Robustness of general dichotomies

Abstract

For a linear equation v ′ = A ( t ) v we consider general dichotomies that may exhibit stable and unstable behaviors with respect to arbitrary asymptotic rates e c ρ ( t ) for some function ρ ( t ) . This includes as a special case the usual exponential behavior when ρ ( t ) = t . We also consider the general case of nonuniform exponential dichotomies. We establish the robustness of the exponential dichotomies in Banach spaces, in the sense that the existence of an exponential dichotomy for a given linear equation persists under sufficiently small linear perturbations. We also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy.