Abstract and Applied Analysis

A k -Dimensional System of Fractional Finite Difference Equations

Dumitru Baleanu, Shahram Rezapour, and Saeid Salehi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We investigate the existence of solutions for a k -dimensional system of fractional finite difference equations by using the Kranoselskii’s fixed point theorem. We present an example in order to illustrate our results.

Article information

Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 312578, 8 pages.

First available in Project Euclid: 2 October 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Baleanu, Dumitru; Rezapour, Shahram; Salehi, Saeid. A $k$ -Dimensional System of Fractional Finite Difference Equations. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 312578, 8 pages. doi:10.1155/2014/312578. https://projecteuclid.org/euclid.aaa/1412279695

Export citation


  • D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, World Scientific, 2012.
  • R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, 2000.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Application of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
  • R. L. Magin, Fractional Calculus in Bioengineering, Begell House, 2006.
  • F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, 2010.
  • K. S. Miller and B. Ross, “Fractional difference čommentComment on ref. [23a?]: We split this reference to[23a, 23b?]. Please check.calculus,” in Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Tokyo, Japan, 1988.
  • K. S. Miller and B. Ross, “Fractional difference calculus,” in Univalent Functions, Fractional Calculus and Their Applications, Ellis Horwood Series in Mathematics and Its Applications, pp. 139–152, Horwood, Chichester, UK, 1989.
  • I. Podlubny, Fractional Differential Equations, Academic Press, 1999.
  • G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
  • F. M. At\ic\i and S. Şengül, “Modeling with fractional difference equations,” Journal of Mathematical Analysis and Applications, vol. 369, no. 1, pp. 1–9, 2010.
  • G.-C. Wu and D. Baleanu, “Discrete fractional logistic map and its chaos,” Nonlinear Dynamics, vol. 75, no. 1-2, pp. 283–287, 2014.
  • F. M. Atici and P. W. Eloe, “Initial value problems in discrete fractional calculus,” Proceedings of the American Mathematical Society, vol. 137, no. 3, pp. 981–989, 2009.
  • P. Awasthi, Boundary value problems for discrete fractional equations [Ph.D. thesis], University of Nebraska-Lincoln, 2013.
  • G. C. Wu and D. Baleanu, “Discrete chaos of fractional sine and standard maps,” Physics Letters A, vol. 378, pp. 484–487, 2013.
  • F. M. Atici and P. W. Eloe, “A transform method in discrete fractional calculus,” International Journal of Difference Equations, vol. 2, no. 2, pp. 165–176, 2007.
  • F. M. At\ic\i and P. W. Eloe, “Discrete fractional calculus with the nabla operator,” Electronic Journal of Qualitative Theory of Differential Equations. Special Edition I, no. 3, pp. 1–12, 2009.
  • B. Ahmad and S. K. Ntouyas, “A boundary value problem of fractional differential equations with anti-periodic type integral boundary conditions,” Journal of Computational Analysis and Applications, vol. 15, no. 8, pp. 1372–1380, 2013.
  • C. S. Goodrich, “Solutions to a discrete right-focal fractional boundary value problem,” International Journal of Difference Equations, vol. 5, no. 2, pp. 195–216, 2010.
  • C. S. Goodrich, “Some new existence results for fractional difference equations,” International Journal of Dynamical Systems and Differential Equations, vol. 3, no. 1-2, pp. 145–162, 2011.
  • C. S. Goodrich, “On a fractional boundary value problem with fractional boundary conditions,” Applied Mathematics Letters, vol. 25, no. 8, pp. 1101–1105, 2012.
  • M. Holm, “Sum and difference compositions in discrete fractional calculus,” Cubo, vol. 13, no. 3, pp. 153–184, 2011.
  • M. Holm, The theory of discrete fractional calculus: development and applications [Ph.D. thesis], University of Nebraska-Lincoln, 2011.
  • Sh. Kang, Y. Li, and H. Chen, “Positive solutions to boundary value problems of fractional difference equation with nonlocal conditions,” Advances in Differential Equations, vol. 2014, article 7, 2014.
  • Y. Pan, Z. Han, S. Sun, and Y. Zhao, “The existence of solutions to a system of discrete fractional boundary value problems,” Abstract and Applied Analysis, vol. 2012, Article ID 707631, 15 pages, 2012.
  • S. N. Elaydi, An Introduction to Difference Equations, Springer, 1996.
  • J. J. Mohan and G. V. S. R. Deekshitulu, “Fractional order difference equations,” International Journal of Differential Equations, vol. 2012, Article ID 780619, 11 pages, 2012.
  • Sh. Rezapour and R. Hamlbarani, “Some notes on the paper, `Cone metric spaces and fixed point theorems of contractive mappings',” Journal of Mathematical Analysis and Applications, vol. 345, pp. 719–724, 2008.
  • R. P. Agarwal, M. Meehan, and D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press, 2001.
  • D. R. Dunninger and H. Wang, “Existence and multiplicity of positive solutions for elliptic systems,” Nonlinear Analysis: Theory, Methods & Applications A: Theory and Methods, vol. 29, no. 9, pp. 1051–1060, 1997.
  • J. Henderson, S. K. Ntouyas, and I. K. Purnaras, “Positive solutions for systems of nonlinear discrete boundary value problems,” Journal of Difference Equations and Applications, vol. 15, no. 10, pp. 895–912, 2009. \endinput