Stable manifolds for delay equations and parameter dependence

We establish the existence of Lipschitz stable invariant manifolds for semiflows generated by a delay equation x′=L(t)xt+f(t,xt,λ)x′=L(t)xt+f(t,xt,λ), assuming that the linear equation x′=L(t)xtx′=L(t)xt admits a nonuniform exponential dichotomy and that ff is a sufficiently small Lipschitz perturbation. We also show that the stable invariant manifolds are Lipschitz in the parameter λλ.