Existence And Properties Of Solutions For Fractional Differential Equations

In recent years increasing interests and large researches have been given to the fractional differential equations both in time and space variables.These are due to the applications of the fractional differential operators to problems in a wide areas of various of science.It is well known that lots of practical problems can be described by the models of fractional differential equations,and the fractional calculus and fractional differential equations have being become an important mathematical tools in dealing the practical problems.Therefore.this dissertation is devoted to study the existence,uniqueness,stability,stabilization and approximate controllability of solutions for some class of fractional order systems,lby applying the theories of Banach fixed point theorem and fractional resolvent operators,in the framework of fractional differential equations.The research work of this thesis is contained in the following five chapters.At the beginning,we try to introduce a brief introduction as well as motivation and main results of this thesis.In Chapter 1,a class of boundary value problems of nonlinear fractional differ-ential equations with multi-fractional derivatives is discussed in finite dimensional spaces.First,expression of the solution for linear fractional differential equation is derived by the method of Green's functions.Then,Some new existence and u-niqueness results are obtained by using standard Banach contraction principle and Krasnoselskii's fixed point theorem,respectively.At last,a class of fractional dif-ferential equations with variable coefficients is also considered.In Chapter 2,we continue our study on the boundary value problems of non-linear fractional differential quations with multi-fractional derivatives and nonlocal integral conditions in finite dimensional spaces.The existence and uniqueness of so-lutions for nonlinear fractional differential equations and differential inclusions with nonlocal integral conditions are gained by using the approaches of Leray-Schauder degree theory,nonlinear alternative of Leray-Schauder,multivalued analysis and fixed point theorems for multivalued maps.Chapter 3 is concerned with the stability and stabilization of a class of fractional-order semi-linear control system and semi-linear feedback control system with Ca-puto derivatives in finite dimensional spaces.New conditions ensuring the asymp-totic stability and stabilization of the fractional system and feedback system with the fractional-order between 0 and 2 are proposed,respectively.The analysis is based on a property of convolution and the asymptotic properties of the Mittag-Leffler functions.At last,numerical simulations of three systems are provided to illustrate the feasibility and validity and reliability of proposed approach.Chapter 4 is devoted to discuss the approximate controllability of fractional control systems governed by differential equations with Riemann-Liouville fractional derivative in general Banach spaces.First,we derive the uniqueness and existence of mild solutions for fractional differential equations by the approach of fixed point and fractional resolvent under more general settings.Then,we present new sufficient conditions for approximate controllability of fractional control system by means of the iterative and approximate method.The final Chapter 5 is concerned with approximate controllability of fraction-al order semilinear delay system.First,the suitable mathematical model for the fractional differential equations with delay is formulated and the definition of mild solutions for this system is given by means of the method of weighted delay.Then,the existence and uniqueness of the mild solutions are proved by the approach of Banach contraction principle and resolvent operators.At last,the approximate controllability results are gained under the assumption that the corresponding lin-ear system is approximately controllable by the transform method and the iterative technique.Our results extend the relative existing ones in this area.