Monotonicity results for fractional difference operators with discrete exponential kernels

We prove that if the Caputo-Fabrizio nabla fractional difference operator (CFRₐ₋₁∇αy)(t) of order 0<α≤1 and starting at $a-1$ is positive for t=a,a+1,… , then $y(t)$ is α-increasing. Conversely, if $y(t)$ is increasing and y(a)≥0 , then (CFRₐ₋₁∇αy)(t)≥0 . 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.