Skip to main content
Log in

Multiple Periodic Solutions of Differential Delay Equations with 2k − 1 Lags

  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

In this paper, we study the periodic solutions to a type of differential delay equations with 2k − 1 lags. The 4k-periodic solutions are obtained by using the variational method and the method of Kaplan-Yorke coupling system. This is a new type of differential delay equations compared with all the previous researches. And this paper provides a theoretical basis for the study of differential delay equations. An example is given to demonstrate our main results.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Benci, V. On critical point theory for indefinite functionals in the presence of symmetries. Trans. Amer. Math. Soc., 274: 533–572 (1982)

    Article  MathSciNet  Google Scholar 

  2. Fannio, L. Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity. Discrete and Cont. Dynamical Sys., 3: 251–264 (1997)

    Article  MathSciNet  Google Scholar 

  3. Fei, G. Multiple periodic solutions of differential delay equations via Hamiltonian systems (I). Nonlinear Analysis, 65: 25–39 (2006)

    Article  MathSciNet  Google Scholar 

  4. Fei, G. Multiple periodic solutions of differential delay equations via Hamiltonian systems (II). Nonlinear Analysis, 65: 40–58 (2006)

    Article  MathSciNet  Google Scholar 

  5. Ge, W. Periodic solutions of the differential delay equation x′(t) = -f(x(t − 1)). Acta Math. Sinica (New Series), 12: 113–121 (1996)

    Article  MathSciNet  Google Scholar 

  6. Ge, W. On the existence of periodic solutions of differential delay equations with multiple lags. Acta Appl. Math. Sinica (in Chinese), 17: 173–181 (1994)

    Google Scholar 

  7. Ge, W. Two existence theorems of periodic solutions for differential delay equations. Chin. Ann. Math., 15: 217–224 (1994)

    MathSciNet  MATH  Google Scholar 

  8. Ge, W. Oscillatory periodic solutions of differential delay equations with multiple lags. Chin. Sci. Bulltin., 42 444–447 (1997)

    Article  MathSciNet  Google Scholar 

  9. Ge, W., Zhang, L. Multiple periodic solutions of delay differential systems with 2k − 1 lags via variational approach. Discrete and Continuous Dynamical Systems, 36: 4925–4943 (2016)

    Article  MathSciNet  Google Scholar 

  10. Guo, Z., Yu, J. Multiple results for periodic solutions to delay differential equations via critical point theory. JDE, 218: 15–35 (2005)

    Article  Google Scholar 

  11. Guo, Z., Yu, J. Multiple results on periodic solutions to higher dimensional differential equations with multiple delays. J. Dyn. Diff. Eqns., 23: 1029–1052 (2011)

    Article  Google Scholar 

  12. Li, L., Xue, C., Ge, W. Periodic orbits to Kaplan-Yorke like differential delay equations with two lags of ratio (2k − 1)/2. Adv. Difference Equ., 247 (2016)

  13. Li, L., Sun, H., Ge, W. On the Number of Periodic Solutions to KaplanCYorke-like High Order Differential Delay Equations with 2k Lags. International Journal of Bifurcation and Chaos, 29 (2019)

  14. Li, J., He, X. Proof and generalization of Kaplan-Yorke conjecture under the condition f′(0) > 0 on periodic solution of differential delay equations. Sci. China Series A., 42: 957–964 (1999)

    Article  MathSciNet  Google Scholar 

  15. Li, J., He, X., Liu, Z. Hamiltonian symmetric group and multiple periodic solutions of differential delay equations. Nonlinear Analysis TMA., 35: 957–964 (1999)

    Article  MathSciNet  Google Scholar 

  16. Li, S., Liu, J. Morse theory and asymptotically linear Hamiltonian systems. JDE, 78: 53–73 (1989)

    Article  Google Scholar 

  17. Kaplan, J., Yorke, J. Ordinary differential equations which yield periodic solution of delay equations. J. Math. Anal. Appl., 48: 317–324 (1974)

    Article  MathSciNet  Google Scholar 

  18. Mawhen, J., Willem, M. Critical Point Theory and Hamiltonian Systems. Springer-Verlag. New Yorke, 1989.

    Book  Google Scholar 

  19. Nussbaum, R. Periodic solutions of special differential delay equations: an example in nonlinear functional analysis. Proc. Royal Soc. Edingburgh., 81: 131–151 (1978)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hua-fei Sun.

Additional information

This subject is supported by the National Natural Science Foundations of China (No. 61179031).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, L., Sun, Hf. & Ge, Wg. Multiple Periodic Solutions of Differential Delay Equations with 2k − 1 Lags. Acta Math. Appl. Sin. Engl. Ser. 36, 390–400 (2020). https://doi.org/10.1007/s10255-020-0946-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-020-0946-z

Keywords

2000 MR Subject Classification

Navigation