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# On the asymptotic behavior of an exponentially bounded, strongly continuous cocycle over a semiflow

• Let $\pi = (\Phi, \sigma)$ be an exponentially bounded, strongly continuous cocycle over a continuous semiflow $\sigma$. We prove that $\pi = (\Phi, \sigma)$ is uniformly exponentially stable if and only if there exist $T>0$ and $c \in(0,1)$, such that for each $\theta \in \Theta$ and $x \in X$ there exists $\tau_{\theta,x} \in (0,T]$ with the property that

$||\Phi(\theta, \tau_{\theta,x})x|| \leq c||x||.$

As a consequence of the above result we obtain generalizations, in both continuous-time and discrete-time, of the the well-known theorems of Datko-Pazy, Rolewicz and Zabczyk for an exponentially bounded, strongly continuous cocycle over a semiflow $\sigma$. A version of the above theorems for the case of the exponential instability is also obtained.

Mathematics Subject Classification: 34D05, 47D06.

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