The Existence Of Homoclinic Solutions Of Several Classes Of Second-order Hamiltonian Systems

The existence of homoclinic orbits of several class of second order Hamiltonian systems is studied by using critical theory and the minimax method in this dissertation. Separately discussing under periodic and non-periodic condition, we apply different ways to the existence of homoclinic orbits. This dissertation is divied into four chapters.In the first chapter, we introduce a survey to the historical situation of second order Hamiltonian systems and the present development con-cernd the problem of homoclinic solutions for second order Hamiltonian system. In addition, the critical point theory and some important Lemmas is present. We summarize main results of the dissertation.In the second chapter, we study the existence of homoclinic orbits of second order Hamiltonian system with a p-Laplace. By constructing a sup-port system and Mountain Pass theorem, it exists 2kT-periodic solution, and then get Homoclinic solutions of original systems.In the third chapter, super-linear condition, being different from the corresponding conditions existing in the literature. The existence of ho-moclinic orbits of second order Hamiltonian system is stuided. Continues to meet the Mountain Pass Theorem, with the definition of Homoclinic Solutions, permit may exist homoclinic solution of the system.In the last chapter, we obtain a different criterion for guaranteeing that the system has one nontrivial homoclinic solution without any peri-odicity or coercivity condition. A homoclinic orbit is obtained as a limit of solutions of a certain sequence of boundary-value problems which are obtained by the minimax methods.