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 $\mathbb {X}$, where $n=1,2,...$, $B$ is a linear continuous operator in $\mathbb {X}$, and $(y(n))$ is a sequence in $\mathbb {X}$ 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.