Existence Of Solutions And Approximate Controllability For Delayed Evolution Systems

In this dissertation, we study the existence and regularity of solutions for a neutral functional integro-differential equation with state-dependent delay in Banach space. And we also study the approximate controllability of semi-linear non-autonomous evolution systems with nonlocal condi-tions. The main tools are fraction power operator, α-norm. fixed point theorems and the theory of resolvent operators and evolution system.This dissertation contains three chapters:In Chapter1we introduce some background knowl-edge of integro-differential equations and approximate controllability. In Chapter2we investigate the existence and regularity of solutions for a neutral functional integro-differential equation with state-dependent delay by fixed point theorem, fraction power operators and theory of resolvent op-erators. In particular, it is not require that F. G be continuously differentiable when studying the regularity of mild solutions, instead, they only satisfy Holder continuous. In Chapter3we study the approximate controllability for the semi-linear non-autonomous evolution system with nonlocal condition by using fixed point theorem and the theory of evolution system. It is worth mentioning that the fraction power theory and α-norm are used to discuss the problems so that the obtained results can apply to the systems in which the nonlinear terms involve derivatives of spatial variables.