A General Version of Carathéodory's Existence and Uniqueness Theorem

2025/05/30 by de Carvalho-Neto, Paulo M., Frota, Cícero L., Torelli, Pedro G. P. · 1 citation
#26A33 #34A08 #34A12 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2505.24516

Abstract

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.

Citations

• The Riemann-Liouville fractional integral in Bochner-Lebesgue spaces III
• On the Initial Boundary Value Problem to the Time-Fractional Wave Equation with Acoustic Boundary Conditions
• The Riemann-Liouville fractional integral in Bochner-Lebesgue spaces II
• The Riemann-Liouville fractional integral in Bochner-Lebesgue spaces I

Cited by

• Energy Estimates for Fractional Evolution Equations

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