Periodic Solutions And Homoclinic Solutions Of Impulsive Differential Equations

In this thesis we study the existence of periodic solutions and homoclinic so-lutions for some class of second-order impulsive differential equations mainly via variational methods,critical point and Mountian Pass lemma, etc. These disserta-tion will motivate the development of qualitative theory of impulsive differential equations. The dissertation is divided into three chapters, the main contents are follows.In Chapter1, we introduce some issues arising from the historical background, status of impulsive differential equations, point out the main work and innovations we do.In Chapter2, we give the existence of nontrivial periodic solution and ho-moclinic solution of a class of second-order impulsive differential equation. Since this equation has a disturbance▽W(t,u)which brings about difficulties for our research. By using variational methods, critical point and Mountian Pass lemma, we obtain a sufficient condition of this problem which possess at least one non-zero periodic solution and at least one non-zero homoclinic solution and these solutions are generated by impulses. The results improve the existing conclusions. At the last of this chapter, we give two examples to illustrate the main results.In Chapter3, we study the existence of nontrivial homoclinic solution of second-order impulsive differential equation. Since this equation has a function V(t,x)=Vi(t,x)-V2(t,x), which brings about difficulties for our research.And improve the condition of the literature.By using Mountain Pass lemma, a weak convergence argument and a weak version of Lieb’s lemma, we obtain this problem existing at least one nontrivial weak homoclinc solution. But at theV2(t,x)=0, that is the existing results. Therefore, our results extend some known results. At the last of this chapter, to illustrate the feasibility of the conclusion, we also give a examples.Finally, we summarize the main results in every chapter in this paper.