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On the Definitions of Nabla Fractional Operators

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  • Thabet Abdeljawad
  • Ferhan M. Atici

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.

Suggested Citation

  • Thabet Abdeljawad & Ferhan M. Atici, 2012. "On the Definitions of Nabla Fractional Operators," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    • Handle: RePEc:wly:jnlaaa:v:2012:y:2012:i:1:n:406757
    DOI: 10.1155/2012/406757
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    References listed on IDEAS

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    1. Fahd Jarad & Thabet Abdeljawad & Dumitru Baleanu & Kübra Biçen, 2012. "On the Stability of Some Discrete Fractional Nonautonomous Systems," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-9, February.
    2. Fahd Jarad & Thabet Abdeljawad & Dumitru Baleanu & Kübra Biçen, 2012. "On the Stability of Some Discrete Fractional Nonautonomous Systems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
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    Cited by:

    1. Chao Wang & Chunlin Gong & Hongtao Yue & Yin Wang, 2025. "Event-Triggered Anti-Synchronization of Fuzzy Delay-Coupled Fractional Memristor-Based Discrete-Time Neural Networks," Mathematics, MDPI, vol. 13(12), pages 1-15, June.
    2. Nikolay D. Dimitrov & Jagan Mohan Jonnalagadda, 2025. "Multiple Positive Solutions of Nabla Fractional Equations with Summation Boundaries," Mathematics, MDPI, vol. 13(19), pages 1-15, October.
    3. Ravi P. Agarwal & Ekaterina Madamlieva, 2025. "Analysis of Mild Extremal Solutions in Nonlinear Caputo-Type Fractional Delay Difference Equations," Mathematics, MDPI, vol. 13(8), pages 1-27, April.

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