Skip to main content
SpringerLink
Log in

Anti-periodic boundary value problems for fractional \(q\)-difference equations

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

Abstract

In this paper, we study the boundary value problem with anti-periodic boundary conditions involving the Caputo fractional \(q\)-derivative. By means of the Banach contraction mapping principle and Scheafer fixed point theorem, some results of the existence and uniqueness of solutions are obtained. At last, examples are presented to illustrate our main 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. Agarwal, R.: Certain fractional \(q\)-integrals and \(q\)-derivatives. Proc. Camb. Philos. Soc. 66, 365–370 (1969)

    Article  MATH  Google Scholar 

  2. Rajković, P., Marinković, S., Stanković, M.: Fractional integrals and derivatives in \(q\)-calculus. Appl. Anal. Discret. Math. 1(1), 311–323 (2007)

    Article  MATH  Google Scholar 

  3. Annaby, M., Mansour, Z.: \(q\)-Fractional Calculus and Equations. Springer-Verlag, Berlin Heidelberg (2012)

    Google Scholar 

  4. Abdel-Gawad, H., Aldailami, A.: On \(q\)-dynamic equations modelling and complexity. Appl. Math. Model. 34(3), 697–709 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Field, C., Joshi, N., Nijhoff, F.: \(q\)-difference equations of KdV type and Chazy-type second-degree difference equations. Phys. A Math. Theor. 41(33), 1–13 (2008)

    MathSciNet  Google Scholar 

  6. Abdi, W.: Certain inversion and representation formulae for \(q\)-Laplace transforms. Math. Z. 83, 238–249 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  7. Abreu, L.: Sampling theory associated with \(q\)-difference equations of the Sturm-Liouville type. J. Phys. A. 38(48), 10311–10319 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ernst, T.: A Comprehensive Treatment of \(q\)-Calculus. Springer, Basel (2012)

    Book  Google Scholar 

  9. Cheng, H.: Canonical quantization of Yang-Mills theories. Perspectives in Mathematical Physics. International Press, Somerville (1994)

    Google Scholar 

  10. Finkelstein, R.: \(q\)-gravity. Lett. Math. Phys. 38(1), 53–62 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cabada, A., Hamdi, Z.: Nonlinear fractional differential equations with integral boundary value conditions. Appl. Math. Comput. 228, 251–257 (2014)

    Article  MathSciNet  Google Scholar 

  12. Lv, Z.: Existence results for \(m\)-point boundary value problems of nonlinear fractional differential equations with \(p\)-Laplacian operator. Adv. Differ. Equ. 2014(69), 1–16 (2014)

    Google Scholar 

  13. Ferreira, R.: Nontrivial solutions for fractional \(q\)-difference boundary value problems. Electron. J. Qual. Theory Differ. Equ. 70, 1–10 (2010)

    Article  Google Scholar 

  14. Li, X., Han, Z., Sun, S.: Existence of positive solutions of nonlinear fractional \(q\)-difference equation with parameter. Adv. Differ. Equ. 2013(260), 1–13 (2013)

    MathSciNet  Google Scholar 

  15. Zhou, W., Liu, H.: Existence solutions for boundary value problem of nonlinear fractional \(q\)-difference equations. Adv. Differ. Equ. 1(113), 1–12 (2013)

    Article  MATH  Google Scholar 

  16. Yang, W.: Anti-periodic bounndary value problems involving nonlinear fractional \(q-\)difference equations. Malaya J. Mat. 4(1), 107–114 (2013)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, F., Liu, Z.: Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order. Adv. Differ. Equ. 2012(116), 1–12 (2012)

    MATH  Google Scholar 

  19. Benchohra, M., Hamidi, N., Henderson, J.: Fractional differential equations with anti-periodic boundary conditions. Numer. Funct. Anal. Optim. 34(4), 404–414 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

Download references

Acknowledgments

This research is supported by the Natural Science Foundation of China (61374074), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2013AL003), also supported by Graduate Innovation Foundation of University of Jinan (YCX13013). The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have lead to the present improved version of the original manuscript.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Jinan, Jinan, 250022, Shandong, People’s Republic of China

    Xinhui Li, Zhenlai Han & Shurong Sun

Authors
  1. Xinhui Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhenlai Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shurong Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhenlai Han.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, X., Han, Z. & Sun, S. Anti-periodic boundary value problems for fractional \(q\)-difference equations. J. Appl. Math. Comput. 50, 243–257 (2016). https://doi.org/10.1007/s12190-015-0868-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-015-0868-8

Keywords

Mathematics Subject Classification

Search

Navigation