一类非线性分数阶微分方程边值解的存在性和唯一性

Existence and Uniqueness of Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equations

本文考虑非线性分数阶微分方程边值问题D0^α＋u（t）＋f（t,u（t）,D0^β＋u（t））=0,0〈t〈1.u（0）=u＇（0）=0 D0^β＋u（1）=μD0＋^β＋u（ξ）的解，其中2〈α≤3,1≤β≤2,并且α-β≥1，0≤μ≤1,0〈ξ〈1,D0^α＋D0^β是标准的Riemann-Liouville分数阶导数.文章研究了Green函数的性质,并利用一些不动点定理得到了解的存在性和唯一性结果.最后给出几个具体例题说明本文结果的应用.

In this paper, the nonlinear fractional differential equation boundary value problem D0^α＋u（t）＋f（t,u（t）,D0^β＋u（t））=0,0〈t〈1.u（0）=u＇（0）=0 D0^β＋u（1）=μD0＋^β＋u（ξ）is considered. Where 2〈α≤3,1≤β≤2,and α-β≥1,0≤μ≤1,0〈ξ〈1,D0^α＋D0^βare the standard Riemann-Liouville fractional order derivative. The properties of Green＇s function are investigated and the existence and uniqueness results of solutions are obtained by using some fixed point theorem. At last some examples are presented to demonstrate the application of our main results.