An Unexpected Property Of Fractional Difference Operators: Finite And Eventual Monotonicity

We consider the relationship between the sign of the fractional difference (Delta(alpha) u) (n) and the positivity or monotonicity of u. Our focus is on the case in which the fractional difference can be negative, and we show that surprisingly, (Delta(alpha) u) (n) > -C, where C > 0 is a constant, can still imply that u is increasing or is positive. We also consider the setting of sequential difference operators such as Delta(beta) o Delta(alpha) for suitable choices of the parameters alpha and beta. As is demonstrated by explicit examples, our results substantially improve some recent results in the literature and, moreover, shed light on some previous observations that heretofore were only able to be investigated by numerical simulations. We also provide applications of our results to an analysis of fractional-order initial value problems.