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Parameter dependence of stable manifolds for difference equations

We establish the existence of C1 invariant stable manifolds for a general nonautonomous dynamics with discrete time, obtained from perturbing a nonuniform exponential dichotomy. We also consider parameter-dependent perturbations, and we show that the stable manifolds are C1 in the parameter. We use the fibre contraction principle to establish the smooth dependence. In addition, our proof includes linear perturbations without any further effort, and thus our results can be readily applied to the robustness problem of nonuniform exponential dichotomies under parameter-dependent linear perturbations.