Parameter dependence of stable manifolds under nonuniform hyperbolicity

For a nonautonomous linear equation v′=A(t)v in a Banach space with a nonuniform exponential dichotomy, we show that the nonlinear equation v′=A(t)v+f(t,v,λ) has stable invariant manifolds Vλ which are Lipschitz in the parameter λ provided that f is a sufficiently small Lipschitz perturbation. Since any linear equation with nonzero Lyapunov exponents has a nonuniform exponential dichotomy, the above assumption is very general. We emphasize that passing from a classical uniform exponential dichotomy to a general nonuniform exponential dichotomy requires a substantially new approach.