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Oscillation of second-order strongly superlinear and strongly sublinear

dynamic equations

Said R. Grace

a

, Ravi P. Agarwal

b

, Martin Bohner

c,*,1

, Donal O’Regan

d a

Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza 12221, Egypt

b

Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA

cDepartment of Economics and Finance, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA dDepartment of Mathematics, National University of Ireland, Galway, Ireland

a r t i c l e

i n f o

Article history:

Received 21 October 2008 Accepted 9 January 2009 Available online 19 January 2009 AMS Subject Classification: 34C10 39A10 PACS: 02.30.Hq 02.30.Ks 02.60.Lj 02.70.Bf Keywords: Dynamic equation Time scale Nonlinear Superlinear Sublinear Oscillation

a b s t r a c t

We establish some new criteria for the oscillation of second-order nonlinear dynamic equations on a time scale. We study the case of strongly superlinear and the case of strongly sublinear equations subject to various conditions.

Ó2009 Elsevier B.V. All rights reserved.

1. Introduction

This paper is concerned with the oscillatory behavior of solutions of second-order nonlinear dynamic equations of the form ðaxDÞDðtÞ þfðt

;xrðtÞÞ ¼0; tPt

0 ð1:1Þ

subject to the following hypotheses:

1007-5704/$ - see front matterÓ2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.01.003

*Corresponding author.

E-mail addresses:srgrace@eng.cu.edu.eg(S.R. Grace),agarwal@fit.edu(R.P. Agarwal),bohner@mst.edu(M. Bohner),donal.oregan@nuigalway.ie(D. O’Regan).

1

Supported by NSF Grant #0624127.

Contents lists available atScienceDirect

Commun Nonlinear Sci Numer Simulat

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(i) ais a positive real-valued rd-continuous function satisfying either Z1 t0

D

s aðsÞ¼ 1 ð1:2Þ or Z1 t0

D

s aðsÞ<1: ð1:3Þ

(ii) f:½t0;1Þ R!Ris continuous satisfying

sgnfðt;xÞ ¼sgnx and fðt;xÞ6fðt;yÞ; x6y; tPt0: ð1:4Þ By a solution of Eq.(1.1), we mean a nontrivial real-valued functionxsatisfying Eq.(1.1)fortPtxPt0. A solutionxof Eq. (1.1)is calledoscillatoryif it is neither eventually positive nor eventually negative; otherwise it is callednonoscillatory. Eq. (1.1)is calledoscillatoryif all its solutions are oscillatory.

The theory of time scales, which has recently received a lot of attention, was introduced by Hilger[19]. Several authors have expounded on various aspects of this new theory, see[5,11,12]. In recent years, there has been much research activity concerning the oscillation and nonoscillation of various equations on time scales, we refer the reader to[2,3,10,13,14,16,17]. Most of the results are obtained for special cases of Eq.(1.1), e.g. whena¼1 andfðt;xÞ ¼qðtÞxora¼1 andfðt;xÞ ¼qðtÞfðxÞ, wherefsatisfies the conditionjfðxÞ=xjPk>0 forx–0, orf0ðxÞPfðxÞ=x>0 forx0, see[15–17]. In[18], the authors con-sidered the second-order Emden–Fowler dynamic equation on time scalesxDDþqðtÞxa¼0, where

a

is the ratio of odd inte-gers, and they used the Riccati transformation technique to obtain several oscillation criteria for this equation. For the continuous case of(1.1), i.e.,

ðax0Þ0ðtÞ þfðt

;xðtÞÞ ¼0;

numerous oscillation and nonoscillation criteria have been established, see e.g.[1,4,6–9]. The main goal of this paper is to establish new oscillation criteria for(1.1)(without employing the Riccati transformation technique). The paper is organized as follows: In Section2, we present some basic preliminaries concerning calculus on time scales and prove some auxiliary results that will be used in the remainder of this paper. In Section3, we present some oscillation criteria when condition(1.2) holds, and Section4is devoted to the study of oscillation of Eq.(1.1)when condition(1.3)holds. In Section5, some appli-cations are discussed.

The results of this paper are presented in a form which is essentially new and of a high degree of generality. The obtained results unify and improve many known oscillation criteria which appeared in the literature.

2. Preliminaries

A time scaleTis an arbitrary nonempty closed subset of the real numbersR. Since we are interested in oscillatory behav-ior, we suppose that the time scale under consideration is unbounded above. The forward jump operator

r

:T!Tis defined by

r

ðtÞ ¼inffs2T:s>tg. For a functionf:T!Rwe writefr¼f

r

, andfDrepresents the delta derivative of the function fas defined for example in[11]. The reader unfamiliar with time scales may think offDas the usual derivativef0ifT¼Rand as the usual forward differenceDfifT¼Z. For the definition of rd-continuity and further details we refer to[11]. Besides the usual properties of the time scales integral, we require in this paper only the use of the chain rule[11, Theorem 1.90]

ðx1aÞD 1

a

¼x D Z 1 0 ½hxrþ ð1hÞxadh ; ð2:1Þ

where

a

>0 andxis such that the right-hand side of(2.1)is well defined. Integration by parts, the product rule, and the quotient rule are not needed in this paper. One consequence of(2.1), that will be used in the proof of four results throughout this paper, is as follows.

Lemma 2.1.SupposejyjDis of one sign on½t0;1Þand

a

>0. Then

jyjD ðjyjrÞa6 ðjyj1aÞD 1

a

6 jyjD jyja on½t0;1Þ:

Proof.Replacing x by jyj in (2.1), we see that either jyjD

>0 so that jyj is increasing and hence jyj6jyjr and thus jyj6hjyjrþ ð1hÞjyj6jyjr for allh2 ½0;1, or otherwisejyjD<0 so thatjyjis decreasing and hence jyjPjyjr and thus jyjPhjyjrþ ð1hÞjyjPjyjr for allh2 ½0;1. h

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Definition 2.2. Eq.(1.1)(or the functionf) is said to bestrongly superlinearif there exists a constantb>1 such that jfðt;xÞj

jxjb 6

jfðt;yÞj

jyjb forjxj6jyj; xy>0; tPt0; ð2:2Þ

and it is said to bestrongly sublinearif there exists a constant

c

2 ð0;1Þsuch that jfðt;xÞj

jxjc P jfðt;yÞj

jyjc forjxj6jyj; xy>0; tPt0: ð2:3Þ

If(2.2)holds withb¼1, then(1.1)is calledsuperlinearand if(2.3)holds with

c

¼1, then(1.1)is calledsublinear.

Lemma 2.3. Condition(1.4)implies that

jfðt;xÞj6jfðt;yÞj forjxj6jyj; xyP0; tPt0: ð2:4Þ

Proof. Assume(1.4). Supposejxj6jyjandxyP0. Then either 06x6yand thus 06fðt;xÞ6fðt;yÞso thatjfðt;xÞj6jfðt;yÞj for alltPt0, ory6x60 and thusfðt;yÞ6fðt;xÞ60 so that againjfðt;xÞj6jfðt;yÞjfor alltPt0. Thus(2.4)holds. h

The following lemma will be used throughout.

Lemma 2.4. Suppose x solves(1.1)and is of one sign on½t0;1Þ. Let u;

v

;tPt0. Then

jxjDð

v

Þ ¼xDð

v

Þsgn

v

Þ ð2:5Þ and jxjðtÞ ¼ jxjðuÞ það

v

ÞjxjD ð

v

Þ Z t u

D

s aðsÞ Z t u 1 aðsÞ Z s v jfð

s

;xð

r

ð

s

ÞÞÞj

D

s

D

s: ð2:6Þ

Proof. Letu;

v

;t2Twithu;

v

;tPt0. Lets2TwithsPt0. Integrate(1.1)from

v

tosand divide the resulting equation by

aðsÞ. Now integrate the resulting equation from utot and multiply the resulting equation with sgnxð

v

Þ. Then observe sgnxð

v

Þ ¼sgnxðtÞ,jxð

v

Þj ¼xð

v

Þsgnxð

v

Þ,jxðtÞj ¼xðtÞsgnxðtÞ, and so(1.4)gives jxjðtÞ ¼ jxjðuÞ það

v

ÞxDð

v

Þsgn

v

Þ Z t u

D

s aðsÞ Z t u 1 aðsÞ Z s v jfð

s

;xð

r

ð

s

ÞÞÞj

D

s

D

s: ð2:7Þ

Now differentiate(2.7)with respect totand then plug int¼

v

to obtain(2.5). Finally use(2.5)in(2.7)to arrive at(2.6). h

3. Criteria under condition(1.2)

In this section, we give some new oscillation criteria for Eq.(1.1)when condition(1.2)holds. We let

AðtÞ:¼

Z t t0

D

s

aðsÞ fortPt0:

The following simple consequence ofLemma 2.4will be used throughout this section.

Lemma 3.1. Assume(1.2). Suppose x solves(1.1)and is of one sign on½t0;1Þ. Then on½t0;1Þ,

jxjD

P0; hencejxjis increasing: ð3:1Þ

Moreover, pick any t1>t0and let

~ c¼xðt0Þ and c¼ jxðt0Þj Aðt1Þþ aðt0ÞjxjDðt0Þ sgnxðt0Þ: Then jxjPj~cj on½t0;1Þ; where~cx>0 ð3:2Þ and jxj6jcAj on½t1;1Þ; where cAx>0: ð3:3Þ

Proof. Using(2.6)withtP

v

¼uPt0, we find

jxðtÞj6jxðuÞj þaðuÞjxjDðuÞ Z t

u

D

s

(4)

which is a contradiction to(1.2)whenjxjD

ðuÞ<0. This completes the proof of(3.1). Next,(3.2)follows from(3.1). Finally, using(2.6)withtPt1andu¼

v

¼t0, we find

jxðtÞj6jxðt0Þj þaðt0ÞjxjDðt0ÞAðtÞ ¼ jxðt0Þj AðtÞ þaðt0Þjxj Dðt 0Þ AðtÞ6jcAðtÞj so that(3.3)follows. h

The first oscillation result about(1.1)is immediate.

Theorem 3.2.Assume(1.2). If

Z1 t0

jfð

s

;cÞj

D

s

¼ 1 for all c–0; ð3:4Þ

then Eq.(1.1)is oscillatory.

Proof.Differentiating(2.6)with respect totand then lettingt¼t0and using(3.2) and (2.4), we find, for all

v

Pt0,

jxjDðt0ÞP 1 aðt0Þ Z v t0 jfð

s

;xð

r

ð

s

ÞÞÞj

D

s

P 1 aðt0Þ Z v t0 jfð

s

;~cÞj

D

s

; which contradicts(3.4)and completes the proof. h

The next result deals with the oscillation of all bounded solutions of Eq.(1.1).

Theorem 3.3.Assume(1.2). If Z1 t0 1 aðsÞ Z 1 s jfð

s

;cÞj

D

s

D

s¼ 1 for all c–0; ð3:5Þ

then all bounded solutions of Eq.(1.1)are oscillatory.

Proof.Letxbe a bounded nonoscillatory solution of(1.1)such thatxis of one sign on½t0;1Þ. Using(3.2) and (2.4), we obtain

jfð

s

;xð

r

ð

s

ÞÞÞjPjfð

s

;~cÞj for all

s

Pt0:

Thus, using(2.6)with

v

PtPt0¼u, together with(3.1), we find

jxðtÞjP Z t t0 1 aðsÞ Z v s jfð

s

;xð

r

ð

s

ÞÞÞj

D

s

D

sP Z t t0 1 aðsÞ Z v s jfð

s

;~cÞj

D

s

D

s; which, sincexis bounded, contradicts(3.5)and completes the proof. h

Theorem 3.4.Assume(1.2). Suppose(1.1)is superlinear. If

lim sup t!1 AðtÞ Z 1 t jfð

s

;cÞj

D

s

>jcj for all c–0; ð3:6Þ

then Eq.(1.1)is oscillatory.

Proof.Letxbe a nonoscillatory solution of(1.1)such thatxis of one sign on½t0;1Þ. Using(3.2) and (2.2)(withb¼1), we obtain

jfð

s

;xð

r

ð

s

ÞÞÞj jxð

r

ð

s

ÞÞj P

jfð

s

;~cÞj

j~cj for all

s

Pt0:

Thus, using(2.6)with

v

PtPt0¼u, along with(3.1), we find

jxðtÞjP Z t t0 1 aðsÞ Z v s jfð

s

;xð

r

ð

s

ÞÞÞj

D

s

D

sP Z t t0 1 aðsÞ Z v s jfð

s

;~cÞj j~cj jxð

r

ð

s

ÞÞj

D

s

D

s P Z t t0 1 aðsÞ Z v t jfð

s

;~cÞj j~cj jxð

r

ð

s

ÞÞj

D

s

D

sP Z t t0 1 aðsÞ Zv t jfð

s

;~cÞj j~cj jxðtÞj

D

s

D

s and hence j~cjPAðtÞ Z v t jfð

s

;~cÞj

D

s

;

which contradicts(3.6)and completes the proof. h

(5)

Theorem 3.5. Assume(1.2). Suppose(1.1)is strongly superlinear. If Z 1 t0 1 aðsÞ Z1 s jfð

s

;cÞj

D

s

D

s¼ 1 for all c–0; ð3:7Þ

then Eq.(1.1)is oscillatory.

Proof. Letxbe a nonoscillatory solution of(1.1)such thatxis of one sign on½t0;1Þ. Using(3.2) and (2.2)(withb>1), we obtain

jfð

s

;xð

r

ð

s

ÞÞÞj jxð

r

ð

s

ÞÞjb P

jfð

s

;~cÞj

j~cjb for all

s

Pt0:

Thus, differentiating(2.6)with respect totand using(3.1)andLemma 2.1(the inequality on the left-hand side), we find, for all

v

Pt, jxjDðtÞ P 1 aðtÞ Zv t jfð

s

;xð

r

ð

s

ÞÞÞj

D

s

P 1 aðtÞ Zv t jfð

s

;~cÞj j~cjb jxð

r

ð

s

ÞÞj b

D

s

P 1 aðtÞ Z v t jfð

s

;~cÞj j~cjb

D

s

jxð

r

ðtÞÞj b P 1 aðtÞ Z v t jfð

s

;~cÞj j~cjb

D

s

ðb1ÞjxjDðtÞ ðjxj1b ÞDðtÞ and hence ðjxj1bÞDðtÞP b1 j~cjbaðtÞ Z v t jfð

s

;~cÞj

D

s

:

Integrating this inequality fromt0totPt0, we obtain

jxðt0Þj1bPjxðtÞj1bþ b1 j~cjb Z t t0 1 aðsÞ Z v s jfð

s

;~cÞj

D

s

D

sPb1 j~cjb Z t t0 1 aðsÞ Z v s jfð

s

;~cÞj

D

s

D

s; which contradicts(3.7)and completes the proof. h

Finally, for the strongly sublinear Eq.(1.1), we have the following result.

Theorem 3.6. Assume(1.2). Suppose(1.1)is strongly sublinear. If

Z 1 t0

jfð

s

;cAð

s

ÞÞj

D

s

¼ 1 for all c–0; ð3:8Þ

then Eq.(1.1)is oscillatory.

Proof. Letxbe a nonoscillatory solution of(1.1)such thatxis of one sign on½t0;1Þ. Using(3.3) and (2.3)(with 0<

c

<1), we obtain

jfð

s

;xð

s

ÞÞj jxð

s

Þjc P

jfð

s

;c

s

ÞÞj

jc

s

Þjc for all

s

Pt1:

Thus, using(2.6)withu¼t0and

v

PtPt1, we find

jxðtÞjP Z t t0 1 aðsÞ Zv s jfð

s

;xð

r

ð

s

ÞÞÞj

D

s

D

sP Z t t0 1 aðsÞ Z v t jfð

s

;xð

r

ð

s

ÞÞÞj

D

s

D

s ¼AðtÞ Z v t jfð

s

;xð

r

ð

s

ÞÞÞj

D

s

PAðtÞ Z v t jfð

s

;xð

s

ÞÞj

D

s

PAðtÞ Z v t jfð

s

;c

s

ÞÞj jc

s

Þjc jxð

s

Þj c

D

s

(where we have used(3.1) and (2.4)in the second last inequality) and hence xðtÞ AðtÞ PzðtÞ; wherezðtÞ:¼ jcjc Z v t jfð

s

;c

s

ÞÞj

s

Þ Að

s

Þ c

D

s

: Thus, usingLemma 2.1(the inequality on the right-hand side),

jzjDð

s

Þ ¼ zDð

s

Þ ¼ jcjc jfð

s

;c

s

ÞÞj

s

Þ Að

s

Þ c Pjcjc jfð

s

;c

s

ÞÞjjzð

s

Þjc Pjcjc jfð

s

;c

s

ÞÞjð1

c

Þðjzj D ð

s

ÞÞ ðjzj1cÞDð

s

Þ

(6)

and hence ðjzj1c

ÞDð

s

ÞP1

c

jcjc jfð

s

;c

s

ÞÞj:

Integrating this inequality fromt1totPt1, we obtain

jzðt1Þj1cPjzðtÞj1cþ 1

c

jcjc Zt t1 jfð

s

;c

s

ÞÞj

D

s

P1

c

jcjc Z t t1 jfð

s

;c

s

ÞÞj

D

s

;

which contradicts(3.8)and completes the proof. h

4. Criteria under condition(1.3)

The purpose of this section is to present criteria for the oscillation of Eq.(1.1)whenfis either strongly superlinear or strongly sublinear and when(1.3)holds. We let

e AðtÞ:¼ Z 1 t

D

s aðsÞ fortPt0:

The following consequence ofLemma 2.4will be used throughout this section.

Lemma 4.1.Assume(1.3). Suppose x solves(1.1)and is of one sign on½t0;1Þ. Then either on½t0;1Þ

jxjD

P0; hencejxjis increasing ð4:1Þ

or there exists t2>t0such that on½t2;1Þ

jxjD 60; hencejxjis decreasing: ð4:2Þ Moreover, let c¼ jxðt0Þj þaðt0ÞjxDðt0ÞjAðte 0Þ n o sgnxðt0Þ and ^c¼ xðt0Þ eAðt0Þ if ð4:1Þholds; aðt2ÞjxDðt2Þjsgnxðt0Þ if ð4:2Þholds: 8 < : Then jxj6jcj on½t0;1Þ; wherecx>0 ð4:3Þ and jxjPj^ceAj on½t2;1Þ; where^cAxe >0: ð4:4Þ

Proof.Assume that(4.1)does not hold. Then there existst2>t0such thatjxjDðt2Þ<0. Now, differentiating(2.6)and using

v

¼t2, we find jxjDðtÞ ¼aðt2Þjxj Dðt 2Þ aðtÞ 1 aðtÞ Z t t2 jfð

s

;xð

r

ð

s

ÞÞÞj

D

s

6aðt2Þ aðtÞjxj D ðt2Þ60

for alltPt2, which proves(4.2). Next, using(2.6)with

v

¼u¼t06tand taking into account(2.5), we find

jxðtÞj6jxðt0Þj þaðt0ÞjxjDðt0Þ Z t t0

D

s aðsÞ6jxðt0Þj þaðt0Þjx Dðt 0Þj Z t t0

D

s

aðsÞ6jcj for alltPt0; which proves(4.3). Finally, to prove(4.4), we consider two cases: If(4.1)holds, then

jxðtÞjPjxðt0Þj ¼ xðt0Þ e Aðt0Þ

eAðt0Þ ¼ j^cjeAðt0ÞPj^cjeAðtÞ:

If(4.2)holds, then, using(2.6)withtPuPt2¼

v

, we find

jxðuÞj ¼ jxðtÞj aðt2ÞjxjDðt2Þ Z t u

D

s aðsÞþ Z t u 1 aðsÞ Z s t2 jfð

s

;xð

r

ð

s

ÞÞÞj

D

s

D

sPj^cj Z t u

D

s aðsÞ; and lettingt! 1showsjxðuÞjPj^cjeAðuÞfor alluPt2. h

(7)

Theorem 4.2. Assume(1.3). If Z 1 t0 1 aðsÞ Zs t0

jfð

s

;ceAð

r

ð

s

ÞÞÞj

D

s

D

s¼ 1 for all c–0; ð4:5Þ

then Eq.(1.1)is oscillatory.

Proof. Letxbe a nonoscillatory solution of(1.1)such thatxis of one sign on½t0;1Þ. Using(4.4) and (2.4), we obtain

jfð

s

;xð

r

ð

s

ÞÞÞjPjfð

s

;^ceAð

r

ð

s

ÞÞÞj for all

s

Pt2: Thus, using(2.6)withu¼

v

¼t26t, we find

jxðtÞj ¼ jxðt2Þj þaðt2ÞjxjDðt2Þ Z t t2

D

s aðsÞ Z t t2 1 aðsÞ Z s t2 jfð

s

;xð

r

ð

s

ÞÞÞj

D

s

D

s 6jxðt2Þj þaðt2ÞjxjDðt2Þ Z t t2

D

s aðsÞ Zt t2 1 aðsÞ Z s t2 jfð

s

;^ceAð

r

ð

s

ÞÞÞj

D

s

D

s; which contradicts(4.5)and completes the proof. h

For the strongly superlinear Eq.(1.1), we present the following result.

Theorem 4.3. Assume(1.3). Suppose(1.1)is strongly superlinear. If

Z 1 t0

jfð

s

;ceAð

r

ð

s

ÞÞÞj

D

s

¼ 1 for all c–0; ð4:6Þ

then Eq.(1.1)is oscillatory.

Proof. Letxbe a nonoscillatory solution of(1.1)such thatxis of one sign on½t0;1Þ. Using(4.4) and (2.2)(withb>1), we obtain

jfð

s

;xð

r

ð

s

ÞÞÞj jxð

r

ð

s

ÞÞjb P

jfð

s

;^ceAð

r

ð

s

ÞÞÞj

j^ceAð

r

ð

s

ÞÞjb for all

s

Pt2:

Thus, using(2.6)withtPuPt2¼

v

, we find

jxðuÞj ¼ jxðtÞj aðt2ÞjxjDðt2Þ Z t u

D

s aðsÞþ Z t u 1 aðsÞ Z s t2 jfð

s

;xð

r

ð

s

ÞÞÞj

D

s

D

s Paðt2ÞjxjDðt2Þ Z t u

D

s aðsÞþ Z t u 1 aðsÞ Z u t2 jfð

s

;xð

r

ð

s

ÞÞÞj

D

s

D

s so that, with(2.4)andb¼aðt2ÞjxDðt2Þj,

jxðuÞjPbeAðuÞ þAðuÞe

Zu t2 jfð

s

;xð

r

ð

s

ÞÞÞj

D

s

PbeAðuÞ þeAðuÞ Z u t2 jfð

s

;^ceAð

r

ð

s

ÞÞÞj j^cAðe

r

ð

s

ÞÞjb jxð

r

ð

s

ÞÞj b

D

s

and hence xðuÞ e AðuÞ

PwðuÞ; wherewðuÞ:¼bþ j^cj

bZ u t2 jfð

s

;^ceAð

r

ð

s

ÞÞÞjxð

r

ð

s

ÞÞ e Að

r

ð

s

ÞÞ b

D

s

:

Thus, usingLemma 2.1(the inequality on the left-hand side), jwjDð

s

Þ ¼wDð

s

Þ ¼ j^cjbjfð

s

;^ce

r

ð

s

ÞÞÞj

r

ð

s

ÞÞ e Að

r

ð

s

ÞÞ b Pj^cjbjf ð

s

;^ceAð

r

ð

s

ÞÞÞjjwð

r

ð

s

ÞÞjbPj^cjbjfð

s

;^ceAð

r

ð

s

ÞÞÞjðb1Þjwj Dð

s

Þ ðjwj1b ÞDð

s

Þ and hence ðjwj1b ÞDð

s

ÞPb1 j^cjb jfð

s

;^ceAð

r

ð

s

ÞÞÞj:

(8)

Integrating this inequality fromt2totPt2, we obtain jwðt2Þj1bPjwðtÞj1bþ b1 j^cjb Z t t2 jfð

s

;^ceAð

r

ð

s

ÞÞÞj

D

s

Pb1 j^cjb Z t t2 jfð

s

;^cAðe

r

ð

s

ÞÞÞj

D

s

; which contradicts(4.6)and completes the proof. h

Finally, for the strongly sublinear Eq.(1.1), we have the following result.

Theorem 4.4.Assume(1.3). Suppose(1.1)is strongly sublinear. If

Z1 t0 1 aðsÞ Z s t0 jfð

s

;cÞj

D

s

D

s¼ 1 for all c–0; ð4:7Þ

then Eq.(1.1)is oscillatory.

Proof.Letxbe a nonoscillatory solution of(1.1)such thatxis of one sign on½t0;1Þ. ByLemma 4.1, either(4.1)or(4.2)holds. In the case of(4.1), we havejxðtÞjPjxðt0Þjfor alltPt0and thus, by(2.6)withu¼

v

¼t06t, together with(2.4),

jxðtÞj ¼ jxðt0Þj þaðt0ÞjxjDðt0Þ Z t t0

D

s aðsÞ Z t t0 1 aðsÞ Z s t0 jfð

s

;xð

r

ð

s

ÞÞÞj

D

s

D

s 6jxðt0Þj þaðt0ÞjxjDðt0Þ Z t t0

D

s aðsÞ Z t t0 1 aðsÞ Z s t0 jfð

s

;xðt0ÞÞj

D

s

D

s;

a contradiction to(4.7). In the case of(4.2), using(4.3) and (2.3)(with 0<

c

<1), we obtain jfð

s

;xð

r

ð

s

ÞÞÞj

jxð

r

ð

s

ÞÞjc P jfð

s

;cÞj

jcjc for all

s

Pt2:

Thus, differentiating(2.6)with respect totand letting

v

¼t26t, we find

jxjDðtÞ ¼aðt2ÞjxjDðt2Þ aðtÞ 1 aðtÞ Z t t2 jfð

s

;xð

r

ð

s

ÞÞÞj

D

s

6jcj c aðtÞ Z t t2 jfð

s

;cÞjjxð

r

ð

s

ÞÞjc

D

s

6jcj c aðtÞ Z t t2 jfð

s

;cÞj

D

s

jxðtÞjc;

where we have used again(4.2)in the last inequality. Now, using(2.1)(the inequality on the right-hand side), we obtain jcjc aðtÞ Z t t2 jfð

s

;cÞj

D

s

6jxj DðtÞ jxðtÞjc 6 ðjxj1c ÞDðtÞ 1

c

: Integrating this inequality fromt2totPt2, we obtain

jxðt2Þj1cPjxðtÞj1cþ 1

c

jcjc Z t t2 1 aðsÞ Z s t2 jfð

s

;cÞj

D

s

D

sP1

c

jcjc Z t t2 1 aðsÞ Z s t2 jfð

s

;cÞj

D

s

D

s; which contradicts(4.7)and completes the proof. h

5. Some applications

We shall apply the obtained results for Eq.(1.1)to the second-order Emden–Fowler dynamic equation on time scales ðaxDÞD

þqðxrÞa¼0; ð5:1Þ

whereaandqare nonnegative rd-continuous functions and

a

is the ratio of positive odd integers. ByTheorems 3.2, 3.4 ,3.5 and 3.6, respectively, we obtain the following.

Theorem 5.1. Let condition(1.2)hold and define AðtÞ ¼Rtt0Ds=aðsÞ. Eq.(5.1)is oscillatory if one of the following conditions holds:

Rt1 0qð

s

ÞD

s

¼ 1, if

a

>0; lim supt!1fAðtÞ R1 t qð

s

ÞD

s

g>c for any c>0, if

a

P1; Rt1 0 1 aðsÞ R1 s qð

s

ÞD

s

Ds¼ 1, if

a

>1; Rt1 0ðAð

s

ÞÞ aqð

s

ÞD

s

¼ 1, if0<

a

<1.

(9)

Theorem 5.2. Let condition(1.3)hold and defineeAðtÞ ¼Rt1Ds=aðsÞ. Eq.(5.1)is oscillatory if one of the following conditions holds: Rt1 0 1 aðsÞ Rs t0qð

s

ÞðeAð

r

ð

s

ÞÞÞ aD

s

Ds¼ 1, if

a

>0; Rt1 0ðAeð

r

ð

s

ÞÞÞ aqð

s

ÞD

s

¼ 1, if

a

>1; Rt10 1 aðsÞ Rs t0qð

s

ÞD

s

Ds¼ 1, if0<

a

<1.

Remark 5.3. From the results of this paper, we can obtain some oscillation criteria for Eq.(1.1)on different types of time scales. IfT¼R, then

r

ðtÞ ¼tandxD¼x0. In this case, the results of this paper are the same as those in[20]. IfT¼Z, then

r

ðtÞ ¼tþ1 andxDðtÞ ¼DxðtÞ ¼xðtþ1Þ xðtÞ:In this case, the results of this paper are the discrete analogues of those in [20]. IfT¼hZwithh>0, then

r

ðtÞ ¼tþhandxDðtÞ ¼D

hxðtÞ ¼ ðxðtþhÞ xðtÞÞ=h. The reformulation of our results are easy

and left to the reader. We may employ other types of time scales, e.g.T¼qN0withq>1,T¼N2

0, and others, see[11,12]. The details are left to the reader.

Remark 5.4. The results of this paper can be extended to dynamic equations of type(1.1)with deviating arguments, e.g. ðaxDÞD

þfðt;xð

s

ðtÞÞÞ ¼0;

where

s

:T!Twith limt!1

s

ðtÞ ¼ 1. The details are left to the reader.

References

[1] Agarwal R, Bohner M, Li W-T. Nonoscillation and oscillation theory for functional differential equations. Monographs and textbooks in pure and applied mathematics. New York: Marcel Dekker; 2004.

[2] Agarwal RP, Bohner M, Grace SR. On the oscillation of second-order half-linear dynamic equations. J Differ Equ Appl 2008 (to appear).

[3] Agarwal RP, Bohner M, Grace SR. Oscillation criteria for first-order forced nonlinear dynamic equations. Can Appl Math Q 2008;15(3) (to appear). [4] Agarwal RP, Bohner M, Grace SR, O’Regan D. Discrete oscillation theory. New York: Hindawi Publishing; 2005.

[5] Agarwal RP, Bohner M, O’Regan D, Peterson A. Dynamic equations on time scales: a survey. J Comput Appl Math 2002;141(1–2):1–26. Agarwal RP, Bohner M, O’Regan D, editors. Special issue on dynamic equations on time scales. Preprint in Ulmer Seminare 5.

[6] Agarwal RP, Grace SR, O’Regan D. Oscillation theory for difference and functional differential equations. Dordrecht: Kluwer Academic Publishers; 2000. [7] Agarwal RP, Grace SR, O’Regan D. Oscillation theory for second order linear, half-linear, superlinear and sublinear dynamic

equations. Dordrecht: Kluwer Academic Publishers; 2002.

[8] Agarwal RP, Grace SR, O’Regan D. On the oscillation of certain second order difference equations. J Differ Equ Appl 2003;9(1):109–119. In honour of Professor Allan Peterson on the occasion of his 60th birthday, Part II.

[9] Agarwal RP, Grace SR, O’Regan D. Oscillation theory for second order dynamic equations. Series in mathematical analysis and applications, vol. 5. London: Taylor & Francis; 2003.

[10] Akın-Bohner E, Bohner M, Saker SH. Oscillation criteria for a certain class of second order Emden–Fowler dynamic equations. Electron Trans Numer Anal 2007;27:1–12.

[11] Bohner M, Peterson A. Dynamic equations on time scales: an introduction with applications. Boston: Birkhäuser; 2001. [12] Bohner M, Peterson A. Advances in dynamic equations on time scales. Boston: Birkhäuser; 2003.

[13] Bohner M, Saker SH. Oscillation criteria for perturbed nonlinear dynamic equations. Math Comput Model 2004;40(3–4):249–60. [14] Bohner M, Saker SH. Oscillation of second order nonlinear dynamic equations on time scales. Rocky Mountain J Math 2004;34(4):1239–54. [15] Erbe L, Peterson A. Boundedness and oscillation for nonlinear dynamic equations on a time scale. Proc Amer Math Soc 2004;132(3):735–44. [16] Erbe L, Peterson A, Rˇehák P. Comparison theorems for linear dynamic equations on time scales. J Math Anal Appl 2002;275(1):418–38.

[17] Erbe L, Peterson A, Saker SH. Oscillation criteria for second-order nonlinear dynamic equations on time scales. J London Math Soc 2003;67(3):701–14. [18] Han Z, Sun S, Shi B. Oscillation criteria for a class of second-order Emden–Fowler delay dynamic equations on time scales. J Math Anal Appl

2007;334(2):847–58.

[19] Hilger S. Analysis on measure chains — a unified approach to continuous and discrete calculus. Results Math 1990;18:18–56.

[20] Kusano T, Ogata A, Usami H. On the oscillation of solutions of second order quasilinear ordinary differential equations. Hiroshima Math J 1993;23(3):645–67.

References

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