Asymptotic solutions of almost diagonal differential and difference systems

New methods for both asymptotic integration of the linear differential systems Y\u27(t) = [D( t) + R(t)]Y( t) and asymptotic summation of the linear difference systems Y(t + 1) = [D(t)+ R(t)]Y(t) are derived. The fundamental solution Y(t) = phi( t)[I+P(t)] for differential and difference systems is constructed in terms of a product. The first matrix function phi(t) is decided by the diagonal matrix D(t) and the second matrix I + P(t) is a perturbation of the identity matrix I. Another fundamental solution Y( t) = [I + Q(t)]phi( t) is also constructed for difference systems. Conditions are given on the matrix [D(t) + R( t)] that allow us to represent I + P( t) or Q(t) + I as an absolutely convergent resolvent series without imposing stringent conditions on R(t). In particular the analogs, in the setting of difference equations, of fundamental theorems of Levison and Hartman-Wintner are shown to follow from one and same theorem in this work