Hyers-Ulam, Rassias, and Mittag-Leffler stability for quantum difference equations in β-calculus

This paper investigates first-order nonlinear quantum difference equations governed by a general β-difference operator, encompassing the Jackson q -difference and Hahn difference operators as special cases. We establish sufficient conditions for the existence and uniqueness of solutions using fixed-point theory and examine their solvability under specific assumptions to ensure well-posedness. Particular attention is given to various notions of stability, including Hyers-Ulam, Hyers-Ulam-Rassias, and Mittag-Leffler type stability. Under suitable Lipschitz conditions, we derive explicit error bounds characterizing each type of stability, with Mittag-Leffler stability demonstrated to be of exponential order α. Several illustrative examples are included to validate the theoretical findings within the framework of quantum calculus and discrete dynamical systems.