On asymptotic equivalence of perturbed linear systems of differential and difference equations ✩

Standard results on asymptotic integration of systems of linear differential equations give sufficient conditions which imply that a system is strongly asymptotically equivalent to its principal diagonal part. These involve certain dichotomy conditions on the diagonal part as well as growth conditions on the offdiagonal perturbation terms. Here, we study perturbations with a triangularly-induced structure and see that growth conditions can be substantially weakened. In addition, we give results for not necessarily triangular perturbations which in some sense “interpolate” between the classical theorems of Levinson and Hartman–Wintner. Some analogous results for systems of linear difference equations are also given. © 2006 Elsevier Inc. All rights reserved.