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Nondegeneracy and Uniqueness of Periodic Solution for a Neutral Differential Equation

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Abstract

We analyze the nondegeneracy of second-order linear neutral differential equation

$$\begin{aligned} (x(t)-cx(t-\tau ))''=a(t)x(t), \end{aligned}$$

where c is a constant. By applications of the nondegeneracy of this linear neutral equation and an extension of Mawhin’s continuation theorem, we obtain existence and uniqueness of periodic solution for the prescribed second-order neutral differential equations. At last, we give two examples to show the applications of the theorems.

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Acknowledgements

The authors would like to thank the referee for invaluable comments and insightful suggestions.

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Authors and Affiliations

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

    Zhibo Cheng

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

    Zhibo Cheng

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  1. Zhibo Cheng
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Correspondence to Zhibo Cheng.

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Research is supported by NSFC Project (No. 11501170), Postdoctoral fund in China (2016M590886), Young backbone teachers of colleges and universities in Henan Province (2017GGJS057) and Fundamental Research Funds for the Universities of Henan Provience (NSFRF170302).

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Cheng, Z. Nondegeneracy and Uniqueness of Periodic Solution for a Neutral Differential Equation. Qual. Theory Dyn. Syst. 19, 92 (2020). https://doi.org/10.1007/s12346-020-00429-7

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  • DOI: https://doi.org/10.1007/s12346-020-00429-7

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