非线性分数阶微分方程四点非局部边值问题

On 4 Problems of Nonlocal Boundary Value of Nonlinear Differential Equations of Fractional Order

从两点到三点到m点再到无穷多点,对常微分方程边值问题的研究最早始于牛顿和莱布尼茨建立微积分的最初阶段。这些常微分方程多点边值问题也常常被称为常微分方程非局部问题。讨论阶数为q∈(1,2)的非线性分数阶微分方程四点非局部边值问题,借助Ascoli-Arzela定理,首先利用压缩映射原理得到解的唯一性,其次利用Krasnoselskii不动点定理得到四点边值问题至少存在一个解,并且举例验证。

From two to three, and then to the infinite points, researches of BVPs（ boundary value problems） of ODE （ ordinary differential equation） were first started in the initial stages of the calculus set up by Newton and Leibniz. These boundary value problems of differential equations are also often referred to as non - local ordinary differential equations problems. In this paper, the existence and uniqueness of solutions ＇to a four - point non - lo- cal boundary value problems of nonlinear differential equations of fractional order q∈ （1,2） is analyzed, with the help of the Ascoli - Arzela theorem, and the use of the uniqueness of the contracting mapping principle solution, followed by Krasnoselskii fixed point theorem for four - point boundary value problem, is at least one solution, and example is provided to illustrate the theory.