A monotonicity result for the $q-$fractional operator

Abstract

In this article we prove that if the $q-$fractional operator $({}_q{\nabla }_{qa}^{\alpha }y)(t)$ of order $0<\alpha \le 1$ , $0<q<1$ and starting at some $qa\in T_q=\{q^k:k\in \mathbb{Z}\}\cup \{0\},a>0$ is positive such that $y(a)\ge 0$, then $y(t)$ is $c_q(\alpha )-$increasing, $c_q(\alpha )=\frac{1-q^{\alpha }}{1-q}{q}^{1-\alpha }$. Conversely, if y(t) is increasing and $y(a)\ge 0$, then $({}_q{\nabla }_{qa}^{\alpha }y)(t)\ge 0$. As an application, we proved a $q-$fractional version of the Mean-Value Theorem.