, Volume 23, Issue 4, pp 779–787 | Cite as

Positive periodic solution to indefinite singular Liénard equation

  • Yun Xin
  • Zhibo ChengEmail author


In this paper, we investigate the existence of a positive periodic solution for the following Liénard equation with a indefinite singularity
$$\begin{aligned} x''+f(x)x'+\frac{b(t)}{x}=p(t), \end{aligned}$$
where \(b\in C({\mathbb {R}},{\mathbb {R}})\) is a T-periodic sign-changing function. The novelty of the present article is that for the first time we show that a indefinite singularity enables the achievement of a new existence criterion of positive periodic solutions through a application of a topological degree theorem by Mawhin. Recent results in the literature are generalized and significantly improved, and we give the existence interval of a positive periodic solution of this equation. At last, an example is given to show applications of the theorems.


Positive periodic solution Indefinite singularity Liénard equation 

Mathematical Subject Classification

34B16 34C25 



The authors would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by National Natural Science Foundation of China (11501170), China Postdoctoral Science Foundation funded Project (2016M590886), Fundamental Research Funds for the Universities of Henan Provience (NSFRF170302), Education Department of Henan Province Project (16B110006), Henan Polytechnic University Outstanding Youth Fund (J2015-02).

Author Contributions

ZC designed the research,YX and ZC wrote the main manuscript, ZC supervised the project. All authors revised the manuscript.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no competing interests concerning the publication of this manuscript.


  1. 1.
    Lazer, A., Solimini, S.: On periodic solutions of nonlinear differential equations with singularities. Proc. Am. Math. Soc. 99, 109–114 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cheng, Z., Ren, J.: Periodic and subharmonic solutions for Duffing equation with singularity. Discrete Contin. Dyn. Syst. A 32, 1557–1574 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cheng, Z., Ren, J.: Studies on a damped differential equation with repulsive singularity. Math. Methods Appl. Sci. 36, 983–992 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chu, J., Torres, P., Zhang, M.: Periodic solution of second order non-autonomous singular dynamical systems. J. Differ. Equ. 239, 196–212 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fonda, A., Manásevich, R., Zanolin, F.: Subharmonics solutions for some second-order differential equations with singularities. SIAM J. Math. Anal. 24, 1294–1311 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hakl, R., Torres, P.: On periodic solutions of second-order differential equations with attractive–repulsive singularities. J. Differ. Equ. 248, 111–126 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hakl, R., Zamora, M.: Periodic solutions to the Liénard type equations with phase attractive singularities. Bound. Value Probl. 2013:47, 20 (2013)zbMATHGoogle Scholar
  8. 8.
    Hakl, R., Zamora, M.: Periodic solutions to second-order indifinite singular equations. J. Differ. Equ. 263, 451–469 (2017)CrossRefzbMATHGoogle Scholar
  9. 9.
    Habets, P., Sanchez, L.: Periodic solution of some Liénard equations with singularities. Proc. Am. Math. Soc. 109, 1035–1044 (1990)zbMATHGoogle Scholar
  10. 10.
    Jiang, D., Chu, J., Zhang, M.: Multiplicity of positive periodic solutions to superlinear repulsive singular equations. J. Differ. Equ. 211, 282–302 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Li, S., Liao, F., Xing, W.: Periodic solutions for Liénard differential equations with singularities. Electron. J. Differ. Equ. 2015:151, 12 (2015)zbMATHGoogle Scholar
  12. 12.
    Lu, S., Wang, Y., Guo, Y.: Existence of periodic solutions of a Liénard equation with a singularity of repulsive type. Bound. Value Probl. 2017:95, 10 (2017)zbMATHGoogle Scholar
  13. 13.
    Lu, S.: A new result on the existence of periodic solutions for Liénard equations with a singularity of repulsive type. J. Inequal. Appl. 2017:37, 13 (2017)zbMATHGoogle Scholar
  14. 14.
    del Pino, M., Manásevich, R.: Infinitely many \(T\)-periodic solutions for a problem ariding in nonlinear elasticity. J. Differ. Equ. 103, 260–277 (1993)CrossRefzbMATHGoogle Scholar
  15. 15.
    Ren, J., Yuan, Q.: Bifurcations of a periodically forced microbial continuous culture model with restrained growth rate. Chaos 27, 083124 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ren, J., Li, X.: Bifurcations in a seasonally forced Predator–Prey model with generalized holling type IV functional response. Int. J. Bifurc. Chaos Appl. Sci. Eng. 26, 1650203 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Torres, P.: Weak singularities may help periodic solutions to exist. J. Differ. Equ. 232, 277–284 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Torres, P.: Mathematical Models with Singularities-A Zoo of Singular Creatures. Atlantis Press (2015)Google Scholar
  19. 19.
    Uren̆a, A.: Periodic solutions of singular equations. Topol. Methods Nonlinear Anal. 47, 55–72 (2016)MathSciNetGoogle Scholar
  20. 20.
    Wang, H.: Positive periodic solutions of singular systems with a parameter. J. Differ. Equ. 249, 2986–3002 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Wang, Z.: Periodic solutions of Liénard equation with a singularity and a deviating argument. Nonlinear Anal. RWA 16, 227–234 (2014)CrossRefzbMATHGoogle Scholar
  22. 22.
    Xia, J., Wang, Z.: Existence and multiplicity of periodic solutions for the Duffing equation with singularity. Proc. R. Soc. Edinb. Sect. A 137, 625–645 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Xin, Y., Cheng, Z.: Positive periodic solution of \(p\)-Laplacian Liénard type differential equation with singularity and deviating argument. Adv. Differ. Equ. 2016:41, 11 (2016)zbMATHGoogle Scholar
  24. 24.
    Zamora, M.: On a periodically forced Liénard differential equation with singular \(\phi \)-Laplacian. Bull. Math. Soc. Sci. Math. Roum. 105, 327–336 (2014)zbMATHGoogle Scholar
  25. 25.
    Zhang, M.: Periodic solutions of Liénard equation singular forces of repusive type. J. Math. Anal. Appl. 203, 254–269 (1996)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mawhin, J.: Topological degree and boundary value problems for nonlinear differental equations. Topol. Methods Ordinary Differ. Equ. 1537, 74–142 (1993)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.College of Computer Science and TechnologyHenan Polytechnic UniversityJiaozuoChina
  2. 2.Department of MathematicsSichuan UniversityChengduChina
  3. 3.School of Mathematics and Information ScienceHenan Polytechnic UniversityJiaozuoChina

Personalised recommendations