A local unstable manifold for differential equations with state-dependent delay

Existence and $C^N$-smoothness of a local unstable manifold at $0$ are shown for the delay differential equation $\dot x(t)=F(x_t)$ with $F:C([-h,0],\mathbb R^n)\to \mathbb R^n$, $h>0$, $F(0)=0$, under the hypotheses: There exist a linear continuous $L$ and a continuous $g$ with $F=L+g$; $0$ is a hyperbolic equilibrium of $\dot y(t)=Ly_t$; the restriction $g|_{C^k([-h,0],\mathbb R^n)}:C^k([-h,0],\mathbb R^n)\to \mathbb R^n$ is $C^k$-smooth for each $k\in$ {$1,\ldots,N$}; $D(g|_{C^1([-h,0],\mathbb R^n)})(0)=0$; in addition, for the derivatives $D^k(g|_{C^k([-h,0],\mathbb R^n)}) $, $k\in${$1,\ldots,N$}, certain extension properties hold. The conditions on $F$ are motivated and are satisfied by a wide class of differential equations with state-dependent delay.