Monotonicity results for h-discrete fractional operators and application

In this article, we formulate nabla fractional sums and differences of order 0 < α ≤ 1 $0 < \alpha \leq 1$ on the time scale h Z $h\mathbb{Z}$ , where 0 < h ≤ 1 $0 < h \leq 1$ . Then, we prove that if the nabla h-Riemann–Liouville (RL) fractional difference operator ( a ∇ h α y ) ( t ) > 0 $({}_{a}\nabla_{h}^{\alpha }y)(t) > 0 $ , then y ( t ) $y(t)$ is α-increasing. Conversely, if y ( t ) $y(t)$ is α-increasing and y ( a ) > 0 $y(a)>0$ , then ( a ∇ h α y ) ( t ) > 0 $({}_{a}\nabla_{h}^{\alpha }y)(t)>0$ . 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 h Z $h\mathbb{Z}$ .