Initial Value Problem for Mixed Differential Equations of Variable Order with Finite Delay

This study presents a fresh perspective on the existence, uniqueness, and stability of solutions for initial value problems involving variable-order differential equations with finite delay. Departing from conventional techniques that utilize generalized intervals and piecewise constant functions, we introduce a novel fractional operator tailored for this specific problem. Our methodology integrates sophisticated mathematical analysis, including the Schauder fixed-point theorem and Banach’s contraction principle, with an examination of the Ulam–Hyers stability of the problem. The strength of our approach is in its simplicity, requiring fewer restrictive assumptions. We conclude with a practical application to illustrate our findings. These results are valuable for understanding complex dynamical systems with time delays, offering applications in diverse fields such as engineering, economics, and medicine, and enhancing numerical methods for solving delay equations.