Smooth invariant manifolds in Banach spaces with nonuniform exponential dichotomy

We establish the existence of smooth stable manifolds for semiflows defined by ordinary differential equations v′=A(t)v+f(t,v) in Banach spaces, assuming that the linear equation v′=A(t)v admits a nonuniform exponential dichotomy. Our proof of the Ck smoothness of the manifolds uses a single fixed point problem in the unit ball of the space of Ck functions with α-Hölder continuous kth derivative. This is a closed subset of the space of continuous functions with the supremum norm, by an apparently not so well-known lemma of Henry (see Proposition 3). The estimates showing that the functions maintain the original bounds when transformed under the fixed-point operator are obtained through a careful application of the Faà di Bruno formula for the higher derivatives of the compositions (see (31) and (35)). 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 up to order k when the vector field is of class Ck.