Bahaaeldin Abdalla

A monotonicity result for the $q-$fractional operator

In this article we prove that if the $q-$fractional operator $(~_{q}\nabla_{qa}^αy)(t)$ of order $0<α\leq 1$ , $0 0$ is positive such that $y(a) \geq 0$, then $y(t)$ is $c_q(α)-$increasing, $c_q(α)=\frac{1-q^α}{1-q}q^{1-α}$. Conversely, if y(t) is increasing and $y(a)\geq 0$, then $(~_{q}\nabla_{qa}^αy)(t)\geq 0$. As an application, we proved a $q-$fractional version of the Mean-Value Theorem.