Existence And Approximate Controllability Of Mild Solutions Of Two Kinds Of Fractional Evolution Equations

In recent years,the study of fractional calculus theory and fractional differential equations have made great progress,among which the study of fractional evolution equations have become a hot issue.In this paper,we mainly study the existence and approximate controllability of mild solutions of two kinds of fractional evolution equations under nonlocal conditions.The first chapter mainly introduces the development history of fractional calculus theory,the research status at home and abroad in recent years,the main work of this paper,and some preliminaries needed in this thesis.In Chapter 2,the definition of mild solution of fractional evolution equation with Riemann-Liouville derivatives under nonlocal conditions is given by using Laplace transform and probability density function.Secondly,the sufficient conditions for the existence of mild solution of subproblem are given by using Schauder’s fixed point theorem.Then,the sufficient conditions for the existence of mild solutions of the fractional evolution equations under nonlocal conditions are given by using the idea of approximation and diagonal argument.Finally,under the assumption that the corresponding linear differential system is approximately controllable,we give the sufficient conditions for the differential system to be approximately controllable.In Chapter 3,we firstly use Laplace transform and probability density function to give the definition of mild solution of fractional evolution equation with Hilfer derivatives under nonlocal conditions.Then sufficient conditions for the existence of mild solution of the problem is obtained by constructing iterative sequences using the noncompactness measure theory and Arzela-Ascoli theorem.Finally,under the assumption that the operator semigroup have differentiability,we give the sufficient conditions for the differential system to be approximately controllable.In Chapter 4,we give a summary of this thesis and a prospect for future research.