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On the asymptotic behavior of an exponentially bounded, strongly continuous cocycle over a semiflow. (English) Zbl 1188.37017

A pair \(\pi=\left( \Phi,\sigma\right) \) is said to be an exponentially bounded, strongly continuous cocycle over a semiflow \(\sigma\) if \(\Phi :\theta\times\mathbb{R}_{+}\rightarrow B\left( X\right) \) has the following properties: (i) \(\Phi\left( \theta,0\right) =I,\) for all \(\theta\in\Theta,\) \(t\in\mathbb{R}_{+},\) where \(I\) is the identity operator in \(X;\) (ii) \(\Phi\left( \theta,\cdot\right) x\) is continuous for any \(\theta\in\Theta,\) \(x\in X;\) (iii) \(\Phi\left( \theta,t+s\right) =\Phi\left( \sigma\left( \theta,t\right) s\right) \Phi\left( \theta,t\right) ,\) for all \(t,s\geq0\) and \(\theta\in\Theta;\) (iv) there exist constants \(M\) and \(\omega\) such that \(\left\| \Phi\left( \theta,t\right) \right\| \leq Me^{\omega t},\) for all \(t\geq0\) and \(\theta\in\Theta.\)
The authors prove that an exponentially bounded, strongly continuous cocycle \(\pi=\left( \Phi,\sigma\right) \) over a semiflow \(\sigma\) is uniformly exponentially stable if and only if there exist a \(T>0\) and a \(c\in\left( 0,1\right) \) such that, for each \(\theta\in\Theta\) and for each \(x\in X,\) there exists a \(\tau_{\theta,x}\in\left( 0,T\right] \) with the property \(\left\| \Phi\left( \theta,\tau_{\theta,x}\right) x\right\| \leq c\left\| x\right\| .\) As a consequence of this main result, generalizations of theorems due to Datko-Pazy, Rolewicz and Zabczyk for exponentially bounded, strongly continuous cocycles over a semiflow \(\sigma\) are derived for continuous and discrete cases. Similar extensions are obtained in the case of exponential instability.

MSC:

37C10 Dynamics induced by flows and semiflows
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
47D06 One-parameter semigroups and linear evolution equations
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