Studies On Existence For Two Kinds Of Fractional Differential Equations

This article first briefly introduces the development history and research subject of fractional calculus, and the definition of fractional calculus operator, applied to the fractional order differential equation.The fractional order calculus equation model is established. Under the inspiration of other literature materias, use fixed point theorem to discuss two types of fractional differential equations with the existence of the mild solution, and introduce the concept of almost automorphic, pseudo almost automorphic and weighted pseudo almost automorphic functions, further discussed with these characteristics moderate the existence of the solution.Fractional order calculus is an extension of integer order differential and integral calculus and its applied range is very wide, used to describe differential equations of science and engineering field, etc. Based on the advantages of practicality is very large, the research of mild solutions to fractional order differential equation especially almost periodic solution has attracted wide attention of many mathematics workers and researchers, emerging out many theories and methods in this aspect.This thesis is divided into four main research topic for discussion, the first part of the fractional order calculus is given from three aspects with its related equations, the first one is background introduction, the second aspect is given the main research contents of this paper, study existence for two kinds of fractional order differential equation with mild solutions. The third aspect is the generalization of research significance. The second part of this paper is given to use the knowledge and the basic method of preparation. The third part mainly discusses the fractional order differential equation model between1and2, gives the definition of mild solutions and some lemma and theorem,which is related to weighted pseudo almost automorphic. Using the method of decomposition structure and Leray Schauder alternative theorem, the theorem proof process is presented. Use mainly the continuity, compactness, compact operator and the completely continuity. The fourth part discusses another kind of fractional order differential equation model, the order number is between0and1with state dependent-delay.and firstly presents the definition of mild solutions for such equations and its associated lemma and theorem, using analysis method with step by step, the continuity, the degree of continuity, completely continuity and choose a fixed point theorem to find the fixed point of operator, which is the mild solution of the equation. And prove the existence of mild solutions. In this part also discuss the neutrality of fractional order differential equation, it still belongs to the fractional order differential equations with state dependent-delay. The proof method is similar to the front, just different on the structure of the operator, needs to operator decomposition through full continuity of Arzela-Ascoli theorem to the operator, thus the original operator is condensing operator, so that we can find a solution to the equation.In this paper, some sufficient conditions the form of theorems are given for two kinds of fractional order differential equations with the existence of the mild solution in the form of theorems, especially existence of the solutions is-given for the first equation with weighted pseudo almost automorphic function. The results are obtained by the fixed point theorem.