Periodic Solutions And Homoclinic Orbits Of Hamiltonian System

This dissertation deals with the existence of periodic solutions and homoclinic orbits of Hamiltonian system by using of the least action principe, Mountain pass lemma, Linking theorem, etc. The results will motivate the development of qualitative theory of Hamiltonian systems. The 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 the up-to-date progress for all the investigated problems together with preliminary tools and main results in this dissertation.In chapter 2, the existence of homoclinic orbits of the second Hamiltonian systemis investigeted, where V(t,x) = -K(t,x) + W(t,x). By using mountain pass lemma, the sufficient condition of the existence of homoclinic solutions of the above system are obtained under the condition that K(t, x) does' t satisfy usual pinnching condition in the literature. Some known results are generlized and improved in the literature.In chapter 3, by using the least action principe, the existence of homoclinic solutions of the second Hamiltonian systemwith forcing potencial is investigeted. The condition F(t, x)≥F(t, 0) + 6|x|~2 is replaced by more general one in this dissertation. Some known results are generalized and improved in the literature.In chapter 4, the existence of homoclinic solutions for the non-autonomous second system with p-Laplacian operator is discussed, where p > 1. We generalize the results in chapter 3.In chapter 5, the existence of periodic solutions of second-order Hamiltonian systemswith potential indefinite in sign is studied. By using Linking theorem, the sufficient conditions of existence of periodic solutions for the above Hamiltonian systems are obtained, some known results in the literature are generalized and improved...