Existence of strictly decreasing positive solutions of linear differential equations of neutral type

Abstract

The paper is concerned with a linear neutral differential equation \begin{document}$ \dot y(t) = -c(t)y(t-\tau(t))+d(t)\dot y(t-\delta(t)) $\end{document} where \begin{document}$ c\colon [t_0,\infty)\to (0,\infty) $\end{document} , \begin{document}$ d\colon [t_0,\infty)\to [0,\infty) $\end{document} , \begin{document}$ t_0\in {\Bbb{R}} $\end{document} and \begin{document}$ \tau, \delta \colon [t_0,\infty)\to (0,r] $\end{document} , \begin{document}$ r\in{\mathbb{R}} $\end{document} , \begin{document}$ r>0 $\end{document} are continuous functions. A new criterion is given for the existence of positive strictly decreasing solutions. The proof is based on the Rybakowski variant of a topological Wazewski principle suitable for differential equations of the delayed type. Unlike in the previous investigations known, this time the progress is achieved by using a special system of initial functions satisfying a so-called sewing condition. The result obtained is extended to more general equations. Comparisons with known results are given as well.