A monotonicity result for discrete fractional difference operators

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.