Doundary Value Problems,Periodic Solutions And Homoclinics For Nonself-adjoint And Self-adjoint Difference Equations

This dissertation contains five chapters, which mainly deals with the existence and multiplicity of boundary value problems(BVP for short), periodical solutions and homoclinics for non self-ajoint and self-adjoint nonlinear second order difference equations.Chapter 1 concentrates on the brief introduction of the historic background and significance for all the investigated problems, primary tools and dissertation's main work.Chapter 2 discusses the BVP of non self-adjoint nonlinear second order difference equations by applying matrix theory and Krasnosel'skii fixed point theorem. This research method is independent of Green's function and variational structure. So it can tackle with the BVP for non self-adjoint and self-adjoint difference equations and overcome the difficulties brought about by the establishment of Green's function, a study of the sign and the estimate of below and above bounded for Green's function. A completely new research way is provided to study the same type problems.Chapter 3 is centered around the existence of periodical solutions for non self-adjoint nonlinear second order difference equations by invoking matrix theory and coincide degree theory. The research method does not rely on Green's function and variational structure through transforming the existence of periodic solutions for difference equations into solving a corresponding operator equation. A new powerful approach is provided to investigate periodical problems which are not of variational structures and difficult in building Green's functions.Chapter 4 is devoted to the existence and multiplicity of homoclinic for self-adjoint nonlinear second order difference equations by using "Mountain Pass " theorem and "Symmetry Mountain Pass "theorem in critical point theory. This is the first time to discuss homoclinic solutions for difference equations, some satisfactory results are obtained. For the homoclinics taking on values in the unbounded domain, in order to apply the usual "Mountain Pass " theorem, A compact embedding theorem is proved.Chapter 5 introduces another technique different from that in Chapter 4 to overcome the lack of the natural compactness resulted from the homoclinics taking on values in the unbounded domain. Firstly by applying "Mountain Pass " theorem, the existence for sub-harmonics is verified, and then the uniformly bounded in speacial function spaces for subhar-monics is established, finally it is proved that a nontrivial homoclinic is just the limit of the subharmonics.