Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line

Consider an evolution family U = (U(t, s))t s 0 on a half-line R+ and an integral equation u(t) = U(t, s)u(s)+ t s U(t, ξ)f (ξ)dξ.We characterize the exponential dichotomy of the evolution family through solvability of this integral equation in admissible function spaces which contain wide classes of function spaces like function spaces of Lp type, the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory. We then apply our results to study the robustness of the exponential dichotomy of evolution families on a half-line under small perturbations. © 2005 Elsevier Inc. All rights reserved.