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Journal of Function Spaces
Volume 2017, Article ID 2785937, 7 pages
https://doi.org/10.1155/2017/2785937
Research Article

The Existence of Solutions to Integral Boundary Value Problems of Fractional Differential Equations at Resonance

1Department of Statistics and Finance, Shandong University of Science and Technology, Qingdao 266590, China
2Shandong Academy of Sciences, Jinan 250014, China

Correspondence should be addressed to Yumei Zou; moc.621@myuozds

Received 10 April 2017; Accepted 10 September 2017; Published 2 November 2017

Academic Editor: Xinguang Zhang

Copyright © 2017 Yumei Zou and Guoping He. 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, Amsterdam, Elsevier, 2006. View at MathSciNet
  2. V. Lakshmikantham, S. Leela, and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009.
  3. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993. View at MathSciNet
  4. 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 MathSciNet · View at Scopus
  5. Y. Cui, “Uniqueness of solution for boundary value problems for fractional differential equations,” Applied Mathematics Letters, vol. 51, pp. 48–54, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  6. R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973–1033, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. B. Ahmad and A. Alsaedi, “Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations,” Fixed Point Theory and Applications, vol. 2010, Article ID 364560, 2010. View at Publisher · View at Google Scholar · View at Scopus
  8. 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
  9. H. Li and Y. Liu, “On the uniqueness of the positive solution for a second-order integral boundary value problem with switched nonlinearity,” Applied Mathematics Letters, vol. 24, no. 12, pp. 2201–2205, 2011. View at Publisher · View at Google Scholar · View at Scopus
  10. H. Li and Y. Liu, “On sign-changing solutions for a second-order integral boundary value problem,” Computers and Mathematics with Applications, vol. 62, pp. 651–656, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  11. L. Liu, X. Hao, and Y. Wu, “Positive solutions for singular second order differential equations with integral boundary conditions,” Mathematical and Computer Modelling, vol. 57, no. 3-4, pp. 836–847, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. L. Liu, F. Sun, X. Zhang, and Y. Wu, “Bifurcation analysis for a singular differential system with two parameters via to degree theory,” Nonlinear Analysis: Modelling and Control, vol. 22, no. 1, pp. 31–50, 2017. View at Google Scholar · View at MathSciNet
  13. L. Liu, H. Li, C. Liu, and Y. Wu, “Existence and uniqueness of positive solutions for singular fractional differential systems with coupled integral boundary conditions,” Journal of Nonlinear Sciences and Applications. JNSA, vol. 10, no. 1, pp. 243–262, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  14. M. ur Rehman and R. A. Khan, “Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1038–1044, 2010. View at Publisher · View at Google Scholar · View at Scopus
  15. F. Sun, L. Liu, X. Zhang, and Y. Wu, “Spectral analysis for a singular differential system with integral boundary conditions,” Mediterranean Journal of Mathematics, vol. 13, no. 6, pp. 4763–4782, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. S. Zhang and X. Su, “The existence of a solution for a fractional differential equation with nonlinear boundary conditions considered using upper and lower solutions in reverse order,” Comput. Math. Appl, vol. 62, pp. 1269–1274, 2011. View at Google Scholar · View at MathSciNet
  17. 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 MathSciNet · View at Scopus
  18. X. Zhang, L. Liu, and Y. Wu, “Existence and uniqueness of iterative positive solutions for singular Hammerstein integral equations,” Journal of Nonlinear Sciences and Applications. JNSA, vol. 10, no. 7, pp. 3364–3380, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  19. D. Zhao and Y. Liu, “Positive solutions for a class of fractional differential coupled system with integral boundary value conditions,” Journal of Nonlinear Sciences and Applications, vol. 9, pp. 2922–2942, 2016. View at Google Scholar · View at MathSciNet
  20. Y. Zou and G. He, “On the uniqueness of solutions for a class of fractional differential equations,” Applied Mathematics Letters, vol. 74, pp. 68–73, 2017. View at Google Scholar · View at MathSciNet
  21. Z. Bai and Y. Zhang, “The existence of solutions for a fractional multi-point boundary value problem,” Computers and Mathematics with Applications, vol. 60, no. 8, pp. 2364–2372, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  22. Y. Cui and Y. Zou, “Existence of solutions for second-order integral boundary value problems,” Lithuanian Association of Nonlinear Analysts. Nonlinear Analysis: Modelling and Control, vol. 21, no. 6, pp. 828–838, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. Y. Cui, “Existence of solutions for coupled integral boundary value problem at resonance,” Publicationes Mathematicae, vol. 89, no. 1-2, pp. 73–88, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  24. N. Kosmatov and W. Jiang, “Resonant functional problems of fractional order,” Chaos, Solitons and Fractals, vol. 91, pp. 573–579, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  25. Q. Sun and Y. Cui, “Existence results for conjugate boundary-value problems with integral boundary conditions at resonance with dim ker L= 2,” Boundary Value Problems, vol. 2017, no. 1, article no. 29, 2017. View at Publisher · View at Google Scholar · View at Scopus
  26. Q. Sun and Y. Cui, “Solvability of (k, n - k) Conjugate Boundary Value Problems with Integral Boundary Conditions at Resonance,” Journal of Function Spaces, vol. 2016, Article ID 3454879, 2016. View at Publisher · View at Google Scholar · View at Scopus
  27. 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 MathSciNet · View at Scopus
  28. Y. Zou, L. Liu, and Y. Cui, “The existence of solutions for four-point coupled boundary value problems of fractional differential equations at resonance,” Abstract and Applied Analysis, vol. 2014, Article ID 314083, 2014. View at Publisher · View at Google Scholar · View at Scopus
  29. Y. Zou and Y. Cui, “Existence results for a functional boundary value problem of fractional differential equations,” Advances in Difference Equations, vol. 2013, article no. 233, 2013. View at Publisher · View at Google Scholar · View at Scopus
  30. J. Mawhin, in Proceedings of the, vol. 40 of NSFCBMS Regional Conference Series in Mathematics, American Mathematical Society. View at MathSciNet