Advances in Difference Equations
Volume 2010 (2010), Article ID 470375, 19 pages
doi:10.1155/2010/470375
Research Article

Existence of Homoclinic Solutions for a Class of Nonlinear Difference Equations

Peng Chen and X. H. Tang

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

Received 5 May 2010; Accepted 2 August 2010

Academic Editor: Jianshe Yu

Copyright © 2010 Peng Chen and X. H. Tang. 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 using the critical point theory, we establish some existence criteria to guarantee that the nonlinear difference equation Δ[p(n)(Δx(n-1))δ]-q(n)(x(n))δ=f(n,x(n)) has at least one homoclinic solution, where n∈ℤ, x(n)∈ℝ, and f:ℤ×ℝ→ℝ is non periodic in n. Our conditions on the nonlinear term f(n,x(n)) are rather relaxed, and we generalize some existing results in the literature.