Monotonicity results for h-discrete fractional operators and application

In this article, we formulate nabla fractional sums and differences of order 0 < α ≤ 1 on the time scale hZ , where 0 < h ≤ 1 . Then, we prove that if the nabla h-Riemann–Liouville (RL) fractional difference operator (ₐ∇ₕαy)(t) > 0 , then $y(t)$ is α-increasing. Conversely, if $y(t)$ is α-increasing and $y(a)>0$ , then (ₐ∇ₕαy)(t)>0 . The monotonicity results for the nabla h-Caputo fractional difference operator are also concluded by using the relation between h-nabla RL and Caputo fractional difference operators. It is observed that the reported monotonicity coefficient is not affected by the step h. We formulate a nabla h-fractional difference initial value problem as well. Finally, we furniture our results by proving a fractional difference version of the Mean Value Theorem (MVT) on hZ .