Periodic Solutions Of Nonlinear Difference Equation And Homoclinic Orbits

This paper deals with the existence of periodic solutions and homoclinic orbits of nonlinear difference equations by using of the least action principle,the saddle point theorem and symmetric mountain pass theorem.This paper contains five chapters.The main contents are follows:In chapter1,a brief introduction is given to the historical background and main results in the paper.In chapter2,some basic knowledge about this paper is introuduced, such as general mathematical symbols,basic concept and basic lemma.In chapter3,the existence of periodic solutions for nonautonomous second-order discrete systems with (q,p)-Laplacian is investigated.By appropriate restrictions on F,▽uF(n,u,v),▽vF(n,u,v) and using the least action principle and the saddle point theorem,the sufficient condition of the existence of solutions of the above system are obtained.In chapter4,the existence of infinitely many homoclinic orbits for a class of second discrete Hamilton system△(p(n)△u(n-1))-q (n)u(n)+f (n,u(n+1),u(n),u(n-1))=0, is studied.under comparatively general conditions,by using a compact embedding conditions and a sysmetric mountain pass theorem,the existence of infinitely many homoclinic is obtained.In chapter5,reviewed the main results about this paper fist,and then puts forward some problems which need further studied.