Solutions Of Impulsive Differential Equations And Their Applications

The focus of this dissertation is to study solutions of existence of impulsive differential equations. In this dissertation, we mainly devote to the fixed point index theory and the fixed point theorems to research the existence of solutions to the given system, which is organized as follows:Firstly, we introduce the research background, and then show definitions and theorems which are used in this dissertation, point out the work and innovations we do, and then systematically analyze development situations of impulsive differ-ential equations.Secondly, we give existent conditions of a class of boundary value problems (BVP) for nonlinear second order impulsive ordinary differential equations. Since this equation has a function p(t) which brings about difficulties for our research. By establishing the expression of system solutions, with integral form and special skills of boundary value problems, we obtain a solution operator A. Next by using the fixed point index theorem, we get a sufficient condition of this problem. The results improve the existing conclusions. We give an example to illustrate the main results.Thirdly, by using two fixed point theory, we give existent conditions and a sufficient condition of a second order impulsive functional differential equations. Since this equation has a function p(t) which brings about difficulties for our research. We promote boundary value problem of impulsive ordinary differential equations to functional differential equations. To illustrate the feasibility of the conclusion, we present an example.Fourthly, we solve existence and uniqueness of mild solutions to fractional impulsive differential equations. It is difficult to study fractional impulsive differ-ential equations, so there are few results in this aspect. We give the expression of solutions following by the above forms, then by using the contraction map-ping principle and semigroup theory, we get the existence and uniqueness of mild solutions. We give an example to illustrate the main results.Finally, we summarize the main results in every chapter in this paper, and note some subjects which need further study.