Journal
CHAOS SOLITONS & FRACTALS
Volume 102, Issue -, Pages 106-110Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.chaos.2017.04.006
Keywords
Discrete fractional derivative; Discrete Mittag-Leffler function; Discrete ABR fractional derivative; alpha-increasing; Discrete fractional mean-value theorem
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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 ((ABR)(a -1) del(alpha)y) (t) of order 0 < alpha < 1/2 and starting at a - 1 is positive, then y(t) is alpha(2)- increasing. That is y (t + 1) >= alpha(2)y(t) for all t is an element of N-a = {a, a + 1,...}. Conversely, if y(t) is increasing and y(a) >= 0, then ((ABR)(a-1)del(alpha)y)(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. (C) 2017 Elsevier Ltd. All rights reserved.
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