Mathematics, Department of


Date of this Version



Erbe et al. Advances in Difference Equations (2016) 2016:43


© 2016 Erbe et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License


In this article we discuss some of the qualitative properties of fractional difference operators. We especially focus on the connections between the fractional difference operator and the monotonicity and convexity of functions. In the integer-order setting, these connections are elementary and well known. However, in the fractional-order setting the connections are very complicated and muddled. We survey some of the known results and suggest avenues for future research. In addition, we discuss the asymptotic behavior of solutions to fractional difference equations and how the nonlocal structure of the fractional difference can be used to deduce these asymptotic properties.