Difference monotonicity analysis on discrete fractional operators with discrete generalized Mittag-Leffler kernels

In this paper, we present the monotonicity analysis for the nabla fractional differences with discrete generalized Mittag-Leffler kernels ((ABR)(alpha-1)del(delta,gamma) y)(.) of order 0 < delta < 0.5, beta = 1, 0 < gamma <= 1 starting at alpha - 1. If ((ABR)(alpha-1)del(delta,gamma) y)(eta) >= 0, then we deduce that y(eta) is delta(2)gamma-increasing. That is, y(eta + 1) >= delta(2)gamma y(eta) for each eta is an element of N-alpha := {alpha, alpha + 1, ... }. Conversely, if y(eta) is increasing with y(alpha) >= 0, then we deduce that ((ABR)(alpha-1)del(delta,gamma) y)(eta) >= 0. Furthermore, the monotonicity properties of the Caputo and right fractional differences are concluded to. Finally, we find a fractional difference version of the mean value theorem as an application of our results. One can see that our results cover some existing results in the literature.