Homoclinic Orbits In Discrete Hamiltonian System

This dissertation deals with the existence and multiplicity of homoclinic or-bits for second order discrete Hamiltonian systems by using of the Mountain pass lemma, the sysmetric mountain pass lemma,the clark therem etc. Under compara-tively general assumptions, we obtain a series of existence principals which greatly generlize and improve some known result in the literature. This dissertation is divided into five chapters. The main contents are follows:In chapter 1, a brief introduction is given to the historical background, status and up-to-date progress for all the investigated programs together with main results and preliminary tools in this dissertation.In chapter 2, the existence of homoclinic orbits for a class of second order periodic discrete Hmailtonian system with force term is investigeted. By using Mountain pass lemma, the existence of subharmonics is verified, and then it is proved that a nontrivial homoclinic is just the limit of the subharmonics by diagonal method. Some known results are generlized and improved in the literature.In chapter 3, the existence of homoclinic orbits for a class of second order non-periodic discrete Hamiltonia system with force term is discussed. For the homoclinics taking on values in the unbounded domain, in order to apply the usual Mountain pass theorem, a compact embedding inequility is proved. Some known rusults are generlized and improved in the literature.In chapter 4, the existence of infinitely many homoclinic orbits for a class of second order non-periodic discrete Hamiltonia system is studied. Under comparatively general conditions, the existence of infinitly many homoclinic is obtained by using a compact embedding theorem and then a sysm-metric Mountain pass lemma. Some know results are generlized and improved in the literature.In chapter 5, by using the Clark theorem in critical point theory, the existence and multiplicity of homoclinic solutions for discrete Hamiltonian system with subquadratic polential W(n, x) is the first time to be discussed.