A General Carathéodory Existence and Uniqueness Theorem for Fractional Differential Equations and an Application to Kinetic Models

In this paper, we establish a general version of Carathéodory's existence and uniqueness theorem for semilinear systems of integro‐differential equations driven by Caputo fractional derivatives of distinct orders. Our proof shows that the integrability exponent of the Carathéodory function must be strictly greater than the maximum of the reciprocals of the fractional orders in the system; otherwise, even the existence of a solution cannot be guaranteed. Our approach combines fine regularity estimates with delicate fractional integral techniques, offering new insights into the interplay between fractional differentiability and nonlinear dynamics in systems with mixed derivative orders. As an illustration, we apply our theorem to a two‐component kinetic model with distinct memory effects, obtaining a complete well‐posedness theory in a regime where classical continuity assumptions fail.