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 ∇ α h y)(t) > 0, then y(t) is α-increasing. Conversely, if y(t) is α-increasing and y(a) > 0, then (a ∇ α h 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 hZ.

In this work, the nabla fractional differences of order 0 < < 1 with discrete exponential kernels are formulated on the time scale hZ, where 0 < h ≤ 1.Hence, the earlier results obtained in Adv. Differ. Equ., 2017, (78) (2017) are generalized. The monotonicity properties of the h-Caputo-Fabrizio (CF) fractional difference operator are concluded using its relation with the nabla h-Riemann-Liouville (RL) fractional difference operator. It is shown that the monotonicity coefficient depends on the step h, and this dependency is explicitly derived. As an application, a fractional difference version of the mean value theorem (MVT) on hZ is proved.

In this work, the nabla discrete new Riemann–Liouville and Caputo fractional proportional differences of order 0<ε<1 on the time scale ℤ are formulated. The differences and summations of discrete fractional proportional are detected on ℤ, and the fractional proportional sums associated to ∇cRχε,ρz with order 0<ε<1 are defined. The relation between nabla Riemann–Liouville and Caputo fractional proportional differences is derived. The monotonicity results for the nabla Caputo fractional proportional difference are proved; specifically, if ∇c−1Rχε,ρz>0 then χz is ερ −increasing, and if χz is strictly increasing on ℕc and χc>0, then ∇c−1Rχε,ρz>0. As an application of our findings, a new version of the fractional proportional difference of the mean value theorem (MVT) on ℤ is proved.