Research Article Oscillation of Second-Order Mixed-Nonlinear Delay Dynamic Equations

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1 Hndaw Publshng Corporaton Advances n Dfference Equatons Volume 200 Artcle ID pages do:0.55/200/38909 Research Artcle Oscllaton of Second-Order Mxed-Nonlnear Delay Dynamc Equatons M. Ünal and A. Zafer 2 Department of Software Engneerng Bahçeşehr Unversty Beşktaş Istanbul Turkey 2 Department of Mathematcs Mddle East Techncal Unversty 0653 Ankara Turkey Correspondence should be addressed to M. Ünal munal@bahcesehr.edu.tr Receved 9 January 200; Accepted 20 March 200 Academc Edtor: Josef Dblk Copyrght q 200 M. Ünal and A. Zafer. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense whch permts unrestrcted use dstrbuton and reproducton n any medum provded the orgnal work s properly cted. New oscllaton crtera are establshed for second-order mxed-nonlnear delay dynamc equatons on tme scales by utlzng an nterval averagng technque. No restrcton s mposed on the coeffcent functons and the forcng term to be nonnegatve.. Introducton In ths paper we are concerned wth oscllatory behavor of the second-order nonlnear delay dynamc equaton of the form rtx Δ t) Δ p0 txτ 0 t p t xτ t α xτ t et t t 0. on an arbtrary tme scale T where α >α 2 > >α m > >α m > >α n > 0 n >m ;.2 the functons r p e: T R are rght-dense contnuous wth r>0 nondecreasng; the delay functons τ : T T are nondecreasng rght-dense contnuous and satsfy τ t t for t T wth τ t as t. We assume that the tme scale T s unbounded above that s sup T and defne the tme scale nterval t 0 T by t 0 T :t 0 T. It s also assumed that the reader s already famlar wth the tme scale calculus. A comprehensve treatment of calculus on tme scales can be found n 3.

2 2 Advances n Dfference Equatons By a soluton of. we mean a nontrval real valued functon x : T R such that x C rd T T and rxδ C rd T T for all T T wth T t 0andthatx satsfes.. A functon x s called an oscllatory soluton of. f x s nether eventually postve nor eventually negatve otherwse t s nonoscllatory. Equaton. s sad to be oscllatory f and only f every soluton x of. s oscllatory. Notce that when T R. s reduced to the second-order nonlnear delay dfferental equaton rtx t ) p0 txτ 0 t p t xτ t α xτ t et t t 0.3 whle when T Z t becomes a delay dfference equaton ΔrkΔxk p 0 kxτ 0 k p k xτ k α xτ k ek k k 0..4 Another useful tme scale s T q N : {q m : m N and q> s a real number} whch leads to the quantum calculus. In ths case. s the q-dfference equaton Δ q rtδq xt ) p 0 txτ 0 t p t xτ t α xτ t et t t 0.5 where Δ q ft fσt ft/μt σt qt andμt q t. Interval oscllaton crtera are more natural n vew of the Sturm comparson theory snce t s stated on an nterval rather than on nfnte rays and hence t s necessary to establsh more nterval oscllaton crtera for equatons on arbtrary tme scales as n T R. Asfar as we know when T R an nterval oscllaton crteron for forced second-order lnear dfferental equatons was frst establshed by El-Sayed 4. In 2003 Sun 5 demonstrated ncely how the nterval crtera method can be appled to delay dfferental equatons of the form x t pt xτt α xτt et α.6 where the potental p and the forcng term e may oscllate. Some of these nterval oscllaton crtera were recently extended to second-order dynamc equatons n 6 0. Further results on oscllatory and nonoscllatory behavor of the second order nonlnear dynamc equatons on tme scales can be found n 23 and the references cted theren. Therefore motvated by Sun and Meng s paper 24 usng smlar technques ntroduced n 7 by Kong and an arthmetc-geometrc mean nequalty we gve oscllaton crtera for second-order nonlnear delay dynamc equatons of the form.. Examples are consdered to llustrate the results.

3 Advances n Dfference Equatons 3 2. Man Results We need the followng lemmas n provng our results. The frst two lemmas can be found n 25 Lemma. Lemma 2.. Let {α } 2...nbe the n-tuple satsfyng α >α 2 > >α m > >α m > > α n > 0. Then there exsts an n-tuple {η η 2...η n } satsfyng α η η < 0 <η <. 2. Lemma 2.2. Let {α } 2...nbe the n-tuple satsfyng α >α 2 > >α m > >α m > > α n > 0. Then there exsts an n-tuple {η η 2...η n } satsfyng α η η 0 <η <. 2.2 The next two lemmas are qute elementary va dfferental calculus; see Lemma 2.3. Let u A and B be nonnegatve real numbers. Then Au γ B γ γ ) /γ A /γ B /γ u γ >. 2.3 Lemma 2.4. Let u A and B be nonnegatve real numbers. Then Cu Du γ γ ) γ γ/ γ C γ/γ D / γ 0 <γ<. 2.4 The last mportant lemma that we need s a specal case of the one gven n 6. For completeness we provde a proof. Lemma 2.5. Let τ : T T be a nondecreasng rght-dense contnuous functon wth τt t and a b T wth a<b.ifx C rd τab T s a postve functon such that rtxδ t s nonncreasng on τab T wth r>0 nondecreasng then xτt Proof. By the Mean Value Theorem 2 Theorem.4 τt τa σt τa t a b T. 2.5 xt xτa x Δ η ) t τa 2.6 for some η τat T for any t τab T. Snce rtx Δ t s nonncreasng and rt s nondecreasng we have rtx Δ t r η ) x Δ η ) rtx Δ η ) t > η 2.7

4 4 Advances n Dfference Equatons and so x Δ t x Δ η t η.now xt xτa x Δ tt τa t τab T. 2.8 Defne μs : xs s τax Δ s s τtσt T t a b T. 2.9 It follows from 2.8 that μs xτa > 0fors τtσt T and t a b T. Thus we have 0 < σt τt μs xsx σ Δs s whch completes the proof. σt τt ) s τa Δ Δs xs σt τa τt τa 2.0 xτt In what follows we say that a functon Ht s : T 2 R belongs to H T f and only f H s rght-dense contnuous functon on {t s T 2 : t s t 0 } havng contnuous Δ-partal dervatves on {t s T 2 : t>s t 0 }wthht t 0 for all t and Ht s / 0 for all t / s.note that n case H R theδ-partal dervatves become the usual partal dervatves of Ht s.the partal dervatves for the cases H Z and H N wll be explctly gven later. Denotng the Δ-partal dervatves H Δ t t s and H Δ s t s of Ht s wth respect to t and s by H t s and H 2 t s respectvely the theorems below extend the results obtaned n 5 to nonlnear delay dynamc equaton on arbtrary tme scales and concde wth them when H 2 t s s replaced by Ht s. Indeed f we set Ht s Ut s then t follows that H t s U t s Uσts Ut s H 2 t s U 2 t s Ut σs Ut s. 2. When T R they become Ht s t Ut s/ t 2 Ut s Ht s Ut s/ s s 2 Ut s 2.2 as n 5. However we prefer usng H 2 t s nstead of Ut s for smplcty. Theorem 2.6. Suppose that for any gven arbtrarly large) T T there exst subntervals a b T and a 2 b 2 T of T T wherea <b and a 2 <b 2 such that p t 0 for t a b T a 2 b 2 T n l et > 0 for t a l b l T l where a l mn { τ j a l : j n } 2.4

5 Advances n Dfference Equatons 5 hold. Let {η η 2...η n } be an n-tuple satsfyng 2. of Lemma 2.. If there exst a functon H H T and numbers c ν b ν T such that cν [ QtH 2 σta H 2 ν rth 2 c ν t Δt bν [ QtH 2 b H 2 ν σt rth 2 2 b ν c ν b νt Δt >0 c ν 2.5 for ν 2 where Qt p 0 t τ 0t τ 0 σt τ 0 k 0 et η 0 k 0 0 η 0 p t ) ) η τ t τ α η σt τ η 2.6 then. s oscllatory. Proof. Suppose on the contrary that x s a nonoscllatory soluton of.. Frst assume that xt and xτ j t j n are postve for all t t for some t t 0 T. Choose a suffcently large so that τ j τ j a t.lett a b T. Defne Usng the delta quotent rule we have wt rt xδ t xt t t. 2.7 w Δ t rtx Δ t ) Δ xt rt x Δ t ) 2 xt rtx Δ t ) Δ rt x Δ t ) 2 xt. 2.8 Notce that [ [ xt xt xt μtx Δ t x 2 t μt wt rt x2 t[ rt μtwt rt 2.9 whch mples rt μtwt rt xσ t xt > Hence we obtan rtx Δ w Δ t ) Δ w 2 t t x σ t rt μtwt. 2.2

6 6 Advances n Dfference Equatons Substtutng 2.2 nto. yelds w Δ t p 0txτ 0 t w 2 t rt μtwt p t xτ t α xτ t et By assumpton we can choose a b t such that p t n and et 0 for all t a b T where a s defned as n 2.4. Clearly the condtons of Lemma 2.5 are satsfed when τ replaced wth τ j for each fxed j n. Therefore from 2.5 we have x τ j t ) τ jt τ j a σt τ j a t a b T 2.23 and takng nto account 2.22 yelds w Δ t p 0 t τ 0t τ 0 a σt τ 0 a w 2 t rt μtwt τ t τ a p k t σt τ a ) α α et Denote Q 0 t : p 0t τ 0t τ 0 a σt τ 0 a ) Q t : p τ t τ a α t σt τ a From 2.24 we have w Δ t Q w 0t 2 t rt μtwt Q txσ t α et Now recall the well-known arthmetc-geometrc mean nequalty see 26 u η 0 u η where η 0 n η and η > n. Settng u 0 η 0 : et u η : Q txσ t α 2.28 n 2.26 yelds w Δ t Q w 0t 2 t rt μtwt u η u 0 η 0 Q w 2 t 0t rt μtwt u η

7 Advances n Dfference Equatons 7 From 2.29 and takng nto account 2.27 weget w Δ t Q w 0t 2 t rt μtwt u η and hence w Δ t Q w 0t 2 t rt μtwt η η 0 et η 0 0 η 0 Q w 2 t 0t rt μtwt η η 0 0 et η 0 Q w 2 t 0t rt μtwt η η 0 0 et η 0 Q t) η α ) η Q t) η η 0 n j α jη j η j Q t) η 2.3 whch yelds w Δ t Q w 2 t 0t rt μtwt η η 0 0 et η 0 Qt w 2 t rt μtwt p t ) ) η τ t τ a α η σt τ a 2.32 where Qt Q 0 t η η 0 0 et η 0 p t ) ) η τ t τ a α η σt τ a Multplyng both sdes of 2.32 by H 2 σta and ntegratng both sdes of the resultng nequalty from a to c a <c <b yeld c a w Δ th 2 σta Δt c a QtH 2 σta Δt c a w 2 th 2 σta Δt rt μtwt Fx s and note that wth 2 Δt t s) H 2 σtsw Δ t H 2 Δt t s) wt H 2 σtsw Δ t H t shσtswt Ht sh t swt 2.35 from whch we obtan H 2 σtsw Δ t wth 2 t s) Δt H t shσtswt Ht sh t swt. 2.36

8 8 Advances n Dfference Equatons Therefore c a w Δ th 2 σta Δt c wth 2 Δt t a ) Δt a c H t a Hσta wt Ht a H t a wtδt. a 2.37 Notce that c a wth 2 t a ) Δt Δt wc H 2 c a wa H 2 a a wc H 2 c a 2.38 snce Ha a 0 and hence we obtan from 2.34 that wc H 2 c a c a QtH 2 σta Δt c a w 2 t rt μtwt H2 σta Δt c H t a Hσta wt Ht a H t a wtδt. a 2.39 On the other hand w 2 th 2 σts rt μtwt wthσtsh t s Ht sh t swt [ wthσts 2 rt μtwth t s rt μtwt 2.40 rt μtwt ) H 2 t s wthσtsh t s Ht sh t swt. Takng nto account that HσtsHt sμth t s we have w 2 th 2 σta rt μtwt wthσta H t a Ht a H t a wt rth 2 t a. 2.4 Usng ths nequalty n 2.39 we have wc H 2 c a c [ QtH 2 σta rth 2 t a Δt. a 2.42

9 Advances n Dfference Equatons 9 Smlarly by followng the above calculaton step by step that s multplyng both sdes of 2.32 ths tme by H 2 b σs after takng nto account that H 2 t σsw Δ s wsh 2 t s) Δs H 2 t sht σsws Ht sh 2 t sws 2.43 one can easly obtan wc H 2 b c b [ QsH 2 b σs rsh 2 2 b s Δs. c 2.44 Addng up 2.42 and 2.44weobtan 0 c [ QtH 2 σta H 2 rth 2 c a t a Δt a b [ QtH 2 b H 2 σt rsh 2 2 b c b t Δt. c 2.45 Ths contradcton completes the proof when xt s eventually postve. The proof when xt s eventually negatve s analogous by repeatng the above arguments on the nterval a 2 b 2 T nstead of a b T. Corollary 2.7. Suppose that for any gven arbtrarly large) T t 0 there exst subntervals a b and a 2 b 2 of T such that p t 0 for t a b a 2 b n l et 0 for t a l b l l where a l mn{τ j a l : j n} holds. Let {η η 2...η n } be an n-tuple satsfyng 2. of Lemma 2.. If there exst a functon H H R and numbers c ν b ν such that for ν 2 where cν [ QtH 2 t a H 2 ν rth 2 c ν t dt bν [ QtH 2 b H 2 ν t rth 2 2 b ν c ν b νt dt > 0 c ν Qt p 0 t τ 0t τ 0 t τ 0 k 0 k 0 et η 0 0 η 0 p t ) η τ t τ t τ η ) α η then.3 s oscllatory.

10 0 Advances n Dfference Equatons Corollary 2.8. Suppose that for any gven arbtrarly large) T t 0 there exst a b a 2 b 2 Z wth T a <b and T a 2 <b 2 such that for each n p t 0 for t {a a a 2...b } {a 2 a 2 a b 2 } l et 0 for t {a l a l a l 2...b l } l where a l mn{τ j a l : j n} holds. Let {η η 2...η n } be an n-tuple satsfyng 2. of Lemma 2.. If there exst a functon H H Z and numbers c ν { 2...b ν } such that c ν [ QtH 2 t a H 2 ν rth 2 c ν t t b ν [ QtH 2 b H 2 ν t rth 2 2 b ν c ν b νt > 0 tc ν 2.50 for ν 2 where H t : Ht Ht Qt p 0 t τ 0t τ 0 t τ 0 k 0 et η 0 k 0 0 H 2 b ν t : Hb ν t Hb ν t η 0 p t ) ) η τ t τ α η t τ η 2.5 then.4 s oscllatory. Corollary 2.9. Suppose that for any gven arbtrarly large) T t 0 there exst a b a 2 b 2 N wth T a <b and T a 2 <b 2 such that for each n p t 0 for t l et 0 { } { } q a q a...q b q a 2 q a2...q b 2 for t { } q a l q al...q b l l where q a l mn{τ j q a l : j n} holds. Let {η η 2...η n } be an n-tuple satsfyng 2. of Lemma 2.. If there exst a functon H H q and numbers q c ν {q aν q aν2...q bν } such that H 2 ) q c ν q c ν q m m H 2 ) q b ν q c ν [ Q q m) H 2 q m q b ν q m mc ν ) r q m) H 2 q m q a ) ν [ Q q m) H 2 q b ν q m) r q m) H 2 2 q b ν q m) >

11 Advances n Dfference Equatons for ν 2 where H q m q a ) ν : H ) q m q H q m q ν) a ) H q q m 2 q b ν q m) : H q b ν q m) H q b ν q m) ) q q m Qt p 0 t τ 0t τ 0 q ) qt τ 0 q ) k 0 et η 0 p t ) η τ t τ q ) α η qt τ q a ν k 0 0 η 0 η 2.54 then.5 s oscllatory. Notce that Theorem 2.6 does not apply f there s no forcng term that s et 0. In ths case we have the followng theorem. Theorem 2.0. Suppose that for any gven arbtrarly large) T T there exsts a subnterval a b T of T T wherea<bsuch that p t 0 for t a b T n 2.55 where a mn{τ j a : j n} holds. Let {η η 2...η n } be an n-tuple satsfyng 2.2 n Lemma 2.2. If there exst a functon H H T and a number c a b T such that H 2 c a c a H 2 b c [QtH 2 σta rth 2 t a Δt b c [ QtH 2 b σt rsh 2 2 b t2 Δt> where Qt p 0 t τ 0t τ 0 a σt τ 0 a k 0 p t ) ) η τ t τ a α η k 0 σt τ a 2.57 then. wth et 0 s oscllatory. Proof. We wll just hghlght the proof snce t s the same as the proof of Theorem 2.6. We should remark here that takng et 0andη 0 0 n proof of Theorem 2.6 we arrve at w Δ t Q w 0t 2 t rt μtwt u η. 2.58

12 2 Advances n Dfference Equatons The arthmetc-geometrc mean nequalty we now need s u η u η 2.59 where n η and η > nare as n Lemma 2.2. Corollary 2.. Suppose that for any gven arbtrarly large) T t 0 there exsts a subnterval a b of T wheret a<bwth a b R such that p t 0 for t a b n 2.60 where a mn{τ j a : j n} holds. Let {η η 2...η n } be an n-tuple satsfyng 2.2 n Lemma 2.2. If there exst a functon H H R and a number c a b such that H 2 c a c a H 2 b c [QtH 2 t a rth 2 t a dt b c [QsH 2 b t rth 2 2 b t dt > where Qt p 0 t τ 0t τ 0 a t τ 0 a k 0 p t ) ) η τ t τ a α η k 0 t τ a 2.62 then.3 wth et 0 s oscllatory. Corollary 2.2. Suppose that for any gven arbtrarly large) T t 0 there exsts a b Z wth T a<bsuch that p t 0 for t {a a...b} n 2.63 where a mn{τ j a : j n} holds. Let {η η 2...η n } be an n-tuple satsfyng 2.2 n Lemma 2.2. If there exst a functon H H Z and a number c {a a 2...b } such that c [QtH 2 t a rth 2 H 2 c a t a ta b [QtH 2 b t rth 2 H 2 2 b c b t > 0 tc 2.64

13 Advances n Dfference Equatons 3 where H t a : Ht a Ht a Qt p 0 t τ 0t τ 0 a t τ 0 a k 0 H 2 b t : Hb t Hb t p t ) ) η τ t τ a α η k 0 t τ a 2.65 then.4 wth et 0 s oscllatory. Corollary 2.3. Suppose that for any gven arbtrarly large) T t 0 there exst a b N wth T a<bsuch that p t 0 for t { q a q a...q b} n 2.66 where q a mn{τ j q a : j n} holds. Let {η η 2...η n } be an n-tuple satsfyng 2.2 n Lemma 2.2. If there exst a functon H H q N and a number q c {q a q a...q b } such that c H 2 q c q a) q m[ Q q m) H 2 q m q a) r q m) H q m q a ) 2 ma b H 2 q b q c) q [Q m q m) H 2 q b q m) r q m) ) 2 H 2 q b q m > 0 mc 2.67 where H q m q a) : H q m q a) H q m q a) ) H q q m 2 q b q m) : H q b q m) H q b q m) ) q q m Qt p 0 t τ 0t τ 0 q a ) qt τ 0 q a ) k 0 p t ) η τ t τ q a ) α η qt τ q a k then.5 wth et 0 s oscllatory. It s obvous that Theorem 2.6 s not applcable f the functons p t are nonpostve for m m 2...n. In ths case the theorem below s vald. Theorem 2.4. Suppose that for any gven arbtrarly large) T T there exst subntervals a b T and a 2 b 2 T of T T wherea <b and a 2 <b 2 such that p t 0 for t a b T a 2 b 2 T n l et > 0 for t a l b l T l

14 4 Advances n Dfference Equatons where a l mn{τ j a l : j n} holds. If there exst a functon H H T postve numbers λ and ν satsfyng m λ ν m 2.70 and numbers c ν b ν T such that cν [ QtH 2 σta H 2 ν rth 2 c ν t Δt bν [ QtH 2 b H 2 ν σt rth 2 2 b ν c ν b νt Δt >0 c ν 2.7 for ν 2 where Qt p 0 t τ 0t τ 0 m ) σt τ 0 μ λ et /α p /α τ t τ t σt τ m ) β ν et /α /α p τ t τ t σt τ 2.72 wth μ α α /α β α α /α p max { p t 0 } 2.73 then. s oscllatory. Proof. Suppose that. has a nonoscllatory soluton. Wthout losss of generalty we may assume that xt and xτ t n are eventually postve on a b T when a s suffcently large. If xt s eventually negatve one may repeat the same proof step by step on the nterval a 2 b 2 T. Rewrtng. for t a b T as rtx Δ t) Δ p0 txτ 0 t m [ p tx α τ t λ et [ p tx α τ t ν et 0 m and applyng Lemma 2.3 to each term n the frst sum we obtan 2.74 rtx Δ t) Δ p0 txτ 0 t m μ λ et /α p /α txτ t [ p tx α τ t ν et 0 m 2.75

15 Advances n Dfference Equatons 5 where μ α α /α for 2...m. Settng wt rt xδ t xt 2.76 yelds rtx Δ w Δ t ) Δ w 2 t t x σ t rt μtwt Substtutng the above last equalty nto 2.75 we have w Δ t p 0 t xτ 0t m m μ λ et /α p /α t xτ t [ p tx α τ t ν et w 2 t rt μtwt It follows from 2.5 that x α τ t xτ 0 t xτ t τ 0t τ 0 a σt τ 0 a τ t τ a σt τ a x α τ t τ t τ a σt τ a Notce that the second sum n 2.78 can be wrtten as [ p tx α τ t ν et m [ p t xα τ t m ν et [ τ t τ a [ ) α ν et σt τ a xτ t p t xτ t m 2.82 and hence applyng the Lemma 2.4 yelds [ τ t τ a [ ) α ν et σt τ a xτ t p t xτ t m [ τ t τ a β ν et /α /α p t σt τ a m 2.83

16 6 Advances n Dfference Equatons where β α α /α and p max{ p t 0} for m m 2...n. Usng and2.78 nto 2.78 weobtan w Δ t p 0 t τ 0t τ 0 a m [ σt τ 0 a τ t τ a σt τ a [ τ t τ a σt τ a m β ν et /α p /α t μ λ et /α p /α t w 2 t rt μtwt Settng Qt p 0 t τ 0t τ 0 a m ) σt τ 0 a μ λ et /α p /α τ t τ a t σt τ a m ) β ν et /α /α p τ t τ a t σt τ a 2.85 we have w Δ t Qt w 2 t rt μtwt The rest of the proof s the same as that of Theorem 2.6 and hence t s omtted. Corollary 2.5. Suppose that for any gven arbtrarly large) T t 0 there exst subntervals a b and a 2 b 2 of T wheret a <b and T a 2 <b 2 such that p t 0 for t a b a 2 b n l et > 0 for t a l b l l where a l mn{τ j a l : j n} holds. If there exst a functon H H R postve numbers λ and ν satsfyng m λ ν m 2.88 and numbers c ν b ν such that cν [ QtH 2 t a H 2 ν rth 2 c ν t dt bν [ QtH 2 b H 2 ν t rth 2 2 b ν c ν b νt dt > 0 c ν 2.89

17 Advances n Dfference Equatons 7 for ν 2 where Qt p 0 t τ 0t τ 0 t τ 0 m ) μ λ et /α p /α τ t τ t t τ ) β ν et /α /α p τ t τ t t τ m 2.90 wth μ α α /α β α α /α p max { p t 0 } 2.9 then.3 s oscllatory. Corollary 2.6. Suppose that for any gven arbtrarly large) T t 0 there exst a b a 2 b 2 Z wth T a <b and T a 2 <b 2 such that for each n p t 0 for t {a a...b } {a 2 a 2...b 2 } l et > 0 for t {a l a l...b l } l where a l mn{τ j a l : j n} holds. If there exst a functon H H Z postve numbers λ and ν satsfyng m λ ν m 2.93 and numbers c ν { 2...b ν } such that c ν [ QtH 2 t a H 2 ν rth 2 c ν t t b ν [ QtH 2 b H 2 ν t rth 2 2 b ν c ν b νt > 0 tc ν 2.94 for ν 2 where H t : Ht Ht H 2 b ν t : Hb ν t Hb ν t Qt p 0 t τ 0t τ 0 m ) t τ 0 μ λ et /α p /α τ t τ t t τ ) β ν et /α /α p τ t τ t t τ m 2.95

18 8 Advances n Dfference Equatons wth μ α α /α β α α /α p max { p t 0 } 2.96 then.4 s oscllatory. Corollary 2.7. Suppose that for any gven arbtrarly large) T t 0 there exst a b a 2 b 2 N wth T a <b and T a 2 <b 2 such that for each n p t 0 for t l et > 0 { } { } q a q a...q b q a 2 q a2...q b 2 for t { } q a l q al...q b l l where q a l mn{τ j q a l : j n} holds. If there exst a functon H H q postve numbers λ and ν satsfyng m λ ν m 2.98 and numbers q c ν {q aν q aν2...q bν } such that H 2 ) q c ν q c ν q m m H 2 ) q b ν q c ν [ Q q m) H 2 q m q b ν q m mc ν ) rth 2 q m q a ) ν [ Q q m) H 2 q b ν q m) rth 2 2 q b ν q m) > for ν 2 where H q m q a ) ν : H ) q m q H q m q ν) a ) H q q m 2 q b ν q m) : H q b ν q m) H q b ν q m) ) q q m Qt p 0 t τ 0t τ 0 q ) qt τ 0 q ) m μ λ et /α p /α t )) β ν et /α /α τ p t τ q t ) qt τ q m )) τ t τ q qt τ q ) 2.00 wth μ α α /α β α α /α p max { p t 0 } 2.0 then.5 s oscllatory.

19 Advances n Dfference Equatons 9 3. Examples In ths secton we gve three examples when n 2 and α 2 α 2 /2 n.. That s we consder x ΔΔ t p 0 txτ 0 t p t xτ t xτ t p 2 t xτ t /2 xτ 2 t For smplcty we take Ht s t s thush t s H 2 t s. Note that η /3 and η 2 2/3 bylemma 2.2. Example 3.. Let A 0andB C > 0 be constants. Consder the dfferental equaton x t Axt B xt 2 xt 2 C xt /2 xt Let a j b j 2 and c j j N. We calculate ) t j Qt A 3 ) t j t j 3 4 B /3 C 2/3 ) 2/3 ) /3 3.3 t j 2 t j and see that 2.6 holds f 4A 9 BC 2) /3 > Snce all condtons of Corollary 2. are satsfed we conclude that 3.2 s oscllatory when 3.4 holds. Example 3.2. Let A 0andB C > 0 be constants. Defne p 0 t A p t B andp 2 t C for t 0j k k j ; otherwse the functons are defned arbtrarly. Consder the dfference equaton Δ 2 xt p 0 txt p t xt 2 xt 2 p 2 t xt /2 xt Let a 0j b 0j 3 and c 0j. We derve Qt A t 0j t 0j BC 2) /3 t 0j ) 4 2/3 ) /3 3.6 t 3j 3 t 0j 4 and see that postvty n 2.64 satsfes f A 9 BC 2) / > Snce all condtons of Corollary 2.2 are satsfed we conclude that 3.5 s oscllatory f 3.7 holds.

20 20 Advances n Dfference Equatons Example 3.3. Let A 0andB C > 0 be constants. Defne p 0 t A p t B and p 2 t C for t 2 0jk k j ; otherwse the functons are defned arbtrarly. Consder the q-dfference equaton q 2 ) ) ) t Δ 2 qxt p 0 tx p t t t 2 4) x x p 2 t t /2 ) t 4 x x Let a 0j b 0j 3 and c 0j. We have Qt A t 20j 4t BC 2) /3 t 2 0j 0j ) 4 8t 2 0j 2/3 ). 6t 2 0j /3 3.9 We see that 2.67 holds for all A 0andB C > 0. Snce all condtons of Corollary 2.2 are satsfed we conclude that 3.8 s oscllatory f A 0andB C > 0arepostve. Acknowledgments The paper s supported n part by the Scentfc and Research Councl of Turkey TUBITAK under Contract 08T688. The authors would lke to thank the referees for ther valuable comments and suggestons. References M. Bohner and A. Peterson Dynamc Equatons on Tme Scales: An Introducton wth Applcatons Brkhäuser Boston Mass USA M. Bohner and A. Peterson Eds. Advances n Dynamc Equatons on Tme Scales Brkhäuser Boston Mass USA V. Lakshmkantham S. Svasundaram and B. Kaymakçalan Dynamc Systems on Measure Chans vol. 370 of Mathematcs and Its Applcatons Kluwer Academc Publshers Dordrecht The Netherlands M. A. El-Sayed An oscllaton crteron for a forced second order lnear dfferental equaton Proceedngs of the Amercan Mathematcal Socety vol. 8 no. 3 pp Y. G. Sun A note on Nasr s and Wong s papers Journal of Mathematcal Analyss and Applcatons vol. 286 no. pp R. P. Agarwal D. R. Anderson and A. Zafer Interval oscllaton crtera for second-order forced delay dynamc equatons wth mxed nonlneartes Computers and Mathematcs wth Applcatons vol. 59 no. 2 pp R. P. Agarwal and A. Zafer Oscllaton crtera for second-order forced dynamc equatons wth mxed nonlneartes Advances n Dfference Equatons vol Artcle ID pages D. R. Anderson Oscllaton of second-order forced functonal dynamc equatons wth oscllatory potentals Journal of Dfference Equatons and Applcatons vol. 3 no. 5 pp D. R. Anderson and A. Zafer Interval crtera for second-order super-half-lnear functonal dynamc equatons wth delay and advanced arguments to appear n Journal of Dfference Equatons and Applcatons. 0 A. F. Güvenlr and A. Zafer Second-order oscllaton of forced functonal dfferental equatons wth oscllatory potentals Computers & Mathematcs wth Applcatons vol. 5 no. 9-0 pp M. Bohner and C. C. Tsdell Oscllaton and nonoscllaton of forced second order dynamc equatons Pacfc Journal of Mathematcs vol. 230 no. pp M. Bohner and S. H. Saker Oscllaton of second order nonlnear dynamc equatons on tme scales The Rocky Mountan Journal of Mathematcs vol. 34 no. 4 pp

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