Existence Of Solutions And Approximate Controllability For Neutral Integro-differential Equations With Unbunded Delay

In this dissertatioin,by using the the theory of resolvent operators and the fraction power opera-tors and the ixed point theorem,we study the existence,regularity and approximate controllability of solutions for a neutral functional differential system with state-dependent delay.Two examples are provided to illustrate the application of the obtained results.The obtained results extend and develop some existing thorems in literature.This dissertation contains there chapters.In Chapter 1 we introduce briefly background and main results of this thesis.In Chapter 2,we first tudy the existence of mild solutions in Ba-nach space by Sadovskii Fixed poiot theprem,the fraction power operator and the respluvent operator under compactness condition for the resolvent operator.Then the strong and the strict solutions are obtained under lipschitz condition and Holder continuity condition respectively,Moreover,based on the local existence result and a piecewise extended method,we obtained a blowup alternative result for the equation.Finally,an example is given to illustrate the applications of the obtained results.We consider in Chapter 3 the approximate controllability of semilinear neutral integro-difterential systems with state-dependent delay in Hilbert space.By using Sadovskii fixed point theprem and the theory of analytic resolvent operator,we also obtain some sufficient conditions for approximate controliability of the considered system under the assumption that the corresponding linear system is approximately controllable.An example is presented to illustrate the application of the obtained results in the end.