Existence And Multiplicity Of Periodic Solutions And Homoclinic Orbits For Hamiltonian System And P-Laplacian Differential System

In this doctoral dissertation, by using methods and techniques of the critical point theory, we study the existence and multiplicity of periodic solutions and homoclinic orbits for Hamiltonian system and p-Laplacian differential system. This dissertation consists of four chapter.In Chapter1, we introduce the historical background, status, the up-to-date progress for the investigated problems, and outline our main work, preliminary facts on critical point theorem used in the dissertation.In Chapter2, by applying the least action principle and mountain pass lemma, we study the existence of periodic solutions for the following second-order impulsive Hamiltonian systemsWhen VF(t, x) is bounded sublinearly, we improve the existing results about the existence of periodic solutions and we get the corresponding results at a weaker con-dition. Besides, suppose that▽F(t, x) is not bounded sublinearly, we obtain the theorem of existence of periodic solutions and generalize the existing results.In Chapter3, we study the existence of homoclinic orbits for the following second-order p-Laplacian differential systems by applying the three critical points theorem of Ricceri. We convert the homoclinic solution of the system to the limit of the2kT-periodic solution for the system, and we obtain the existence of the homoclinic solutions.In Chapter4, we investigate the existence of homoclinic solutions and multiplicity of periodic solutions for three classes of impulsive differential systems. Firstly, by using three critical point theorem, we discuss the existence of homoclinic solutions for the following second-order impulsive p-Laplacian differential systems Secondly, by constructing the corresponding variational framework, we obtain the existence of infinitely many periodic solutions for the following impulsive p-Laplacian differential systemsThirdly, by applying B.Ricceri’s variational method, we obtain two new theorems for the sufficient conditions of the multiplicity of periodic solutions for the following impulsive (p, q)-Laplacian differential systems...