About this Journal Submit a Manuscript Table of Contents
Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 104276, 9 pages
http://dx.doi.org/10.1155/2013/104276
Research Article

Existence Results for Fractional Difference Equations with Three-Point Fractional Sum Boundary Conditions

1Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology, North Bangkok, Bangkok, Thailand
2Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Received 3 June 2013; Accepted 7 August 2013

Academic Editor: Eric R. Kaufmann

Copyright © 2013 Thanin Sitthiwirattham et al. 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.

Linked References

  1. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at Zentralblatt MATH · View at MathSciNet
  2. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993. View at Zentralblatt MATH · View at MathSciNet
  3. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. View at Zentralblatt MATH · View at MathSciNet
  4. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK, 2009.
  6. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston, Mass, USA, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. K. Diethelm, The Analysis of Fractional Differential Equations. An Application-oriented Exposition Using Differential Operators of Caputo Type, vol. 2004 of Lecture Notes in Mathematics, Springer, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. A. Guezane-Lakoud and R. Khaldi, “Solvability of a three-point fractional nonlinear boundary value problem,” Differential Equations and Dynamical Systems, vol. 20, no. 4, pp. 395–403, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. A. Guezane-Lakoud and R. Khaldi, “Positive solution to a higher order fractional boundary value problem with fractional integral condition,” Romanian Journal of Mathematics and Computer Science, vol. 2, no. 1, pp. 41–54, 2012. View at MathSciNet
  10. E. R. Kaufmann, “Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem,” Discrete and Continuous Dynamical Systems A, vol. 2009, pp. 416–423, 2009. View at Zentralblatt MATH · View at MathSciNet
  11. J. Wang, H. Xiang, and Z. Liu, “Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations,” International Journal of Differential Equations, vol. 2010, Article ID 186928, 12 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 916–924, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. W. Sudsutad and J. Tariboon, “Boundary value problems for fractional differential equations with three-point fractional integral boundary conditions,” Advances in Difference Equations, vol. 2012, article 93, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  14. S. K. Ntouyas, “Boundary value problems for nonlinear fractional differential equations and inclusions with nonlocal and fractional integral boundary conditions,” Opuscula Mathematica, vol. 33, no. 1, pp. 117–138, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  15. A. Guezane-Lakoud and R. Khaldi, “Solvability of a fractional boundary value problem with fractional integral condition,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 2692–2700, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. B. Ahmad, S. K. Ntouyas, and A. Assolami, “Caputo type fractional differential equations with nonlocal Riemann-Liouville integral boundary conditions,” Journal of Applied Mathematics and Computing, vol. 41, no. 1-2, pp. 339–350, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  17. F. M. Atici and P. W. Eloe, “Two-point boundary value problems for finite fractional difference equations,” Journal of Difference Equations and Applications, vol. 17, no. 4, pp. 445–456, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. C. S. Goodrich, “Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions,” Computers & Mathematics with Applications, vol. 61, no. 2, pp. 191–202, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. C. S. Goodrich, “Continuity of solutions to discrete fractional initial value problems,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3489–3499, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. 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. View at MathSciNet
  21. C. S. Goodrich, “Existence of a positive solution to a system of discrete fractional boundary value problems,” Applied Mathematics and Computation, vol. 217, no. 9, pp. 4740–4753, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. C. S. Goodrich, “On a discrete fractional three-point boundary value problem,” Journal of Difference Equations and Applications, vol. 18, no. 3, pp. 397–415, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. F. Chen, X. Luo, and Y. Zhou, “Existence results for nonlinear fractional difference equation,” Advances in Difference Equations, vol. 2011, Article ID 713201, 12 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. Y. Pan, Z. Han, S. Sun, and Z. Huang, “The existence and uniqueness of solutions to boundary value problems of fractional difference equations,” Mathematical Sciences, vol. 6, article 7, 7 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. T. Abdeljawad, “On Riemann and Caputo fractional differences,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1602–1611, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. T. Sitthiwirattham, J. Tariboon, and S. K. Ntouyas, “Boundary value problems for fractional difference equations with three-point fractional sum boundary conditions,” Advances in Difference Equations. In press.
  27. V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, vol. 2 of International Series in Nonlinear Mathematics: Theory, Methods and Applications, Pergamon Press, Oxford, UK, 1981. View at MathSciNet
  28. W. Rudin, Functional Analysis, McGraw-Hill, New York, NY, USA, 1973. View at MathSciNet
  29. M. A. Krasnoselskii, “Two remarks on the method of successive approximations,” Uspekhi Matematicheskikh Nauk, vol. 10, pp. 123–127, 1955. View at MathSciNet
  30. A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, NY, USA, 2003. View at MathSciNet