Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 463480, 14 pages
http://dx.doi.org/10.1155/2011/463480
Research Article

Existence of Periodic Solutions for a Class of Discrete Hamiltonian Systems

Qiongfen Zhang, X. H. Tang, and Qi-Ming Zhang

School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, China

Received 9 April 2011; Accepted 12 June 2011

Academic Editor: Mingshu Peng

Copyright © 2011 Qiongfen Zhang 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

By applying minimax methods in critical point theory, we prove the existence of periodic solutions for the following discrete Hamiltonian systems Δ 2 𝑢 ( 𝑡 − 1 ) + ∇ 𝐹 ( 𝑡 , 𝑢 ( 𝑡 ) ) = 0 , where 𝑡 ∈ ℤ , 𝑢 ∈ ℝ 𝑁 , 𝐹 ∶ ℤ × ℝ 𝑁 → ℝ , 𝐹 ( 𝑡 , 𝑥 ) is continuously differentiable in 𝑥 for every 𝑡 ∈ ℤ and is 𝑇 -periodic in 𝑡 ; 𝑇 is a positive integer.