A General Version of Carath\'{e}odory's Existence and Uniqueness Theorem

TL;DR

This paper extends Carathéodory's theorem to a broad class of semilinear integro-differential systems involving Caputo fractional derivatives, establishing conditions for solution existence and uniqueness.

Contribution

It provides a generalized theorem for systems with multiple fractional orders, specifying integrability conditions for solution existence.

Findings

Solutions exist if the integrability order exceeds the maximum reciprocal of fractional orders.

The theorem guarantees uniqueness under specified conditions.

It applies to systems with multiple Caputo fractional derivatives.

Abstract

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.