Applications Of Critical Point Theory In Impulsive Differential Equations

In this dissertation, we mainly consider the existence of some special solutions(include periodic solutions and homoclinic solutions) for some classes of impulsive differential equations by critical point theory. For the different kind of impulsive differential equations, the corresponding functional is constructed under the appropriated impulsive conditions. Then the existence of solutions for the impulsive differential system is attained from the existence of critical points for the functional. This dissertation consists of six chapters.Chapterâ… introduces the research background and the development for the impulsive differen-tial equations, as well as the basic knowledge of critical point theory and the main results in this dissertation.Chapterâ…¡studies the existence of periodic solutions for a class of impulsive differential equa-tions with a nonnegative linear term, which extends the results in literature. Firstly the existence of nonzero periodic solutions is got by the Mountain Pass Lemma and Linking Theorem. Then the ex-istence of multiple periodic solutions is established by Clark Lemma and the number of solutions is determined by the linear coefficient.Chapterâ…¢studies the existence of periodic and homoclinic solutions for the superlinear im-pulsive differential equations. Under the appropriated impulsive conditions, the existence of nonzero periodic solutions, multiple periodic solutions and homoclinic solutions which are all generated by impulses is attained, where the number of multiple periodic solutions is determined by the impulses.Chapterâ…£studies the existence of periodic and homoclinic solutions for the sublinear impul-sive differential equations. In the same way as Chapterâ…¢, we established the existence of nonzero periodic solutions and homoclinic solutions generated by impulses by strengthening the impulsive conditions.Chapterâ…£studies the existence of periodic solutions generated by impulses for the p-Laplace differential equations with the impulsive conditions. And in the last chapter, we established the exis-tence of nonzero periodic solutions and multiple periodic solutions for the delay differential equations with the impulsive conditions, where the number of multiple periodic solutions is determined by the impulses.