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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, Hindawi, vol. 2012, pages 1-13, October.
    • Handle: RePEc:hin:jnlaaa:406757
    DOI: 10.1155/2012/406757
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    Cited by:

    1. Rashid, Saima & Sultana, Sobia & Jarad, Fahd & Jafari, Hossein & Hamed, Y.S., 2021. "More efficient estimates via ℏ-discrete fractional calculus theory and applications," Chaos, Solitons & Fractals, Elsevier, vol. 147(C).
    2. Rashid, Saima & Sultana, Sobia & Hammouch, Zakia & Jarad, Fahd & Hamed, Y.S., 2021. "Novel aspects of discrete dynamical type inequalities within fractional operators having generalized ℏ-discrete Mittag-Leffler kernels and application," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    3. Pshtiwan Othman Mohammed & Thabet Abdeljawad & Faraidun Kadir Hamasalh, 2021. "On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis," Mathematics, MDPI, vol. 9(11), pages 1-17, June.
    4. Jiraporn Reunsumrit & Thanin Sitthiwirattham, 2020. "On the Nonlocal Fractional Delta-Nabla Sum Boundary Value Problem for Sequential Fractional Delta-Nabla Sum-Difference Equations," Mathematics, MDPI, vol. 8(4), pages 1-13, March.
    5. Suwan, Iyad & Abdeljawad, Thabet & Jarad, Fahd, 2018. "Monotonicity analysis for nabla h-discrete fractional Atangana–Baleanu differences," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 50-59.
    6. Gu, Yajuan & Wang, Hu & Yu, Yongguang, 2020. "Synchronization for fractional-order discrete-time neural networks with time delays," Applied Mathematics and Computation, Elsevier, vol. 372(C).
    7. Zhang, Xiao-Li & Li, Hong-Li & Kao, Yonggui & Zhang, Long & Jiang, Haijun, 2022. "Global Mittag-Leffler synchronization of discrete-time fractional-order neural networks with time delays," Applied Mathematics and Computation, Elsevier, vol. 433(C).
    8. Yao, Yu & Wu, Li-Bing, 2022. "Backstepping control for fractional discrete-time systems," Applied Mathematics and Computation, Elsevier, vol. 434(C).
    9. Li, Ruoxia & Cao, Jinde & Xue, Changfeng & Manivannan, R., 2021. "Quasi-stability and quasi-synchronization control of quaternion-valued fractional-order discrete-time memristive neural networks," Applied Mathematics and Computation, Elsevier, vol. 395(C).
    10. Qiushuang Wang & Run Xu, 2022. "On Hilfer Generalized Proportional Nabla Fractional Difference Operators," Mathematics, MDPI, vol. 10(15), pages 1-16, July.
    11. Pshtiwan Othman Mohammed & Hari Mohan Srivastava & Dumitru Baleanu & Rashid Jan & Khadijah M. Abualnaja, 2022. "Monotonicity Results for Nabla Riemann–Liouville Fractional Differences," Mathematics, MDPI, vol. 10(14), pages 1-14, July.
    12. Baogui Xin & Wei Peng & Yekyung Kwon, 2019. "A fractional-order difference Cournot duopoly game with long memory," Papers 1903.04305, arXiv.org.
    13. Cui, Xueke & Li, Hong-Li & Zhang, Long & Hu, Cheng & Bao, Haibo, 2023. "Complete synchronization for discrete-time fractional-order coupled neural networks with time delays," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).

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