Robustness of nonuniform exponential dichotomies

We give in this chapter conditions for the robustness of nonuniform exponential dichotomies in Banach spaces, in the sense that the existence of an exponential dichotomy for a linear equation v´ = A(t)v persists under suf- ficiently small linear perturbations. We also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy and the “angles” between the stable and unstable subspaces. The proofs exhibit (implicitly) the exponential dichotomies of the perturbed equations in terms of fixed points of certain contractions. We emphasize that we do not need the notion of admissibility (with respect to bounded nonlinear perturbations). We also establish related results in the case of strong nonuniform exponential dichotomies. All the results are obtained in Banach spaces. The presentation follows closely [18].