非自治Hamilton系统多重周期解的存在性

On Existence of Multiple Periodic Solutions for Nonautonomous Hamiltonian Systems

本文讨论如下形式的非自治Hamilton系统周期解的存在性: JZ=G’_x(t,z). (*)我们对对应的变分泛函用若干个因子加以修正,构造特殊的极小极大值序列并给出增长阶的精细估计,由此证明了当G(t,z)=H(z)+F(t,z),其中H'_z在Z=0为次线性,F’_x为超线性,并且二者的指数满足一定关系时,(*)存在无穷多个周期解.

We discuss in this paper the existence of periodic solutions of nonautonomous Hamiltonian systems as follows :Jz=G'z(t,z). (*)We firstly modify the corresponding variational function with several factors,construct some special minimax value series and give the sharp estimates of their growth orders . Then we prove that the equation ( * ) possesses infinite periodic solutions when G(t ,z) =H(z)+F(t, z), where H'z(z) is sublinear at z = 0 and F'z(t,z) superlinear, and their indexes satisfy specific relations.