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arXiv · 2505.24516

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

Paulo M. de Carvalho-Neto·Cícero L. Frota·Pedro G. P. Torelli
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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.

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Paulo M. de Carvalho-Neto, Cícero L. Frota, Pedro G. P. Torelli. 2025-05-30. A General Version of Carath\'{e}odory's Existence and Uniqueness Theorem. https://arxiv.org/abs/2505.24516

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