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

Abstract

The periodic problem is studied in this paper for the neutral Lienard 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.