| In this paper, the existence of periodic solutions of Hamiltonian sys-tems with a p-Laplacian operator are studied through the least action principle and the Saddle Point Theorem. This dissertation is divided into four chapters. The main contents are as follows:In Chapter l,it is given that a brief introductions 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.Chapter 2 gives some prepared knowledges needed in this text. At the same time, some sufficient conditions are given to guarantee that the corresponding variational functional of the above problem is coercive and hence a minimum critical point is obtained. Then combining with Fundamental Lemma about weak derivatives, we know that this minimal critical point is just the weak solution (that is the periodic solution) of the Hamiltonian systems we considered. The prepared knowledges also contain some important inequalities, the corresponding variational frame of the above problem, the minimizing sequence, the least action principle (PS) condition and the Saddle Point Theorem.Through the use of the least action principle and the Saddle Point Theorem, Chapter 3 and Chapter 4 proves the existence of periodic solutions for Hamiltonian systems with a p-Laplacian operator, and the existence of some sufficient conditions are obtained, and some examples are given. Especially, in the case of p= 2, we get the more accurate results with the better estimation of the inequation. These results will popularize and improve the achievements in some documents available. |
