A new proof for a Rolewicz's type theorem: An evolution semigroup approach (Q2723232)
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scientific article; zbMATH DE number 1614312
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A new proof for a Rolewicz's type theorem: An evolution semigroup approach |
scientific article; zbMATH DE number 1614312 |
Statements
9 July 2001
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bounded linear operator
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evolution family
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evolution operator semigroup
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strongly continuous
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exponentially bounded
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uniformly exponentially stable
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0.8950769
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0.8777445
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0.8718734
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0.8686722
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0.86768764
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0.8648089
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0.86118793
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A new proof for a Rolewicz's type theorem: An evolution semigroup approach (English)
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Let \(\varphi\) be a positive and non-decreasing function on \([0,\infty)\) and \({\mathcal U}=\{U(t,s): t\geq s\geq 0\}\) be a strongly continuous and exponentially bounded evolution family of bounded linear operators acting on a Banach space \(X\). If for all \(x\in X\), \(\|x\|\leq 1\), NEWLINE\[NEWLINE\sup_{s\geq 0}\int^\infty_s\varphi(\|U(t,s)x\|) dt =M_\varphi<\infty,NEWLINE\]NEWLINE then \({\mathcal U}\) is uniformly exponentially stable. When \(\varphi\) is continuous, this result is due to S. Rolewicz.
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