Periodic Solutions to a Kind of Second Order Neutral Functional Differential Equation in the Critical Case

In this paper, the authors consider the existence of periodic solutions for a kind of second neutral functional differential equation as follows:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ {\left( {x{\left( t \right)} - cx{\left( {t - \tau } \right)}} \right)}^{{\prime \prime }} = g{\left( {t,x{\left( {t - \mu {\left( t \right)}} \right)}} \right)} + e{\left( t \right)}, $$\end{document} in the critical case |c| = 1. By employing Mawhin's continuation theorem and some analysis techniques, some new results are obtained.