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

Abstract

Existence and $C^N$-smoothness of a local unstable manifold at $0$ are shownfor 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 foreach $k\in$ {$1,\ldots,N$}; $D(g|_{C^1([-h,0],\mathbb R^n)})(0)=0$;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 differentialequations with state-dependent delay.