Center manifolds and contractions on a scale of Banach spaces

We give a new proof for the existence of a C^k-center manifold at a nonhyperbolic equilibrium point of a finite-dimensional vector field of class C^k. The problem is reduced to a fixed point problem on a scale of Banach spaces; these Banach spaces consist of mappings with a certain maximal exponential growth at infinity. We give conditions under which there is a unique fixed point depending differentiably on the parameters; the main difficulty is that the mappings under consideration become only differentiable after composition with appropriate embeddings on the scale of Banach spaces.