Skip to main content
SpringerLink
Log in

Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order

  • Applied mathematics
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this paper, the existence of solutions of an anti-periodic fractional boundary value problem for nonlinear fractional differential equations is investigated. The contraction mapping principle and Leray-Schauder’s fixed point theorem are applied to establish the results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications. Gordon & Breach, Yverdon (1993)

    MATH  Google Scholar 

  2. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  3. Lakshmikantham, V., Leela, S., Vasundhara Devi, J.: Theory of Fractional Dynamic Systems. Cambridge Scientific, Cambridge (2009)

    MATH  Google Scholar 

  4. Lakshmikantham, V., Vatsala, A.S.: Basic theory of fractional differential equations. Nonlinear Anal. 69, 2677–2682 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Laksmikantham, V., Leela, S.: Nagumo-type uniqueness result for fractional differential equations. Nonlinear Anal. 8, 2886–2889 (2009)

    Article  Google Scholar 

  6. Ahmad, B., Nieto, J.J.: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58, 1838–1843 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chang, Y.K., Nieto, J.J.: Some new existence results for fractional differential inclusions with boundary conditions. Math. Comput. Model. 49, 605–609 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Li, C.F., Luo, X.N., Zhou, Y.: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl. 59, 1363–1375 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Byszewski, L.: Theorems about existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162, 494–505 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal. 11, 4465–4475 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Henderson, J., Ouahab, A.: Impulsive differential inclusions with fractional order. Comput. Math. Appl. 59, 1191–1226 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Wang, X.H.: Impulsive boundary value problem for nonlinear differential equations of fractional order. Comput. Math. Appl. 62, 2383–2391 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Wang, X.H.: Existence of solutions for nonlinear impulsive higher order fractional differential equations. Electron. J. Qual. Theory Differ. Equ. 80, 1–12 (2011)

    Google Scholar 

  14. Wang, F.: Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order. Adv. Differ. Equ. (2012). doi:10.1186/1687-1847-2012-116

    Google Scholar 

  15. Cernea, A.: On the existence of solutions for fractional differential inclusions with anti-periodic boundary conditions. J. Appl. Math. Comput. 38, 133–143 (2012)

    Article  MathSciNet  Google Scholar 

  16. Chen, Y., Nieto, J.J., O’Regan, D.: Anti-periodic solutions for evolution equations associated with maximal monotone mappings. Appl. Math. Lett. 24, 302–307 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pan, L., Cao, J.: Anti-periodic solution for delayed cellular neural networks with impulsive effects. Nonlinear Anal. 12, 3014–3027 (2011)

    MathSciNet  MATH  Google Scholar 

  18. Franco, D., Nieto, J.J., O’Regan, D.: Anti-periodic boundary value problem for nonlinear first order ordinary differential equations. Math. Inequal. Appl. 6, 477–485 (2003)

    MathSciNet  MATH  Google Scholar 

  19. Ahmad, B., Nieto, J.J.: Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions. Nonlinear Anal. 69, 3291–3298 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Luo, Z.G., Shen, J.H., Nieto, J.J.: Antiperiodic boundary value problem for first-order impulsive ordinary differential equations. Comput. Math. Appl. 49, 253–261 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang, K.Z.: A new existence result for nonlinear first-order anti-periodic boundary value problems. Appl. Math. Lett. 21, 1149–1154 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Luo, Z.G., Wang, W.B.: Existence of solutions to anti-periodic boundary value problems for second order differential equations. Acta Math. Appl. Sin. 29, 1111–1117 (2006)

    MathSciNet  Google Scholar 

  23. Wang, K.Z., Li, Y.: A note on existence of (anti)-periodic and heteroclinic solutions for a class of second-order odes. Nonlinear Anal. 70, 1711–1724 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Aftabizadeh, A.R., Aizicovici, S., Pavel, N.H.: Anti-periodic boundary value problems for higher order differential equations in Hilbert spaces. Nonlinear Anal. 18, 253–267 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ahmad, B., Otero-Espinar, V.: Existence of solutions for fractional differential inclusions with anti-periodic boundary conditions. Bound. Value Probl. 2009, 625347 (2009)

    MathSciNet  Google Scholar 

  26. Ahmad, B.: Existence of solutions for fractional differential equations of order q∈(2,3] with anti-periodic boundary conditions. J. Appl. Math. Comput. 34, 385–391 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ahmad, B., Nieto, J.J.: Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Topol. Methods Nonlinear Anal. 35, 295–304 (2010)

    MathSciNet  MATH  Google Scholar 

  28. Alsaedi, A.: Existence of solutions for integro-differential equations of fractional order with antiperiodic boundary conditions. Int. J. Differ. Equ. (2009). doi:10.1155/2009/41706

    MathSciNet  Google Scholar 

  29. Agarwal, R.P., Ahmad, B.: Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl. 62, 1200–1214 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ahmad, B., Nieto, J.J.: Existence of solutions for impulsive anti-periodic boundary value problems of fractional order. Taiwan. J. Math. 15, 981–993 (2011)

    MathSciNet  MATH  Google Scholar 

  31. Ahmad, B., Nieto, J.J.: Anti-periodic fractional boundary value problems. Comput. Math. Appl. 62, 1150–1156 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Smart, D.R.: Fixed Point Theorems. Cambridge University Press, Cambridge (1980)

    MATH  Google Scholar 

  33. Bai, Z.B., Lü, H.S.: Positive solutions of boundary value problems of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495–505 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank the referee for his or her careful reading and some comments on improving the presentation of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Baoshan College, Baoshan, Yunnan, 678000, P.R. China

    Xuhuan Wang & Xiuqing Guo

  2. Zhangjiagang Campus, Jiangsu University of Science and Technology, Zhangjiagang, Jiangsu, 215600, P.R. China

    Guosheng Tang

Authors
  1. Xuhuan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiuqing Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Guosheng Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xuhuan Wang.

Additional information

Project supported by NNSF of China Grant No. 11271087 and No. 61263006.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, X., Guo, X. & Tang, G. Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order. J. Appl. Math. Comput. 41, 367–375 (2013). https://doi.org/10.1007/s12190-012-0613-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-012-0613-5

Keywords

Mathematics Subject Classification (2000)

Search

Navigation