Asymptotic Behavior of Solutions of Higher-Order Dynamic Equations on Time Scales
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Research Article Asymptotic Behavior of Solutions of Higher-Order Dynamic Equations on Time Scales Taixiang Sun,1 Hongjian Xi,2 and Xiaofeng Peng1 1 2
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, [email protected] 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 ft, xt, 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: 1 limt → ∞ xΔ t 0; 2 there exist n−1 0, such that limt → ∞ xt/ i0 ai hn−i−1 t, t0 1, where constants ai 0 ≤ i ≤ n − 1 with a0 / hi t, t0 0 ≤ i ≤ n − 1 are as in Main Results.
1. Introduction In this paper, we investigate the asymptotic behavior of solutions of the following higherorder dynamic equation n n−1 xΔ t f t, xt, xΔ t, . . . , xΔ 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 infinity, we assume that sup T ∞, and define the time scale interval t0 , ∞T {t ∈ 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.
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Advances in Difference Equations
The theory of time scales, which has recently received a lot of attention, was introduced by Hilger’s landmark paper 1 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 2 . 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 3
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