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Positive periodic solution to indefinite singular Liénard equation

  • Yun Xin
  • Zhibo Cheng


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.


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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

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