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

Abstract

In this paper, we present the monotonicity analysis for the nabla fractional differences with discrete generalized Mittag-Leffler kernels ( {}^{ABR}_{a-1}{nabla }^{delta ,gamma }y )(eta ) of order 0<delta <0.5, beta =1, 0<gamma leq 1 starting at a-1. If ({}^{ABR}_{a-1}{nabla }^{delta ,gamma }y ) ( eta )geq 0, then we deduce that y(eta ) is delta ^{2}gamma -increasing. That is, y(eta +1)geq delta ^{2} gamma y(eta ) for each eta in mathcal{N}_{a}:={a,a+1,ldots}. Conversely, if y(eta ) is increasing with y(a)geq 0, then we deduce that ({}^{ABR}_{a-1}{nabla }^{delta ,gamma }y )(eta ) geq 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.