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Volume 19, Issue 2

## Volume 1 (2001)

Lucas Backes
• Departamento de Matemática, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, CEP 91509-900, Porto Alegre, RS, Brazil
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/ Davor Dragičević
Published Online: 2018-10-31 | DOI: https://doi.org/10.1515/ans-2018-2033

## Abstract

We prove that whenever a sequence of bounded operators ${\left({A}_{m}\right)}_{m\in ℤ}$ acting on a Banach space X admits an exponential dichotomy and a sequence of differentiable maps ${f}_{m}:X\to X$, $m\in ℤ$, has bounded and Hölder derivatives, the nonautonomous dynamics given by ${x}_{m+1}={A}_{m}{x}_{m}+{f}_{m}\left({x}_{m}\right)$, $m\in ℤ$, has various shadowing properties. Hence, we extend recent results of Bernardes Jr. et al. in several directions. As a nontrivial application of our results, we give a new proof of the nonautonomous Grobman–Hartman theorem.

MSC 2010: 37C50; 34D09; 34D10

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Revised: 2018-08-22

Accepted: 2018-09-17

Published Online: 2018-10-31

Published in Print: 2019-05-01

L. Backes was partially supported by a CAPES-Brazil postdoctoral fellowship under Grant No. 88881.120218/2016-01 at the University of Chicago. D. Dragičević was supported in part 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.

Citation Information: Advanced Nonlinear Studies, Volume 19, Issue 2, Pages 425–436, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365,

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