A generalization of the concept
of asymptotic equivalence of two systems of ordinary differential equations is
investigated. This extension of asymptotic equivalence is novel in two ways.
First, the dimensions of the linear asymptotic subspaces of the differential
equations are utilized. Secondly, the two Banach spaces L∞ and L0∞, that
are implicitly used in the usual definition of asymptotic equivalence, are
replaced by two (arbitrary) Banach spaces that are stronger that L(X). The
main theorem establishes a functional asymptotic relationship between the
solutions of two perturbed linear differential equations that utilizes the above
modifications.
