Global stability for price models with delay

This is a joint work with Tibor Krisztin. We consider the delay differential equation \begin{align} \dot x(t)=a\int_0^r x(t-s)\,d\eta(s)-g(x(t))\label{eq:balazs:1} \end{align} and the neutral differential equation \begin{align} \dot y(t)=a\int_0^r \dot y(t-s)\,d\mu(s)-g(y(t)),\label{eq:balazs:2} \end{align} where $a>0$, $ug(u)>0$ for all $u\in\mathbb{R}\setminus\{0\}$, and some further conditions hold. Both equations can be interpreted as price models. Global asymptotic stability of $y=0$ is obtained, in case $a\in(0,1)$, for \eqref{eq:balazs:2} by using a Lyapunov functional. Then this result is applied to get global asymptotic stability of $x=0$ for \eqref{eq:balazs:1} provided $a\in (0,1)$. As particular cases, two related global stability conjectures are solved, with an affirmative answer.