Abstract
A proof of a sufficient condition for a strongly continuous semigroup $\{ T(t)\}_{t\ge 0}$ on a Banach space $X$ to be uniformly exponentially stable is given. This result is a simplification of an earlier theorem by van Neerven, and concludes that a semigroup is uniformly exponentially stable provided $\sup\nolimits_{||x||\le 1}J(||T(\cdot)x||)<\infty$ ; here $J$ is a certain nonlinear functional with certain natural properties. A non-autonomous version of this theorem for evolution families is also given. This implies the well-known Datko-Pazy and Rolewicz Theorems. This result is connected to the uniform asymptotic stability of the well-posed linear and non-autonomous abstract Cauchy problem \begin{equation*} \left\{ \begin{array}{rcl} \dot{u}(t)& = & A(t)u(t),\quad t\geq s\geq 0, \\ u(s) & = & x\in X. \end{array} \right. \end{equation*}
Citation
N.S. Barnett. C. Buşe. P. Cerone. S.S. Dragomir. "Integral Characterizations For Exponential Stability Of Semigroups And Evolution Families On Banach Spaces." Bull. Belg. Math. Soc. Simon Stevin 13 (2) 345 - 353, June 2006. https://doi.org/10.36045/bbms/1148059469
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