# American Institute of Mathematical Sciences

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 and Applied Analysis, 2009, 8 (5) : 1637-1645. doi: 10.3934/cpaa.2009.8.1637
 [1] Mateusz Krukowski. Arzelà-Ascoli's theorem in uniform spaces. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 283-294. doi: 10.3934/dcdsb.2018020 [2] Zhihua Liu, Pierre Magal. Functional differential equation with infinite delay in a space of exponentially bounded and uniformly continuous functions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (6) : 2271-2292. doi: 10.3934/dcdsb.2019227 [3] Sigurdur Freyr Hafstein. A constructive converse Lyapunov theorem on exponential stability. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 657-678. doi: 10.3934/dcds.2004.10.657 [4] Jacques Féjoz. On "Arnold's theorem" on the stability of the solar system. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3555-3565. doi: 10.3934/dcds.2013.33.3555 [5] Fatiha Alabau-Boussouira, Piermarco Cannarsa. A constructive proof of Gibson's stability theorem. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 611-617. doi: 10.3934/dcdss.2013.6.611 [6] C.P. Walkden. Solutions to the twisted cocycle equation over hyperbolic systems. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 935-946. doi: 10.3934/dcds.2000.6.935 [7] Samir EL Mourchid. On a hypercylicity criterion for strongly continuous semigroups. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 271-275. doi: 10.3934/dcds.2005.13.271 [8] Pedro Roberto de Lima, Hugo D. Fernández Sare. General condition for exponential stability of thermoelastic Bresse systems with Cattaneo's law. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3575-3596. doi: 10.3934/cpaa.2020156 [9] Antoine Perasso. Global stability and uniform persistence for an infection load-structured SI model with exponential growth velocity. Communications on Pure and Applied Analysis, 2019, 18 (1) : 15-32. doi: 10.3934/cpaa.2019002 [10] Gilbert Peralta. Uniform exponential stability of a fluid-plate interaction model due to thermal effects. Evolution Equations and Control Theory, 2020, 9 (1) : 39-60. doi: 10.3934/eect.2020016 [11] Artur Avila, Carlos Matheus, Jean-Christophe Yoccoz. The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces. Journal of Modern Dynamics, 2019, 14: 21-54. doi: 10.3934/jmd.2019002 [12] Angela A. Albanese, Xavier Barrachina, Elisabetta M. Mangino, Alfredo Peris. Distributional chaos for strongly continuous semigroups of operators. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2069-2082. doi: 10.3934/cpaa.2013.12.2069 [13] John Hubbard, Yulij Ilyashenko. A proof of Kolmogorov's theorem. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 367-385. doi: 10.3934/dcds.2004.10.367 [14] Rabah Amir, Igor V. Evstigneev. On Zermelo's theorem. Journal of Dynamics and Games, 2017, 4 (3) : 191-194. doi: 10.3934/jdg.2017011 [15] Jinxin Xue. Continuous averaging proof of the Nekhoroshev theorem. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3827-3855. doi: 10.3934/dcds.2015.35.3827 [16] Najwa Najib, Norfifah Bachok, Norihan Md Arifin, Fadzilah Md Ali. Stability analysis of stagnation point flow in nanofluid over stretching/shrinking sheet with slip effect using buongiorno's model. Numerical Algebra, Control and Optimization, 2019, 9 (4) : 423-431. doi: 10.3934/naco.2019041 [17] Stefan Meyer, Mathias Wilke. Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces. Evolution Equations and Control Theory, 2013, 2 (2) : 365-378. doi: 10.3934/eect.2013.2.365 [18] Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623 [19] Karim Boulabiar, Gerard Buskes and Gleb Sirotkin. A strongly diagonal power of algebraic order bounded disjointness preserving operators. Electronic Research Announcements, 2003, 9: 94-98. [20] Daniel Coronel, Andrés Navas, Mario Ponce. On bounded cocycles of isometries over minimal dynamics. Journal of Modern Dynamics, 2013, 7 (1) : 45-74. doi: 10.3934/jmd.2013.7.45

2020 Impact Factor: 1.916