Homoclinic Orbits For Some Class Of Second Order Discrete Hamiltonian Systems

This dissertation deals with the existence and multiplicity of homoclinic orbits for several second order discrete Hamiltonian systems by using of variational method. We get the corresponding conclusions after weakening control conditions of the nonlinear term, and greatly generalize and improve the some know result in the literature. By constructing a variable structure and working space, and seeking the approximate critical point sequence, we can prove that existence and multiplicity of solutions for the Hamiltonian systems.This dissertation is divided into three parts. The main contents of the paper are as follows:At the first part, 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.At the second part, the existence and multiplicity of homoclinic orbits for a class of second order superquadratic discrete Hamiltonian system is discussed under the assumptions that L(t) is allowed to be not global positive definite, b(t) is a real number with sign-changing and W(t,u(t)) is superquadratic in u. We establish results on the existence of infinitely many homoclinic orbits by use of a standard minimizing argument in critical point theory.At the third part, the existence and multiplicity of homoclinic orbits for a class of second order subquadratic discrete Hamiltonian systems is discussed under the assumptions that L(t) is allowed to be global positive definite and W(t,u(t)) is subquadratic in u. New results about the existence of homoclinic orbits are obtained by new assumptions, we establish results on the existence of homoclinic orbits by use of a standard minimizing argument in critical point theory.