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Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 237219, 14 pages doi:10.1155/2011/237219 Research Article Asymptotic Behavior of Solutions of Higher-Order Dynamic Equations on Time Scales Taixiang Sun,1 Hongjian Xi,2 and Xiaofeng Peng1 College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi 530003, China Correspondence should be addressed to Taixiang Sun, stx1963@163.com Received 18 November 2010; Accepted 23 February 2011 Academic Editor: Abdelkader Boucherif Copyright q 2011 Taixiang Sun 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 We investigate the asymptotic behavior of solutions of the following higher-order dynamic n n−1 equation xΔ t f t, x t , xΔ t , , xΔ t 0, on an arbitrary time scale T, where the n function f is defined on T × R We give sufficient conditions under which every solution x of n−1 this equation satisfies one of the following conditions: limt → ∞ xΔ t 0; there exist n−1 constants ≤ i ≤ n − with a0 / 0, such that limt → ∞ x t / i hn−i−1 t, t0 1, where hi t, t0 ≤ i ≤ n − are as in Main Results Introduction In this paper, we investigate the asymptotic behavior of solutions of the following higherorder dynamic equation xΔ t n f t, x t , xΔ t , , xΔ n−1 t 0, 1.1 on an arbitrary time scale T, where the function f is defined on T × Rn Since we are interested in the asymptotic and oscillatory behavior of solutions near {t ∈ T : infinity, we assume that sup T ∞, and define the time scale interval t0 , ∞ T t ≥ t0 }, where t0 ∈ T By a solution of 1.1 , we mean a nontrivial real-valued function n x ∈ Crd Tx , ∞ T , R , Tx ≥ t0 , which has the property that xΔ t ∈ Crd Tx , ∞ T , R and satisfies 1.1 on Tx , ∞ T , where Crd is the space of rd-continuous functions The solutions vanishing in some neighborhood of infinity will be excluded from our consideration A solution x of 1.1 is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is called nonoscillatory 2 Advances in Difference Equations The theory of time scales, which has recently received a lot of attention, was introduced by Hilger’s landmark paper in order to create a theory that can unify continuous and discrete analysis The cases when a time scale is equal to the real numbers 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 many applications see Not only the new theory of the so-called “dynamic equations” unifies the theories of differential equations and difference equations but also extends these classical cases to cases “in between,” for example, to the so-called q-difference equations when T qN0 , which has important applications in quantum theory see On a time scale T, the forward jump operator, the backward jump operator, and the graininess function are defined as σ t inf{s ∈ T : s > t}, sup{s ∈ T : s < t}, ρ t μ t σ t − t, 1.2 respectively We refer the reader to 2, for further results on time scale calculus Let p ∈ Crd T, R with μ t p t / 0, for all t ∈ T, then the delta exponential function ep t, t0 is defined as the unique solution of the initial value problem yΔ p t y, y t0 1.3 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 5–18 Recently, Erbe et al 19–21 considered the asymptotic behavior of solutions of the third-order dynamic equations a t r t xΔ t Δ xΔΔΔ t a t r t xΔ t Δ p t f x t p t x t Δ γ 0, 0, 1.4 Δ f t, x t 0, respectively, and established some sufficient conditions for oscillation Karpuz 22 studied the asymptotic nature of all bounded solutions of the following higher-order nonlinear forced neutral dynamic equation x t At x α t Δn f t, x β t , x γ t 1.5 ϕt Chen 23 derived some sufficient conditions for the oscillation and asymptotic behavior of the nth-order nonlinear neutral delay dynamic equations a tΨ x t x t p t x τ t Δn−1 α−1 x t p t x τ t Δn−1 γ Δ λF t, x δ t 0, 1.6 Advances in Difference Equations on an arbitrary time scale T Motivated by the above studies, in this paper, we study 1.1 and give sufficient conditions under which every solution x of 1.1 satisfies one of the following n−1 0; there exist constants ≤ i ≤ n − with a0 / 0, such conditions: limt → ∞ xΔ t n−1 1, where hi t, t0 ≤ i ≤ n − are as in Section that limt → ∞ x t / i hn−i−1 t, t0 Main Results Let k be a nonnegative integer and s, t ∈ T, then we define a sequence of functions hk t, s as follows: ⎧ ⎪1 ⎪ ⎨ hk t, s if k t ⎪ h ⎪ ⎩ k−1 τ, s Δτ 0, 2.1 if k ≥ s To obtain our main results, we need the following lemmas Lemma 2.1 Let n be a positive integer, then there exists Tn > t0 , such that hk t, t0 − hk t, t0 ≥ for t ≥ Tn , ≤ k ≤ n − Proof We will prove the above by induction First, if k h1 t, t0 − h0 t, t0 t t, t0 − hm t, t0 Thus, t − t0 − ≥ for t ≥ T1 Next, we assume that there exists Tm > t0 , such that hk ≤ k ≤ m with ≤ m < n − 1, then hm 0, then we take T1 ≥ t0 2.2 2.3 t, t0 − hk t, t0 ≥ for t ≥ Tm and hm τ, t0 − hm−1 τ, t0 Δτ t0 Tm hm τ, t0 − hm−1 τ, t0 Δτ t t0 ≥ Tm t hm τ, t0 − hm−1 τ, t0 Δτ Tm hm τ, t0 − hm−1 τ, t0 Δτ Δτ Tm t0 Tm 2.4 hm τ, t0 − hm−1 τ, t0 Δτ t − Tm , t0 from which it follows that there exists Tm > Tm , such that hk and ≤ k ≤ m The proof is completed t, t0 −hk t, t0 ≥ for t ≥ Tm Advances in Difference Equations Lemma 2.2 see 24 Let p ∈ Crd T, 0, ∞ , then t t p s Δs p s Δs ≤ ep t, t0 ≤ e t0 2.5 t0 Lemma 2.3 see Let y, p ∈ Crd T, 0, ∞ t y t ≤A and A ∈ 0, ∞ , then y τ p τ Δτ, ∀t ∈ T 2.6 t0 implies y t ≤ Aep t, t0 , ∀t ∈ T 2.7 Lemma 2.4 see Let n be a positive integer Suppose that x is n times differentiable on T Let n−1 α ∈ T κ and t ∈ T, then n−1 x t ρn−1 t hk t, α xΔ α k hn−1 t, σ τ xΔ τ Δτ n 2.8 α k Lemma 2.5 see Assume that f and g are differentiable on T with limt → ∞ g t exists T > t0 , such that g t > 0, g Δ t > 0, ∞ If there ∀t ≥ T, 2.9 then fΔ t t → ∞ gΔ t r or ∞ implies lim lim f t t t→∞g r or ∞ 2.10 Lemma 2.6 see 23 Let x be defined on t0 , ∞ T , and x t > with xΔ t ≤ for t ≥ t0 and not eventually zero If x is bounded, then n limt → ∞ xΔ t i −1 i Δn−i x for ≤ i ≤ n − 1, t > for all t ≥ t0 and ≤ i ≤ n − Now, one states and proves the main results Theorem 2.7 Assume that there exists t1 > t0 , such that the function f t, u0 , , un−1 satisfies f t, u0 , , un−1 ≤ n−1 i pi t |ui |, ∀ t, u0 , , un−1 ∈ t1 , ∞ T × Rn , 2.11 Advances in Difference Equations ≤ i ≤ n − are nonnegative functions on t1 , ∞ where pi t T and lim eq t, t1 < ∞, 2.12 t→∞ with q t conditions: n−1 i t ≥ t1 , then every solution x of 1.1 satisfies one of the following pi t hn−i−1 t, t0 limt → ∞ xΔ n−1 t 0, there exist constants ≤ i ≤ n − with a0 / 0, such that x t lim n−1 i t→∞ hn−i−1 t, t0 2.13 Proof Let x be a solution of 1.1 , then it follows from Lemma 2.4 that for ≤ m ≤ n − 1, x Δm n−m−1 hk t, t1 x t Δk m ρn−m−1 t t1 hn−m−1 t, σ τ xΔ τ Δτ n for t ≥ t1 2.14 t1 k By 2.11 and Lemma 2.1, we see that there exists T > t1 , such that for t ≥ T and ≤ m ≤ n − 1, xΔ t m n−m−1 ≤ hn−m−1 t, t0 xΔ k m t n−1 t1 pi τ xΔ τ Δτ i 2.15 t1 i k Then we obtain xΔ t m ≤ hn−m−1 t, t0 F t for t ≥ T, ≤ m ≤ n − 1, 2.16 where t n−1 F t A pi τ xΔ τ Δτ, i 2.17 T i with n−m−1 A max 0≤m≤n−1 xΔ k m T n−1 t1 pi τ xΔ τ Δτ i 2.18 t1 i k Using 2.16 and 2.17 , it follows that F t ≤A t n−1 T i pi τ hn−i−1 τ, t0 F τ Δτ for t ≥ T 2.19 Advances in Difference Equations By Lemma 2.3, we have F t ≤ Aeq t, T ∀t ≥ T, 2.20 n−1 with q t i pi t hn−i−1 t, t0 Hence from 2.12 , there exists a finite constant c > 0, such that F t ≤ c for t ≥ T Thus, inequality 2.20 implies that xΔ t m ≤ hn−m−1 t, t0 c for t ≥ T, ≤ m ≤ n − 2.21 By 1.1 , we see that if t ≥ T , then xΔ n−1 t xΔ n−1 t T − f τ, x τ , xΔ τ , , xΔ n−1 τ Δτ 2.22 T Since condition 2.12 and Lemma 2.2 implies that t n−1 lim t→∞ pi τ hn−i−1 τ, t0 Δτ < ∞, 2.23 T i we find from 2.11 and 2.21 that the sum in 2.22 converges as t → ∞ Therefore, n−1 n−1 a0 If a0 / 0, then it follows limt → ∞ xΔ t exists and is a finite number Let limt → ∞ xΔ t from Lemma 2.5 that x t t → ∞ hn−1 t, t0 lim xΔ lim t→∞ n−1 t a0 , 2.24 and x has the desired asymptotic property The proof is completed Theorem 2.8 Assume that there exist functions pi : t0 , ∞ T → 0, ∞ ≤ i ≤ n , and nondecreasing continuous functions gi : 0, ∞ → 0, ∞ ≤ i ≤ n − , and t1 > t0 such that ≤ f t, u0 , , un−1 n−1 pi t gi i |ui | hn−i−1 t, t0 pn t for t ≥ t1 , 2.25 with ∞ pi t Δt Pi < ∞ for ≤ i ≤ n, t1 ∞ ε ds n−1 i gi s 2.26 ∞ for any ε > 0, Advances in Difference Equations then every solution x of 1.1 satisfies one of the following conditions: limt → ∞ xΔ n−1 t 0, there exist constants ≤ i ≤ n − with a0 / such that x t lim n−1 i t→∞ hn−i−1 t, t0 2.27 Proof Let x be a solution of 1.1 , then it follows from Lemma 2.4 that for ≤ m ≤ n − 1, xΔ t m n−m−1 hk t, t1 xΔ k m ρn−m−1 t t1 hn−m−1 t, σ τ xΔ τ Δτ n for t ≥ t1 2.28 t1 k By Lemma 2.1 and 2.25 , we see that there exists T > t1 , such that for t ≥ T and ≤ m ≤ n − 1, ⎡ xΔ t m n−m−1 ≤ hn−m−1 t, t0 ⎣ xΔ k m ⎡ t ⎣ t1 t1 k n−1 ⎛ pi τ gi ⎝ i xΔ τ i hn−i−1 τ, t0 ⎞ ⎤ ⎠ pn τ ⎦Δτ ⎦ ⎤ 2.29 Then, we obtain xΔ t m ≤ hn−m−1 t, t0 F t , for t ≥ T, ≤ m ≤ n − 1, 2.30 where t n−1 F t A ⎛ pi τ gi ⎝ T i ⎞ xΔ τ i ⎠Δτ, hn−i−1 τ, t0 2.31 with n−m−1 A max 0≤m≤n−1 xΔ k m T n−1 t1 t1 i k ⎛ pi τ gi ⎝ xΔ τ i hn−i−1 τ, t0 ⎞ ⎠Δτ Pn 2.32 Using 2.30 and 2.31 , it follows that F t ≤A t n−1 T i pi τ gi F τ Δτ for t ≥ T 2.33 Advances in Difference Equations Write t n−1 u t A pi τ gi F τ Δτ for t ≥ T, 2.34 T i y G y A ds n−1 i gi s , 2.35 then Δ G u t uΔ t G hu t − h uσ t dh n−1 pi t gi F t i n−1 i pi n−1 i ≤ ≤ dh n−1 i − h uσ t gi hu t 2.36 t gi u t gi u t n−1 pi t , i from which it follows that G u t t n−1 ≤G u T n−1 pi τ Δτ ≤ G u T Pi T i 2.37 i ∞ and G y is strictly increasing, there exists a constant c > 0, such that Since limy → ∞ G y u t ≤ c for t ≥ T By 2.30 , 2.33 , and 2.34 , we have xΔ t m ≤ hn−m−1 t, t0 c for t ≥ T, ≤ m ≤ n − 2.38 It follows from 1.1 that if t ≥ T , then xΔ n−1 t xΔ n−1 T − t T f τ, x τ , xΔ τ , , xΔ n−1 τ Δτ 2.39 Advances in Difference Equations Since 2.38 and condition 2.25 implies that t f τ, x τ , xΔ τ , , xΔ T ⎡ t ≤ ⎣ T ≤ n−1 ⎛ pi τ gi ⎝ i n−1 Δτ τ ⎞ xΔ τ i ⎠ hn−i−1 τ, t0 ⎤ pn τ ⎦Δτ 2.40 n−1 Pi g i c Pn i M < ∞, we see that the sum in 2.39 converges as t → ∞ Therefore, limt → ∞ xΔ t exists and is a n−1 a0 If a0 / 0, then it follows from Lemma 2.5 that finite number Let limt → ∞ xΔ t n−1 x t t, t0 lim lim xΔ t → ∞ hn−1 n−1 t→∞ a0 , t 2.41 and x has the desired asymptotic property The proof is completed Theorem 2.9 Assume that there exist positive functions p : t0 , ∞ T → 0, ∞ , and nondecreasing continuous functions gi : 0, ∞ → 0, ∞ ≤ i ≤ n − , and t1 > t0 , such that ≤p t f t, u0 , , un−1 n−1 gi i |ui | hn−i−1 t, t0 for t ≥ t1 , 2.42 with ∞ p t Δt P < ∞, t1 ∞ ε 2.43 ds n−1 i gi s ∞, for any ε > 0, then every solution x of 1.1 satisfies one of the following conditions: limt → ∞ xΔ n−1 t 0, there exist constants ≤ i ≤ n − with a0 / 0, such that lim t→∞ x t n−1 i hn−i−1 t, t0 2.44 10 Advances in Difference Equations Proof Arguing as in the proof of Theorem 2.8, we see that there exists T > t1 , such that for t ≥ T and ≤ m ≤ n − 1, ⎡ xΔ t m n−m−1 ≤ hn−m−1 t, t0 ⎣ xΔ t n−1 k m t1 ⎛ p τ gi ⎝ t1 i k xΔ τ i hn−i−1 τ, t0 ⎞ ⎤ ⎠Δτ ⎦, 2.45 from which we obtain xΔ t m ≤ hn−m−1 t, t0 F t for t ≥ T, ≤ m ≤ n − 1, 2.46 where t n−1 F t A ⎛ p τ gi ⎝ T i n−m−1 A max 0≤m≤n−1 xΔ i t0 ⎛ p τ gi ⎝ t1 i k ⎠, hn−i−1 τ, t0 T n−1 k m ⎞ xΔ τ 2.47 xΔ τ i hn−i−1 τ, t0 ⎞ ⎠ 2.48 Using 2.46 and 2.47 , it follows that t n−1 F t ≤A p τ gi F τ Δτ for t ≥ T 2.49 p τ gi F τ Δτ for t ≥ T, 2.50 T i Write t n−1 u t A T i y ds G y n−1 i gi A s , 2.51 then G u t Δ uΔ t G hu t − h uσ t dh n−1 p t gi F t i ≤ n−1 i p t gi n−1 i gi u p t , u t t dh n−1 i gi hu t − h uσ t 2.52 Advances in Difference Equations 11 from which it follows that t ≤G u T G ut p τ Δτ ≤ G u T 2.53 P T The rest of the proof is similar to that of Theorem 2.8, and the details are omitted The proof is completed Theorem 2.10 Assume that the function f t, u0 , , un−1 satisfies f t, u0 , , un−1 p t F u0 , , un−1 for all t, u0 , , un−1 ∈ t0 , ∞ ∞ t0 p t ≥ for t ≥ t0 and hn−1 τ, t0 p τ Δτ T × Rn , ∞, u0 F u0 , , un−1 > for u0 / and F u0 , , un−1 is continuous at u0 , 0, , with u0 / 0, then (1) if n is even, then every bounded solution of 1.1 is oscillatory; (2) if n is odd, then every bounded solution x t of 1.1 is either oscillatory or tends monotonically to zero together with i xΔ t ≤ i ≤ n − Proof Assume that 1.1 has a nonoscillatory solution x on t0 , ∞ , then, without loss of generality, there is a t1 ≥ t0 , sufficiently large, such that x t > for t ≥ t1 It follows from n 1.1 that xΔ t ≤ for t ≥ t1 and not eventually zero By Lemma 2.6, we have lim xΔ t i −1 for ≤ i ≤ n − 1, 0, t→∞ 2.54 i Δn−i x t > ∀t ≥ t1 , ≤ i ≤ n − 1, and x t is eventually monotone Also xΔ t > for t ≥ t1 if n is even and xΔ t < for t ≥ t1 if n is odd Since x t is bounded, we find limt → ∞ x t c ≥ Furthermore, if n is even, then c > We claim that c If not, then there exists t2 > t1 , such that F x t , xΔ t , , xΔ n−1 t > F c, 0, , >0 for t ≥ t2 , 2.55 since F is continuous at c, 0, , by the condition From 1.1 and 2.55 , we have xΔ t n p t F c, 0, , ≤ 0, for t ≥ t2 2.56 Multiplying the above inequality by hn−1 t, t0 , and integrating from t2 to t, we obtain t t2 hn−1 τ, t0 xΔ τ Δτ n t hn−1 τ, t0 p τ t2 F c, 0, , Δτ ≤ 0, for t ≥ t2 2.57 12 Advances in Difference Equations Since t hn−1 τ, t0 x Δn t n τ Δτ ≥ t2 −1 i hn−i τ, t0 x Δn−i τ i t2 n ≥ −1 i hn−i t2 , t0 xΔ n−i 2.58 −1 t2 n x t , i we get A −1 n t x t F c, 0, , Δτ ≤ 0, hn−1 τ, t0 p τ t2 for t ≥ t2 , 2.59 ∞ i n Δ t2 Thus, t2 hn−1 τ, t0 p τ Δτ < ∞ since x t is bounded, where A i −1 hn−i t2 , t0 x which gives a contradiction to the condition The proof is completed n−i Examples Example 3.1 Consider the following higher-order dynamic equation: xΔ t βi h n−i−1 t, t0 0t i n−1 xΔ t n i where t ≥ t1 > t0 > and βi > ≤ i ≤ n − Let pi t 3.1 0, ≤ i ≤ n − and 1/ tβi hn−i−1 t, t0 n−1 f t, u0 , , un−1 i ui , βi h n−i−1 t, t0 0t 3.2 then we have ≤ f t, u0 , , un−1 n−1 pi t |ui |, ∀ t, u0 , , un−1 ∈ t1 , ∞ T × Rn , i e n−1 i pi t hn−i−1 t, t1 3.3 e n−1 i t, t1 1/tβi ≤e t t1 n−1 i 1/τ βi Δτ < ∞, by Example 5.60 in Thus, it follows from Theorem 2.7 that if x is a solution of 3.1 n−1 with limt → ∞ xΔ t / 0, then there exist constants ≤ i ≤ n − with a0 / 0, such that n−1 limt → ∞ x t / i hn−i−1 t, t0 Example 3.2 Consider the following higher-order dynamic equation: x Δn n−1 t i βi 0t xΔ t hn−i−1 t, t0 i αi tβ n 0, 3.4 Advances in Difference Equations 13 where t > t0 > 0, αi ∈ 0, ≤ i ≤ n − , and βi > ≤ i ≤ n Let gi u 1/tβi ≤ i ≤ n , and pi t n−1 f t, u0 , , un−1 i βi 0t αi ui uαi ≤ i ≤ n − , tβ n hn−i−1 t, t0 3.5 It is easy to verify that f t, u0 , , un−1 satisfies the conditions of Theorem 2.8 Thus, it follows n−1 that if x is a solution of 3.4 with limt → ∞ xΔ t / 0, then there exist constants ≤ i ≤ n − with a0 / 0, such that limt → ∞ x t / n−1 hn−i−1 t, t0 i Example 3.3 Consider the following higher-order dynamic equation: xΔ t n tβ n−1 i xΔ t hn−i−1 t, t0 i where t > t0 > 0, αi ∈ 0, ≤ i ≤ n − with < i≤n−1 , p t 1/tβ , and n−1 f t, u0 , , un−1 i tβ n−1 i αi 3.6 0, αi < and β > Let gi u ui hn−i−1 t, t0 uαi 0≤ αi 3.7 It is easy to verify that f t, u0 , , un−1 satisfies the conditions of Theorem 2.9 Thus, it follows n−1 that if x is a solution of 3.6 with limt → ∞ xΔ t / 0, then there exist constants ≤ i ≤ n − with a0 / 0, such that limt → ∞ x t / n−1 hn−i−1 t, t0 i Acknowledgment This paper was supported by NSFC no 10861002 and NSFG no 2010GXNSFA013106, no 2011GXNSFA018135 and IPGGE 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Equations on Time Scales, Birkhă user, Boston, Mass, a USA, 2003 M Bohner and S H Saker, “Oscillation of second order nonlinear dynamic equations on time scales,” The Rocky Mountain Journal of. .. nonlinear dynamic equations, ” Advances in Difference Equations, vol 2008, Article ID 739602, 17 pages, 2008 15 T S Hassan, “Oscillation of third order nonlinear delay dynamic equations on time scales,”