Research On The Existence Of Solutions For Some Classes Of Boundary Value Problems Of Nonlinear Fractional Differential Equations

Fractional calculus has widely used in various fields of science as an important branch of calculus theory whose theoretical study have drawn the attention of the researchers at home and abroad along with the increasingly rapid development of science and technology. As the basis of fluid mechanics and fractal theory, fractional calculus especially fractional differential equations has become a focus of research scholars and many scholars make efforts in different areas, obtaining many results.In this paper, we study the existence of solutions for three kinds of boundary value problems of nonlinear fractional differential equations. The paper is divided into the following five parts.In Chapter 1, we introduce the current situation and research background of fractional differential equations, give a brief description of other scholars’ research results of fractional differential equations field. In the end, we provide an introduction of the research purpose and content of this paper.In Chapter 2, we list the definitions and properties of the integro-differential equations of fractional order and some relevant fixed point theorems, which will be used in our research.In Chapter 3, we study a class of integral boundary value problems for nonlinear fractional differential equation with p-Laplacian operator. To obtain existence of the positive solutions by means of Avery-Peterson fixed point theorem, we point out the properties of the related Green’s function, the definition of the cone and the integral operator. At last, an example is given to show the effectiveness of the obtained result.In Chapter 4, we are concerned with the existence of positive solutions for the boundary value problem of fractional impulsive differential equation. By using fixed point theorem of cone expansion and compression of norm type, some results about positive solutions for the boundary value problem are obtained.In Chapter 5, we consider the existence of solutions for the boundary value problems of fractional impulsive integro-differential equations. By using Banach contraction mapping principle and Krasnoselskii’s fixed point theorem, some sufficient conditions for the existence of at least one solution about the problem are gained.