Existence And Controllability For Fractional Evolution Equations And Inclusions With Nonlocal Conditions

In recent years,scholars have paid more and more attention to the application of frac-tional calculus theory on evolution equations and evolution inclusions research field.Among them,the research on the problems about qualitative theory of fractional order evolution e-quations and inclusions has become hot issues in this field.This thesis mainly studies the exist,ence?exact controllability and approximate controllability for several kinds of fractional evolution differential equations or inclusions.These systems considered in the thesis are influ-enced by the nonlocal conditions?impulse or delay and so on.This Ph.D thesis contains 6 chapters.Chapter 1 is concerned with the background and significance of the issue,recent research general situation and the main work of the thesis.In Chapter 2,we introduce some relevant preliminary knowledge about some notations and function spaces,the related theory of fractional order calculus?semigroup of operators and multi-valued analysis that we will need in this thesis.In physical science,It has been found that the nonlocal initial condition is more general to describe the nature phenomena than the classical initial condition.It is also more extensive in practice,since more boundary conditions are taken into account,such as initial value,integral,multi-point average,period and inverse period etc.In Chapter 3,firstly we apply Laplace transform?probability density function and the operator spectrum theorem to give and testify a new definition of mild solution for a new class of fractional neutral integro-differential evolution equations with infinite delay and nonlocal conditions;Secondly,by using a concrete type of nonlocal function,we delate the compactness and Lipschitz continuity of nonlocal function,only assume that the coefficients of nonlocal function satisfy the weaker conditions in this thesis;Thirdly,we establish and demonstrate suitable sufficient conditions for the exact controllability of the fractional neutral nonlocal control system with infinite delay by relying on measure of noncompactness and Monch fixed point theorem in addition new phase space axioms.So the conclusions of this chapter generalize the results of related literatures completely.Hilfer fractional derivative interpolates Riemann-Liouville fractional derivative and Ca-puto fractional derivative.And it has important applications in practice.However,the research on the qualitative problem of Hilfer fractional dynamical systems is rare.Based on Chapter 3,in Chapter 4,we present a new definition of mild solution for Hilfer fractional differential inclu-sions by using of fractional calculus?Laplace transform?operator spectrum theorem?measure of noncompactness in combination with multi-valued analysis at first.Secondly,by combining with o'Regan-Precup fixed point theorem which is the promotion of Monch fixed point theorem,the suitable sufficient conditions for exact controllability of this Hilfer fractional differential in-clusions are formulated and proved.Finally,an example is given to illustrate the application of abstract conclusions.In view of mathematical point,approximate controllability is more general than exact controllability.Based on Chapter 4,we consider the existence and approximate controllability for Hilfer fractional differential evolution inclusions with impulse in Chapter 5.Firstly we establish a new definition of PC1-v-mild solution using operator semigroup theory?probability density function in combination with impulsive condition;Secondly we give the proof on the existence of mild solution by fractional calculus?multi-valued analysis and suitable fixed point theorem;Thirdly we demonstrate the approximate controllability of the relevant system under the hypothesis that its linear system is controllability;At last,an example is presented to explain the abstract conclusions of the theorem.Taking advantage of fractional calculus,fractional-order damping with a viscoelastic damping element provides a better model to describe a damping system.So it's very im-portant and necessary to discuss the qualitative problem of fractional damping system.In Chapter 6,we research the existence and approximate controllability for fractional neutral damping systems with delay and order in(1,2).Firstly,we provide the representation of mild solutions for fractional neutral damping systems with delay and order in(1,2)by applying the method of Laplace transform and the theory of(p,q)-regularized families of operators.Next,we explore the existence and uniqueness of this system under some suitable sufficient conditions by using Banach contraction mapping principle.Furthermore,under certain hypotheses,the approximate controllability is obtained by using the approximate sequence method.