Invariant manifolds of partial functional differential equations

Abstract

This paper is concerned with the existence, smoothness and attractivity of invariant manifolds for evolutionary processes on general Banach spaces when the nonlinear perturbation has a small global Lipschitz constant and locally Ck-smooth near the trivial solution. Such a nonlinear perturbation arises in many applications through the usual cut-off procedure, but the requirement in the existing literature that the nonlinear perturbation is globally Ck-smooth and has a globally small Lipschitz constant is hardly met in those systems for which the phase space does not allow a smooth cut-off function. Our general results are illustrated by and applied to partial functional differential equations for which the phase space C([-r, 0], X) (where r>0 and X being a Banach space) has no smooth inner product structure and for which the validity of variation-of-constants formula is still an interesting open problem. ?? 2003 Elsevier Inc. All rights reserved.