Periodic Solutions For A Class Of Second-Order Hamiltonian System Under Sub-Linear Condition

The motion that is described by Hamiltion system is the easiest periodic motion in the motion. The motion trajectory of celestial body can be described by periodic solution for Hamiltonian systems. So, studying the existence of periodic solution for Hamiltonian systems is an important problem.In this dissertation, some rencent results are introduced and the existence of periodic solution for a class of second order Hamiltonian systems are studied through the least action principle and the Saddle Point Theorem. The main contents are as follows:1. In the present paper, under the assumptions that the nonlinearity▽F(t, x) satisfied the sublinear conditions, that is, there exists f(t), g(t)∈L1(0,T;R+) and a nonnegative function h∈([0,+∞),[0,+∞)) such that|▽F(t,x)≤f(t)h(|x|)＋g(t), the existence of periodic solution for a class of second order Hamiltonian systems are studied with this conditions, and the existence of some sufficient conditions are obtained, and some examples are given.2. The existence of periodic solution for a class of second order Hamiltonian systems are studied with F(t,x)=G(x)+H(t,x).In particular, under the assumptions that the nonlinearity VH(t,x) satisfied the sublinear conditions, that is, there exists f(t), g(t)∈L1(0,T;R+) and a∈[0,1) such that|▽H(t,x)|≤f(t)|x|α+g(t). The existence of periodic solution for a class of second order Hamiltonian systems are studied by the least action principle and the Saddle Point Theorem with this sublinear conditions, and the existence of two sufficient conditions are obtained, and some examples are given.