Variational Methods For Fourth Order Difference Boundary Value Problem And Second Order Difference System

In this paper, utilizing variational approach, we consider existence of multiple solutions of fourth order difference boundary value problem with parameter, and, existence and nonexistence of periodic solutions for difference system. In chapter one, we introduce the background, present study of our problem that are to be investigated, some preliminaries and our main results. In chapter two, we discuss existence of multiple solutions of the following fourth order difference equation we establish suitable function space, in which we build variational functional according to this problem, translating existence of solutions for this problem to existence of critical points of functional in this function space. Then, making use of Mountain pass theorem, Clark theorem and Linking theorem, we obtain the sufficient condition of existence of solutions for this problem. Finally, we illustrate several examples to verify our results.In chapter three, we discuss the existence and nonexistence of nontrivial T periodic solutions for the following difference system (P2) using Descartes product, we set higher dimensional function space, and build functional according to this difference system on the function space directly. Then we gain the sufficient condition of existence and nonexistence of periodic solutions for this difference system.