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Attractive singularity problems for superlinear Liénard equation

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Abstract

In this paper, we consider the following quasilinear Liénard equation with a singularity

$$\begin{aligned} (\phi _p(x'(t)))'+f(x(t))x'(t)+g(t,x(t))=e(t), \end{aligned}$$

where g has a attractive singularity at the origin and satisfies superlinear condition at \(x=+\infty \). By using Manásevich–Mawhin continuous theorem, we prove that this equation has at least one positive T-periodic solution. We solve a difficulty to estimate it a priori bounds of a periodic solution for quasilinear Liénard equation in the case that superlinear condition. At last, example and numerical solution (phase portrait and time series portrait of the positive periodic solution of example) are given to show applications of the theorem.

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Acknowledgements

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 Province (NSFRF170302).

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Authors and Affiliations

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

    Zhibo Cheng, Xiaoxiao Cui & Zhonghua Bi

  2. Department of Mathematics, Sichuan University, Chengdu, 610064, China

    Zhibo Cheng

Authors
  1. Zhibo Cheng
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  2. Xiaoxiao Cui
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  3. Zhonghua Bi
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Contributions

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

Corresponding author

Correspondence to Zhibo Cheng.

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The authors declare that they have no conflict of interest concerning the publication of this manuscript.

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Cheng, Z., Cui, X. & Bi, Z. Attractive singularity problems for superlinear Liénard equation. Positivity 23, 431–444 (2019). https://doi.org/10.1007/s11117-018-0615-0

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  • DOI: https://doi.org/10.1007/s11117-018-0615-0

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