Survey of the qualitative properties of fractional difference operators: monotonicity, convexity, and asymptotic behavior of solutions

Authors

Lynn Erbe, University of Nebraska-LincolnFollow
Christopher S. Goodrich, Creighton Preparatory School
Baoguo Jia, Sun Yat-sen University
Allan C. Peterson, University of Nebraska-LincolnFollow

Document Type

Article

Date of this Version

2016

Citation

Advances in Difference Equations (2016) 2016:43

Comments

© 2016 Erbe et al.

This is an open access article

DOI 10.1186/s13662-016-0760-3

Abstract

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.