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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
##### Keywords:
delay differential equation; stability
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##### References:
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