一类具有积分边界条件的分数阶微分方程的解

Solutions for A Class of Fractional Differential Equations with Integral Boundary Value Conditions

主要对一类带有双积分边界条件的分数阶微分方程进行分析和研究。首先应用分数阶微积分的相关性质给出此类方程的等价方程。然后通过构建Green函数,在Banach空间中给出算子T的定义,将此等价方程的求解问题转换为T在Banach空间中的不动点问题。再由Green函数的有关特性分析算子T,在不同的条件下,分别利用Banach压缩映像原理和Krasnoselskii不动点定理,得到算子T不动点的存在唯一性和存在性,即原边值问题解的存在唯一性和存在性。最后给出一个例子来说明所得结果的应用性。

In this paper,a class of fractional differential equations with double integral boundary value problems are investigated.Firstly,the equivalent equations of this kind of equations are obtained via the related properties of fractional calculus.Then,by constructing Green's function and the operator T in Banach space,the solutions of this equivalent equations can be transformed into the fixed points of T in Banach space.The operator T is analyzed by the properties of Green's function.The uniqueness and existence of the fixed point about the operator T are given,by using Banach's contraction mapping principle and Krasnoselskii's fixed point theorem respectively,and therefore the uniqueness and existence of the solution of the kind of boundary value problems are obtained.An illustrative example is given to show the applicability of the results in this paper.