The Existence Of Solutions For Nonlinear Boundary Value Problem Of Fractional Differential Equations

Fractional differential equations is an important branch of differential equations, which is a theory studies arbitrary noninteger order differential equation, including arbitrary real and plural number. Research on fractional differential equations is based on the theory of integer order differential equations. Qualitative analysis which includes existence, uniqueness, stability, oscillation and other properties of solutions for fractional differential equations, especially the existence and uniqueness have concerned by scholar.In this paper, we study two types of fractional differential equations with nonlinear boundary conditions. Firstly, we mainly investigate the existence and uniqueness of solutions for a type of Caputo fractional differential equation with nonlinear boundary conditions. Secondly, we study the existence, uniqueness and multiple solutions for a type of Riemann-Liouville fractional differential equation. Finally, we get some new theory, and cite an example to prove our conclusions. This article is divided into the following four parts:In chapter one, we introduce the history, significance and the status at home and abroad of fractional differential equations, and provide an overview of this major work.In chapter two, we give some basic notions and lemmas about fractional differential equation and that will be used in our paper.In chapter three, when1 <d <2, we discuss a type of Caputo fractional differential equation with nonlinear boundary conditions. By constructing comparison theory and upper and lower solutions with monotone iterative techniques, we deserve the theory of existence and uniqueness for solutions, and the extremum solution. At last, an example is given to illustrate our results.In chapter four, we still use upper and lower solutions with monotone iterative technique to study a type of fractional differential equations with delay and obtain the existence and uniqueness of solutions. Then by using Leray-Schauder degree, we get the existence of multiple solutions.