The Existence And Multiplicity Of Periodic Solutions For Second-order Nonautonomous Hamiltonian Systems

In this paper,consider the following second-order nonautonomous Hamiltonian system some existence and multiplicity of periodic solutions for it are studied by using the least action principle and the minimax methods.According to the content,this paper is divided into six chapters:In Chapter 1,we briefly introduce the development of variational method and the theory of Hamiltonian system.In Chapter 2,we give the preliminary knowledge to prove the results in this paper.In Chapter 3,we mainly study the Hamiltonian system in a class of mixed potential under the existence of periodic solutions,where potential F(t,x)=F1(t,x)+F2(t,x),F1(t,x)is(?,?)-subconvex,i.e.,a functional G:RN?R called(?,?)-subconvex,if exist ?,? and (?)x,y?RN,such that G(?(x+y))??(G(x)+G(y)),and nonlinearity ?F2(t,x)is the generalize sublinear terms(or the generalize linear terms),that is,there exist nonnegative control function h ? C([0,?),[0,?)),which is sublinear(or linear),and exist f,g L1([0,T],R+),such that|?F2(t,x)|?f(t)h(|x|)+g(t).In Chapter 4,we mainly study the Hamiltonian system in a class of mixed potential under the existence and multiplicity of periodic solutions,where potential F(t,x)=F1(t,x)+F2(t,x),nonlinearity ?F2(t,x)is the generalize linear terms,and(?)??L1([0,T],R),(?)??L1([0,T],RN),for(?)x ? RN,such that F1(t,x)?(?(t),x)+?(t).In Chapter 5,we mainly study the Hamiltonian system in a class of periodic potential under the existence and multiplicity of periodic solutions,where potential F(t,x)=F1(t,x)+F2(t,x)is periodic,that is,there exist T>0,for (?)?RN,such that F1(t+T,x)=F1(t,x),F2(t+T,x)=F2(t,x)and F1,F2?C1(R,RN,R).In Chapter 6,we mainly give examples that satisfy the main results of this paper.