一类分数阶微分方程解的性质探讨
Exploration on the Nature of Solutions for a Differential Equation of Fractional Order

DOI: 10.12677/PM.2016.61009, PP. 56-64

Keywords: 分数阶微分方程，Caputo微分，Schauder不动点定理，压缩映象原理
Differential Equation of Fractional Order, Caputo Derivative, Schauder Fixed Point Theorem, Contraction Mapping Principle

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Abstract:

本文主要证明了一类分数阶非线性微分方程解的存在性和唯一性。文中用到的微分算子是Caputo分数阶微分算子。因这类方程的可解性是与一类Volterra型的积分方程的可解性等价，所以我们主要研究了与之等价的积分方程解的存在性和唯一性。我们通过Schauder不动点定理证明了积分方程解的存在性，用压缩映象原理证明了解的唯一性。
We prove existence and uniqueness of the solution of a nonlinear differential equation of fractional order. The differential operator is the Caputo fractional derivative. For the solvability of the equation is equivalent to a class of Volterra integral equation, we study the existence and uniqueness of the integral equation. We prove the existence of the solution of integral equation by Schau- der fixed point theorem and the uniqueness of the solution by contraction mapping principle.

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