The Same Solution Of The Second-order Hamiltonian System

This paper is divided into two parts:(i) The existences of Homoclinic solutions for the second order Hamiltonian system.We make a detailed discussion about the existences of Homoclinic solutions for the second order Hamiltonian systems with perturbation.(ii) The problem of index theory for the first order Hamiltonian systems.where , In is identity matrix and J satisfies:We discuss the relationship between some indices defined in literatures with some special cases.In chapter one, we recall some basic concepts and results,Which are a basis of this paper. Meanwhile, we give the main results of this paper.In chapter two, we mainly propose new conditions different from [10] and under these conditions we prove the system has non-trivial homoclinic solution, which by making use of variational principle, mountain theorem, and so on.In chapter three, we discuss these indices defined in the literatures of [6] and[7]. Even though they define morse index, relative morse index in different forms, for some special cases, through our calculations , they are closely linked indeed.