On the Ulam-Hyers-Rassias-Mittag-Leffler Stability of the Solution to a \(\Psi\)-Hilfer Abstract Fractional Differential Equation

Abstract

This study devises an appropriate mathematical framework to explore the stability of the solution to \(\Psi\)-Hilfer abstract fractional differential equations. Schauder’s fixed point theorem serves as a cornerstone in establishing the existence of the solution for such equations. Building upon this foundation, we elegantly demonstrate the Ulam–Hyers–Mittag–Leffler stability as well as the Ulam–Hyers–Rassias–Mittag–Leffler stability pertaining to these equations. By leveraging fixed point theory and generalized Grönwall’s inequality, we develop a rigorous framework that guarantees the existence and stability of the solution. This study demonstrates how resilient and consistent the solutions remain in the face of disruptions.