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Nieto, Thabet Abdeljawad","submitted_at":"2016-02-24T21:12:06Z","abstract_excerpt":"In this article we prove that if the $q-$fractional operator $(~_{q}\\nabla_{qa}^\\alpha y)(t)$ of order $0<\\alpha\\leq 1$ , $0<q<1$ and starting at some $qa \\in T_q=\\{q^k: k \\in \\mathbb{Z}\\}\\cup \\{0\\},~~a>0$ is positive such that $y(a) \\geq 0$, 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 $y(a)\\geq 0$, then $(~_{q}\\nabla_{qa}^\\alpha y)(t)\\geq 0$. 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