l (t) = 1,

C2a (n-1)(a-jg)/a (t) ,

cl > 0 is any constant, when/3 > a,

when/3 = a,

c2 > 0 is any constant, when/3 < ci.

COROLLARY 7.2. Let n be odd and assume that (ii)-(v) hold. exists a function a C cl([ to , oe), R+), such that (3.2) holds.

(oi). Suppose that g(t) <_ t, for t >_ to and

S± ~ du

o p i . (~-------5 < ~"

Moreover, assume that there

Then, for all large t _> to, the condition,

i ~ (s) (s) (s) = o o , o.n--2 G ! Q1/. ds

where

i (s; (L) ) g' (') (~ _ g ( , ) ) . . - ~ ~/<. (.) V m V ) ' q ( r ) d r d~ d8 = ~¢


(02). Suppose that f (x)sgn x _> Ix[ ~, for x ~ 0. Then, for some constant A E (0, 1) and all large t, the conditions,

[ /- ] aA a n-2 a' Q( .+I)I . 11o~ lim._+s~p c, " -~ (t) Q (t) + (n - 2)------'7. (s) (s) (s) ds > (n - 1)!,


where /; l imsup [~'(t)-g(s)](n-D~q(s) ds > ( ( n - 1)!) ~ t--*oo (t)



f~ du o F (~1 . ) < ~ '

where F is as in (3.36). Then, for all large t, the conditions,

7 ~ q(8) f (gn--1 (S)) US = O0

and

/' ( ) l imsup q(s) f [r (t) -- g (s)] n-1 ds>O,


(04). Suppose that a _> 1 and f(x)sgn x > [xl #, for x ¢ 0, where fl is the quotient of positive odd integers. Equation (7.1;1) is oscillatory if, for some constant ), E (0, 1) and all large t, the equation,

/ / (n - 2)! ((n - 1)!)"-1 r (t) , , . , '~' --,:-r y t~) t -~-~--~-~+-~--~(t)a i t ) ) + n q ( t l Y ( t ) = O ,


[ 1, when fl = a, (t) r

t ca(n-1)(~-#) (t), c > 0 is any constant, when fl < a,

and

f t

l imsup IT (t) -- g (S)] (n-1)~ q (S) ds > ((n - 1)!) ~ , Jr(t)

l imsup f ' [r( t) -g(s)](n-1)Zq(s) ds > 0, t--*oo Jr(t)

when fl = a,

when fl < a.

COROLLARY 7.3. Let n be odd and assume that Oi)-(v) hold. exists a function a 6 Cl([to, c~), R+), such that (3.2) holds.

(J1)- Suppose that

f +~ du yl i a (u-------~ < c~,

and that there exists a function ~(t) E Cl([to, c~),R+), such that


~ ' ( t ) > 0 , f o r t > t o , and F ( t.<,:-, (s)<.. ( , ) ) i Then, the conditions,

f ~ ¢ (s) q (s) ds = ~¢


and

L, J

ids) d (4

(g (s) - thy3

s (n-3)! (p) d7)l/a duds=o%

are sufficient for equation (7.1; -1) to be oscillatory.

(Jz). Suppose that f(z)sgn 2 2 ]zla, for 2 # 0. Equation (7.1; -1) is oscillatory if, for all large t and some constant X E (0, l), the first-order equation,

( 1 (I

Y’ (t> + (n ! l)! d---l (9 4 w Y b WI = 07

is oscillatory, and

J

P(t) lim t--r00 t

[g(s) - p (t)]‘“-l’” q (s) ds > ((n - l)!)” .


J fm du rto F (da) < O”’


s O” q(s) f (g’+-l (s)) ds = oo

and

are sufficient for equation (7.1; -1) to be oscillatory.

(J4), Suppose that 0 < o 5 1 and

f (~1 w 2 2 14 P , for 2 # 0,

where p is the quotient of positive odd integers. Equation (7.1; -1) is oscillatory if, for some constant X E (0,l) and all large t, the equation,

(n - 2)! r (t) Q(a-1)/a (t)

x [A (1 - A)]“-” P--2 (t) o’ (t) y’(t) ‘+Pdt)Y(t) =o,


r (t) = c, c > 0 is any constant, when ,!9 > o,

1, when ,L3 = CY,

Q(t) = St” q(s) ds > 0 and

s

p(t) lim sup [g (s) - s]@-l) q (s) ds > 0, when p > cx,

t+cc t P(t) lim sup J [g (s) - p(t)]‘“-“” q (s) ds > ((n - l)!)” , when 0 = (Y.

t--Km t

458 R, P. AGARWAL et al.

COROLLARY 7.4. Let n be even and assume that (ii)-(iv) hold. exists a function cre Cl([to, oo),R+), such that (3.2) holds.

(M1). Suppose tha t either (3.32) or (3.35) holds and

f k~ du o F(u 1/~) < oo,


and


~ - 2 (s) a' (s) Q1/~ (s) ds = oo, Q (t) = q (s) ds,

/A q ( s ) f ( [ g ( s ) - s ] ~ - l ) d s = c % g

/: ( ) l imsup q (s) f [r (t) - g (s)] "-1 ds > O, t--~oo (t)


(M2). Suppose tha t f (x)sgn x > ]xl ~, for x ~ 0. Then, for some constant ), E (0, 1) and all large t, the conditions,

limsupt~ ~----1~ Q ( t ) + ~ (s) (s) (s) ds > 1,

where

and

l imsup ?/t [ r ( t ) - g ( s ) ] ( n - i ) a q ( s ) ds > ( ( n - 1)!) ~ t--*oo Jr( t )

l imsup fp(t) [g (s) - p (t)] ('~-1)'~ q (s) ds > ((n - 1)!) '~ , t--+ oo J t


and

f oo q (S) f (gn--1 iS ) ) ds = ~ ,

l imsup q(s) f [r(t)-g(s)] ~-~ d s = ~ , t ~ o o (t)

£qls): sl °-:) ds _-_


f oo du

o F(ul/~)




(Md). Suppose that cy 2 1 and f(z)sgn z 1 ]x]~, for z # 0. Equation (7.1; -1) is oscillatory if, for some constant X E (0,l) and all large t, the equation,

( 1

Y’ (4 ‘+ >

XOcY &-lb++2 (t) cl’ (t) (n - 2)! ((n - q!)“-1 q WY @) = Ov

is oscillatory,

lim sup s

t (7(t) -g (s)]+‘I” q (s) ds > ((n - 1))” t-K%3 r(t)

s P(t) lim sup [g(s) - p (t)]‘“-l’o q (s) ds > ((n - l)!)O . t-+co t

(MS). Suppose that 0 < o < 1 and f(z)sgn z 2 ]z]~, f or z # 0. Equation (7.1; -1) is oscillatory if, for some constant X E (0,l) and all large t, the equation,

( y&--2 (t) CT’ (tf Q(l-a),cx @)Y (t) >

I + $Y2)! Q (t) y (9 = O,


lim sup J

t [I -g (s)](n-l)a q (s) ds > ((n - I)!)” t-em T(t)

and

J p(t)

lim sup [g (s) - p (t)]‘“-l’* q (s) ds > ((n - l)‘)Q * . t--too t

8. GENERAL REMARKS

1. The results of this paper are presented in a form which is essentially new and are of a higher degree of generality. In fact, one can easily extract more criteria than those presented, for the oscillation of equation (1.1; S) and/or related equations as well as some special cases of equation (1.1; 6) and also for the mixed equations of the form,

Lx (t) + 6 2 qj w fj (x 1% @)I> = 0, j=l

(8.1;6)

where L, is as in equation (1.1;6), qj E C([te,oo),R+), gj E C((to,co),R), and fj E C(n,a),zfj(z)>O,fj’(~)LO,forz#Oandlimt,,gj(t)=~,j=1,2,.,.,m.

The formulation of such results are left to the reader. 2. The results of this paper can be extended to equations of type (1.1; S) where f need not be

a monotonic function. As in [26, Lemma 31, the function f has the following decomposition (f is locally of bounded variation on [a, b] C R): f(x) = G(x)H(x), where G : (-oo, --cro]U

[Qo,cQ> = RI, -+ Iw+, o. 2 0, nondecreasing on (-co, -QO] and nonincreasing on [WI, co) and H : B,, + R and nondecreasing on IF&.

3. The results of this paper can be extended to forced equations of the form,

Lx (t> + h (t) f (x 19 WI> = e 0) , (8.2;6)

where e E C([te, oo), IR) as well as neutral functional differential equations of the type

Ln (x (t> + c (t> x [h (a> + &I w f (x 19 (a> = 07 (8.3;6)

where c and h E C([tc, oo),W).

460 R .P . AGARWAL et al.

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Oscillation of functional differential equations