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

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