Skip to main content
SpringerLink
Log in

Positive Periodic Solution for a Second-Order Damped Singular Equation via Fixed Point Theorem in Cones

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

The aim of this paper is to show that fixed point theorem in cones can be applied to singular equations. Using the positivity of Green’s function and the external force e(t), we prove the existence of a positive periodic solution for a damped singular equation with sub-linearity, semi-linearity and super-linearity conditions, and these results are applicable to weak and strong singularities. As applications, we consider the existence of a positive periodic solution for nonlinear elasticity model and Ermakov–Pinney equation

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. Cheng, Z., Yuan, Q.: Damped superlinear Duffing equation with strong singularity of repulsive type. J. Fixed Point Theory Appl. 22, 1–18 (2020)

    Article  MathSciNet  Google Scholar 

  2. Cheng, Z., Ren, J.: Studies on a damped differential equation with repulsive singularity. Math. Methods Appl. Sci. 36, 983–992 (2013)

    Article  MathSciNet  Google Scholar 

  3. Cheng, Z., Ren, J.: Periodic solution for second order damped differential equations with attractive-repulsive singularities. Rocky Mt. J. Math. 48, 753–768 (2018)

    Article  MathSciNet  Google Scholar 

  4. Cheng, Z., Li, F.: Positive periodic solutions for a kind of second-order neutral differential equations with variable coefficient and delay. Mediterr. J. Math. 15, 1–19 (2018)

    Article  MathSciNet  Google Scholar 

  5. Chu, J., Torres, P., Zhang, M.: Periodic solutions of second order nonautonomous singular dynamical systems. J. Differ. Equ. 239, 196–212 (2007)

    Article  Google Scholar 

  6. Chu, J., Torres, P.: Applications of Schauder’s fixed point theorem to singular differential equations. Bull. Lond. Math. Soc. 39, 653–660 (2007)

    Article  MathSciNet  Google Scholar 

  7. Chu, J., Fan, N., Torres, P.: Periodic solutions for second order singular damped differential equations. J. Math. Anal. Appl. 388, 665–675 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Deimling, K.: Nonlinear Functional Analysis. Springer, New York (1985)

    Book  Google Scholar 

  10. Fonda, A., Manásevich, R., Zanolin, F.: Subharmonic solutions for some second-order differential equations with singularities. SIAM J. Math. Anal. 24, 1294–1311 (1993)

    Article  MathSciNet  Google Scholar 

  11. Hakl, R., Torres, P.: On periodic solutions of second-order differential equations with attractive-repulsive singularities. J. Differ. Equ. 248, 111–126 (2010)

    Article  MathSciNet  Google Scholar 

  12. Hakl, R., Zamora, M.: Existence and uniqueness of a periodic solution to an indefinite attractive singular equation. Ann. Mat. Pura Appl. 4(195), 995–1009 (2016)

    Article  MathSciNet  Google Scholar 

  13. Hakl, R., Zamora, M.: Periodic solutions to second-order indefinite singular equations. J. Differ. Equ. 263, 451–469 (2017)

    Article  MathSciNet  Google Scholar 

  14. Han, W., Ren, J.: Some results on second-order neutral functional differential equations with infinite distributed delay. Nonlinear Anal. 70, 1393–1406 (2009)

    Article  MathSciNet  Google Scholar 

  15. Jiang, D., Chu, J., Zhang, M.: Multiplicity of positive periodic solutions to superlinear repulsive singular equations. J. Differ. Equ. 211, 282–302 (2005)

    Article  MathSciNet  Google Scholar 

  16. Kong, F., Lu, S.: Existence of positive periodic solutions of fourth-order singular p-Laplacian neutral functional differential equations. Filomat 31, 5855–5868 (2017)

    Article  MathSciNet  Google Scholar 

  17. Lazer, A., Solimini, S.: On periodic solutions of nonlinear differential equations with singularities. Proc. Am. Math. Soc. 99, 109–114 (1987)

    Article  MathSciNet  Google Scholar 

  18. Li, S., Wang, Y.: Multiplicity of positive periodic solutions to second order singular dynamical systems. Mediterr. J. Math. 14, 202 (2017)

    Article  MathSciNet  Google Scholar 

  19. Li, X., Zhang, Z.: Periodic solutions for damped differential equations with a weak repulsive singularity. Nonlinear Anal. TMA 70, 2395–2399 (2009)

    Article  MathSciNet  Google Scholar 

  20. Ma, R., Chen, R., He, Z.: Positive periodic solutions of second-order differential equations with weak singularities. Appl. Math. Comput. 232, 97–103 (2014)

    MathSciNet  MATH  Google Scholar 

  21. Del Pino, M., Manásevich, R.: Infinitely many \(T\)-periodic solutions for a problem arising in nonlinear elasticity. J. Differ. Equ. 103, 260–277 (1993)

    Article  MathSciNet  Google Scholar 

  22. Torres, P.: Weak singularities may help periodic solutions to exist. J. Differ. Equ. 232, 277–284 (2007)

    Article  MathSciNet  Google Scholar 

  23. Torres, P.: Mathematical Models with Singularities—A Zoo of Singular Creatures. Atlantis Briefs in Differential Equations. Atlantis Press, Paris (2015)

    Google Scholar 

  24. Torres, P.: Existence of one signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed theorem. J. Differ. Equ. 190, 643–662 (2003)

    Article  MathSciNet  Google Scholar 

  25. Wang, H.: Positive periodic solutions of singular systems with a parameter. J. Differ. Equ. 249, 2986–3002 (2010)

    Article  MathSciNet  Google Scholar 

  26. Wang, H.: Positive periodic solutions of singular systems of first order ordinary differential equations. Appl. Math. Comput. 218, 1605–1610 (2011)

    MathSciNet  MATH  Google Scholar 

  27. Wang, Y., Lian, H., Ge, W.: Periodic solutions for a second order nonlinear functional differential equation. Appl. Math. Lett. 20, 110–115 (2007)

    Article  MathSciNet  Google Scholar 

  28. Xin, Y., Cheng, Z.: Positive periodic solution to indefinite singular Liénard equation. Positivity 23, 779–787 (2019)

    Article  MathSciNet  Google Scholar 

  29. Xin, Y., Yao, S.: Positive periodic solution for p-Laplacian Rayleigh equation with weak and strong singularities of repulsive type. J. Fixed Point Theory Appl. 22, 1–9 (2020)

    Article  MathSciNet  Google Scholar 

  30. Zhang, M.: Periodic solutions of Liénard equations with singular forces of repulsive type. J. Math. Anal. Appl. 203, 254–269 (1996)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454000, China

    Zhibo Cheng & Xiaoxiao Cui

Authors
  1. Zhibo Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoxiao Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhibo Cheng.

Additional information

Communicated by Syakila Ahmad.

Publisher's Note

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

Research is supported by NSFC project (11501170), Technological Innovation Talents in Universities of Henan Province (21HASTIT025) and Fundamental Research Funds for the Universities of Henan Province (NSFRF170302).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cheng, Z., Cui, X. Positive Periodic Solution for a Second-Order Damped Singular Equation via Fixed Point Theorem in Cones. Bull. Malays. Math. Sci. Soc. 44, 2675–2691 (2021). https://doi.org/10.1007/s40840-021-01083-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-021-01083-1

Keywords

Mathematics Subject Classification

Search

Navigation