Geodesic
A theorem of Rolewicz's type for measurable evolution families in Banach spaces
Electronic journal of differential equations, Tome 2001 (2001)
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},
     year = {2001},
     volume = {2001},
     zbl = {0991.47024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2001__2001__a154/}
}
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%A Dragomir,  Sever S.
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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/