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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 406757, 13 pages
http://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.

Citations to this Article [26 citations]

The following is the list of published articles that have cited the current article.

  • Thabet Abdeljawad, “Dual identities in fractional difference calculus within Riemann,” Advances in Difference Equations, vol. 2013, 2013. View at Publisher · View at Google Scholar
  • Thabet Abdeljawad, Dumitru Baleanu, Fahd Jarad, and Ravi P. Agarwal, “Fractional Sums and Differences with Binomial Coefficients,” Discrete Dynamics in Nature and Society, vol. 2013, pp. 1–6, 2013. View at Publisher · View at Google Scholar
  • Thabet Abdeljawad, “On Delta and Nabla Caputo Fractional Differences and Dual Identities,” Discrete Dynamics in Nature and Society, vol. 2013, pp. 1–12, 2013. View at Publisher · View at Google Scholar
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  • T. Abdeljawad, “On conformable fractional calculus,” Journal of Computational and Applied Mathematics, 2014. View at Publisher · View at Google Scholar
  • Jaganmohan Jonnalagadda, “Solutions of Perturbed Linear Nabla Fractional Difference Equations,” Differential Equations and Dynamical Systems, vol. 22, no. 3, pp. 281–292, 2014. View at Publisher · View at Google Scholar
  • Carlos Lizama, “lp-maximal regularity for fractional difference equations on UMD spaces,” Mathematische Nachrichten, 2015. View at Publisher · View at Google Scholar
  • Luciano Abadias, and Carlos Lizama, “Almost automorphic mild solutions to fractional partial difference-differential equations,” Applicable Analysis, 2015. View at Publisher · View at Google Scholar
  • Jaganmohan Jonnalagadda, “Analysis of a system of nonlinear fractional nabla difference equations,” International Journal of Dynamical Systems and Differential Equations, vol. 5, no. 2, pp. 149–174, 2015. View at Publisher · View at Google Scholar
  • Thabet Abdeljawad, and Dumitru Baleanu, “Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels,” Advances in Difference Equations, vol. 2016, no. 1, 2016. View at Publisher · View at Google Scholar
  • Carlos Lizama, and M. Pilar Velasco, “Weighted bounded solutions for a class of nonlinear fractional equations,” Fractional Calculus and Applied Analysis, vol. 19, no. 4, pp. 1010–1030, 2016. View at Publisher · View at Google Scholar
  • Guo-Cheng Wu, Dumitru Baleanu, and He-Ping Xie, “Riesz Riemann–Liouville difference on discrete domains,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 26, no. 8, pp. 084308, 2016. View at Publisher · View at Google Scholar
  • Wei Nian Li, and Weihong Sheng, “Sufficient conditions for oscillation of a nonlinear fractional nabla difference system,” SpringerPlus, vol. 5, no. 1, 2016. View at Publisher · View at Google Scholar
  • Thabet Abdeljawad, and Delfim F.M. Torres, “Symmetric duality for left and right Riemann–Liouville and Caputo fractional differences,” Arab Journal of Mathematical Sciences, 2016. View at Publisher · View at Google Scholar
  • Weihong Sheng, and Wei Nian Li, “Forced oscillation for solutions of boundary value problems of fractional partial difference equations,” Advances in Difference Equations, vol. 2016, no. 1, 2016. View at Publisher · View at Google Scholar
  • Luciano Abadias, Pedro J. Miana, Carlos Lizama, and M. Pilar Velasco, “Cesàro sums and algebra homomorphisms of bounded operators,” Israel Journal of Mathematics, vol. 216, no. 1, pp. 471–505, 2016. View at Publisher · View at Google Scholar
  • Jagan Mohan Jonnalagadda, “Solutions of fractional nabla difference equations -existence and uniqueness,” Opuscula Mathematica, vol. 36, no. 2, pp. 215–238, 2016. View at Publisher · View at Google Scholar
  • M. Ganji, and F. Gharari, “The Discrete Delta and Nabla Mittag-Leffler Distributions,” Communications in Statistics - Theory and Methods, pp. 0–0, 2017. View at Publisher · View at Google Scholar
  • Thabet Abdeljawad, and Dumitru Baleanu, “On Fractional Derivatives with Exponential Kernel and their Discrete Versions,” Reports on Mathematical Physics, vol. 80, no. 1, pp. 11–27, 2017. View at Publisher · View at Google Scholar
  • Thabet Abdeljawad, and Dumitru Baleanu, “Monotonicity analysis of a nabla discrete fractional operator with discrete Mittag-Leffler kernel,” Chaos, Solitons & Fractals, 2017. View at Publisher · View at Google Scholar
  • Thabet Abdeljawad, and Dumitru Baleanu, “Monotonicity results for fractional difference operators with discrete exponential kernels,” Advances in Difference Equations, vol. 2017, no. 1, 2017. View at Publisher · View at Google Scholar
  • Guo-Cheng Wu, Dumitru Baleanu, and Wei-Hua Luo, “Lyapunov functions for Riemann–Liouville-like fractional difference equations,” Applied Mathematics and Computation, vol. 314, pp. 228–236, 2017. View at Publisher · View at Google Scholar
  • Mohammed Al-Refai, and Thabet Abdeljawad, “Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems,” Complexity, vol. 2017, pp. 1–7, 2017. View at Publisher · View at Google Scholar