On Positive Periodic Solutions for Nonlinear Delayed Differential Equations

We prove the existence of positive ω-periodic solutions for the delayed differential equation x′(t)=a(t)g(x(t))x(t)-λb(t)f(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}$$x^{\prime}(t) = a(t)g(x(t))x(t) - \lambda b(t)f(x(t - \tau (t))),$$\end{document}where λ is a positive parameter, a,b,τ∈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}$${a,b,\tau \in C(\mathbb{R},\mathbb{R})}$$\end{document} are ω-periodic functions with a,b≥0,a,b≢0,f,g∈C([0,∞),[0,∞))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a,b\geq 0,a,b \not \equiv 0,f,g\in C([0,\infty ),[0,\infty ))}$$\end{document}, g does not need to be bounded above or bounded away from 0, and g(0) = 0 is allowed.