Higher regularity of invariant manifolds for nonautonomous equations

For semiflows generated by ordinary differential equations v' = A(t)v admitting a nonuniform exponential dichotomy, we show that for any sufficiently small perturbation f (t, v) of class C-m, there exist C-m stable and unstable invariant manifolds for the perturbed equation v' = A(t)v + f(t, v). We emphasize that we do not need to assume the existence of a uniform exponential dichotomy. Our proof of the smoothness of the invariant manifolds is based on a geometric induction argument by considering a linear extension of the vector field, which in particular avoids any lengthy computation related to the higher order derivatives and the induction process. As a consequence, we obtain, in a direct manner, not only the exponential decay of solutions along the stable manifolds but also of their derivatives.