Asymptotic Behavior of a Poincaré Recurrence System Original Research Article

We establish asymptotic formulae for the solutions of thefirst order recurrence systemxn+1=(A+Bn) xn,whereAandBn(n=0, 1, …) are square matrices and ∑∞n=0 ‖Bn‖2<∞. As a consequence, we confirma recent conjecture about the asymptotic behavior of the solutionsof the higher order scalar equationu(n+1)=∑ki=0 (ci+di(n)) u(n−i).