Stability of positive steady-state solutions to a time-delayed system with some applications

<p style='text-indent:20px;'>In this paper, we study a general nonlinear retarded system:</p><p style='text-indent:20px;'><disp-formula> <label>1</label> <tex-math id="E1"> \begin{document}$ \begin{equation} y'(t) = a(t)F(y(t),y(t-\tau)), \; \; t\geq 0, \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \tau&gt;0 $\end{document}</tex-math></inline-formula> is a constant, <inline-formula><tex-math id="M2">\begin{document}$ a(t) $\end{document}</tex-math></inline-formula> is a positive value function defined on <inline-formula><tex-math id="M3">\begin{document}$ [0,\infty) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ F(y,z) $\end{document}</tex-math></inline-formula> is continuous in <inline-formula><tex-math id="M5">\begin{document}$ \mathscr{D} = \mathbb{R}_+^2 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{R_+} = (0,+\infty) $\end{document}</tex-math></inline-formula>. Sufficient conditions for stability of the unique positive equilibrium are established. Our results show that if <inline-formula><tex-math id="M7">\begin{document}$ F_z(y,z)&gt;0 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M8">\begin{document}$ y,z\in \mathbb{R_+} $\end{document}</tex-math></inline-formula>, then the unique positive equilibrium of (1) which denoted by <inline-formula><tex-math id="M9">\begin{document}$ \bar{y} $\end{document}</tex-math></inline-formula> is globally stable for any positive initial value and all <inline-formula><tex-math id="M10">\begin{document}$ \tau&gt;0 $\end{document}</tex-math></inline-formula>; if <inline-formula><tex-math id="M11">\begin{document}$ F(y,z) $\end{document}</tex-math></inline-formula> is decreasing in <inline-formula><tex-math id="M12">\begin{document}$ y $\end{document}</tex-math></inline-formula>, then <inline-formula><tex-math id="M13">\begin{document}$ \bar{y} $\end{document}</tex-math></inline-formula> is globally stable for small <inline-formula><tex-math id="M14">\begin{document}$ \tau $\end{document}</tex-math></inline-formula>. Some applications are given.</p>