About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 410505, 8 pages
http://dx.doi.org/10.1155/2013/410505
Research Article

A Study of Nonlinear Fractional -Difference Equations with Nonlocal Integral Boundary Conditions

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 12 July 2013; Revised 9 September 2013; Accepted 9 September 2013

Academic Editor: Dumitru Baleanu

Copyright © 2013 Ahmed Alsaedi 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. 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 MathSciNet
  2. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  3. J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Eds., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007.
  4. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993. View at MathSciNet
  5. D. Băleanu and O. G. Mustafa, “On the global existence of solutions to a class of fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1835–1841, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. D. Băleanu, O. G. Mustafa, and R. P. Agarwal, “On Lp-solutions for a class of sequential fractional differential equations,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 2074–2081, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  7. R. P. Agarwal and B. Ahmad, “Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1200–1214, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. B. Ahmad and J. J. Nieto, “Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions,” Boundary Value Problems, vol. 2011, article 36, 2011. View at Zentralblatt MATH · View at MathSciNet
  9. S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, and R. Magin, “Fractional Bloch equation with delay,” Computers & Mathematics with Applications, vol. 61, no. 5, pp. 1355–1365, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. F. T. Akyildiz, H. Bellout, K. Vajravelu, and R. A. Van Gorder, “Existence results for third order nonlinear boundary value problems arising in nano boundary layer fluid flows over stretching surfaces,” Nonlinear Analysis. Real World Applications, vol. 12, no. 6, pp. 2919–2930, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. B. Ahmad and J. J. Nieto, “Sequential fractional differential equations with three-point boundary conditions,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3046–3052, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. R. Graef, L. Kong, and Q. Kong, “Application of the mixed monotone operator method to fractional boundary value problems,” Fractional Differential Calculus, vol. 2, pp. 554–567, 2011.
  13. R. P. Agarwal, “Certain fractional q-integrals and q-derivatives,” Proceedings of the Cambridge Philosophical Society, vol. 66, pp. 365–370, 1969. View at MathSciNet
  14. W. A. Al-Salam, “Some fractional q-integrals and q-derivatives,” Proceedings of the Edinburgh Mathematical Society, vol. 15, pp. 135–140, 1966. View at Publisher · View at Google Scholar · View at MathSciNet
  15. F. M. Atici and P. W. Eloe, “Fractional q-calculus on a time scale,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 3, pp. 333–344, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  16. F. M. Atici and P. W. Eloe, “Linear systems of fractional nabla difference equations,” The Rocky Mountain Journal of Mathematics, vol. 41, no. 2, pp. 353–370, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. F. M. Atıcı 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. M. El-Shahed and F. M. Al-Askar, “Positive solutions for boundary value problem of nonlinear fractional q-difference equation,” ISRN Mathematical Analysis, vol. 2011, Article ID 385459, 12 pages, 2011. View at MathSciNet
  19. R. A. C. Ferreira, “Positive solutions for a class of boundary value problems with fractional q-differences,” Computers & Mathematics with Applications, vol. 61, no. 2, pp. 367–373, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  20. R. A. C. Ferreira, “Nontrivial solutions for fractional q-difference boundary value problems,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 70, pp. 1–10, 2010. View at MathSciNet
  21. 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
  22. J. R. Graef and L. Kong, “Positive solutions for a class of higher order boundary value problems with fractional q-derivatives,” Applied Mathematics and Computation, vol. 218, no. 19, pp. 9682–9689, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  23. J. Ma and J. Yang, “Existence of solutions for multi-point boundary value problem of fractional q-difference equation,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 92, pp. 1–10, 2011. View at MathSciNet
  24. G. C. Wu and D. Baleanu, “New applications of the variational iteration methodfrom differential equations to q-fractional difference equations,” Advances in Difference Equations, vol. 2013, article 21, 2013.
  25. F. Jarad, T. Abdeljawad, and D. Baleanu, “Stability of q-fractional non-autonomous systems,” Nonlinear Analysis. Real World Applications, vol. 14, no. 1, pp. 780–784, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  26. B. Ahmad, S. K. Ntouyas, and I. K. Purnaras, “Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations,” Advances in Difference Equations, vol. 2012, article 140, 2012.
  27. P. M. Rajković, S. D. Marinković, and M. S. Stanković, “Fractional integrals and derivatives in q-calculus,” Applicable Analysis and Discrete Mathematics, vol. 1, no. 1, pp. 311–323, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  28. M. H. Annaby and Z. S. Mansour, q-Fractional Calculus and Equations, vol. 2056 of Lecture Notes in Mathematics, Springer, Heidelberg, Germany, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  29. P. M. Rajković, S. D. Marinković, and M. S. Stanković, “On q-analogues of Caputo derivative and Mittag-Leffler function,” Fractional Calculus & Applied Analysis, vol. 10, no. 4, pp. 359–373, 2007. View at MathSciNet
  30. A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics, Springer, New York, NY, USA, 2005. View at MathSciNet
  31. D. R. Smart, Fixed Point Theorems, Cambridge University Press, 1980. View at MathSciNet