A Katznelson-Tzafriri type theorem for difference equations and applications

Minh, Nguyen; Matsunaga, Hideaki; Huy, Nguyen; Luong, Vu Trong

doi:10.1090/proc/15722

A Katznelson-Tzafriri type theorem for difference equations and applications
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by Nguyen Van Minh, Hideaki Matsunaga, Nguyen Duc Huy and Vu Trong Luong

Proc. Amer. Math. Soc. 150 (2022), 1105-1114

DOI: https://doi.org/10.1090/proc/15722

Published electronically: December 7, 2021

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Abstract:

We consider a Katznelson-Tzafriri type theorem for linear difference equations of the form $x(n+1)=Tx(n)+y(n)\ (*)$, where $T$ is a bounded operator in a Banach space $\mathbb {X}$ and $\{y(n)\}_{n=1}^\infty$ is a bounded sequence that is asymptotically constant, that is, $\lim _{n\to \infty } [y(n+1)-y(n)]=0$. A sequence $\{u(n)\}_{n=1}^\infty$ is said to be an asymptotic solution to $(*)$ if $u(n+1)=Tu(n)+y(n)+\epsilon (n)$, where $\lim _{n\to \infty }\epsilon (n)=0$. We show that if $1$ is either not in $\sigma (T)\cap \{z\in \mathbb {C}:\ |z|=1\}$, or is its isolated element, then if $(*)$ has a bounded asymptotic solution, it has an asymptotic solution that is asymptotically constant. Furthermore, if $\sigma (T)\cap \{z\in \mathbb {C}:\ |z|=1\}\subset \{ 1\}$, then every asymptotic solution of $(*)$ is asymptotic constant. An application to the evolution periodic equations is given.

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