Affiliation:
1. School of Mathematics and Statistics UNSW Sydney Sydney New South Wales Australia
2. Departamento de Matemática y Ciencia de la Computación Universidad de Santiago de Chile Santiago Chile
Abstract
We consider the relationship between the sign of the fractional difference
and the positivity or monotonicity of
. Our focus is on the case in which the fractional difference can be negative, and we show that surprisingly,
, where
is a constant, can still imply that
is increasing or is positive. We also consider the setting of sequential difference operators such as
for suitable choices of the parameters
and
. 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.