Normalized Nabla Fractional Difference Operators with Discrete Exponential and Mittag-Leffler Kernels

Fractional calculus provides an effective framework for modeling memory effects, but classical discrete fractional operators may exhibit structural inconsistencies at the initial node. In this paper, we introduce normalized nabla Atangana-Baleanu and Caputo-Fabrizio fractional operators together with their associated normalized fractional sums. The proposed normalization preserves the nonlocal memory effects while ensuring well-posedness. We establish generalized nabla Gröwall-type inequalities and comparison principles involving discrete Mittag-Leffler kernels, leading to sufficient conditions for the existence, uniqueness, stability, and boundedness of solutions. Explicit formulations are derived for representative normalized fractional initial value problems, including a discrete recurrence formula and an equivalent continuous Volterra integral equation. Numerical results confirm the theoretical analysis and show that both discrete and continuous solutions approach their respective classical limits as the fractional order increases.