The Existence Of Hamiltonian System Periodic Solution With P-the Laplace Operator

This dissertation deals with the existence of periodic solutions of Hamiltonian systems with a p-Laplacian operator by applying the Saddle Point Theorem,the least action principle etc,where p＞1 is a constant. These results will motivate the development of qualitative theory of Hamiltonian systems with a p-Laplacian operator.This dissertation divided into three chapters.The main contents are as follows:Chapter 1 gives 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.In the Chapter 2,the existence of periodic solutions for Hamiltonian systems with a p-Laplacian operator is studied by using the Saddle Point Theorem.In general,the Saddle Point Theorem holds under the Palais-Smale condition.By using a weaker compact condition presented by Cerami in 1978(that is(C) condition) and combining with the fact that the validity of Palais-Smale condition can guarantee the establishment of(C) condition,we conclude that the Saddle Point Theorem also holds under the(C) condition.As a result,the existence of periodic solutions is proved when the systems satisfy(C) condition by using the Saddle Point Theorem,so that a series of meaningful results are obtained.Through the use of the least action principle,Chapter 2 proves the existence of periodic solutions for Hamiltonian systems with a p-Laplacian operator.Firstly,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.When p=2,the above systems degenerate to the second order Hamiltonian systems.Therefore,our conclusions obtained in this dissertation cover some results about existence of periodic solution for the second order Hamiltonian systems.