Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under(μ,ν)-Dichotomies

Abstract

We obtain the existence of stable invariant manifolds for the nonlinear equationx′=L(t)xt+f(t,xt,λ)provided that the linear delay equationx′=L(t)xtadmits a nonuniform(μ,ν)-dichotomy andfis a sufficiently small Lipschitz perturbation. We show that the stable invariant manifolds are dependent on parameterλ. Namely, the stable invariant manifolds are Lipschitz in the parameterλ. In addition, we also show that nonuniform(μ,ν)-contraction persists under sufficiently small nonlinear perturbations.