Periodic Solutions And Homoclinic Orbits Of A Class Of Hamiltonian Systems

In this paper,the existence of periodic solution and homoclinic orbits of the Hamiltonian system with a p-Laplacian operator and perturbation term are studied.The paper is divided into five chapters.In chapter 1,the historical background and significance of Hamilton sys-tem,the development of variational methods and research status of the problem,as well as the main work of this paper are introduced.In chapter 2,some mathematical symbols,basic definitions and basic the-orems are given.In chapter 3,we study the following Hamiltonian systems:where p>1,V:[0,T]× RN?R,f?L1([0,T];RN),?V(t,x)is the gradient of V with respect to x.Using the variational method and the Saddle Point The-orem of critical point theory,we prove the existence of periodic solutions of the system(HS1)under a locally asymptotic p-quadratic condition.It generalize some existing results.In chapter 4,we consider the following system:where p>1,t ? R,u ? Rn,V:R x Rn? R,f:R ? Rn,?V(t,x)is the gradient of V with respect to x.Using the variational method and the Mountain Pass Theorem of critical point theory,we prove the existence of a non-trivial homoclinic orbit of the system(HS2)under general conditions.It promote some existing results.In chapter 5,the main contents of this paper are summarized and the future research is prospected.