`Abstract and Applied AnalysisVolume 2011, Article ID 390543, 16 pageshttp://dx.doi.org/10.1155/2011/390543`
Research Article

## Positive Solutions to Boundary Value Problems of Nonlinear Fractional Differential Equations

1School of Science, University of Jinan, Jinan, Shandong 250022, China
2Department of Mathematics and Statistics, Missouri University of Science and Technology Rolla, MO 65409-0020, USA

Received 23 September 2010; Revised 5 November 2010; Accepted 6 December 2010

Copyright © 2011 Yige Zhao 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.

1. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equation, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.
2. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
3. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
4. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integral and Derivative: Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993.
5. Q. Li and S. Sun, “On the existence of positive solutions for initial value problem to a class of fractional differential equation,” in Proceedings of the 7th Conference on Biological Dynamic System and Stability of Differential Equation, pp. 886–889, World Academic Press, Chongqing, China, 2010.
6. Q. Li, S. Sun, M. Zhang, and Y. Zhao, “On the existence and uniqueness of solutions for initial value problem of fractional differential equations,” Journal of University of Jinan, vol. 24, pp. 312–315, 2010.
7. Q. Li, S. Sun, Z. Han, and Y. Zhao, “On the existence and uniqueness of solutions for initial value problem of nonlinear fractional differential equations,” in Proceedings of the 6th IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications (MESA '10), pp. 452–457, Qingdao, China, 2010.
8. D. Delbosco and L. Rodino, “Existence and uniqueness for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 204, no. 2, pp. 609–625, 1996.
9. S. Zhang, “The existence of a positive solution for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 804–812, 2000.
10. 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.
11. H. Jafari and V. Daftardar-Gejji, “Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method,” Applied Mathematics and Computation, vol. 180, no. 2, pp. 700–706, 2006.
12. S. Zhang, “Positive solutions for boundary-value problems of nonlinear fractional differential equations,” Electronic Journal of Differential Equations, vol. 36, pp. 1–12, 2006.
13. T. Qiu and Z. Bai, “Existence of positive solutions for singular fractional differential equations,” Electronic Journal of Differential Equations, vol. 146, pp. 1–9, 2008.
14. Z. Bai and H. Lü, “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.
15. M. Zhang, S. Sun, Y. Zhao, and D. Yang, “Existence of positive solutions for boundary value problems of fractional differential equations,” Journal of University of Jinan, vol. 24, pp. 205–208, 2010.
16. Y. Zhao and S. Sun, “On the existence of positive solutions for boundary value problems of nonlinear fractional differential equations,” in Proceedings of the 7th Conference on Biological Dynamic System and Stability of Differential Equation, pp. 682–685, World Academic Press, Chongqing, China, 2010.
17. Y. Zhao, S. Sun, Z. Han, and M. Zhang, “Existence on positive solutions for boundary value problems of singular nonlinear fractional differential equations,” in Proceedings of the 6th IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications, pp. 480–485, Qingdao, China, 2010.
18. Y. Zhao, S. Sun, Z. Han, and Q. Li, “The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 4, pp. 2086–2097, 2011.
19. Y. Yu and D. Jiang, “Multiple Positive Solutions for the Boundary Value Problem of A Nonlinear Fractional Differential Equation,” Northeast Normal University, 2009.
20. 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, Amsterdam, The Netherlands, 2006.
21. M. A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, The Netherlands, 1964.
22. W. Ge, Boundary Value Problem of Nonlinear Ordinary Differential Equations, Science Press, Beijing, China, 2007.