Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 314083, 8 pages
http://dx.doi.org/10.1155/2014/314083
Research Article

The Existence of Solutions for Four-Point Coupled Boundary Value Problems of Fractional Differential Equations at Resonance

1Department of Statistics and Finance, Shandong University of Science and Technology, Qingdao 266590, China
2School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China
3Department of Mathematics, Shandong University of Science and Technology, Qingdao, Shandong 266590, China

Received 18 December 2013; Accepted 14 February 2014; Published 23 March 2014

Academic Editor: Xinguang Zhang

Copyright © 2014 Yumei Zou 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. V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK, 2009.
  3. J. J. Nieto and J. Pimentel, “Positive solutions of a fractional thermostat model,” Boundary Value Problems, vol. 2013, article 5, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  5. W. Wu and X. Zhou, “Eigenvalue of fractional differential equations with P-Laplacian operator,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 137890, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  6. X. Zhang, L. Liu, Y. Wu, and Y. Lu, “The iterative solutions of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 219, no. 9, pp. 4680–4691, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  7. X. Zhang, L. Liu, and Y. Wu, “The uniqueness of positive solution for a singular fractional differential system involving derivatives,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 6, pp. 1400–1409, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. X. Zhang, L. Liu, and Y. Wu, “The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8526–8536, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. X. Zhang, L. Liu, and Y. Wu, “Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 1420–1433, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  10. X. Zhang, L. Liu, B. Wiwatanapataphee, and Y. Wu, “Positive solutions of eigenvalue problems for a class of fractional differential equations with derivatives,” Abstract and Applied Analysis, vol. 2012, Article ID 512127, 16 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. X. Zhang, L. Liu, and Y. Wu, “Multiple positive solutions of a singular fractional differential equation with negatively perturbed term,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1263–1274, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. K. Deng, “Blow-up rates for parabolic systems,” Zeitschrift für Angewandte Mathematik und Physik, vol. 47, no. 1, pp. 132–143, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. K. Deng, “Global existence and blow-up for a system of heat equations with non-linear boundary conditions,” Mathematical Methods in the Applied Sciences, vol. 18, no. 4, pp. 307–315, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  14. L. Zhigui and X. Chunhong, “The blow-up rate for a system of heat equations with nonlinear boundary conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 34, no. 5, pp. 767–778, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. D. G. Aronson, “A comparison method for stability analysis of nonlinear parabolic problems,” SIAM Review, vol. 20, no. 2, pp. 245–264, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M. Pedersen and Z. Lin, “Blow-up analysis for a system of heat equations coupled through a nonlinear boundary condition,” Applied Mathematics Letters, vol. 14, no. 2, pp. 171–176, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. N. A. Asif and R. A. Khan, “Positive solutions to singular system with four-point coupled boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 386, no. 2, pp. 848–861, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. C. Yuan, D. Jiang, D. O'Regan, and R. P. Agarwal, “Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions,” Electronic Journal of Qualitative Theory of Differential Equations, no. 13, p. 17, 2012. View at Google Scholar · View at MathSciNet
  19. Y. Cui and J. Sun, “On existence of positive solutions of coupled integral boundary value problems for a nonlinear singular superlinear differential system,” Electronic Journal of Qualitative Theory of Differential Equations, no. 41, pp. 1–13, 2012. View at Google Scholar · View at MathSciNet
  20. Z. Bai and Y. Zhang, “Solvability of fractional three-point boundary value problems with nonlinear growth,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 1719–1725, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. Y. Cui, “Solvability of second-order boundary-value problems at resonance involving integral conditions,” Electronic Journal of Differential Equations, no. 45, pp. 1–9, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. Z. Hu and W. Liu, “Solvability for fractional order boundary value problems at resonance,” Boundary Value Problems, vol. 2011, article 20, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. W. Jiang, “The existence of solutions to boundary value problems of fractional differential equations at resonance,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 5, pp. 1987–1994, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. N. Kosmatov, “Multi-point boundary value problems on time scales at resonance,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 253–266, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. G. Wang, W. Liu, S. Zhu, and T. Zheng, “Existence results for a coupled system of nonlinear fractional 2m-point boundary value problems at resonance,” Advances in Difference Equations, vol. 2011, article 80, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  26. J. R. L. Webb, “Remarks on nonlocal boundary value problems at resonance,” Applied Mathematics and Computation, vol. 216, no. 2, pp. 497–500, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. J. R. L. Webb and M. Zima, “Multiple positive solutions of resonant and non-resonant non-local fourth-order boundary value problems,” Glasgow Mathematical Journal, vol. 54, no. 1, pp. 225–240, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. X. Zhang, M. Feng, and W. Ge, “Existence result of second-order differential equations with integral boundary conditions at resonance,” Journal of Mathematical Analysis and Applications, vol. 353, no. 1, pp. 311–319, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. X. Zhang, C. Zhu, and Z. Wu, “Solvability for a coupled system of fractional differential equations with impulses at resonance,” Boundary Value Problems, vol. 2013, artice 80, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  30. Z. Zhao and J. Liang, “Existence of solutions to functional boundary value problem of second-order nonlinear differential equation,” Journal of Mathematical Analysis and Applications, vol. 373, no. 2, pp. 614–634, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. Y. Zou and Y. Cui, “Existence results for a functional boundary value problem of fractional differential equations,” Advances in Difference Equations, vol. 2013, article 25, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  32. J. Mawhin, “Topological degree and boundary value problems for nonlinear differential equations,” in Topological Methods for Ordinary Differential Equations, P. M. Fitzpertrick, M. Martelli, J. Mawhin, and R. Nussbaum, Eds., vol. 1537 of Lecture Notes in Mathematics, pp. 74–142, Springer, Berlin, Germany, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, vol. 40 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1979. View at MathSciNet
  34. Y. Zhang and Z. 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 · View at Zentralblatt MATH · View at MathSciNet