基于Picard迭代的Conformable分数阶微分方程初值问题的存在唯一性研究

Existence and Uniqueness of Solutions for Initial Value Problems of Conformable Fractional Differential Equations via Picard Iteration

摘要: 本文利用常微分方程理论中典型的Picard迭代方法研究了一类conformable分数阶微分方程的初值问题连续解的存在唯一性结论。利用conformable分数阶导数的基本性质，我们得到了一个积分方程的连续解一定是该conformable分数阶微分方程的初值问题的连续解的结论，并基于此结论通过积分方程构造了Picard迭代序列。通过证明该迭代序列在给定区间上一致收敛，我们得到了该conformable分数阶微分方程的初值问题存在唯一的连续解的结论。

Abstract: This paper employs the classical Picard iteration method in the theory of ordinary differential equations to study the existence and uniqueness of continuous solutions to the initial value problem for a class of conformable fractional differential equations. Using the basic properties of the conformable fractional derivative, we show that any continuous solution to the corresponding integral equation must be a continuous solution to the initial value problem of the conformable fractional differential equation. On this basis, the Picard iterative sequence is constructed via the integral equation. By proving that the iterative sequence converges uniformly on a given interval, we establish the existence and uniqueness of continuous solutions to the initial value problem of the conformable fractional differential equation.