Existence of positive periodic solutions for a neutral Liénard equation with a singularity of repulsive type

  • Shiping LuEmail author
  • Xingchen Yu


The periodic problem is studied in this paper for the neutral Liénard equation with a singularity of repulsive type
$$\begin{aligned} (x(t)-cx(t-\sigma ))''+f(x(t))x'(t)+\varphi (t)x(t-\tau )-\frac{r(t)}{x^{\mu }(t)}=h(t), \end{aligned}$$
where \(f:[0,+\infty )\rightarrow R\) is continuous, \(r: R\rightarrow (0,+\infty )\) and \(\varphi :R \rightarrow R\) are continuous with T-periodicity in the t variable, \(c,\mu ,\sigma ,\tau \) are constants with \(|c|>1,\mu >1,0<\sigma ,\tau <T\). Many authors obtained the existence of periodic solutions under the condition \(|c|<1\) , and we extend their results to the case of \(|c|>1\). The proof of the main result relies on a continuation theorem of coincidence degree theory established by Mawhin.


Neutral functional differential equation periodic solution singularity continuation theorem 

Mathematics Subject Classification

34K13 34B16 34C25 



The authors are grateful to the referee for the careful reading of the paper and for useful suggestions. The authors gratefully acknowledge support from NSF of China (no. 11271197).


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsNanjing University of Information Science and TechnologyNanjingPeople’s Republic of China

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