Skip to main content
SpringerLink
Log in

Positive solutions for fractional (p, q)-difference boundary value problems

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

Abstract

In this paper, we investigate the boundary value problem of a class of fractional (p, q)-difference equations involving the Riemann–Liouville fractional derivative. Based on the generalization of Banach contraction principle, we obtain a sufficient condition for existence and uniqueness of solutions of the problem. By applying a fixed point theorem in cones, we establish a sufficient condition for the existence of at least one positive solution of the problem. As an application, some examples are presented to illustrate the 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. Jackson, F.: On \(q\)-difference equations. Am. J. Math. 32, 305–314 (1910)

    Article  Google Scholar 

  2. Liu, H.: Application of the variational iteration method to strongly nonlinear \(q\)-difference equations. J. Appl. Math. 1, 1–12 (2012)

    Article  MathSciNet  Google Scholar 

  3. Riyasat, M., Khan, S., Nahid, T.: \(q\)-difference equations for the composite \(2D\)\(q\)-Appell polynomials and their applications. Cogent Math. 4, 1–23 (2017)

    Article  MathSciNet  Google Scholar 

  4. Ernst, T.: On several \(q\)-special matrices, including the \(q\)-Bernoulli and \(q\)-Euler matrices. Linear Algebra Appl. 542, 422–440 (2018)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  6. Chakrabarti, R., Jagannathan, R.: A \((p, q)\)-oscillator realization of two-parameter quantum algebras. J. Phys. A Math. General 24, 5683–5701 (1991)

    Article  MathSciNet  Google Scholar 

  7. Kamsrisuk, N., Promsakon, C., Ntouyas, S., Tariboon, J.: Nonlocal boundary value problems for \((p, q)\)-difference equations. Differ. Equ. Appl. 10, 183–195 (2018)

    MathSciNet  MATH  Google Scholar 

  8. Promsakon, C., Kamsrisuk, N., Ntouyas, S.: On the second-order quantum \((p, q)\)-difference equations with separated boundary conditions. Adv. Math. Phys. 1–9 (2018)

  9. Sadjang, P.N.: On the function theorem of \((p, q)\)-calculus and some \((p, q)\)-Taylor formulas. Results Math. 73, 1–21 (2013)

    Google Scholar 

  10. Mursaleen, M., Ansari, K., Khan, A.: Some approximation results by \((p, q)\)-analogue of Bernstein-Stancu operators. Appl. Math. Comput. 264, 1–4 (2015)

    MathSciNet  MATH  Google Scholar 

  11. Al-Salam, W.: Some fractional \(q\)-integrals and \(q\)-derivatives. Proc. Edinburgh Math. Soc. 15(1966), 135–140 (1966)

    Article  MathSciNet  Google Scholar 

  12. Agarwal, R.: Certain fractional \(q\)-integrals and \(q\)-derivatives. Math. Proc. Cambridge Philos. Soc. 66(1969), 365–370 (1969)

    Article  MathSciNet  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  MathSciNet  Google Scholar 

  14. Ferreira, R.: Positive solutions for a class of boundary value problems with fractional \(q\)-differences. Comput. Math. Appl. 61, 367–373 (2011)

    Article  MathSciNet  Google Scholar 

  15. Ma, K., Sun, S., Han, Z.: Existence of solutions of boundary value problems for singular fractional \(q\)-difference equations. J. Appl. Math. Comput. 54, 23–40 (2017)

    Article  MathSciNet  Google Scholar 

  16. Li, X., Han, Z., Sun, S., et al.: Eigenvalue problems of fractional \(q\)-difference equations with generalized \(p\)-Laplacian. Appl. Math. Lett. 57, 46–53 (2016)

    Article  MathSciNet  Google Scholar 

  17. Mao, J., Zhao, Z., Wang, C.: The unique iterative positive solution of fractional boundary vulue problme with \(q\)-difference. Appl. Math. Lett. 100, 1–8 (2020)

    Article  Google Scholar 

  18. Soontharanon, J., Sitthiwirattham, T.: Existence results of nonlocal Robin boundary value problems for fractional \((p, q)\)-integrodifference equations. Adv. Differ. Equ. 342, 1–17 (2020)

    MathSciNet  MATH  Google Scholar 

  19. Soontharanon, J., Sitthiwirattham, T.: On fractional \((p, q)\)-calculus. Adv. Differ. Equ. 35, 1–18 (2020)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led 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

    Zhongyun Qin & Shurong Sun

Authors
  1. Zhongyun Qin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shurong Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shurong Sun.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research is supported by Shandong Provincial Natural Science Foundation (ZR2020MA016), also supported by the Natural Science Foundation of China (62073153).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qin, Z., Sun, S. Positive solutions for fractional (p, q)-difference boundary value problems. J. Appl. Math. Comput. 68, 2571–2588 (2022). https://doi.org/10.1007/s12190-021-01630-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-021-01630-w

Keywords

Mathematics Subject Classification

Search

Navigation