  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.
