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We prove that if the Caputo-Fabrizio nabla fractional difference operator (a−1CFR∇αy)(t)$({}^{\mathrm{CFR}}_{a-1}\nabla^{\alpha}y)(t)$ of order 0<α≤1$0<\alpha\leq1$ and starting at a−1$a-1$ is positive for t=a,a+1,…$t=a,a+1,\ldots$ , then y(t)$y(t)$ is α-increasing. Conversely, if y(t)$y(t)$ is increasing and y(a)≥0$y(a)\geq0$, then (a−1CFR∇αy)(t)≥0$({}^{\mathrm{CFR}}_{a-1}\nabla^{\alpha}y)(t)\geq0$. A monotonicity result for the Caputo-type fractional difference operator is proved as well. As an application, we prove a fractional difference version of the mean-value theorem and make a comparison to the classical discrete fractional case.

Monotonicity results for fractional difference operators with discrete exponential kernels