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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.

34G10Linear ODE in abstract spaces
34D20Stability of ODE
Full Text: DOI
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