Table of Contents Author Guidelines Submit a Manuscript
International Journal of Differential Equations
Volume 2017 (2017), Article ID 4683581, 7 pages
https://doi.org/10.1155/2017/4683581
Research Article

Existence and Uniqueness of Solutions for BVP of Nonlinear Fractional Differential Equation

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China

Correspondence should be addressed to Jian-Ping Sun; nc.tul@nuspj

Received 11 October 2016; Revised 14 December 2016; Accepted 15 December 2016; Published 29 January 2017

Academic Editor: Julio D. Rossi

Copyright © 2017 Cheng-Min Su 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 MathSciNet
  2. D. Jiang and C. Yuan, “The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 710–719, 2010. View at Publisher · View at Google Scholar · View at Scopus
  3. S. Zhang, “Positive solutions for boundary-value problems of nonlinear fractional differential equations,” Electronic Journal of Differential Equations, vol. 2006, no. 36, pp. 1–12, 2006. View at Google Scholar
  4. Z. Bai and T. Qiu, “Existence of positive solution for singular fractional differential equation,” Applied Mathematics and Computation, vol. 215, no. 7, pp. 2761–2767, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. J.-R. Yue, J.-P. Sun, and S. Zhang, “Existence of positive solution for BVP of nonlinear fractional differential equation,” Discrete Dynamics in Nature and Society, vol. 2015, Article ID 736108, 6 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. R. P. Agarwal, M. Benchohra, and S. Hamani, “Boundary value problems for fractional differential equations,” Georgian Mathematical Journal, vol. 16, no. 3, pp. 401–411, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. K. Zhao and P. Gong, “Existence of positive solutions for a class of higher-order Caputo fractional differential equation,” Qualitative Theory of Dynamical Systems, vol. 14, no. 1, pp. 157–171, 2015. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. W. Yang, “Positive solutions for nonlinear Caputo fractional differential equations with integral boundary conditions,” Journal of Applied Mathematics and Computing, vol. 44, no. 1-2, pp. 39–59, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. X. Zhang, L. Wang, and Q. Sun, “Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter,” Applied Mathematics and Computation, vol. 226, pp. 708–718, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. A. Guezane-Lakoud and R. Khaldi, “Existence results for a fractional boundary value problem with fractional Lidstone conditions,” Journal of Applied Mathematics and Computing, vol. 49, no. 1-2, pp. 261–268, 2015. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. A. Guezane-Lakoud and S. Bensebaa, “Solvability of a fractional boundary value problem with fractional derivative condition,” Arabian Journal of Mathematics, vol. 3, no. 1, pp. 39–48, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. R. Li, “Existence of solutions for nonlinear singular fractional differential equations with fractional derivative condition,” Advances in Difference Equations, vol. 2014, article 292, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. Z.-W. Lv, “Positive solutions of m-point boundary value problems for fractional differential equations,” Advances in Difference Equations, vol. 2011, Article ID 571804, 2011. View at Publisher · View at Google Scholar
  14. Y. Ji, Y. Guo, J. Qiu, and L. Yang, “Existence of positive solutions for a boundary value problem of nonlinear fractional differential equations,” Advances in Difference Equations, vol. 2015, article 13, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. C. F. Li, X. N. Luo, and Y. Zhou, “Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1363–1375, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. J. Xu, Z. Wei, and W. Dong, “Uniqueness of positive solutions for a class of fractional boundary value problems,” Applied Mathematics Letters, vol. 25, no. 3, pp. 590–593, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. S. K. Ntouyas, “Existence results for first order boundary value problems for fractional differential equations and inclusions with fractional integral boundary conditions,” Journal of Fractional Calculus and Applications, vol. 3, no. 9, pp. 1–14, 2012. View at Google Scholar
  18. 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 Scopus
  19. W. Feng, S. Sun, X. Li, and M. Xu, “Positive solutions to fractional boundary value problems with nonlinear boundary conditions,” Boundary Value Problems, 2014:225, 15 pages, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. W. Xie, J. Xiao, and Z. Luo, “Existence of extremal solutions for nonlinear fractional differential equation with nonlinear boundary conditions,” Applied Mathematics Letters, vol. 41, pp. 46–51, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, San Diego, Calif, USA, 1988. View at MathSciNet
  22. A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics, Springer, New York, NY, USA, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  23. 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