September  2009, 8(5): 1637-1645. doi: 10.3934/cpaa.2009.8.1637

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

1. 

Department of Mathematics, University of California, Los Angeles, CA 90095

2. 

Department of Mathematics, West University of Timişoara, Timişoara 300223, Romania, Romania

Received  September 2008 Revised  January 2009 Published  April 2009

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.

Citation: Ciprian Preda, Petre Preda, Adriana Petre. On the asymptotic behavior of an exponentially bounded, strongly continuous cocycle over a semiflow. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1637-1645. doi: 10.3934/cpaa.2009.8.1637
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