Attraction property of local center-unstable manifolds for differential equations with state-dependent delay

In the present paper we consider local center-unstable manifolds at a stationary point for a class of functional differential equations of the form $\dot{x}(t)=f(x_{t})$ under assumptions that are designed for application to differential equations with state-dependent delay. Here, we show an attraction property of these manifolds. More precisely, we prove that, after fixing some local center-unstable manifold $W_{cu}$ of $\dot{x}(t)=f(x_{t})$ at some stationary point $\varphi$, each solution of $\dot{x}(t)=f(x_{t})$ which exists and remains sufficiently close to $\varphi$ for all $t\geq 0$ and which does not belong to $W_{cu}$ converges exponentially for $t\to\infty$ to a solution on $W_{cu}$.