Change Index Of The Existence Of Periodic Solutions For Second Order Hamilton System Research

In this paper, we investigate two kinds of second order Hamilton systems with variable exponent operators, with p(t)-Laplace operator and (p(t),q(t))-Laplace operator respectively. The periodic solutions of these two kinds of the second order Hamilton systems are studied through the least action principle and saddle point theory in the critical point theory. This paper contains four chapters, the main contents are follows:In Chapter1, introduce the history and the latest research of the calculus of variations and the main work of this paper.In Chapter2, some essential concepts for this paper are presented, mainly involving, saddle point theory, the least action principle and some important lemmas and so on.In Chapter3, the existence of solutions of the second order Hamilton system is investigated. We separate F into two functions.We discussed the solu-tions of this kind of system under sub-convex and sub-linear conditions. Some sufficient conditions of the existence of periodic solutions of this kind of second order Hamilton system are obtained.In Chapter4, the existence of solutions of the second order Hamilton system is discussed. By appropriate restrictions on F,▽Fu1(t,u1(t),u2(t))and▽Fu2(t,u1(t),u2(t)), and used the least action principle and the saddle point theorem, the sufficient conditions of the existence of solutions of the above system are obtained. We extended the constant exponent case of (q,p)-Laplacian to the case of variable exponent (p(t),q(t))-Laplacian.This chapter is the extentions and applications of the results we obtained in Chapter3.