Geodesic
A new proof for a Rolewicz's type theorem: An evolution semigroup approach
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 continuous and exponentially bounded evolution family of bounded linear operators acting on a Banach space. We prove that if $\varphi$ and $\cal U$ satisfy a certain integral condition (see the relation (2) below) then $\cal U$ is uniformly exponentially stable. For $\varphi$ continuous, this result is due to S. Rolewicz.
Classification : 47A30, 93D05, 35B35, 35B40, 46A30
Keywords: evolution family of bounded linear operators, evolution operator semigroup, rolewicz's theorem 
@article{EJDE_2001__2001__a44,
     author = {Bu\c{s}e,  C. and Dragomir,  S.S.},
     title = {A new proof for a {Rolewicz's} type theorem: {An} evolution semigroup approach},
     journal = {Electronic journal of differential equations},
     year = {2001},
     volume = {2001},
     zbl = {0985.47031},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2001__2001__a44/}
}
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Buşe,  C.; Dragomir,  S.S. A new proof for a Rolewicz's type theorem: An evolution semigroup approach. Electronic journal of differential equations, Tome 2001 (2001). http://geodesic.mathdoc.fr/item/EJDE_2001__2001__a44/