Nontrivial Solution And Homoclinic Orbit Of A Fourth-order System

In this paper we discusses the fourth-order system in two parts.(i)The existences of the following fourth-order system:q(4)-Vq(t,q)=0 q(0)=q(1)=q'(0)=q'(1)=0(ii)Using the knowledge of critical point theory,the homodinic orbit of fourth-order system is discussed under super-second condition.-q(4)+Cq"-L(t)q+Vq(t,q)=0The full text is divided into three chapters to introduce the four-order system:In chapter 1,we give the basic concept and main results of this paper.In Chapter 2,by using the main results of the operator equation in[4],we prove the existence of solutions for the fourth-order system with boundary conditions.In Chapter 3,the variational approximation method of[3]is applied to the proof of the existence of nontrivial homoclinic orbits for fourth-order systems.