An abstract approach to smoothness of invariant manifolds

Abstract

We establish a general abstract C^m-fixed-point theorem for a finite scale of Banach spaces, extending to the C^m-case a previous C¹-fixed-point theorem of Vanderbauwhede and Van Gils. This result gives short proofs of C^m-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 L^P-spaces. In the present paper we present applications to invariant manifolds for evolution equations with time-dependent nonlinearities.

Keywords:

• Scales of Banach spaces
• Frechet derivatives
• evolution equations
• center-
• center-unstable
• inertial manifolds

AMS:

• 35B99
• 58C20