Solutions Of Initial Value Problems For Second Order Impulsive Integro-differential Equations In Banach Spaces

The theory of nonlinear impulsive integro-differential equations is a new and important branch of differential equation, which originates from some mathematical model of biology, medicine. Because all the structure of its emergence has deep physical background, the research on nonlinear impulsive integro-differential equations is very meaningful. The present dissertation is devoted to initial value problems for nonlinear second order impulsive integro-differential equations in Banach spaces. The paper is mainly divided into five chapters.In the first chapter, we introduce some important concepts, notations, auxiliary results and accept certain basic principles of linear functional analysis and nonlinear functional analysis, which are quoted here for easier references until they are applied in later chapters.In the second chapter, we consider the initial value problems for nonlinear second order impulsive integro-differential equations of mixed type in Banach spaces. Under a weaker compactness-type condition, using M?nch fixed point theorem and a new comparison result, the existence theorem of solutions for nonlinear second order impulsive integro-differential equations of mixed type in Banach spaces is obtained in Banach spaces. The result is new.In the third chapter, through solving the problem step-step and by applying Banach fixed-point theorem, under simple conditions, we obtain existence and uniqueness and iterative approximation of solutions for initial value problems of nonlinear second order impulsive integral-differential equations in Banach spaces, which improve the corresponding results of [15], [16]. As an application of our results, the solutions of the infinite system of nonlinear second order impulsive integro-differential equations be obtained.In the fourth chapter, by using recurrence method, the locally convex topology and M?nch fixed point theorem, the existence theorem of solutions for initial value problems of the second order impulsive integro-differential equations on unbounded domains in Banach spaces is achieved, which improves the related results obtained in [22]. The fifth chapter is the summary of this paper.