## Abstract and Applied Analysis

### A $k$-Dimensional System of Fractional Finite Difference Equations

#### Abstract

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

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

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412279695

Digital Object Identifier
doi:10.1155/2014/312578

Mathematical Reviews number (MathSciNet)
MR3198176

Zentralblatt MATH identifier
07022142

#### Citation

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

#### References

• 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