Asymptotic stability of nonuniform behaviour

Dragičević, Davor; Zhang, Weinian

doi:10.1090/proc/14444

Asymptotic stability of nonuniform behaviour
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by Davor Dragičević and Weinian Zhang

Proc. Amer. Math. Soc. 147 (2019), 2437-2451

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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