Skip to main content
SpringerLink
Log in

Existence and Uniqueness of Periodic Solutions for a Class of Higher Order Differential Equations

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we study the existence, non-existence and uniqueness of periodic solutions for a class of higher order differential equations. The proof is based on the Mawhin’s continuation theorem and averaging method. Finally, two examples are given to illustrate the applicability of the conclusions of this paper.

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

Fig. 1
Fig. 2

Similar content being viewed by others

Data Availability Statement

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

References

  1. Boscaggin, A., Feltrin, G., Zanolin, F.: Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case. Proc. R. Soc. Edinb. Sect. A 146(3), 449–474 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boscaggin, A., Feltrin, G., Zanolin, F.: Uniqueness of positive solutions for boundary value problems associated with indefinite \(\phi \)-Laplacian-type equations. Open Math. 19(1), 163–183 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cen, X., Llibre, J., Zhang, M.: Periodic solutions and their stability of some higher-order positively homogenous differential equations. Chaos Solit. Fract. 106(1), 285–288 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chu, J., Torres, P.J., Wang, F.: Twist periodic solutions for differential equations with a combined attractive-repulsive singularity. J. Math. Appl. Anal. 437(2), 1070–1083 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cid, J.A., Infante, G., Tvrdý, M., Zima, M.: A topological approach to periodic oscillations related to the Liebau phenomenon. J. Math. Appl. Anal. 423(2), 1546–1556 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cid, J.A., Mawhin, J., Zima, M.: An abstract averaging method with applications to differential equations. J. Differ. Equ. 274, 231–250 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ding, T., Zanolin, F.: Periodic solutions of Duffing’s equations with superquadratic potential. J. Differ. Equ. 97(2), 328–378 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fonda, A., Sfecci, A.: Periodic solutions of weakly coupled superlinear systems. J. Differ. Equ. 260(3), 2150–2162 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gaines, R.E., Mawhin, J.: Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin (1977)

    Book  MATH  Google Scholar 

  10. Hakl, R., Torres, P.J., Zamora, M.: Periodic solutions of singular second order differential equations: the repulsive case. Topol. Methods Nonlinear Anal. 39(2), 199–220 (2012)

    MathSciNet  MATH  Google Scholar 

  11. Han, X., Yang, H.: Existence and multiplicity of periodic solutions for a class of second-order ordinary differential equations. Monatsh. Math. 193(4), 829–843 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  12. Han, X., Yang, H.: Existence of periodic solutions for two classes of second order p-Laplacian differential equations. J. Appl. Anal. Comput. 13(1), 81–94 (2023)

    MathSciNet  Google Scholar 

  13. Kiguradze, I.: On periodic solutions of \(n\)th order differential equations. Nonlinear Anal. 40, 309–321 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kiguradze, I., Lomtatidze, A.: Periodic solutions of nonautonomous ordinary differential equations. Monatsh. Math. 159(3), 235–252 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kong, Q., Wang, M.: Positive solutions of even order system periodic boundary value problems. Nonlinear Anal. 72(3–4), 1778–1791 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, Y., Fan, H.: Existence of positive periodic solutions for higher-order ordinary differential equations. Comput. Math. Appl. 62(4), 1715–1722 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, X., Lu, S.: Periodic solutions for a kind of high-order \(p\)-Laplacian differential equation with sign-changing coefficient ahead of nonlinear term. Nonlinear Anal. 70(2), 1011–1022 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Llibre, J., Makhlouf, A.: Periodic solutions of some classes of continuous second-order differential equations. Discrete Contin. Dyn. Syst. Ser. B 22(2), 477–482 (2017)

    MathSciNet  MATH  Google Scholar 

  19. Lu, S., Yu, X.: Periodic solutions for second order differential equations with indefinite singularities. Adv. Nonlinear Anal. 9(1), 994–1007 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  20. Makhlouf, A., Debbabi, D.: Periodic solutions of some classes of continuous third-order differential equations. Chaos Solit. Fract. 94(1), 112–118 (2017)

    MathSciNet  MATH  Google Scholar 

  21. Morris, G.R.: An infinite class of periodic solutions of \(\ddot{x}+2x^{3}=p(t)\). Proc. Camb. Philos. Soc. 61(1), 157–164 (1965)

    Article  MATH  Google Scholar 

  22. Novaes, D.D., Silva, F.: Higher order analysis on the existence of periodic solutions in continuous differential equations via coincidence degree. SIAM J. Math. Anal. 53(2), 2476–2490 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ortega, R.: The number of stable periodic solutions of time-dependent Hamiltonian systems with one degree of freedom. Ergod. Theory Dyn. Syst. 18(4), 1007–1018 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Torres, P.J., Cheng, Z., Ren, J.: Non-degeneracy and uniqueness of periodic solutions for 2\(n\)-order differential equations. Discrete Contin. Dyn. Syst. 33(5), 2155–2168 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wang, G., Sui, S.: Even periodic solutions of higher order duffing differential equations. Czechoslov. Math. J. 57(1), 331–343 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wang, F., Cid, J.A., Li, S., Zima, M.: Lyapunov stability of periodic solutions of Brillouin type equations. Appl. Math. Lett. 101, 106057 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang, F., Cid, J.A., Li, S., Zima, M.: Existence results for damped regular equations under periodic or Neumann boundary conditions. J. Math. Anal. Appl. 509(2), 125978 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  28. Yang, H., Han, X.: Existence of periodic solutions for the forced pendulum equations of variable length. Qual. Theory Dyn. Syst. 22(1), 1–13 (2023)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors warmly thank the anonymous referee for his/her careful reading of the article and many pertinent remarks that lead to various improvements to this article.

Author information

Authors and Affiliations

  1. Department of Mathematics, Northwest Normal University, Lanzhou, 730070, Gansu, People’s Republic of China

    Hujun Yang & Xiaoling Han

Authors
  1. Hujun Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoling Han
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

HY and XH wrote the main manuscript text, HY and XH prepared Figs. 1 and 2. All authors reviewed the manuscript.

Corresponding author

Correspondence to Xiaoling Han.

Ethics declarations

Conflict of Interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Supported by NSF of China with No.12161079.

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

Yang, H., Han, X. Existence and Uniqueness of Periodic Solutions for a Class of Higher Order Differential Equations. Mediterr. J. Math. 20, 282 (2023). https://doi.org/10.1007/s00009-023-02477-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-023-02477-0

Keywords

Mathematics Subject Classification

Search

Navigation