Parameter Dependence of Stable Manifolds for Delay Equations with Polynomial Dichotomies

Abstract

We establish the existence of Lipschitz stable invariant manifolds for semiflows generated by a delay equation x′ = L(t)x t + f (t, x t , λ), assuming that the linear equation x′ = L(t)x t admits a polynomial dichotomy and that f is a sufficiently small Lipschitz perturbation. Moreover, we show that the stable invariant manifolds are Lipschitz in the parameter λ. We also consider the general case of nonuniform polynomial dichotomies.