Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học được đăng trên tạp chí toán học quốc tế đề tài: Homoclinic solutions of some second-order nonperiodic discrete systems | Long Advances in Difference Equations 2011 2011 64 http content 2011 1 64 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access Homoclinic solutions of some second-order nonperiodic discrete systems Yuhua Long Correspondence longyuhua214@ College of Mathematics and Information Sciences Guangzhou University Guangzhou 510006 P. R. China Springer Abstract In this article we discuss how to use a standard minimizing argument in critical point theory to study the existence of non-trivial homoclinic solutions of the following second-order non-autonomous discrete systems A2xn-1 AAxn L n xn VW n xn 0 n e Z without any periodicity assumptions. Adopting some reasonable assumptions for A and L we establish that two new criterions for guaranteeing above systems have one non-trivial homoclinic solution. Besides that in some particular case for the first time the uniqueness of homoclinic solutions is also obtained. MSC 39A11. Keywords homoclinic solution variational functional critical point subquadratic second-order discrete system 1. Introduction The theory of nonlinear discrete systems has widely been used to study discrete models appearing in many fields such as electrical circuit analysis matrix theory control theory discrete variational theory etc. see for example 1 2 . Since the last decade there have been many literatures on qualitative properties of difference equations those studies cover many branches of difference equations see 3-7 and references therein. In the theory of differential equations homoclinic solutions namely doubly asymptotic solutions play an important role in the study of various models of continuous dynamical systems and frequently have tremendous effects on the dynamics of nonlinear systems. So homoclinic solutions have extensively been studied since the time of Poincaré see 8-13 . Similarly we give the following definition if xn is a solution of a discrete system xn will be called
