Exponential trichotomies and continuity of invariant manifolds

Abstract

In this work, we consider the invariant manifolds for the family of equations

x  = Ax + f(", x), where A the is generator of a strongly continuous semigroup of linear operators in a Banach space X and f(", ·) : X ! X is continuous. The existence of stable (unstable) and center-stable (center-unstable) manifolds for a large class of these equations has been proved in [2]. We prove here that, if A admits a exponential trichotomy and f satisfies some suitable regularity hypotheses, then those manifolds are continuous with respect to the parameter".