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Asymptotic stability of nonuniform behaviour


Authors: Davor Dragičević and Weinian Zhang
Journal: Proc. Amer. Math. Soc. 147 (2019), 2437-2451
MSC (2010): Primary 34D09; Secondary 37D25
DOI: https://doi.org/10.1090/proc/14444
Published electronically: March 7, 2019
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Abstract: This paper is devoted to exponential dichotomies of nonautono-
mous difference equations. Under the assumptions that $ (A_m)_{m\in \mathbb{Z}}$ is a sequence of bounded operators acting on an arbitrary Banach space $ X$ that admits a uniform exponential dichotomy and that $ (B_m)_{m\in \mathbb{Z}}$ is a sequence of compact operators such that $ \lim _{\lvert m\rvert \to \infty } \lVert B_m\rVert =0$, D. Henry proved that either the sequence $ (A_m+B_m)_{m\in \mathbb{Z}}$ admits a uniform exponential dichotomy or there exists a bounded nonzero sequence $ (x_m)_{m\in \mathbb{Z}}\subset X$ such that $ x_{m+1}=(A_m+B_m)x_m$ for each $ m\in \mathbb{Z}$. In this paper we prove Henry's result in the setting of nonuniform exponential dichotomies. Then we obtain a result on roughness of the nonuniform exponential dichotomy and give stability of Lyapunov exponents. In addition, we establish corresponding results for dynamics with continuous time.


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Additional Information

Davor Dragičević
Affiliation: Department of Mathematics, University of Rijeka, 51000, Rijeka, Croatia
Email: ddragicevic@math.uniri.hr

Weinian Zhang
Affiliation: School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, People’s Republic of China
Email: matzwn@126.com

DOI: https://doi.org/10.1090/proc/14444
Received by editor(s): June 5, 2018
Published electronically: March 7, 2019
Additional Notes: The first author was supported by the Croatian Science Foundation under the project IP-2014-09-2285 and by the University of Rijeka under the project number 17.15.2.2.01.
The second author was supported in part by NSFC grants #11771307, #11726623, #11831012, and #11521061. He is the corresponding author.
Communicated by: Wenxian Shen
Article copyright: © Copyright 2019 American Mathematical Society