Asymptotic Behavior of Individual Orbits of Discrete Systems

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 → ∞. An obtained result with an elementary proof says that if σ(B) ∩ {|z| = 1} ⊂ {1}, then every bounded solution x(n) has the property that limn→∞(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.