Banach Space Differential Equations Moderate Solution Existence,

Abstract differential equations in Banach spaces is an important branch in theory of nonlinear analysis, It has developed fast in the last decades because of its extensive practical applications in many areas of applied mathematics, including engineering, economics, optimal control and optimization theory.Nonlocal initial problems extend and improve the classical Cauchy problems. The study of abstract nonlocal semilinear initial value problems was initiated by Byszewski. Among his several papers, he proves the existence and uniqueness of mild solutions under different conditions. Subsequently, many authors are devoted to the study of various nonlocal Cauchy problems because it is demonstrated that the nonlocal problems have better effects in application than the classical Cauchy problems.The theory of impulsive neutral differential equations has become an important area of investigation in recent years, stimulated by their numerous applications to problems arising in mechanics, engineering, medicine, biology, ecology, etc. The reason for this applicability arises from the fact that impulsive differential problems are an appropriate model for describing process which at certain moments change their state rapidly and which cannot be described using classical differential problems.There have been many results on such problems such as the existence, uniqueness and stability of solutions, asymptotical behavior of solutions and topological properties of solution sets. This paper is concerned with the semilinear integrodifferential equations with nonlocal condition and neutral impulsive partial differential equations in Banach spaces. The existence of mild solutions for the equations are obtained without the assumption of compactness on associated semigroups. This paper is mainly divided into two chapters.In chapter 1, we study the existence of solutions to the following semilinear integordifferential equation with nonlocal conditions in Banach space X .We use the theory and methods of semilinear differential equations in Banach space, the properties of Hausdorff's measure of noncompactness and the fixed point techniques as basic tools. The existence of mild solutions for the equations in different case is obtained without assumption on the compactness of the semigroup T ( t )or function f , and the separation of the spaces X is also not necessary in our results. So we improves and generalizes some previous results in this area.The purpose of the chapter 2 is to study existence of mild solutions for impulsive partial neutral functional equations modeled in the form where A is the infinitesimal generator of an analytic semigroup of linear operators defined on a Banach space X . Our basic tools are the methods and results for differential equations in Banach spaces, the properties of Hausdorff's measure of noncompactness, the theory of analytic and the fixed point techniques. We gain the existence of mild solution for the equations, which extend and improve many results about this problem.