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

Thabet Abdeljawad and Dumitru Baleanu

Chaos, Solitons & Fractals, 2017, vol. 102, issue C, 106-110

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.

Keywords: Discrete fractional derivative; Discrete Mittag-Leffler function; Discrete ABR fractional derivative; α−increasing; Discrete fractional mean-value theorem (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (15)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:102:y:2017:i:c:p:106-110

DOI: 10.1016/j.chaos.2017.04.006

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