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.

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In this article, the results of two-dimensional reduced differential transform method is extended to three-dimensional case for solving three dimensional Volterra integral equation. Using the described method, the exact solution can be obtained after a few number of iterations. Moreover, examples on both linear and nonlinear Volterra integral equation are carried out to illustrate the efficiency and the accuracy of the presented method.

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.