`Abstract and Applied AnalysisVolume 2012 (2012), Article ID 406757, 13 pageshttp://dx.doi.org/10.1155/2012/406757`
Research Article

## On the Definitions of Nabla Fractional Operators

1Department of Mathematics, Çankaya University, 06530 Ankara, Turkey
2Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101, USA

Received 12 April 2012; Accepted 4 September 2012

Academic Editor: Dumitru Bǎleanu

Copyright © 2012 Thabet Abdeljawad and Ferhan M. Atici. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We show that two recent definitions of discrete nabla fractional sum operators are related. Obtaining such a relation between two operators allows one to prove basic properties of the one operator by using the known properties of the other. We illustrate this idea with proving power rule and commutative property of discrete fractional sum operators. We also introduce and prove summation by parts formulas for the right and left fractional sum and difference operators, where we employ the Riemann-Liouville definition of the fractional difference. We formalize initial value problems for nonlinear fractional difference equations as an application of our findings. An alternative definition for the nabla right fractional difference operator is also introduced.