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Hindawi Publishing Corporation Advances in Difference Equations Volume 2009, Article ID 756171, 13 pages doi:10.1155/2009/756171 Research Article Oscillation for Second-Order Nonlinear Delay Dynamic Equations on Time Scales Zhenlai Han,1, Tongxing Li,2 Shurong Sun,2 and Chenghui Zhang1 School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China School of science, University of Jinan, Jinan, Shandon 250022, China Correspondence should be addressed to Zhenlai Han, hanzhenlai@163.com Received December 2008; Revised 27 February 2009; Accepted 25 May 2009 Recommended by Alberto Cabada By means of Riccati transformation technique, we establish some new oscillation criteria for the γ Δ second-order nonlinear delay dynamic equations rtxΔ t ptfxτt on a time scale T; here γ > is a quotient of odd positive integers with r and p real-valued positive rdcontinuous functions defined on T Our results not only extend some results established by Hassan in 2008 but also unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation Copyright q 2009 Zhenlai Han et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Introduction The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D Thesis in 1988 in order to unify continuous and discrete analysis see Hilger 1 Several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al 2 and references cited therein A book on the subject of time scales, by Bohner and Peterson 3 , summarizes and organizes much of the time scale calculus We refer also to the last book by Bohner and Peterson 4 for advances in dynamic equations on time scales For the notation used hereinafter we refer to the next section that provides some basic facts on time scales extracted from Bohner and Peterson 3 A time scale T is an arbitrary closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations Many other interesting time scales exist, and they give rise to plenty of applications, among them the study of population dynamic models which are discrete in season and may follow a difference scheme with variable step-size or often modeled Advances in Difference Equations by continuous dynamic systems, die out, say in winter, while their eggs are incubating or dormant, and then in season again, hatching gives rise to a nonoverlapping population see Bohner and Peterson 3 Not only does the new theory of the so-called “dynamic equations” unify the theories of differential equations and difference equations but also it extends these classical cases to cases “in between”, for example, to the so-called q-difference equations when T qN0 {qt : t ∈ N0 , q > 1} which has important applications in quantum theory and can be applied on different types of time scales like T hN, T N2 , and T Tn the space of the harmonic numbers In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various equations on time scales, and we refer the reader to Bohner and Saker 5 , Erbe 6 , and Hassan 7 However, there are few results dealing with the oscillation of the solutions of delay dynamic equations on time scales 8–15 Following this trend, in this paper, we are concerned with oscillation for the secondorder nonlinear delay dynamic equations γ Δ ptfxτt 0, rt xΔ t t ∈ T 1.1 We assume that γ > is a quotient of odd positive integers, r and p are positive, real : valued rd-continuous functions defined on T, τ : T → R is strictly increasing, and T τT ⊂ T is a time scale, τt ≤ t and τt → ∞ as t → ∞, f ∈ CR, R which satisfies for some positive constant L, fx/xγ ≥ L, for all nonzero x For oscillation of the second-order delay dynamic equations, Agarwal et al 8 considered the second-order delay dynamic equations on time scales xΔΔ t ptxτt 0, t ∈ T, 1.2 and established some sufficient conditions for oscillation of 1.2 Zhang and Shanliang 15 studied the second-order nonlinear delay dynamic equations on time scales xΔΔ t ptfxt − τ 0, t ∈ T, 1.3 and the second-order nonlinear dynamic equations on time scales xΔΔ t ptfxσt 0, t ∈ T, 1.4 where τ ∈ R and t − τ ∈ T, f : R → R is continuous and nondecreasing f u > k > 0, and ufu > for u / 0, and they established the equivalence of the oscillation of 1.3 and 1.4, from which obtained some oscillation criteria and comparison theorems for 1.3 However, the results established in 15 are valid only when the graininess function μt is bounded which is a restrictive condition Also the restriction f u > k > is required Sahiner 13 considered the second-order nonlinear delay dynamic equations on time scales xΔΔ t ptfxτt 0, t ∈ T, 1.5 Advances in Difference Equations where f : R → R is continuous, ufu > for u / and |fu| ≥ L|u|, τ : T → T, τt ≤ t and limt → ∞ τt ∞, and he proved that if there exists a Δ-differentiable function δt such that 2 t σs δΔ s τs σ psδ s − Δs ∞, lim sup σs 4Lk2 δσ sτs t→∞ t0 1.6 then every solution of 1.5 oscillates Now, we observe that the condition 1.6 depends on an additional constant k ∈ 0, 1 which implies that the results are not sharp see Erbe et al 10 Han et al 12 investigated the second-order Emden-Fowler delay dynamic equations on time scales xΔΔ t ptxγ τt 0, t ∈ T, 1.7 established some sufficient conditions for oscillation of 1.7, and extended the results given in 8 Erbe et al 10 considered the general nonlinear delay dynamic equations on time scales ptxΔ t Δ qtfxτt 0, t ∈ T, 1.8 where p and q are positive, real-valued rd-continuous functions defined on T, τ : T → T is rd-continuous, τt ≤ t and τt → ∞ as t → ∞, and f ∈ CR, R such that satisfies for some positive constant L, |fx| ≥ L|x|, xfx > for all nonzero x, and they extended the generalized Riccati transformation techniques in the time scales setting to obtain some new oscillation criteria which improve the results given by Zhang and Shanliang 15 and Sahiner 13 Clearly, 1.2, 1.3, 1.5, and 1.8 are the special cases of 1.1 In this paper, we consider the second-order nonlinear delay dynamic equation on time scales 1.1 As we are interested in oscillatory behavior, we assume throughout this paper that the given time scale T is unbounded above We assume t0 ∈ T, and it is convenient to assume t0 > We define the time scale interval of the form t0 , ∞T by t0 , ∞T t0 , ∞ ∩ T We shall also consider the two cases ∞ t0 ∞ t0 rt rt 1/γ 1/γ Δt ∞, 1.9 Δt < ∞ 1.10 The paper is organized as follows In Section 2, we intend to use the Riccati transformation technique, a simple consequence of Keller’s chain rule, and an inequality to obtain some sufficient conditions for oscillation of all solutions of 1.1 In Section 3, we give an example in order to illustrate the main results 4 Advances in Difference Equations Main Results In this section, we give some new oscillation criteria for 1.1 In order to prove our main results, we will use the formula xtγ Δ 1 γ hxσ t 1 − hxt γ−1 xΔ tdh, 2.1 where xt is delta differentiable and eventually positive or eventually negative, which is a simple consequence of Keller’s chain rule see Bohner and Peterson 3, Theorem 1.90 Also, we need the following auxiliary result For convenience, we note that d t max{0, dt}, d− t max{0, −dt}, Rt : r 1/γ t t t1 Rt , αt : Rt μt βt : αt, < γ ≤ 1; βt : α t, γ Δs r 1/γ s , 2.2 γ > Lemma 2.1 Assume that 1.9 holds and assume further that xt is an eventually positive solution of 1.1 Then there exists a t1 ∈ t0 , ∞T such that xΔ t > 0, xt > RtxΔ t, γ Δ < 0, rt xΔ t xt > αt, xσ t t ∈ t1 , ∞T 2.3 The proof is similar to that of Hassan 7, Lemma 2.1 , and so is omitted : τT ⊂ T is Lemma 2.2 Chain Rule Assume that τ : T → R is strictly increasing and T Δ Δ a time scale, τσt στt Let x : T → R If τ t, and let x τt exist for t ∈ Tκ , then xτtΔ exist, and xτtΔ xΔ τtτ Δ t 2.4 −1 Proof Let < ε < be given and define ε∗ ε1 |τ Δ t| |xΔ τt| Note that < ε∗ < According to the assumptions, there exist neighborhoods N1 of t and N2 of τt such that τσt − τs − σt − sτ Δ t ≤ ε∗ |σt − s|, s ∈ N1 , xστt − xr − στt − rxΔ τt ≤ ε∗ |στt − r|, r ∈ N2 2.5 Advances in Difference Equations Put N N1 τ −1 N2 and let s ∈ N Then s ∈ N1 and τs ∈ N2 and xτσt − xτs − σt − sxΔ τtτ Δ t xτσt − xτs − στt − τsxΔ τt στt − τs − σt − sτ Δ t xΔ τt xστt − xτs − στt − τsxΔ τt τσt − τs − σt − sτ Δ t xΔ τt ≤ ε∗ |στt − τs| ε∗ |σt − s| xΔ τt ≤ ε∗ στt − τs − σt − sτ Δ t |σt − s| τ Δ t |σt − s| xΔ τt ε∗ τσt − τs − σt − sτ Δ t |σt − s| τ Δ t |σt − s| xΔ τt ≤ ε∗ ε∗ |σt − s| |σt − s| τ Δ t |σt − s| xΔ τt ε∗ |σt − s| ε∗ τ Δ t xΔ τt ≤ ε∗ |σt − s| τ Δ t xΔ τt 2.6 ε|σt − s| The proof is completed Theorem 2.3 Assume that 1.9 holds and τ ∈ Crd t0 , ∞T , T, τσt στt Furthermore, assume that there exists a positive function δ ∈ Crd t0 , ∞T , R and for all sufficiently large t1 , one has ⎞ Δ γ1 rτs δ s lim sup ⎝Lpsαγ τsδσ s − γ ⎠Δs ∞ γ1 σ t→∞ t1 γ 1 δ sβτsτ Δ s t ⎛ 2.7 Then 1.1 is oscillatory on t0 , ∞T Proof Suppose that 1.1 has a nonoscillatory solution xt on t0 , ∞T We may assume without loss of generality that xt > and xτt > for all t ∈ t1 , ∞T , t1 ∈ t0 , ∞T We shall consider only this case, since the proof when xt is eventually negative is similar In view of Lemma 2.1, we get 2.3 Define the function ωt by ωt δt γ rt xΔ t xτtγ , t ∈ t1 , ∞T 2.8 Advances in Difference Equations Then ωt > In view of 1.1 and 2.8 we get γ Δ δσ trt xΔ t xτtγ δΔ t xτtγ ω t ≤ −Lptδ t ωt − δt xτ σ tγ xτtγ xτ σ tγ Δ σ 2.9 When < γ ≤ 1, using 2.1 and 2.4, we have xτt γ Δ γ 1 γ−1 hxτtσ 1 − hxτt xτtΔ dh ≥γ 1 hxτ σ t 1 − hxτ σ t γ−1 xτtΔ dh 2.10 γxτ σ tγ−1 xΔ τtτ Δ t So, by 2.9 γ γδσ trt xΔ t xΔ τtτ Δ t δΔ t xτtγ ωt − , ω t ≤ −Lptδ t δt xτ σ tγ xτtγ xτ σ t Δ σ 2.11 hence, we get δΔ t xτtγ ωt γ δt xτ σ t γ1 Δ γδσ trt xΔ t τ t xτt xΔ τt − , xτ σ t xΔ t xτtγ1 ωΔ t ≤ −Lptδσ t 2.12 γ Δ and using Lemma 2.1, by rtxΔ t < 0, we find xΔ τt/xΔ t ≥ rt/rτt1/γ , by xt/xσ t > αt, we get xτt/xτ σ t > ατt, and so we obtain ωΔ t ≤ −Lptδσ tαγ τt − γδσ t δΔ t ωt δt xτt τ Δ t ωtγ1/γ xτ σ t r 1/γ τtδtγ1/γ δΔ t ωt ≤ −Lptδσ tαγ τt δt − γδσ tατt τ Δ t r 1/γ τtδtγ1/γ ωtγ1/γ 2.13 Advances in Difference Equations When γ > 1, using 2.1 and 2.4, we have xτtγ Δ γ 1 hxτtσ 1 − hxτt γ−1 xτtΔ dh ≥γ 1 hxτt 1 − hxτt γ−1 xτtΔ dh 2.14 γxτtγ−1 xΔ τtτ Δ t So, we get δΔ t xτtγ ωt γ δt xτ σ t γ1 γ rt xΔ t x τt xΔ τt Δ σ − γδ t τ t γ xτtγ1 x τ σ t xΔ t ωΔ t ≤ −Lptδσ t δΔ t ωt ≤ −Lptδσ tαγ τt δt − γδσ tαγ τt τ Δ t r 1/γ τtδtγ1/γ 2.15 ωtγ1/γ So by the definition of βt, we obtain, for γ > 0, δΔ t ω t ≤ −Lptδ tα τt δt Δ σ γ ωt − γδσ tβτt τ Δ t ωλ t, r λ−1 τtδλ t 2.16 where λ : γ 1/γ Define A ≥ and B ≥ by λ A : γδσ tβτtτ Δ t δtλ r λ−1 τt ω t, λ B λ−1 r λ−1/λ τt δΔ t : 1/λ λ γδσ tβτtτ Δ t 2.17 Then, using the inequality λABλ−1 − Aλ ≤ λ − 1B λ , λ≥1 2.18 yields δΔ t δt γ1 rτt δΔ t τ Δ t λ ωt − γδ tβτt λ−1 ω t ≤ γ γ1 σ r τtδλ t γ 1 δ tβτtτ Δ t σ 2.19 Advances in Difference Equations From 2.16, we find γ1 rτt δΔ t ω t ≤ −Lptδ tα τt γ γ1 σ γ 1 δ tβτtτ Δ t Δ σ γ 2.20 Integrating the inequality 2.20 from t1 to t, we obtain ⎞ γ1 rτs δΔ s ⎝Lpsα τsδ s − γ ⎠Δs ≤ ωt1 − ωt ≤ ωt1 , γ1 σ t1 γ 1 δ sβτsτ Δ s 2.21 t ⎛ γ σ which contradicts 2.7 The proof is completed Remark 2.4 From Theorem 2.3, we can obtain different conditions for oscillation of all solutions of 1.1 with different choices of δt t0 , ∞T , T, τσt στt Furthermore, Theorem 2.5 Assume that 1.9 holds and τ ∈ Crd assume that there exist functions H, h ∈ Crd D, R, where D ≡ {t, s : t ≥ s ≥ t0 } such that Ht, t 0, t ≥ t0 , Ht, s > 0, t > s ≥ t0 , 2.22 and H has a nonpositive continuous Δ-partial derivation H Δs t, s with respect to the second variable and satisfies H Δs σt, s Hσt, σs δΔ s ht, s − Hσt, σsγ/γ1 , δs δs 2.23 and for all sufficiently large t1 , lim sup Hσt, t1 t→∞ σt Kt, sΔs ∞, 2.24 rτsh− t, sγ1 γ γ1 σ γ 1 δ sβτsτ Δ s 2.25 t1 where δt is a positive Δ-differentiable function and Kt, s LHσt, σsαγ τsδσ sps − Then 1.1 is oscillatory on t0 , ∞T Proof Suppose that 1.1 has a nonoscillatory solution xt on t0 , ∞T We may assume without loss of generality that xt > and xτt > for all t ∈ t1 , ∞T , t1 ∈ t0 , ∞T Advances in Difference Equations We proceed as in the proof of Theorem 2.3, and we get 2.16 Then from 2.16 with δ Δ t replaced by δΔ t, we have Lptδσ tαγ τt ≤ −ωΔ t τ Δ t δΔ t ωt − γδσ tβτt λ−1 ωλ t δt r τtδλ t 2.26 Multiplying both sides of 2.26, with t replaced by s, by Hσt, σs, integrating with respect to s from t1 to σt, we get σt Hσt, σsLpsδσ sαγ τsΔs t1 ≤− σt Δ Hσt, σsω sΔs t1 − σt σt Hσt, σs t1 Hσt, σsγδσ sβτs t1 δΔ s ωsΔs δs 2.27 τ Δ s ωλ sΔs r λ−1 τsδλ s Integrating by parts and using 2.22 and 2.23, we obtain σt Hσt, σsLpsδσ sαγ τsΔs t1 ≤ Hσt, t1 ωt1 σt H Δs σt, sωsΔs t1 − σt σt Hσt, σs t1 Hσt, σsγδσ sβτs t1 τ Δ s r λ−1 τsδλ s δΔ s ωsΔs δs ωλ sΔs ≤ Hσt, t1 ωt1 σt ht, s − Hσt, σs1/λ ωs δs t1 τ Δ s λ −Hσt, σsγδ sβτs λ−1 ω s Δs r τsδλ s σ ≤ Hσt, t1 ωt1 σt t1 h− t, s Hσt, σs1/λ ωs δs τ Δ s λ −Hσt, σsγδ sβτs λ−1 ω s Δs r τsδλ s σ 2.28 10 Advances in Difference Equations Again, let λ : γ 1/γ, define A ≥ and B ≥ by Aλ : Hσt, σsγδσ sβτs B λ−1 τ Δ s r λ−1 τsδsλ ωλ s, h− t, sr λ−1/λ τs : 1/λ , λ γδσ sβτsτ Δ s 2.29 and using the inequality λABλ−1 − Aλ ≤ λ − 1B λ , λ ≥ 1, 2.30 we find h− t, s τ Δ s ωλ s Hσt, σs1/λ ωs − Hσt, σsγδσ sβτs λ−1 δs r τsδλ s rτsh− t, sγ1 ≤ γ γ1 σ γ 1 δ sβτsτ Δ s 2.31 Therefore, from the definition of Kt, s, we obtain σt Kt, sΔs ≤ Hσt, t1 ωt1 , 2.32 t1 and this implies that Hσt, t1 σt Kt, sΔs ≤ ωt1 , 2.33 t1 which contradicts 2.24 This completes the proof Now, we give some sufficient conditions when 1.10 holds, which guarantee that every solution of 1.1 oscillates or converges to zero on t0 , ∞T Theorem 2.6 Assume 1.10 holds and τ ∈ Crd t0 , ∞T , T, τσt στt Furthermore, t0 , ∞T , R , for all sufficiently large t1 , such assume that there exists a positive function δ ∈ Crd t0 , ∞T , R, that 2.7 or 2.22, 2.23, and 2.24 hold If there exists a positive function η ∈ Crd Δ η t ≥ 0, such that ∞ t0 ηtrt t 1/γ η spsΔs σ Δt ∞, t0 then every solution of 1.1 is either oscillatory or converges to zero on t0 , ∞T 2.34 Advances in Difference Equations 11 Proof We proceed as in Theorem 2.3 or Theorem 2.5, and we assume that 1.1 has a nonoscillatory solution such that xt > 0, and xτt > 0, for all t ∈ t1 , ∞T From the proof of Lemma 2.1, we see that there exist two possible cases for the sign of xΔ t The proof when xΔ t is an eventually positive is similar to that of the proof of Theorem 2.3 or Theorem 2.5, and hence it is omitted Next, suppose that xΔ t < for t ∈ t1 , ∞T Then xt is decreasing and limt → ∞ xt b ≥ We assert that b If not, then xτt ≥ xt ≥ xσt ≥ b > for t ∈ t1 , ∞T Since fxτt ≥ Lxτtγ , there exists a number t2 ∈ t1 , ∞T such that fxτt ≥ Lbγ for t ≥ t2 γ Defining the function ut ηtrtxΔ t , we obtain from 1.1 γ γ Δ uΔ t ηΔ trt xΔ t ησ t rt xΔ t ≤ −ησ tptfxτt ≤ −Lησ tptxτtγ ≤ −Lbγ ησ tpt, t ∈ t2 , ∞T 2.35 Hence, for t ∈ t2 , ∞T , we have ut ≤ ut2 − Lb γ t η spsΔs ≤ −Lb σ γ t2 t 2.36 ησ spsΔs, t2 γ because of ut2 ηt2 rt2 xΔ t2 < So, we have xt − xt2 t Δ x sΔs ≤ −L t2 1/γ t b t2 ηsrs s 1/γ η τpτΔτ σ Δs 2.37 t2 By condition 2.34, we get xt → −∞ as t → ∞, and this is a contradiction to the fact that xt > for t ≥ t1 Thus b and then xt → as t → ∞ The proof is completed Application and Example Hassan 7 considered the second-order half-linear dynamic equations on time scales γ Δ rt xΔ t ptxγ t 0, t ∈ T, 3.1 where γ > is a quotient of odd positive integers, and r and p are positive, real-valued rdcontinuous functions defined on T, and he established some new oscillation criteria of 3.1 For example 12 Advances in Difference Equations Theorem 3.1 Hassan 7, Theorem 2.1 Assume that 1.9 holds Furthermore, assume that there t0 , ∞, R such that exists a positive function δ ∈ Crd Δ γ1 ⎞ rs δ s lim sup ⎝psαγ sδσ s − γ ⎠Δs ∞ γ1 σ t→∞ t0 γ 1 δ sβs t ⎛ 3.2 Then 3.1 is oscillatory on t0 , ∞T We note that 1.1 becomes 3.1 when fx ≡ xγ , τt ≡ t, and Theorem 2.3 becomes Theorem 3.1, and so Theorem 2.3 in this paper essentially includes results of Hassan 7, Theorem 2.1 Similarly, Theorem 2.5 includes results of Hassan 7, Theorem 2.2 , and Theorem 2.6 includes results of Hassan 7, Theorem 2.4 One can easily see that nonlinear delay dynamic equations 1.8 considered by Erbe et al 10 are the special cases of 1.1, and the results obtained in 10 cannot be applied in 1.1, and so our results are new Example 3.2 Consider the second-order delay dynamic equations xΔ t γ Δ στtγ γ x τt 0, tτ γ t t ∈ t0 , ∞T , 3.3 where γ > is a quotient of odd positive integers, τt ≤ t, and τT ⊂ T is a time scale, limt → ∞ τt ∞, τ Δ t > 0, τσt στt Let rt 1, pt στtγ /tτ γ t, fx xγ , L 1, t ∈ t0 , ∞T Then condition 1.9 holds, Rt t − t1 , and we can find < k < such that αt Rt/Rt μt t − t1 /σt − t1 ≥ kt/σt, ατt ≥ kτt/στt for t ≥ tk > t1 ≥ t0 Take δt By Theorem 2.3, we obtain ⎞ γ1 rτs δΔ s lim sup ⎝Lpsα τsδ s − γ ⎠Δs γ1 σ t→∞ t1 γ 1 δ sβτsτ Δ s t ⎛ lim sup t→∞ γ t t1 σ psαγ τsΔs t 3.4 στsγ τ γ s ≥ kγ lim sup Δs sτ γ s στsγ t→∞ tk t Δs γ k lim sup ∞ t→∞ tk s We conclude that 3.3 is oscillatory Acknowledgments The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have lead to the present improved version of the original manuscript Advances in Difference Equations 13 This research is supported by the Natural Science Foundation of China 60774004, China Postdoctoral Science Foundation Funded Project 20080441126, Shandong Postdoctoral Funded Project 200802018 and supported by Shandong Research Funds Y2008A28, and also supported by the University of Jinan Research Funds for Doctors B0621, XBS0843 References 1 S Hilger, “Analysis on measure chains—a unified approach to continuous and 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second-order delay differential equations on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol 63, no 5–7, pp e1073–e1080, 2005 14 Y Sahiner, “Oscillation of second-order neutral delay and mixed-type dynamic equations on time scales,” Advances in Difference Equations, vol 2006, Article ID 65626, pages, 2006 15 B G Zhang and Z Shanliang, “Oscillation of second-order nonlinear delay dynamic equations on time scales,” Computers & Mathematics with Applications, vol 49, no 4, pp 599–609, 2005
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