The Existence And Multiplicity Of Solutions Of The Second-order Hamiltonian System

In this paper we will first establish an index theory for second order Hamiltonian system:x"+B(t)x=0,t?[0,1],(1)x(0)cos ?-x'(0)sin ?=0,(2)x(1)cos?-x'(l)sin?=0.(3)Then we will study the existence and multiplicity of solutions of the second order Hamiltonian system under asymptotically linear conditions.x"+V'(t,x)=0,t?[0,1],(4)x(0)cos ?-x'(0)sin ?=0,(5)x(1)cos?-x'(1)sin?=0.(6)where B(t)? L2([0,1],(?)s(Rn)),? ?[0,?),??(0,?),V ? C1([0,1]× Rn,R),Vl(t,x)denotes the gradient of V with respect to x.In the first chapter,we will give an overview the background of the issues studied in this paper,and briefly introduce the main work of this paper,and the relevant preliminary knowledge,and finally give the main results of this paper.In the second chapter,we set up the index theory for the linear Hamiltonian system(1)-(3)under the new conditions.First we defined the index and discussed its nature,then we introduce the concept of the relative Morse,and gives the exact expression of i(B2)-i(B1)(B2>B1).In the third chapter,we prove the existence and multiplicity of solutions of(4)-(6)in the asymptotic condition.In the process of proof we mainly use Leray-Schauder degree theory,Critical point theory and so on.