Abstract
In this article, we formulate nabla fractional sums and differences of order 0 < alpha leq 1 on the time scale hmathbb{Z}, where 0 < h leq 1. Then, we prove that if the nabla h-Riemann–Liouville (RL) fractional difference operator ({}_{a}nabla_{h}^{alpha }y)(t) > 0 , then y(t) is α-increasing. Conversely, if y(t) is α-increasing and y(a)>0, then ({}_{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 hmathbb{Z}.
Highlights
Due to their successful applications in many branches of science and engineering, techniques of fractional calculus have been under focus by many researchers in the past and in the present decades [1,2,3,4,5,6,7,8,9,10]
The monotonicity results for the nabla h-Caputo fractional difference operator are concluded by using the relation between h-nabla RL and Caputo fractional difference operators
We furniture our results by proving a fractional difference version of the Mean Value Theorem (MVT) on hZ
Summary
Due to their successful applications in many branches of science and engineering, techniques of fractional calculus have been under focus by many researchers in the past and in the present decades [1,2,3,4,5,6,7,8,9,10]. Definition 2.6 (Nabla h-Caputo fractional differences) Assume that 0 < α ≤ 1, 0 < h ≤ 1, a, b ∈ R, and a < b, f is defined on Na,h = {a, a + h, a + 2h, . Using Proposition 2.2 and Theorem 3.1, we can state the following h-Caputo fractional difference monotonicity result. Theorem 3.4 Let a function y : Na–h,h → R satisfy y(a) > 0 and be strictly increasing on Na,h, where 0 < α ≤ 1 and 0 < h ≤ 1. Proof Let g : Na–h,h → R be a function such that g(t) = –y(t), a–h∇hαg (t) = a–h∇hα(–y) (t) = – a–h∇hαy (t) ≥ 0. Theorem 3.6 Let a function y : Na–h,h → R satisfy y(a) ≤ 0 and be decreasing on Na,h.
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