Parameter dependence of stable manifolds for nonuniform (µ, .)-dichotomies

We construct stable invariant manifolds for semiflows generated by the nonlinear impulsive differential equation with parameters x. = A(t)x + f(t, x, .), t . . i and x(. +i) = B i x(. i ) + g i (x(. i ), .), i . . in Banach spaces, assuming that the linear impulsive differential equation x. = A(t)x, t . . i and x(. i +) = B i x(. i ), i . . admits a nonuniform (µ, .)-dichotomy. It is shown that the stable invariant manifolds are Lipschitz continuous in the parameter . and the initial values provided that the nonlinear perturbations f, g are sufficiently small Lipschitz perturbations.