Solutions Of Initial Value And Boundary Value Problems For Impulsive Integro-Differential Equations

The present dissertation is devoted to initial value problems and boundary value problems for second order impulsive integro-differential equations in Banach spaces. Especially, when the initial conditions are different, we study the existence of solutions and establish the existence theorems of solutions. This paper is mainly divided into three 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, by established comparison results, through introducing the equivalent norm , by applying Banach fixed theorem and using the monotone iterative technique, we obtain one sufficient condition of existence of maximal and minimal solutions to initial value problem for second order impulsive integro-differential equation in Banach spaces. And we give another sufficient condition of existence of maximal and minimal solutions based on the Kuratowski measure of noncompactness. At the same time, under weaker conditions, we establish the existence theorems of solutions by using Gronwall inequality.In the third chapter, we investigate three-point boundary value problems for second order impulsive integro-differential equations in Banach spaces. By using different methods, we obtain the different prior estimates with their values fixed at one end point and different boundary conditions. Thus, we establish the existence theorems of solutions by employing Leray-Schauder fixed theorem.