About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 829052, 8 pages
http://dx.doi.org/10.1155/2014/829052
Research Article

Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems

1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
2Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China
3School of ELectrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798

Received 3 December 2013; Accepted 10 January 2014; Published 19 February 2014

Academic Editor: Jifeng Chu

Copyright © 2014 Ziheng 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

We are concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systems , (HS) where , , is a continuous bounded function, and the potential has a singularity at , and is the gradient of at . The novelty of this paper is that, for the case that and (HS) is nonautonomous (neither periodic nor almost periodic), we show that (HS) possesses at least one nontrivial homoclinic solution. Our main hypotheses are the strong force condition of Gordon and the uniqueness of a global maximum of . Different from the cases that (HS) is autonomous or (HS) is periodic or almost periodic, as far as we know, this is the first result concerning the case that (HS) is nonautonomous and . Besides the usual conditions on , we need the assumption that for all to guarantee the existence of homoclinic solution. Recent results in the literature are generalized and significantly improved.