Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014, Article ID 419514, 15 pages
http://dx.doi.org/10.1155/2014/419514
Research Article

Solvability for a Fractional Order Three-Point Boundary Value System at Resonance

School of Mathematics and Physics, University of South China, Hengyang 421001, China

Received 31 March 2014; Revised 15 May 2014; Accepted 16 May 2014; Published 11 June 2014

Academic Editor: Chuangxia Huang

Copyright © 2014 Zigen Ouyang and Hongliang Liu. 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. Y. Chen and X. Tang, “Solvability of sequential fractional order multi-point boundary value problems at resonance,” Applied Mathematics and Computation, vol. 218, no. 14, pp. 7638–7648, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  2. Y. Chen and Z. Lv, “Solvability of fractional-order multi-point boundary-value problems at resonance on the half-line,” Electronic Journal of Differential Equations, vol. 2012, no. 230, pp. 1–14, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. C. Huang, C. Peng, X. Chen, and F. Wen, “Dynamics analysis of a class of delayed economic model,” Abstract and Applied Analysis, vol. 2013, Article ID 962738, 12 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. C. Huang, X. Gong, X. Chen, and F. Wen, “Measuring and forecasting volatility in Chinese stock market using HAR-CJ-M model,” Abstract and Applied Analysis, vol. 2013, Article ID 143194, 13 pages, 2013. View at Publisher · View at Google Scholar
  5. C. Huang, Z. Yang, T. Yi, and X. Zou, “On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities,” Journal of Differential Equations, vol. 256, no. 7, pp. 2101–2114, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. F. Wen, Z. Li, C. Xie, and D. Shaw, “Study on the fractal and chaotic features of the Shanghai composite index,” Fractals, vol. 20, no. 2, pp. 133–140, 2012. View at Publisher · View at Google Scholar · View at Scopus
  7. F. Wen and Z. Liu, “A copula-based correlation measure and its application in chinese stock market,” International Journal of Information Technology & Decision Making, vol. 8, no. 4, pp. 787–801, 2009. View at Publisher · View at Google Scholar · View at Scopus
  8. P. Amster, M. K. Kwong, and C. Rogers, “On a Neumann boundary value problem for the Painlevé II equation in two-ion electro-diffusion,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 9, pp. 2897–2907, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. C. Z. 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 · View at Scopus
  10. D. Delbosco, “Fractional calculus and function spaces,” Journal of Fractional Calculus, vol. 6, pp. 45–53, 1994. 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, The Netherlands, 2006. View at Zentralblatt MATH · View at MathSciNet
  12. V. Lakshmikantham, S. Leela, and J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientiffic, 2009.
  13. V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2677–2682, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. C. Z. Bai, “Existence of three solutions for a nonlinear fractional boundary value problem via a critical points theorem,” Abstract and Applied Analysis, vol. 2012, Article ID 963105, 13 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. V. Daftardar-Gejji, “Positive solutions of a system of non-autonomous fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 302, no. 1, pp. 56–64, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. M. Belmekki and M. Benchohra, “Existence results for fractional order semilinear functional differential equations with nondense domain,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 925–932, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  17. 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 Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. V. D. Gejji, “Positive solutions of a system of non-autonomous fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 302, no. 1, pp. 56–64, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. H. Jafari and V. D. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. 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, no. 3, pp. 1–11, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. 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 Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. S. H. Liang and J. H. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 11, pp. 5545–5550, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  23. Z. G. Ouyang and C. H. Ou, “Existence of solutions for two nonlinear three-point boundary value problems with resonance,” Communications in Applied Analysis, vol. 17, Article ID 47C60, 2013. View at Google Scholar
  24. S. Q. Zhang, “Positive solutions for boundary-value problems of nonlinear fractional differential equations,” Electronic Journal of Differential Equations, vol. 2006, pp. 1–12, 2006. View at Google Scholar · View at Scopus
  25. Y. Zhou and Y. Xu, “Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations,” Journal of Mathematical Analysis and Applications, vol. 320, no. 2, pp. 578–590, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  26. Z. B. Bai, “On positive solutions of 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 Zentralblatt MATH · View at MathSciNet
  27. Z. B. Bai, “On solutions of some fractional m-point boundary value problems at resonance,” Electronic Journal of Qualitative Theory of Differential Equations, no. 37, pp. 1–15, 2010. View at Google Scholar · View at MathSciNet
  28. Y. H. Zhang and Z. B. Bai, “Existence of solutions for nonlinear fractional three-point boundary value problems at resonance,” Journal of Applied Mathematics and Computing, vol. 36, no. 1-2, pp. 417–440, 2011. View at Publisher · View at Google Scholar
  29. B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,” Computers & Mathematics with Applications, vol. 58, no. 9, pp. 1838–1843, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  30. J. J. Nieto and J. Pimentel, “Positive solutions of a fractional thermostat model,” Boundary Value Problems, vol. 2013, article 5, 11 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  31. J. J. Nieto, “Existence of a solution for a three-point boundary value problem for a second-order differential equation at resonance,” Boundary Value Problems, vol. 2013, article 130, 7 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  32. J. R. L. Webb and M. Zima, “Multiple positive solutions of resonant and non-resonant nonlocal boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 3-4, pp. 1369–1378, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. R. P. Agarwal, M. Meehan, and D. O'Regan, Fixed Point Theorey and Applictions, Cambridge Unicersity Press, Cambridge, UK, 2001.
  34. Z. G. Ouyang and G. Z. Li, “Existence of the solutions for a class of nonlinear fractional order three-point boundary value problems with resonance,” Boundary Value Problems, vol. 2012, article 68, 2012. View at Publisher · View at Google Scholar · View at Scopus