Positive Periodic Solutions for a Kind of Second-Order Neutral Differential Equations with Variable Coefficient and Delay

In this article, we discuss a type of second-order neutral differential equations with variable coefficient and delay: (x(t)-c(t)x(t-τ(t)))′′+a(t)x(t)=f(t,x(t-δ(t))),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (x(t)-c(t)x(t-\tau (t)))''+a(t)x(t)=f(t,x(t-\delta (t))), \end{aligned}$$\end{document}where c(t)∈C(R,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c(t)\in C({\mathbb {R}},{\mathbb {R}})$$\end{document} and |c(t)|≠1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|c(t)|\ne 1$$\end{document}. By employing Krasnoselskii’s fixed-point theorem and properties of the neutral operator (Ax)(t):=x(t)-c(t)x(t-τ(t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(Ax)(t):=x(t)-c(t)x(t-\tau (t))$$\end{document}, some sufficient conditions for the existence of periodic solutions are established.