Periodic Solution Of Nonlinear Differential Equations, Homoclinic Orbits And The Boundary Value Problem

Many nonlinear differential and difference problems resulted from mathematics, physics, chemistry, biology and economics and so on have increasingly brought people's attention. Now, the existence and multiplicity of solution for nonlinear differential and difference problems have been studied extensively by various methods including variational methods, topology degree method, monotone iterative method and Kaplan-Yorke coupled system method.Application of variational methods to differential equations is to change the existence of solutions into the existence of critical points of some variational functional. The variational theory has a rich content. In this paper, we mainly use variational methods, including the least action principle, minimax theorem, linking theorem, mountain pass lemma and three critical point theorem to deal with the existence of periodic and homoclinic solutions of nonlinear difference equa-tion(system).The whole paper consists of seven chapters.In Chapter 1, we give an introduction to the background of the field we concerned.In Chapter 2, by using linking theorem, we study the existence of periodic solutions for second-order p-Laplacian difference equation. When the nonlinearity is superlinear, the corresponding variational functional is constructed, and obtained at least two nontrivial periodic solutions;In Chapter 3, we deal with the existence of periodic solutions for higher-order coupled difference system on condition that the perturbed term contains superlinear and sublinear nonlinearity. By means of linking theorem on product space, we present three existence results and extend Guo's work;In Chapter 4, we study the existence of homoclinic orbits for self-adjoint difference system with potential changing sign, we establish the existence and multiplicity of homoclinic orbit by using of the mountain pass lemma and symmetric mountain pass lemma of critical point theorem; Chapter 5 introduces another technique different from that in Chapter 4 to overcome the lack of the natural compactness resulted from the homoclinic orbits taking on values in the unbounded domain. By the thought of approximation and technique of function decomposition, we study the existence of of homoclinic and even homoclinic orbits about a class of second-order non self-adjoint difference equation;In Chapter 6, we study the boundary problems of discrete p-Laplacian difference equations by using of three critical point theorem, existence and multiplicity of solutions on condition that nonlinearity is continuous and discontinuous is considered;The Chapter 7 is conclusion and expectation.