Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2010, Article ID 127363, 13 pages
http://dx.doi.org/10.1155/2010/127363
Review Article

Existence of Positive Solutions for a Functional Fractional Boundary Value Problem

Department of Mathematics, Huaiyin Normal University, Huaian, Jiangsu 223300, China

Received 18 February 2010; Accepted 29 April 2010

Academic Editor: Dumitru Baleanu

Copyright © 2010 Chuanzhi Bai. 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. B. Ahmad and J. J. Nieto, “Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions,” Boundary Value Problems, vol. 2009, Article ID 708576, 11 pages, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. C. Bai and J. Fang, “The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 150, no. 3, pp. 611–621, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. C. Bai, “Positive solutions for nonlinear fractional differential equations with coefficient that changes sign,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 4, pp. 677–685, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. Z. Bai and H. Lu, “Positive 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. 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 MathSciNet
  6. M. Benchohra, A. Cabada, and D. Seba, “An existence result for nonlinear fractional differential equations on Banach spaces,” Boundary Value Problems, vol. 2009, Article ID 628916, 11 pages, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. B. Bonilla, M. Rivero, L. Rodrguez-Germa, and J. J. Trujillo, “Fractional differential equations as alternative models to nonlinear differential equations,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 79–88, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. M. El-Shahed, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Abstract and Applied Analysis, vol. 2007, Article ID 10368, 8 pages, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. H. Jafari and S. Seifi, “Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 2006–2012, 2009. View at Publisher · View at Google Scholar · View at Scopus
  10. E. R. Kaufmann and E. Mboumi, “Positive solutions of a boundary value problem for a nonlinear fractional differential equation,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2008, no. 3, pp. 1–11, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. 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, Amsterdam, Holland, 2006.
  12. 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 MathSciNet
  13. 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
  14. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  15. S. Zhang, “Existence of positive solution for some class of nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 278, no. 1, pp. 136–148, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. L. H. Erbe and Q. Kong, “Boundary value problems for singular second-order functional-differential equations,” Journal of Computational and Applied Mathematics, vol. 53, no. 3, pp. 377–388, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. L. J. Grimm and K. Schmitt, “Boundary value problems for differential equations with deviating arguments,” Aequationes Mathematicae, vol. 4, pp. 176–190, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. G. B. Gustafson and K. Schmitt, “Nonzero solutions of boundary value problems for second order ordinary and delay-differential equations,” Journal of Differential Equations, vol. 12, pp. 129–147, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. C. Bai and J. Fang, “Existence of multiple positive solutions for functional differential equations,” Computers & Mathematics with Applications, vol. 45, no. 12, pp. 1797–1806, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. C. Bai and J. Ma, “Eigenvalue criteria for existence of multiple positive solutions to boundary value problems of second-order delay differential equations,” Journal of Mathematical Analysis and Applications, vol. 301, no. 2, pp. 457–476, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. G. L. Karakostas, K. G. Mavridis, and P. Ch. Tsamatos, “Multiple positive solutions for a functional second-order boundary value problem,” Journal of Mathematical Analysis and Applications, vol. 282, no. 2, pp. 567–577, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. J. J. Nieto and R. Rodriguez-Lopez, “Boundary value problems for a class of impulsive functional equations,” Computers & Mathematics with Applications, vol. 55, no. 12, pp. 2715–2731, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. Y. Liu, “Studies on Neumann-type boundary value problems for second-order nonlinear p-Laplacian-like functional differential equations,” Nonlinear Analysis: Real World Applications, vol. 10, no. 1, pp. 333–344, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  24. V. Lakshmikantham, “Theory of fractional functional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 10, pp. 3337–3343, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. M. Benchohra, J. Henderson, S. K. Ntouyas, and A. Ouahab, “Existence results for fractional order functional differential equations with infinite delay,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1340–1350, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  26. Y. Zhou, “Existence and uniqueness of fractional functional differential equations with unbounded delay,” International Journal of Dynamical Systems and Differential Equations, vol. 1, no. 4, pp. 239–244, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. M. Krasnoselski, Positive Solutions of Operator Equations, Noordhoff, Groningen, The Netherlands, 1964. View at MathSciNet