Skip to main content
SpringerLink
Log in

Multiple periodic solutions of a second-order partial difference equation involving p-Laplacian

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

Abstract

In this paper, we investigate a second-order partial difference equation involving p-Laplacian. We establish series of criteria to study multiple nontrivial periodic solutions by Mountain Pass Lemma and Linking Theorem. Our results generalize and improve some known results. Moreover, examples and numerical simulations are presented to illustrate applications of our 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

Data Availibility Statement

Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

References

  1. Yu, J.S., Li, J.: Discrete-time models for interactive wild and sterile mosquitoes with general time steps. Math. Biosci. 346, 108797 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  2. Long, Y.H., Wang, L.: Global dynamics of a delayed two-patch discrete SIR disease model. Commun. Nonlinear Sci. Numer. Simul. 83, 105117 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  3. Guo, Z.M., Yu, J.S.: Existence of periodic and subharmonic solutions for second-order superlinear difference equations. Sci. China Ser. A-Math. 46, 506–515 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Mei, P., Zhou, Z.: Homoclinic solutions of discrete prescribed mean curvature equations with mixed nonlinearities. Appl. Math. Lett. 130, 108006 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ma, M.J., Guo, Z.M.: Homoclinic orbits and subharmonics for nonlinear second order difference equations. Nonlinear Anal. 67, 1737–1745 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Zhang, H., Long, Y.H.: Multiple existence results of nontrivial solutions for a class of second-order partial difference equations. Symmetry 15, 6 (2023)

    Article  Google Scholar 

  7. Zhang, H., Zhou, Y., Long, Y.H.: Results on multiple nontrivial solutions to partial difference equations. AIMS Math. 8, 5413–5431 (2023)

    Article  MathSciNet  Google Scholar 

  8. Long, Y.H., Zhang, H.: Three nontrivial solutions for second-order partial difference equation via morse theory. J. Funct. Spaces 2022, 1564961 (2022)

    MathSciNet  MATH  Google Scholar 

  9. Long, Y.H., Deng, X.Q.: Existence and multiplicity solutions for discrete Kirchhoff type problems. Appl. Math. Lett. 126, 107817 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  10. Long, Y.H.: Multiple results on nontrivial solutions of discrete Kirchhoff type problems. J. Appl. Math. Comput. 69, 1–17 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  11. Long, Y.H.: Nontrivial solutions of discrete Kirchhoff type problems via Morse theory. Adv. Nonlinear Anal. 11, 1352–1364 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  12. Long, Y.H., Zhang, Q.Q.: Infinitely many large energy solutions to a class of nonlocal discrete elliptic boundary value problems. Comm. Pure Appl. Math. (2023). https://doi.org/10.3934/cpaa.2023037

    Article  MATH  Google Scholar 

  13. Mei, P., Zhou, Z.: Homoclinic solutions for partial difference equations with mixed nonlinearities. J. Geom. Anal. 33, 117 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wang, S.H., Zhou, Z.: Periodic solutions for a second-order partial difference equation. J. Appl. Math. Comput. 69, 731–752 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  15. Long, Y.H., Li, D.: Multiple nontrivial periodic solutions to a second-order partial difference equation. Electron. Res. Arch. 31, 1596–1612 (2023)

    Article  MathSciNet  Google Scholar 

  16. He, T.S., Chen, W.G.: Periodic solutions of second order discrete convex systems involving the p-Laplacian. Appl. Math. Comput. 206, 124–132 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Chen, P., Fang, H.: Existence of periodic and subharmonic solutions for second-order p-Laplacian difference equations. Adv. Difference Equ. 2007, 042530 (2007)

    Article  MathSciNet  Google Scholar 

  18. He, Z.M.: On the existence of positive solutions of p-Laplacian difference equations. J. Comput. Appl. Math. 161, 193–201 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kuang, J.H.: Existence of homoclinic solutions for higher-order periodic difference equations with p-Laplacian. J. Math. Anal. Appl. 417, 904–917 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mei, P., Zhou, Z.: Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations. Open Math. 16, 1435–1444 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. Du, S.J., Zhou, Z.: Multiple solutions for partial discrete Dirichlet problems involving the p-Laplacian. Mathematics 8, 2030 (2020)

    Article  Google Scholar 

  22. Wang, S.H., Zhou, Z.: Three solutions for a partial discrete Dirichlet boundary value problem with p-Laplacian. Bound. Value Probl. 2021, 39 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. American Mathematical Society, Providence (1986)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to thank the handling editor and the referees for their valuable comments and suggestions.

Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

    Yuhua Long & Dan Li

  2. Center for Applied Mathematics, Guangzhou University, Guangzhou, 510006, China

    Yuhua Long & Dan Li

Authors
  1. Yuhua Long
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuhua Long.

Ethics declarations

Conflicts of interest

Authors state no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Long, Y., Li, D. Multiple periodic solutions of a second-order partial difference equation involving p-Laplacian. J. Appl. Math. Comput. 69, 3489–3508 (2023). https://doi.org/10.1007/s12190-023-01891-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-023-01891-7

Keywords

Mathematics Subject Classification

Search

Navigation