Table of Contents
ISRN Mathematical Analysis
Volume 2011, Article ID 385459, 12 pages
http://dx.doi.org/10.5402/2011/385459
Research Article

Positive Solutions for Boundary Value Problem of Nonlinear Fractional π‘ž -Difference Equation

1Department of Mathematics, Faculty of Arts and Sciences, Qassim-Unizah 51911, P.O. Box 3771, Saudi Arabia
2Department of Mathematics, Majmaah University, Al-Majmaah 11952, P.O. Box 566, Saudi Arabia

Received 17 January 2011; Accepted 24 February 2011

Academic Editor: G. L. Karakostas

Copyright © 2011 Moustafa El-Shahed and Farah M. Al-Askar. 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. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. View at Zentralblatt MATH
  2. M. El-Shahed, β€œExistence of solution for a boundary value problem of fractional order,” Advances in Applied Mathematical Analysis, vol. 2, no. 1, pp. 1–8, 2007. View at Google Scholar Β· View at Zentralblatt MATH
  3. S. Zhang, β€œExistence of solution for a boundary value problem of fractional order,” Acta Mathematica Scientia, vol. 26, no. 2, pp. 220–228, 2006. View at Google Scholar
  4. M. El-Shahed and F. M. Al-Askar, β€œOn the existence of positive solutions for a boundary value problem of fractional order,” International Journal of Mathematical Analysis, vol. 4, no. 13–16, pp. 671–678, 2010. View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  5. Z. Bai and H. Lü, β€œPositive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet Β· View at Scopus
  6. F. H. Jackson, β€œOn qfunctions and a certain difference operator,” Transactions of the Royal Society Edinburgh, vol. 46, pp. 253–281, 1908. View at Google Scholar
  7. R. Jackson, β€œOn qdefinite integrals,” Quarterly Journal of Pure and Applied Mathematics, vol. 41, pp. 193–203, 1910. View at Google Scholar
  8. V. Kac and P. Cheung, Quantum Calculus, Springer, New York, NY, USA, 2002.
  9. W. A. Al-Salam, β€œSome fractional qintegrals and qderivatives,” Proceedings of the Edinburgh Mathematical Society, vol. 15, no. 2, pp. 135–140, 1967. View at Google Scholar
  10. R. P. Agarwal, β€œCertain fractional qintegrals and qderivatives,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 66, pp. 365–370, 1969. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  11. P. M. Rajković, S. D. Marinković, and M. S. Stanković, β€œOn qanalogues of caputo derivative and Mittag-Leffler function,” Fractional Calculus & Applied Analisys, vol. 10, no. 4, pp. 359–373, 2007. View at Google Scholar Β· View at Zentralblatt MATH
  12. M. El-Shahed and H. A. Hassan, β€œPositive solutions of qdifference equation,” Proceedings of the American Mathematical Society, vol. 138, no. 5, pp. 1733–1738, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  13. R. A.C. Ferreira, β€œNontrivial solutions for fractional qdifference boundary value problems,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 70, pp. 1–10, 2010. View at Google Scholar Β· View at Zentralblatt MATH
  14. R. A.C. Ferreira, β€œPositive solutions for a class of boundary value problems with fractional qdifferences,” Computers and Mathematics with Applications, vol. 61, no. 2, pp. 367–373, 2011. View at Publisher Β· View at Google Scholar
  15. M. S. Stanković, P. M. Rajković, and S. D. Marinković, β€œOn q-fractional deravtives of Riemann-Liouville and Caputo type,” 2009, http://arxiv.org/abs/0909.0387.
  16. D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, San Diego, Calif, USA, 1988.
  17. M. A. Krasnoselskii, Positive Solutions of Operators Equations, Noordhoff, Groningen, The Netherlands, 1964.
  18. H. Gauchman, β€œIntegral Inequalities in qcalculus,” Computers and Mathematics with Applications, vol. 47, no. 2-3, pp. 281–300, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet Β· View at Scopus