On Ulam type stability of the solution to a <i>ψ</i>-Hilfer abstract fractional functional differential equation

Abstract This article explores the stability of the solutions to a ψ -Hilfer abstract fractional functional differential equation under feasible hypotheses. By utilizing the Banach fixed point theorem and generalized Grönwall’s inequality, the existence, uniqueness, and stability of the solutions are rigorously established. The analysis distinguishes between Ulam-Hyers stability, which enures bounded deviations under constant perturbations, and Ulam-Hyers-Rassias stability, which accounts for state-dependent perturbations, offering greater adaptability for dynamic systems. To contextualize the problem, we highlight the significance of fractional-order systems in capturing memory effects and hereditary dynamics, which are essential for modeling complex real-world phenomena in biological, physical, and engineering domains. Numerical experiments are performed to examine solution trajectories under varying fractional orders and weight functions, demonstrating the flexibility and robustness of the fractional framework. The examples and the plots authenticate the theoretical findings and emphasize the applicability of the proposed model in addressing practical challenges.