Existence Of Solutions For Initial Boundary Value Problems Of Nonlinear Fractional Evolution Equations

Nonlinear functional analysis is a important branch of mathematics, which mainly in-clude topology degree theory, fixed point theory, partial order method and so on. Nonlinear functional analysis has many important theory and advanced methods to study the nonlinear problems.Nonlinear functional analysis has important application on the nonlinear evolution e-quation in Banach space. It has been wildly applied on the physics, chemistry, finance and optimum control etc.. In recent years, the existence of mild solutions to initial value, bound-ary value problems for nonlinear evolution equations attracts more and more researchers attention, and series of good results have been obtained. Fractional calculus has been suc-cessfully applied on the fractal, porous medium diffusion, finance and so on. The fractional differential operator has the properties of singular and nonlocal, which make the fractional evolution equation more complex than the integer order evolution equation. Therefore, it has important significance on the theoretical and the practical application to study the initial value and boundary value problems of the fractional evolution equation.In this paper, we consider the existence of the mild solutions for some class of nonlinear fractional evolution equations. By semigroup (resolvent operator) theory, measure of non-compactness, fixed point theorems and so on, we obtain many new results on the existence of mild solutions for the fractional evolution equation. Many results have been published on?Appl. Math. Lett.?(SCI) and?Comput. Math. Appl.?(SCI).This paper includes five chapters. In Chapter I, the application and the evolutionary his-tory of fractional calculus has been introduced, Riemann-Liouville fractional integral (differ-ential) operator?Caputo's fractional differential operator?some preliminary definitions ?properties and lemmas of nonlinear function analysis are given. We also introduce the appli-cation and the research status of the evolution equation with non-instantaneous impulsive.In Chapter ?, by using the resolvent operator theory? measure of noncompactness?fixed point theory and Banach contraction mapping principle, we investigate the existence and uniqueness of global and local mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay. For t in finite interval, we study the existence of mild solu-tions for the nonlinear fractional reaction-diffusion equations under the compact resolvent operator and noncompact resolvent operator. For t in infinite interval, we study the existence of global and local mild solutions for the nonlinear fractional reaction-diffusion equations under the compact resolvent operator. Our main results improve and generalize some exist-ing results.In Chapter ?, by using the measure of noncompactness and fixed point theory, we con-sider the existence of local and global mild solutions for a class of semilinear fractional evolution equation. In addition, we study the existence of mild solution for a class of frac-tional mixed type evolution equation by the same method as the Theory 3.3.2. At the last of this Chapter, an example is given to illustrate the application of our main results.In Chapter IV, we consider the semilinear fractional integro-differential equation with non-instantaneous impulsive and delay. By resolvent operator theory and fixed point theory,we obtain some new results on the existence of mild solutions for the semilinear fractional integro-differential equation with non-instantaneous impulsive and delay. At the last of this Chapter, we give an example to illustrate the application of our main results.In Chapter V, we consider periodic boundary value problems for the semilinear fraction-al integro-differential equation with non-instantaneous impulsive. By using the resolvent operator theory?measure of noncompactness and fixed point theory, we obtain some new results on the existence of mild solutions for the semilinear fractional integro-differential equation with non-instantaneous impulsive. At the last of this Chapter, we give an example to illustrate the application of our main results.In Chapter VI, we consider the fractional non-autonomous evolution equations with im-pulses and delay. By the generalized Banach fixed point theorem, we obtain some new results on the existence and uniqueness of the mild solution. An explicit iterative scheme for the mild solution and an error estimate of the approximation sequence for the initial value problem are also derived. Moreover, the unique mild solution of the problem is continuously dependent on the initial value.