标题： Carathéodory存在性和唯一性定理的一般形式 显示英文标题

标题： A General Version of Carathéodory's Existence and Uniqueness Theorem

摘要： 本文建立了Carathéodory关于半线性积分-微分方程组存在性和唯一性定理的一个广义版本，该方程组来源于具有不同Caputo分数阶导数阶次的微分方程。 我们工作的主要结果表明，Carathéodory函数 $f$ 的可积阶次必须至少大于系统中所有导数阶次的倒数的最大值；否则，即使解的存在性也无法保证。 显示英文摘要

摘要： In this paper, we establish a general version of Carath\'{e}odory's existence and uniqueness theorem for a semilinear system of integro-differential equations arising from differential equations with distinct orders of Caputo fractional derivative. The main result of our work demonstrates that the integrability order of the Carath\'{e}odory function $f$ must be at least greater than the maximum of the reciprocals of all differentiation orders in the system; otherwise, even the existence of a solution cannot be guaranteed.