We consider systems of differential equations of the form (1) , for , x in some domain and (a̠ a fixed real number). We assume that the solution of (1) defined for satisfies for small enough, for some constant and h a continuous positive function defined in [a,. We give conditions for the perturbed system (g to have the same type of stability as (1).
Since the paper appeared, we have obtained further general results about asymptotic integration for perturbed systems (2). Moreover, the corresponding results for linear systems give new insights about Levinson’s theorem on asymptotic integration. In this way using these systems (which we call h-systems) we get a uniform treatment for the concept of stability and we also verify that several classical theorems of stability take a precise form by the asymptotic formulae.