Geodesic
A theorem of Rolewicz's type for measurable evolution families in Banach spaces
Electronic Journal of Differential Equations, Tome 2001 (2001).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let $\varphi$ be a positive and non-decreasing function defined on the real half-line and ${\cal U}$ 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 ${\cal U}$ satisfy a certain integral condition (see the relation ref0.1 from Theorem 1 below) then ${\cal U}$ 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.
Classification : 47A30, 93D05, 35B35, 35B40, 46A30
Keywords: evolution family of bounded linear operators, evolution operator semigroup, rolewicz's theorem, exponential stability 
@article{EJDE_2001__2001__a154,
     author = {Bu\c{s}e, Constantin and Dragomir, Sever S.},
     title = {A theorem of {Rolewicz's} type for measurable evolution families in {Banach} spaces},
     journal = {Electronic Journal of Differential Equations},
     publisher = {mathdoc},
     volume = {2001},
     year = {2001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2001__2001__a154/}
}
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Buşe, Constantin; Dragomir, Sever S. A theorem of Rolewicz's type for measurable evolution families in Banach spaces. Electronic Journal of Differential Equations, Tome 2001 (2001). http://geodesic.mathdoc.fr/item/EJDE_2001__2001__a154/