Roughness of (h, k)-Dichotomies

In this paper we prove a roughness theorem for a class of dichotomies of linear systems of differential equations, the so-called (h,k)-dichotomies. These include a great variety of dichotomies, among them the ordinary and exponential dichotomies, exponential-ordinary dichotomies. Levinson dichotomies for diagonal systems, etc. We prove that under general uniform conditions upon the functions h and k, these dichotomies are not destroyed by integrable perturbations. This result generalizes the property of roughness of ordinary dichotomies under integrable perturbations.