Asymptotic Behavior of Individual Orbits of Discrete Systems

TL;DR

This paper investigates the long-term behavior of solutions to certain discrete linear systems in Banach spaces, extending existing theorems and providing new insights into their stability and asymptotic properties.

Contribution

It extends the Katznelson-Tzafriri theorem to broader conditions and introduces elementary proofs for the asymptotic behavior of solutions in Banach spaces.

Findings

Bounded solutions satisfy _{n o\u221e} (x(n+1)-x(n))=0 under spectral conditions.

The techniques are applied to study individual stability of solutions.

The paper discusses potential extensions of the results.

Abstract

We consider the asymptotic behavior of bounded solutions of the difference equations of the form $x(n+1)=Bx(n) + y(n)$ in a Banach space $\X$, where $n=1,2,...$, $B$ is a linear continuous operator in $\X$, and $(y(n))$ is a sequence in $\X$ converging to 0 as $n\to\infty$. An obtained result with an elementary proof says that if $\sigma (B) \cap \{|z|=1\} \subset \{1\}$, then every bounded solution $x(n)$ has the property that $\lim_{n\to\infty} (x(n+1)-x(n)) =0$. This result extends a theorem due to Katznelson-Tzafriri. Moreover, the techniques of the proof are furthered to study the individual stability of solutions of the discrete system. A discussion on further extensions is also given.