A monotonicity result for discrete fractional difference operators |
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Authors: | Rajendra Dahal Christopher S Goodrich |
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Institution: | 1. Department of Mathematics and Statistics, Coastal Carolina University, Conway, SC, 29526, USA 2. Department of Mathematics, University of Rhode Island, Kingston, RI, 02881, USA
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Abstract: | In this note we demonstrate that if y(t) ≥ 0, for each t in the domain of ${t \mapsto y(t)}$ , and if, in addition, ${\Delta_0^{\nu}y(t) \geq 0}$ , for each t in the domain of ${t \mapsto \Delta_0^{\nu}y(t)}$ , with 1 < ν < 2, then it holds that y is an increasing function of t. This demonstrates that, in some sense, the positivity of the νth order fractional difference has a strong connection to the monotonicity of y. Furthermore, we provide a dual result in case ${\Delta_0^{\nu}y(t) \leq 0}$ and y is nonpositive on its domain. We conclude the note by mentioning some implications of these results. |
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