| In this paper,the variational structure of boundary value problems for a class of fractional differential equations is obtained by using the variational method.Combined with the critical point theory,new criteria for the existence of infinite pairs of solutions and multiple periodic solutions for two classes of second-order Hamiltonian systems are obtained.This paper is divided into six chapters.The specific contents are as follows.In chapter 1,we introduce the research background and current situation,briefly describe the basic ideas of variational method and critical point theory,and expound the difficulties and innovations of the research.In chapter 2,we introduce some basic knowledge needed in this paper,including basic definitions,properties and theorems.In chapter 3,a class of boundary value problems for fractional differential equations with instantaneous and non-instantaneous impulse conditions are studied.The main method is the variational method.The variational structure of the original problem in the case of b,d>0 and bd=0 is obtained.In chapter 4,we study a class of second-order Hamiltonian systems with instantaneous pulse condition,non-instantaneous pulse condition and Sturm-Liouville boundary condition.By using the variational method,we obtain the variational structure of this kind of problem,and combined with the critical point theorem,we obtain a new criterion for the existence of infinite pairs of solutions.In chapter 5,the existence of multiple periodic solutions for a class of second-order Hamiltonian systems is studied.By decomposing the space direct sum into four subspaces with different properties and skillfully making superquadratic growth and subquadratic growth assumptions for nonlinear terms,a new criterion for the existence of multiple periodic solutions is obtained by using variational method,space decomposition technology,the latest second critical point theorem and third critical point theorem.In chapter 6,we summarize the work of this paper and look forward to the work that can be done in the future. |
