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

Fractional differential equations have gained considerable importance due to their application in various sciences, many mathematical modellings in vari-ous science are discribed on basis of fractional differential equation. In the past decade, boundary value problem of fractional differential equation has developed very rapidly. With the further study of this direction, the studing contents and methods continuously get abundant. A lot of results have been obtained for nonlinear fractional equations. This dissertation focuses on the research of the existence of solution of the boundary value problem for nonlinear fractional dif-ferential equations and fractional functional differential equations. The paper consists of four chapters. Main contents are as follows:In Chapter 1, we give a survey to the development and situation of the boundary value problem for nonlinear fractional differential equations and frac-tional functional differential equations and introduces main results in this paper. We also introduces some preliminary material, including some basic concepts from the fractional differential equation, some results from fractional calculus and some importment fixed point theorems.In Chapter 2, we first extends the well-known rule for the differentiation of an integral depending on a parameter with the upper limit depending on the same parameter. We discuss the two points boundary value problem with fractional boundary conditions. We obtain the existence and multiplicity results of positive solutions by using some fixed theorems. Our results improve and extend results in the literature.In Chapter 3, we discuss the three points boundary value problem with frac-tional boundary conditions. We derive first the Green function and its propety. consequently, boundary value problem is deduced to a equivalent integral equa-tion. Next, by using some fixed-point theorems, we obtain the the existence and multiplicity results of positive solutions.In Chapter 4, we begin with the study of boundary value problem for two classes of fractional functional differential equations and give a series of sufficient conditions for the existence of solutions of the boundary value problem. Our results improve and extend a number of results in the literature.