Solvability Of Fractional Differential Inclusions With Nonlocal Initial Conditions

The main aim of this thesis is to investigate the solvability of two kinds of fractional differential inclusions with a nonlocal initial condition in Banach spaces. This thesis is divided into four chapters.In Chapter 1, we introduce the backgrounds and recent development of the differential inclusions, nonlocal initial condition, fractional calculus, resolvent family and give the main aim of this thesis.In Chapter 2, we recall some preliminary results which will be needed throughout this thesis, including some notations, basic definitions and fundamental lemmas.In Chapter 3, we mainly investigate the existence of mild solutions of the following fractional differential inclusions with nonlocal initial conditions under different conditions:where 0 < α < 1, Jtβv(t)= ∫0tGβ(t-s)v(s)ds for v ∈L1(J,X), Gβ(t) =tβ-1/г(β) for β > 0, t >0, and Γ(·) stands for the Gamma function, and гwhere 1 <α< 2, Dtα is understood in Caputo sense, x0,x1 ∈X, F : J×X→P(X), p, q are suitable continuous functions.By theories of resolvent family of operators, multivalued analysis and fixed point approach et al, we establish the existence results of mild solutions to the above equations under the different three conditions, respectively.In Chapter 4, we conclude the main contents of this thesis.