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

In this paper, we study a general nonlinear retarded system:1 \begin{document}$ \begin{equation} y'(t) = a(t)F(y(t),y(t-\tau)), \; \; t\geq 0, \end{equation} $\end{document}where \begin{document}$ \tau>0 $\end{document} is a constant, \begin{document}$ a(t) $\end{document} is a positive value function defined on \begin{document}$ [0,\infty) $\end{document} , \begin{document}$ F(y,z) $\end{document} is continuous in \begin{document}$ \mathscr{D} = \mathbb{R}_+^2 $\end{document} , where \begin{document}$ \mathbb{R_+} = (0,+\infty) $\end{document} . Sufficient conditions for stability of the unique positive equilibrium are established. Our results show that if \begin{document}$ F_z(y,z)>0 $\end{document} for \begin{document}$ y,z\in \mathbb{R_+} $\end{document} , then the unique positive equilibrium of (1) which denoted by \begin{document}$ \bar{y} $\end{document} is globally stable for any positive initial value and all \begin{document}$ \tau>0 $\end{document} ; if \begin{document}$ F(y,z) $\end{document} is decreasing in \begin{document}$ y $\end{document} , then \begin{document}$ \bar{y} $\end{document} is globally stable for small \begin{document}$ \tau $\end{document} . Some applications are given.