Hamilton Differential Systems With Variable Index Periodic Solution

Variational method of Critical point theory is a universal method in the world of the nature, and many problems can be reduced to the problems of critical points of a special functional under some specific conditions, of course including the differential systems with a p(t)-Laplace operator. The existence of periodic solutions for differential systems with a p(t)-Laplace operator is studied by using the Minimization principle and the saddle-point theorem, which will motivate the development of the theory of differential systems.This paper is divided into four chapters, the main structure is as follows:The first chapter is the introduction part of paper. It focuses on the background, the status and progress of the periodic solutions of differential systems; moreover in this chapter the main results of the article are described.The second chapter is the prior knowledge part of paper. Some essential prior knowledge about differential systems with p(t)-Laplace are introduced, such as the space of Lp(t) and w1,p(t), some specific methods for researching.The third chapter is the main part of this paper. We know a lot of papers have studied on the existence of periodic solutions of smooth systems. But the article, which is based on previous studies, has expanded the issues of existence of periodic solutions to non-smooth systems with certain restrictions added to potential function F, by using minimization principle, saddle-point theorem and some other methods.The fourth chapter is a summary and outlook. More systematic summary of these results obtained in this chapter, and we have presented some problems which need further research and discussion.