An abstract approach to smoothness of invariant manifolds

Abstract

We establish a general abstract Cm-fixed-point theorem for a finite scale of Banach spaces, extending to the Cm-case a previous C1-fixed-point theorem of Vanderbauwhede and Van Gils. This result gives short proofs of Cm-smoothness for center-, center-unstable, inertial and other center-like invariant manifolds for various types of differential and difference equations in finite and infinite dimensions. It is also applicable to fixed-point equations on LP-spaces. In the present paper we present applications to invariant manifolds for evolution equations with time-dependent nonlinearities.