Monotonicity results for fractional difference operators with discrete exponential kernels

Abstract

We prove that if the Caputo-Fabrizio nabla fractional difference operator $({}^{\mathrm{CFR}}_{a-1}\nabla^{\alpha}y)(t)$ of order $0<\alpha\leq1$ and starting at $a-1$ is positive for $t=a,a+1,\ldots$ , then $y(t)$ is α-increasing. Conversely, if $y(t)$ is increasing and $y(a)\geq0$ , then $({}^{\mathrm{CFR}}_{a-1}\nabla^{\alpha}y)(t)\geq0$ . 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.