(Existence And Multiplicity Of Periodic And Subharmonic Solutions For Several Kinds Of Differential Equations And Inclusion Systems)

This doctoral dissertation deals with the existence and multiplicity of periodic solutions and subharmonic solutions for three classes of differential systems. The first system is the second order Hamilton system with variable exponent; the second system is the second order Hamilton inclusion system with variable exponent; the third is a class of fractional differential system with variational structre. This thesis is made up by four sections.In Chapter1, a brief introduction is given to the historic background, status and the up-to-date progress for all the investigated problems. Some preliminaries are also introduced in this chapter.In Chapter2, by using the least action principle, saddle point theorem, linking theorems, the existence of solutions and subharmonic solutions for the following second order Hamiltonian system is investigated. Some sufficient conditions are obtained which generalize and im-prove some known results. By using symmetric mountain pass lemma, Fountain theorem and Dual Fountain theorem, the existence of infinitely many periodic solutions for the second order Hamiltonian system is studied. Some new theorems are obtained under the potential satisfying su-perquadratic condition and other conditions.In Chapter3, by building variational structure, the existence of solutions and subharmonic solutions for the following two second order Hamiltonian inclusion system is considered by the least action principle, saddle point theorem, linking theo-rems in nonsmooth critical point theory. The results in some known literatures are generalized; by using symmetric mountain pass lemma, Fountain theorem in nonsmooth critical point theory, the existence of infinitely many solutions for the second order Hamiltonian inclusion system is considered, and some new theorems are obtained.In Chapter4, the existence and multiplicity of solutions for the following fractional differential inclusion system is considered by nonsmooth critical point theory. We expand the methods main-ly used in the literature to deal with the boundary value problem for fractional differential inclusion system, that is, the. nonsmooth critical point theory change the situation that the fixed point theory of multivalue map only give the results of existence for solutions.