Solvability To Two Classes Of Fractional Differential Equations

The main aim of this thesis is to investigate the solvability of an abstract fractional integro-differential equation with a nonlocal initial condition and a functional differential inclusions with a state-dependent delay in Banach spaces.This thesis is divided into four parts.The first chapter is introduction,we introduce the research backgrounds and the main work of this paper.The second chapter preliminaries,we list some fundamental facts and some notations which are used throughout this thesis.The third chapter,is mainly focused upon the following solvability for fractional integro-differential equation with a nonlocal initial condition in Banach spaces of the formwhere ^cD_t^? is the Caputo fractional derivative of order??(0,1],the state x?·?takes value in a Banach Space X.Here A is the infinitesimal generator of an analytic operator S??t?,t>0.Let K:D?R,D = {?t,s??J×J:t?s} and f,g are given functions to be specified later,and x0 is an element of X,under certain assumptions,the solvability conclusion is obtained by fixed point theorem.As an application,we also consider the following controllability problem of an ab-stract fractional integro-differential evolution equation with a nonlocal initial conditionwhere the control function u?·?is given in L₂?J,U?,with U as a Banach Space.B is a bounded linear operator from U into X.Finally,a simple example is provided to illustrate Theorem 3.1.3.The fourth chapter is mainly concerned with the solvability for the following an abstract fractional functional differential inclusion with a state-dependent delay in Banach spaces of the formwhere A defined on a Banach space X is the infinitesimal generate of a Co-semigroup;the nonlinear multivalued term F takes closed and convex values for which F?t,·?is upper semicontinuous for a.e.t?[0,b]and F?·,v?has a Lp-integral selection for each v?C(?-?,0];X?.The functions us:(-?,b]?X,us???= u?s +??,belongs to some abstract phase space B described axiomatically with a seminorm ||·||_B;?:J×B?(-?,b]is appropriate function.Some sufficient conditions to guarantee the solvability are estab-lished via condensing maps for multivaled maps.