American Institute of Mathematical Sciences

A nonlinear cyclic system with delay and the overall negative feedback is considered. The characteristic equation of the linearized system is studied in detail. Sufficient conditions for the oscillation of all solutions and for the existence of monotone solutions are derived in terms of roots of the characteristic equation.

The long-time behavior of positive solutions of a differential equation with state-dependent delay \begin{document}$ \dot{y}(t) = -c(t)y(t-\tau(t,y(t))) $\end{document}, where \begin{document}$ c $\end{document} is a positive coefficient, is considered. Sufficient conditions are given for the existence of positive solutions bounded from below and from above by functions of exponential type. As a consequence, criteria for the existence of positive solutions are derived and their lower bounds are given. Relationships are discussed with the existing results on the existence of positive solutions for delayed differential equations.

We consider a real-valued differential equation

\begin{document}$ \begin{equation*} x'(t) = f(x(t - d(x_t))), \end{equation*} $\end{document}

with strictly monotonic negative feedback and state-dependent delay, that has a nontrivial periodic solution \begin{document}$ q $\end{document} for which the planar map \begin{document}$ q_t \mapsto (q(t),q(t - d(q_t))) $\end{document} is not injective on the orbit of \begin{document}$ q $\end{document} in phase space. This solution demonstrates that Mallet-Paret and Sell's version of the Poincaré-Bendixson theorem for delay equations with constant delay and monotonic feedback does not carry over entirely to the state-dependent delay case.

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 Ważewski 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.

We present generalised Lyapunov-Razumikhin techniques for establishing global asymptotic stability of steady-state solutions of scalar delay differential equations. When global asymptotic stability cannot be established, the technique can be used to derive bounds on the persistent dynamics. The method is applicable to constant and variable delay problems, and we illustrate the method by applying it to the state-dependent delay differential equation known as the sawtooth equation, to find parameter regions for which the steady-state solution is globally asymptotically stable. We also establish bounds on the periodic orbits that arise when the steady-state is unstable. This technique can be readily extended to apply to other scalar delay differential equations with negative feedback.