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Schäffer spaces and uniform exponential stability of linear skew-product semiflows. (English) Zbl 1081.34056

Let ${\Theta }$ be a locally compact metric space and $X$ a Banach space. A mapping $\sigma :{\Theta }×{ℝ}_{+}\to {\Theta }$ is called a semiflow if it is continuous and such that

$\sigma \left(\theta ,0\right)=\theta \phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\theta \in {\Theta },\phantom{\rule{2.em}{0ex}}\left(1\right)$
$\sigma \left(\theta ,t+s\right)=\sigma \left(\sigma \left(\theta ,s\right),t\right)\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}t,s\ge 0\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\theta \in {\Theta }·\phantom{\rule{2.em}{0ex}}\left(2\right)$

Let $\sigma$ be a semiflow on ${\Theta }$ and let $\varphi :{\Theta }×{ℝ}_{+}\to B\left(X\right)$ be a continuous mapping satisfying some extra conditions (for example, $\varphi \left(\theta ,0\right)=I$ – the identity operator on $X$; $\varphi \left(\theta ,t+s\right)=\varphi \left(\sigma \left(\theta ,t\right),s\right)\varphi \left(\theta ,t\right)$ for $t,s\ge 0$ and $\theta \in {\Theta }\right)$. The pair $\pi =\left(\varphi ,\sigma \right)$ is called linear strongly continuous skew-product semiflow on $X×{\Theta }$. A strongly continuous skew-product semiflow $\pi =\left(\varphi ,\sigma \right)$ is called uniformly exponentially stable if there exist $N>0$ and $\nu >0$ such that $\parallel \varphi \left(\theta ,t\right)\parallel \le Nexp\left(-\nu t\right)$ for $t\ge 0$ and $\theta \in {\Theta }$.

Moreover, the authors used the concept that a pair $\left(E,F\right)$ of Schäffer spaces is admissible to $\pi$. The main result of the paper asserts that $\pi$ is uniformly exponentially stable if and only if there exists a pair $\left(E,F\right)$ of Schäffer spaces admissible to $\pi$ and satisfying some additional condition. This nice paper contains also some other results and interesting examples explaining the basic concepts and illustrating the obtained results.

##### MSC:
 34G10 Linear ODE in abstract spaces 34D20 Stability of ODE