The Properties Of Impulsive Integro-Differential Equations

Impulsive integro-differential equations can be successfully used for mathematical simulation in theoretical physics,chemistry,biotechnology,medicine,population dynamics, optimal control,and in other processes and phenomena in science and technology.The stability theory of impulsive differential equations has been developed by a large number of mathematicians,and their studies have attracted much attention.They have been successful in different approaches based on Lyapunov direct method and comparison technique.This method has generality from the theoretical standpoint,but it is not convenient for practical use sometimes. The main difficulty lies in constructing the Lyapunov functional.In this paper,we employ the iterative method which is very concrete to obtain the existence,uniqueness and stability of integro-differential equations with impulses or impulses and delays for periodic boundary value problems,anti-periodic boundary value problems and two points boundary value problems.The paper is divided into six chapters.In the chapter one,we mainly introduce the developing history,the present situation and the significance of differential systems and the main contents of this paper.The second chapter introduces some basic theorems and definitions (Picards existence and uniqueness theorems,stability definitions,as well as the iterative analysis method used in this paper).In chapter three,by means of iterative analysis method,the existence, uniqueness and stability of periodic boundary value problems and anti-periodic boundary value problems and the existence and uniqueness of two points boundary value problems for first order integro-differential systems with impulses are considered.In chapter four,by means of iterative analysis method,the existence,uniqueness and stability of periodic boundary value problems and anti-periodic boundary value problems and the existence and uniqueness of two points boundary value problems for first order integro-differential systems with impulses and delays are considered.In fifth chapter,iterative analysis method is applied to cellular neural networks with impulsive effects and distributed delays,the existence and uniqueness and of periodic solution and anti-periodic solution and stability of the equilibrium point for which are considered.In chapter six,by means of iterative analysis method,the existence,uniqueness and stability of periodic boundary value problems for second order integro-differential systems with impulses are considered. From the discussion in the paper,we can clearly see that the existence and stabilities of the boundary value problems which we studied are all connected with the impulsive conditions and delay variable closely.