Monotonicity analysis of a nabla discrete fractional operator with discrete Mittag-Leffler kernel

Abstract

Discrete fractional calculus is one of the new trends in fractional calculus both from theoretical and applied viewpoints. In this article we prove that if the nabla fractional difference operator with discrete Mittag-Leffler kernel (a−1ABR∇αy)(t) of order 0<α<12 and starting at a−1 is positive, then y(t) is α2−increasing. That is y(t+1)≥α2y(t) for all t∈Na={a,a+1,…}. Conversely, if y(t) is increasing and y(a) ≥ 0, then (a−1ABR∇αy)(t)≥0. The monotonicity properties of the Caputo and right fractional differences are concluded as well. As an application, we prove a fractional difference version of mean-value theorem. Finally, some comparisons to the classical discrete fractional case and to fractional difference operators with discrete exponential kernel are made.