Asymptotic Behavior of Individual Orbits of Discrete Systems

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.