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July  2003, 9(4): 993-1028. doi: 10.3934/dcds.2003.9.993

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

Tibor Krisztin 1,

1. 

Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary

Received  March 2002 Revised  November 2002 Published  April 2003

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.
Keywords: variation-of-constans formula, Functional differential equation, substitution operator., state-dependent delay, Lyapunov–Perron method, unstable manifold.
Mathematics Subject Classification: Primary: 34K19; Secondary: 37D1.
Citation: Tibor Krisztin. A local unstable manifold for differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 993-1028. doi: 10.3934/dcds.2003.9.993
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