Impulsive Integro - Differential Equations Of The Overall Solution And Closed Operator Mp Inverse Perturbation Analysis

In this thesis,we discuss the existence of global solutions for initial value problems of first order implusive integro-differential equations in Banach spaces and the perturbation problem for nonbounded generalized inverses of bounded linear operators. It is consisted of two chapters.In the first chapter, we consider the initial value problems of mixed type first order implusive integro-differential equations (IVP) in Banach space, and it is divided into three parts:Initial value problems for impulsive integro-differential equation are very important in practical applications such as physics, astronomy, bio-engineering and applied mathematics. In recent years, many attempts have been made to study the existence of solutions of first-order initial value problems with impulses, and achieved some good results. In this chapter, the main tools are the theory of Hausdorff's measure of noncompactness and the generalized Darbo's fixed point theorem. Through solving equations step by step, with f is continuous but non-uniform, the existence of global solutions of these issues is obtained .Chapter two deals with the perturbation problem of Moore-Penrose generalized inverse of densely defined closed linear operator in Hilbert space:The theory of generalized inverse has numerous applications in mathematic fields such as numerical linear algebra, numerical analysis, optimization, control theory, mathematical statistics, differential equations. The perturbation problem of generalized inverse of densely defined closed operator in Banach space has already been studied by Ma Jipu, Wang Yuwen, Zhang Hao, Huang Qianglian and so on. In this chapter, we first introduce the definition of Moore-Penrose generalized inverse of densely defined closed linear operator in Hilbert space, and then give the relational expression: . And then by making using this result, we obtain that if there has the bounded Moore - Penrose generalized inverse T^? of the densely defined closed operator T , and the linear perturbation operatorδT is T-bounded with and R (T +δT )∩N (T^?) ={0}, then the operator (T|-) = T +δT has Moore- Penrose generalized inverse (T|-)^?, andWhere P_N(T|-) is the unique norm-preserve extension of I-T^? (I+δTT^?)^-1(T|-) on space X .