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On the problem of linearization for state-dependent delay differential equations. (English) Zbl 0844.34075
The purpose of the paper is twofold: First of all the authors want to present a local linearization \(\dot x(t) = Lx_t\) at the equilibrium 0 of the general delay differential equation of the form \((*)\) \(\dot x(t) = f(x_t, \int^0_{-r} d_\eta (s) g(x_t (- \tau (x_t) + s))\), \(t \geq 0\), where \(f(0, \int^0_{-r} d_\eta (s) g(0)) = 0\) and the delay \(\tau\) is a continuous function of \(x_t\). Second, they show the following stability results:
i) If \(\sup \{\text{Re} \lambda : \text{det} (\lambda I - Le^{\lambda \cdot}) = 0\} = - \alpha < 0\), then for each small \(\varepsilon > 0\), there is a \(K > 0\) and a neighborhood \(V\) of the origin of the phase-space such that for \(\varphi \in V\) the solution \(x_t (\varphi)\) of \((*)\) is defined for \(t \geq 0\) and it satisfies \(|x_t (\varphi) |\leq Ke^{- (\alpha - \varepsilon) t} |\varphi |\).
ii) If \(\sup \{\text{Re} \lambda : \text{det} (\lambda I - Le^{\lambda \cdot}) = 0\} = \alpha > 0\), then the zero solution of \((*)\) is unstable.

MSC:
34K20 Stability theory of functional-differential equations
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