A General Version of Carathéodory's Existence and Uniqueness Theorem

In this paper, we establish a general version of Carathéodory’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 Carathéodory 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.