A General Version of Carath\'{e}odory's Existence and Uniqueness Theorem
Classical Analysis and ODEs
2025-06-02 v1 Functional Analysis
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 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.
Cite
@article{arxiv.2505.24516,
title = {A General Version of Carath\'{e}odory's Existence and Uniqueness Theorem},
author = {Paulo M. de Carvalho-Neto and Cícero L. Frota and Pedro G. P. Torelli},
journal= {arXiv preprint arXiv:2505.24516},
year = {2025}
}