Non-trivial Solutions Of Linear Second-order Asymptotic Indefinite Hamiltonian System

In this paper we mainly study the nontrivial solutions for asymptotically linear second order indefinite Hamiltonian system: Px+V’(t,x)=0, x(0)=0=x(1), where P=(100-1),V∈C1([0,1], R2). We first introduce an index theory of the linear second order indefinite Hamiltonian system: Px+B(t)x＝0, x(0)=0=x(1), and then investigate the asymptotically linear second order indefinite Hamiltonian sys-tem. The main method in the discussion is the classification theory of linear homogenous equations and some results of Leray-Schauder degree and variational methods.