The Solution Of Fractional Differential Equations With Integral Boundary Conditions

The theory of the fractional differential equation is an important branch of the non-linear functional analysis field. In recent years, the theory of the fractional differential equation obtained more and more attention, and its development is gradual perfecter.Fractional differential is the extension and expansion of integer differential, the devel-opment of it almost synchronous with integer order differential equation. Because of the theoretical significance and the value of its practical research, many specialists has began to study it. Its development has extensive on, more and more researchers to join in this area.In this note, we study the solution for two classes of fractional differential equations with integral boundary conditions. The paper is divided into three chapters: The chapter 1, we introduce the history and evolution of the relevant definitions of integral boundary value together with some basic definitions and properties are given.The chapter 2, we study the following fractional differential equations with integral boundary condition and multi-point boundary conditionIn previous studies, the boundary value conditions are generally one of integral boundary value conditions and multi-point boundary value conditions. In this chap-ter I will take the sum of them, and the boundary value conditions is u(i)(1)=?01 g(s)u(s)ds +?j=1m ?ju(i)(?j). Referring to the methods of article [6][7][8][9] and using the Schauder fixed point theorem and monotone iterative method to get the existence and uniqueness of the solution of equation (2.1.1).The chapter 3, we study the following fractional differential equations with Riemann-Stieltjes integral boundary conditionThis chapter is on the basis of the equation in article [2]. I changed the boundary value conditions to u(1) = ?01u(s)dA(s), and changed the second order to the original n order; Changing the equation in article [12] and taking the parameter instead of Riemann-Stieltjes integral boundary value; Referring to the methods of article [12][13][14],I used the fixed point index and Guo-Krasnoselskii fixed point theorem to get the existence of the solution of equation (3.1.1).