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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 $\sigma: \Theta\times\bbfR_+\to\Theta$ is called a semiflow if it is continuous and such that $$\sigma(\theta, 0)= \theta\text{ for }\theta\in\Theta,\tag1$$ $$\sigma(\theta, t+s)= \sigma(\sigma(\theta, s),t)\text{ for all }t,s\ge 0\text{ and }\theta\in \Theta.\tag2$$ Let $\sigma$ be a semiflow on $\Theta$ and let $\phi:\Theta\times\bbfR_+\to B(X)$ be a continuous mapping satisfying some extra conditions (for example, $\phi(\theta, 0)= I$ -- the identity operator on $X$; $\phi(\theta, t+s)= \phi(\sigma(\theta, t), s)\phi(\theta, t)$ for $t,s\ge 0$ and $\theta\in\Theta)$. The pair $\pi= (\phi,\sigma)$ is called linear strongly continuous skew-product semiflow on $X\times\Theta$. A strongly continuous skew-product semiflow $\pi= (\phi,\sigma)$ is called uniformly exponentially stable if there exist $N> 0$ and $\nu> 0$ such that $\Vert\phi(\theta, t)\Vert\le N\exp(-\nu t)$ for $t\ge 0$ and $\theta\in\Theta$. Moreover, the authors used the concept that a pair $(E,F)$ 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 $(E,F)$ 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:
34G10Linear ODE in abstract spaces
34D20Stability of ODE
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References:
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