Monotonicity and Composition Analyses for Fractional Differences Involving Generalized Mittag-Leffler Kernels

This study establishes a rigorous monotonicity and composition analysis for fractional difference operators defined via the generalized Mittag-Leffler kernel. A fundamental criterion linking the sign of the discrete Atangana–Baleanu–Riemann (ABR) fractional difference to a new generalized form of monotonicity (w(1-w)−monotonicity) of the function is derived at first. Subsequently, a crucial composition rule for the ABR difference and its associated sum is proved, providing an explicit formula that elegantly incorporates initial conditions. The use of our theoretical findings is demonstrated practically by its application in solving linear discrete initial value problems, transforming them into explicit solutions by employing the composition theorem. The outcomes reveal the underlying dynamics of discrete models, significantly enhancing the theoretical frameworks and applications.