Advances in Difference Equations
Volume 2011 (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

1College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China
2Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi 530003, China

Received 18 November 2010; Accepted 23 February 2011

Academic Editor: Abdelkader Boucherif

Copyright © 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.

Abstract

We investigate the asymptotic behavior of solutions of the following higher-order dynamic equation 𝑥 Δ 𝑛 ( 𝑡 ) + 𝑓 ( 𝑡 , 𝑥 ( 𝑡 ) , 𝑥 Δ ( 𝑡 ) , … , 𝑥 Δ 𝑛 − 1 ( 𝑡 ) ) = 0 , on an arbitrary time scale 𝐓 , where the function 𝑓 is defined on 𝐓 × 𝐑 𝑛 . We give sufficient conditions under which every solution 𝑥 of this equation satisfies one of the following conditions: (1) l i m 𝑡 → ∞ 𝑥 Δ 𝑛 − 1 ( 𝑡 ) = 0 ; (2) there exist constants 𝑎 𝑖 ( 0 ≤ 𝑖 ≤ 𝑛 − 1 ) with 𝑎 0 ≠ 0 , such that l i m 𝑡 → ∞ ∑ 𝑥 ( 𝑡 ) / 𝑛 − 1 𝑖 = 0 𝑎 𝑖 ℎ 𝑛 − 𝑖 − 1 ( 𝑡 , 𝑡 0 ) = 1 , where ℎ 𝑖 ( 𝑡 , 𝑡 0 ) ( 0 ≤ 𝑖 ≤ 𝑛 − 1 ) are as in Main Results.