Fixed Point Analysis for Cauchy-Type Variable-Order Fractional Differential Equations With Finite Delay

This paper presents a comprehensive analysis of the existence, uniqueness, and Ulam–Hyers stability of solutions for a class of Cauchy-type nonlinear fractional differential equations with variable order and finite delay. The motivation for this study lies in the increasing importance of variable-order fractional calculus in modeling real-world systems with time-dependent memory and hereditary properties, where the order of differentiation evolves with time or state. Moreover, incorporating finite delay allows the model to capture the influence of past states on current dynamics, making it applicable to a wider range of physical and biological processes. By employing tools from fixed point theory, we derive new sufficient conditions for the existence and uniqueness of solutions using both the Banach contraction principle and Schauder’s fixed point theorem. Furthermore, we establish the Ulam–Hyers stability of the proposed system under suitable assumptions on the nonlinear term. To illustrate the validity of our theoretical findings, a numerical example is provided, demonstrating how variations in the fractional order affect the system’s behavior. The obtained results enrich the theoretical framework of variable-order fractional calculus and extend its applicability to delayed systems. These findings may serve as a mathematical foundation for future research on more general models, including neutral or implicit fractional differential equations of variable order, which are of growing relevance in applied sciences and engineering.