Periodic Solutions And Periodic Boundary Value Problems Of Some Impulsive Differential Equations

The theory of impulsive differential is one of important branches of differential equations. In the field of modern applied mathematics, it has made considerable head way in recent years, because all the structure of its emergence has deep physical background. As an important branch of impulsive differential theory, periodic solutions and periodic boundary value problems have always been an active field which attract many scholars' interesting, and have acquired many good results. There are at ot of ways to study these problems, for example, the method of upper and lower solutions and the iterative technique, comparison principle, fixed point theorem, coincidence degree theory, etc.This thesis of Master is composed of three chapters, which mainly studied the existence of positive solutions for first order periodic boundary value problem with impulses and the existence of periodic solutions for the Lienard equation with impulses.In chapter one, we first introduce the development of impulsive differential theory and the latest advancement, then study some results of impulsive periodic boundary value problems, which given by some experts in the inner and outer country.In the second chapter, first we construct the Green's function of the problem, then define a corresponding cone. By using a completely continuous operator on the cone and the fixed point theorems in cones, we study the existence of positive solutions for first order periodic boundary value problem with impulses, and some sufficient conditions will be given. An example shows that our results are more simple and more easily verifiable.In the third chapter, by means of the coincidence degree theory, estimating the range of the periodic solutions, we study the existence of periodic solutions for the Lienard equation with impulses, and some sufficient conditions will be established.In the last we make a conclusion of this paper and give a ptan of the future work.