A Few Kinds Of Fractional Order Calculus Equation In Banach Space Of Mild Solution Existence

Fractional calculus is a new branch of mathematical analysis, which studies on the theory and application of the differential and integral of arbitrary order. Although the definition of fractional integral and fractional differential had appeared in the seventeenth century, the research of the theory of the fractional calculus mainly was in progress in the field of pure mathematics in the past three century. Because of the lack of specific application, it seemed that only the mathematicians were interested in it so that it advanced slowly during this period. However, in the last few decades many authors point out that fractional calculus are very suitable for the description of memory and hereditary properties of various materials and processes, which are often neglected in classical models. Nowadays, fractional differential equations are increasingly used to deal with problems in acoustics and thermal systems, rheology and modelling of materials and mechanical systems, signal processing and systems identification, control and robotics, and other areas of application. Therefore, the researches of fractional calculus and fractional differential equations have very important theoretical significance and practical applied value, and fractional differential equations have become research focus. The main work of this paper includes the following four parts. In the first chapter of introduction, we introduce the history and research background of fractional calculus and fractional differential equations, and briefly introduce some important results on the solution of existence and uniqueness of initial value problem of fractional differential equations. In Chapter2, the contents are about preliminaries, mainly including the definitions, some properties of fractional integral and derivative and the basic definitions and theorems of sectorial operator and measure of noncompactness. In Chapter3, by using of the theory of sectorial operator and measure of noncompactnes and the related fixed point theorems, we can obtain some sufficient conditions of existence of the mild solution to semilinear fractional integro-differential equations with infinite delay without supposing compact operator. In Chapter4, we mainly study the boundary value problems for nonlinear fractional differential equations in Banach Spaces. By using Green functions, measure of noncompactness and related fixed point theorem, we obtain some sufficient conditions of existence of the mild solution to such equations. The results obtained in this thesis improve and generalize some previous results.