Ulam–Hyers–Mittag–Leffler Stability for a Class of Nonlinear Fractional Reaction–Diffusion Equations with Delay

Abstract

In this paper, we introduce and analyze two new results on the Mittag–Leffler–Ulam–Hyers stability for a class of nonlinear fractional reaction–diffusion equations with delay. We demonstrate that these equations exhibit Ulam–Hyers–Mittag–Leffler stability on a compact interval with respect to the Chebyshev and Bielecki norms, using fixed–point method. These results extend and encompass many previous findings while offering notable improvements. To illustrate the practical applications of our results, we provide three examples. Despite extensive literature on the Lyapunov, Ulam, and Mittag–Leffler stability of fractional equations with and without delays, there is limited research on the Mittag–Leffler–Ulam–Hyers stability of fractional equations with delay. Hence, a key aim of this work is to address this gap by exploring a class of nonlinear fractional reaction–diffusion equations with delay and establishing new results on their Mittag–Leffler–Ulam–Hyers stability.