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

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, access via your institution.

References

  1. 1.

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

    MathSciNet  Article  Google Scholar 

  2. 2.

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

    MathSciNet  Article  Google Scholar 

  3. 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)

    MathSciNet  Article  Google Scholar 

  4. 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)

    MathSciNet  Article  Google Scholar 

  5. 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. 6.

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

    MathSciNet  Article  Google Scholar 

  7. 7.

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

    MathSciNet  Article  Google Scholar 

  8. 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)

    MathSciNet  Article  Google Scholar 

  9. 9.

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

    Google Scholar 

  10. 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)

    MathSciNet  Article  Google Scholar 

  11. 11.

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

    MathSciNet  Article  Google Scholar 

  12. 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)

    MathSciNet  Article  Google Scholar 

  13. 13.

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

    MathSciNet  Article  Google Scholar 

  14. 14.

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

    MathSciNet  Article  Google Scholar 

  15. 15.

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

    MathSciNet  Article  Google Scholar 

  16. 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)

    MathSciNet  Article  Google Scholar 

  17. 17.

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

    MathSciNet  Article  Google Scholar 

  18. 18.

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

    MathSciNet  Article  Google Scholar 

  19. 19.

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

    MathSciNet  Article  Google Scholar 

  20. 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. 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)

    MathSciNet  Article  Google Scholar 

  22. 22.

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

    MathSciNet  Article  Google Scholar 

  23. 23.

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

    Google Scholar 

  24. 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)

    MathSciNet  Article  Google Scholar 

  25. 25.

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

    MathSciNet  Article  Google Scholar 

  26. 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. 27.

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

    MathSciNet  Article  Google Scholar 

  28. 28.

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

    MathSciNet  Article  Google Scholar 

  29. 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)

    MathSciNet  Article  Google Scholar 

  30. 30.

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

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Zhibo Cheng.

Additional information

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).

Communicated by Syakila Ahmad.

Rights and permissions

Reprints and Permissions

About this article

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. (2021). https://doi.org/10.1007/s40840-021-01083-1

Download citation

Keywords

  • Periodic solution
  • Singular equation
  • Fixed point theorem in cones
  • External force

Mathematics Subject Classification

  • 34B16
  • 34B18
  • 34C25