Research On The Existence Of Solutions For Impulsive Differential Systems And Discrete Hamiltonian Systems

This dissertation mainly deals with the existence and multiplicity of solutions for impulsive differential systems and discrete Hamiltonian systems by using variational methods and critical point theory and some new results are obtained, which greatly improve and generalize some existing results in the literature. It is consists of three chapters.In Chapter1, the background and significance, status, up-to-date progress, main works of this dissertation and some preliminary tools are introducedIn Chapter2, we discuss the existence and multiplicity of solutions for three types of boundary value problems of impulsive differential systems. In section1, we deal with the existence of multiplicity of periodic solutions for a class of second-order impulsive Hamiltonian systems by using three critical points theorems and establish some new existence criteria, which improve and generalize some existing results in the literature. In section2, we deal with the existence of infinitely many periodic solutions for another kind of second-order impulsive Hamiltonian systems by using two recent critical point theorems and establish some new existence criteria, which improve some existing results in the literature. In section3, we study the existence of infinitely many solutions for a class of for second-order impulsive differential equations with Dirichlet boundary conditions by using fountain theorem and establish a new existence criterion, which improve and generalize some existing results in the literature.In Chapter3, we discuss the existence and multiplicity of homoclinic solutions for second-order discrete Hamiltonian systems. In section1, we deal with the existence and infinitely many of homoclinic solutions for a class of superquadratic second-order discrete Hamiltonian systems by using fountain theorem and mountain pass lemma and establish some new existence criteria, which improve and generalize some existing results in the literature. In section2, we deal with the existence of multiplicity of homoclinic solutions for second-order discrete Hamiltonian systems with asymmetric potentials by using two recent critical point theorems and obtain some new results. In section3, we deal with the existence of infinitely many homoclinic solutions for a class of local subquadratic second-order discrete Hamiltonian systems by using dual fountain theorem and establish a new existence criterion, which improve and generalize some existing results in the literature. In section4, we study the existence of infinitely many homoclinic solutions for a class of asymptotically quadratic second- order discrete Hamiltonian systems by using dual fountain theorem and obtain a new result.