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 aDepartment 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
(i) ais a positive real-valued rd-continuous function satisfying either Z1 t0
D
s aðsÞ¼ 1 ð1:2Þ or Z1 t0D
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 forx–0, 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 byr
ðtÞ ¼inffs2T:s>tg. For a functionf:T!Rwe writefr¼fr
, 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 ; ð2:1Þwhere
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. ThenjyjD 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
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;xÞjjxjc 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. ThenjxjDð
v
Þ ¼xDðv
Þsgnxðv
Þ ð2:5Þ and jxjðtÞ ¼ jxjðuÞ þaðv
ÞjxjD ðv
Þ Z t uD
s aðsÞ Z t u 1 aðsÞ Z s v jfðs
;xðr
ðs
ÞÞÞjD
s
D
s: ð2:6ÞProof. Letu;
v
;t2Twithu;v
;tPt0. Lets2TwithsPt0. Integrate(1.1)fromv
tosand divide the resulting equation byaðsÞ. Now integrate the resulting equation from utot and multiply the resulting equation with sgnxð
v
Þ. Then observe sgnxðv
Þ ¼sgnxðtÞ,jxðv
Þj ¼xðv
Þsgnxðv
Þ,jxðtÞj ¼xðtÞsgnxðtÞ, and so(1.4)gives jxjðtÞ ¼ jxjðuÞ þaðv
ÞxDðv
Þsgnxðv
Þ Z t uD
s aðsÞ Z t u 1 aðsÞ Z s v jfðs
;xðr
ðs
ÞÞÞjD
s
D
s: ð2:7ÞNow 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). h3. 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
saðsÞ 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 findjxðtÞj6jxðuÞj þaðuÞjxjDðuÞ Z t
u
D
swhich 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 findjxðtÞj6jxðt0Þj 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
;cÞjD
s
¼ 1 for all c–0; ð3:4Þthen 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
;xðr
ðs
ÞÞÞjD
s
P 1 aðt0Þ Z v t0 jfðs
;~cÞjD
s
; which contradicts(3.4)and completes the proof. hThe 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
;cÞjD
s
D
s¼ 1 for all c–0; ð3:5Þthen 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
;xðr
ðs
ÞÞÞjPjfðs
;~cÞj for alls
Pt0:Thus, using(2.6)with
v
PtPt0¼u, together with(3.1), we findjxðtÞjP Z t t0 1 aðsÞ Z v s jfð
s
;xðr
ðs
ÞÞÞjD
s
D
sP Z t t0 1 aðsÞ Z v s jfðs
;~cÞjD
s
D
s; which, sincexis bounded, contradicts(3.5)and completes the proof. hTheorem 3.4.Assume(1.2). Suppose(1.1)is superlinear. If
lim sup t!1 AðtÞ Z 1 t jfð
s
;cÞjD
s
>jcj for all c–0; ð3:6Þthen 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
;xðr
ðs
ÞÞÞj jxðr
ðs
ÞÞj Pjfð
s
;~cÞjj~cj for all
s
Pt0:Thus, using(2.6)with
v
PtPt0¼u, along with(3.1), we findjxðtÞjP Z t t0 1 aðsÞ Z v s jfð
s
;xðr
ðs
ÞÞÞjD
s
D
sP Z t t0 1 aðsÞ Z v s jfðs
;~cÞj j~cj jxðr
ðs
ÞÞjD
s
D
s P Z t t0 1 aðsÞ Z v t jfðs
;~cÞj j~cj jxðr
ðs
ÞÞjD
s
D
sP Z t t0 1 aðsÞ Zv t jfðs
;~cÞj j~cj jxðtÞjD
s
D
s and hence j~cjPAðtÞ Z v t jfðs
;~cÞjD
s
;which contradicts(3.6)and completes the proof. h
Theorem 3.5. Assume(1.2). Suppose(1.1)is strongly superlinear. If Z 1 t0 1 aðsÞ Z1 s jfð
s
;cÞjD
s
D
s¼ 1 for all c–0; ð3:7Þthen 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
;xðr
ðs
ÞÞÞj jxðr
ðs
ÞÞjb Pjfð
s
;~cÞjj~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 jfðs
;xðr
ðs
ÞÞÞjD
s
P 1 aðtÞ Zv t jfðs
;~cÞj j~cjb jxðr
ðs
ÞÞj bD
s
P 1 aðtÞ Z v t jfðs
;~cÞj j~cjbD
s
jxðr
ðtÞÞj b P 1 aðtÞ Z v t jfðs
;~cÞj j~cjbD
s
ðb1ÞjxjDðtÞ ðjxj1b ÞDðtÞ and hence ðjxj1bÞDðtÞP b1 j~cjbaðtÞ Z v t jfðs
;~cÞjD
s
:Integrating this inequality fromt0totPt0, we obtain
jxðt0Þj1bPjxðtÞj1bþ b1 j~cjb Z t t0 1 aðsÞ Z v s jfð
s
;~cÞjD
s
D
sPb1 j~cjb Z t t0 1 aðsÞ Z v s jfðs
;~cÞjD
s
D
s; which contradicts(3.7)and completes the proof. hFinally, 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
;cAðs
ÞÞjD
s
¼ 1 for all c–0; ð3:8Þthen 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 obtainjfð
s
;xðs
ÞÞj jxðs
Þjc Pjfð
s
;cAðs
ÞÞjjcAð
s
Þjc for alls
Pt1:Thus, using(2.6)withu¼t0and
v
PtPt1, we findjxðtÞjP Z t t0 1 aðsÞ Zv s jfð
s
;xðr
ðs
ÞÞÞjD
s
D
sP Z t t0 1 aðsÞ Z v t jfðs
;xðr
ðs
ÞÞÞjD
s
D
s ¼AðtÞ Z v t jfðs
;xðr
ðs
ÞÞÞjD
s
PAðtÞ Z v t jfðs
;xðs
ÞÞjD
s
PAðtÞ Z v t jfðs
;cAðs
ÞÞj jcAðs
Þjc jxðs
Þj cD
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
;cAðs
ÞÞjxðs
Þ Aðs
Þ cD
s
: Thus, usingLemma 2.1(the inequality on the right-hand side),jzjDð
s
Þ ¼ zDðs
Þ ¼ jcjc jfðs
;cAðs
ÞÞjxðs
Þ Aðs
Þ c Pjcjc jfðs
;cAðs
ÞÞjjzðs
Þjc Pjcjc jfðs
;cAðs
ÞÞjð1c
Þðjzj D ðs
ÞÞ ðjzj1cÞDðs
Þand hence ðjzj1c
ÞDð
s
ÞP1c
jcjc jfðs
;cAð
s
ÞÞj:Integrating this inequality fromt1totPt1, we obtain
jzðt1Þj1cPjzðtÞj1cþ 1
c
jcjc Zt t1 jfðs
;cAðs
ÞÞjD
s
P1c
jcjc Z t t1 jfðs
;cAðs
ÞÞjD
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 jfðs
;xðr
ðs
ÞÞÞjD
s
6aðt2Þ aðtÞjxj D ðt2Þ60for alltPt2, which proves(4.2). Next, using(2.6)with
v
¼u¼t06tand taking into account(2.5), we findjxðtÞj6jxðt0Þj þaðt0ÞjxjDðt0Þ Z t t0
D
s aðsÞ6jxðt0Þj þaðt0Þjx Dðt 0Þj Z t t0D
saðsÞ6jcj 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 findjxðuÞj ¼ jxðtÞj aðt2ÞjxjDðt2Þ Z t u
D
s aðsÞþ Z t u 1 aðsÞ Z s t2 jfðs
;xðr
ðs
ÞÞÞjD
s
D
sPj^cj Z t uD
s aðsÞ; and lettingt! 1showsjxðuÞjPj^cjeAðuÞfor alluPt2. hTheorem 4.2. Assume(1.3). If Z 1 t0 1 aðsÞ Zs t0
jfð
s
;ceAðr
ðs
ÞÞÞjD
s
D
s¼ 1 for all c–0; ð4:5Þthen 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
;xðr
ðs
ÞÞÞjPjfðs
;^ceAðr
ðs
ÞÞÞj for alls
Pt2: Thus, using(2.6)withu¼v
¼t26t, we findjxðtÞj ¼ jxðt2Þj þaðt2ÞjxjDðt2Þ Z t t2
D
s aðsÞ Z t t2 1 aðsÞ Z s t2 jfðs
;xðr
ðs
ÞÞÞjD
s
D
s 6jxðt2Þj þaðt2ÞjxjDðt2Þ Z t t2D
s aðsÞ Zt t2 1 aðsÞ Z s t2 jfðs
;^ceAðr
ðs
ÞÞÞjD
s
D
s; which contradicts(4.5)and completes the proof. hFor 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
;ceAðr
ðs
ÞÞÞjD
s
¼ 1 for all c–0; ð4:6Þthen 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
;xðr
ðs
ÞÞÞj jxðr
ðs
ÞÞjb Pjfð
s
;^ceAðr
ðs
ÞÞÞjj^ceAð
r
ðs
ÞÞjb for alls
Pt2:Thus, using(2.6)withtPuPt2¼
v
, we findjxðuÞj ¼ jxðtÞj aðt2ÞjxjDðt2Þ Z t u
D
s aðsÞþ Z t u 1 aðsÞ Z s t2 jfðs
;xðr
ðs
ÞÞÞjD
s
D
s Paðt2ÞjxjDðt2Þ Z t uD
s aðsÞþ Z t u 1 aðsÞ Z u t2 jfðs
;xðr
ðs
ÞÞÞjD
s
D
s so that, with(2.4)andb¼aðt2ÞjxDðt2Þj,jxðuÞjPbeAðuÞ þAðuÞe
Zu t2 jfð
s
;xðr
ðs
ÞÞÞjD
s
PbeAðuÞ þeAðuÞ Z u t2 jfðs
;^ceAðr
ðs
ÞÞÞj j^cAðer
ðs
ÞÞjb jxðr
ðs
ÞÞj bD
s
and hence xðuÞ e AðuÞPwðuÞ; wherewðuÞ:¼bþ j^cj
bZ u t2 jfð
s
;^ceAðr
ðs
ÞÞÞjxðr
ðs
ÞÞ e Aðr
ðs
ÞÞ bD
s
:Thus, usingLemma 2.1(the inequality on the left-hand side), jwjDð
s
Þ ¼wDðs
Þ ¼ j^cjbjfðs
;^cAðer
ðs
ÞÞÞjxðr
ðs
ÞÞ e Aðr
ðs
ÞÞ b Pj^cjbjf ðs
;^ceAðr
ðs
ÞÞÞjjwðr
ðs
ÞÞjbPj^cjbjfðs
;^ceAðr
ðs
ÞÞÞjðb1Þjwj Dðs
Þ ðjwj1b ÞDðs
Þ and hence ðjwj1b ÞDðs
ÞPb1 j^cjb jfðs
;^ceAðr
ðs
ÞÞÞj:Integrating this inequality fromt2totPt2, we obtain jwðt2Þj1bPjwðtÞj1bþ b1 j^cjb Z t t2 jfð
s
;^ceAðr
ðs
ÞÞÞjD
s
Pb1 j^cjb Z t t2 jfðs
;^cAðer
ðs
ÞÞÞjD
s
; which contradicts(4.6)and completes the proof. hFinally, 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
;cÞjD
s
D
s¼ 1 for all c–0; ð4:7Þthen 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 jfðs
;xðr
ðs
ÞÞÞjD
s
D
s 6jxðt0Þj þaðt0ÞjxjDðt0Þ Z t t0D
s aðsÞ Z t t0 1 aðsÞ Z s t0 jfðs
;xðt0ÞÞjD
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 jfðs
;xðr
ðs
ÞÞÞjjxð
r
ðs
ÞÞjc P jfðs
;cÞjjcjc for all
s
Pt2:Thus, differentiating(2.6)with respect totand letting
v
¼t26t, we findjxjDðtÞ ¼aðt2ÞjxjDðt2Þ aðtÞ 1 aðtÞ Z t t2 jfð
s
;xðr
ðs
ÞÞÞjD
s
6jcj c aðtÞ Z t t2 jfðs
;cÞjjxðr
ðs
ÞÞjcD
s
6jcj c aðtÞ Z t t2 jfðs
;cÞjD
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
;cÞjD
s
6jxj DðtÞ jxðtÞjc 6 ðjxj1c ÞDðtÞ 1c
: Integrating this inequality fromt2totPt2, we obtainjxðt2Þj1cPjxðtÞj1cþ 1
c
jcjc Z t t2 1 aðsÞ Z s t2 jfðs
;cÞjD
s
D
sP1c
jcjc Z t t2 1 aðsÞ Z s t2 jfðs
;cÞjD
s
D
s; which contradicts(4.7)and completes the proof. h5. 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
ÞDs
¼ 1, ifa
>0; lim supt!1fAðtÞ R1 t qðs
ÞDs
g>c for any c>0, ifa
P1; Rt1 0 1 aðsÞ R1 s qðs
ÞDs
Ds¼ 1, ifa
>1; Rt1 0ðAðs
ÞÞ aqðs
ÞDs
¼ 1, if0<a
<1.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
ÞðeAðr
ðs
ÞÞÞ aDs
Ds¼ 1, ifa
>0; Rt1 0ðAeðr
ðs
ÞÞÞ aqðs
ÞDs
¼ 1, ifa
>1; Rt10 1 aðsÞ Rs t0qðs
ÞDs
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, thenr
ðtÞ ¼tþ1 andxDðtÞ ¼DxðtÞ ¼xðtþ1Þ xðtÞ:In this case, the results of this paper are the discrete analogues of those in [20]. IfT¼hZwithh>0, thenr
ðtÞ ¼tþhandxDðtÞ ¼DhxðtÞ 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!1s
ðtÞ ¼ 1. The details are left to the reader.References
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