Boundary Value Problem On The Nth-order Nonlinear Impulsive Integro-differential Equations In Banach Spaces

Nonlinear functional analysis has been one of the most important branches of learning in modern mathematics at present. It provides an effect theoretical tool for studing many nonlinear problems which result from physics , chemistry , mathematics , biology , medicine , economics , cybernetics and so on. It plays an important role in dealing with nonlinear integral equations and differential equations arsing in applied mathematics. Among them,the theory of nonlinear impulsive differential equtions has been emerging as an important area of investigation in rencent years.It has deep physical background and realistic mathematical model in nature. It is at active fields in analysis mathematics.The paper is divided into two chapters.In the first chapter,we give the preface, introducing briefly the results of investigation about nonlinear functional analysis and ordinary differential equations in abstract spaces in rencent years.In the second chapter, we investigate the exsitence of solutions for a boundary value problem of nth-order nonlinear impulsive integro-differential equations. In rencent papers [27-29], Professor Dajun Guo has obtained the exsitence of multiple positive solutions for a boundary value problem of integro-differential equations in Banach spaces by means of fixed point index theory. In paper[30],the author studies a kind of boundary value problem of first order impulsive differential equations in a Banach space by using Monch fixed point theorem. The existence of a solution is obtained. But in papers[27-29], the nonlinear term in any bounded set of E is relatively compact in E for any variant t in J,where J=[0,∞);in paper[30], the equations are first order,and the nonlinear term has no integral operators.In this paper, we shall use the Kuratowski measure of noncompactness and Monch fixed point theorem to investigate the exsitence of a solution for a boundary value problem of nth-order nonlinear impulsive integro-differential equations on an infinite interval in a Banach space. The equations have no requirement which is mentioned in papers[27-29].We obtain the exsitence of a sulution by means of transforming the integro-differential equations into a integral equation,using the measure of noncompactness and Monch fixed point theorem.Finally, an example for infinite system of scalar second order impulsive integro-differential equations is offered.