A theorem of Rolewicz's type for measurable evolution families in Banach spaces
Let $varphi$ be a positive and non-decreasing function defined on the real half-line and ${mathcal U}$ be a strongly measurable, exponentially bounded evolution family of bounded linear operators acting on a Banach space and satisfing a certain measurability condition as in Theorem 1 below. We prove...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Texas State University
2001-11-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2001/70/abstr.html |
| Summary: | Let $varphi$ be a positive and non-decreasing function defined on the real half-line and ${mathcal U}$ be a strongly measurable, exponentially bounded evolution family of bounded linear operators acting on a Banach space and satisfing a certain measurability condition as in Theorem 1 below. We prove that if $varphi$ and ${mathcal U}$ satisfy a certain integral condition (see the relation ef{0.1} from Theorem 1 below) then ${mathcal U}$ is uniformly exponentially stable. For $varphi$ continuous and $mathcal U$ strongly continuous and exponentially bounded, this result is due to Rolewicz. The proofs uses the relatively recent techniques involving evolution semigroup theory. |
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| ISSN: | 1072-6691 |