Stable manifolds for difference equations with infinite delay

Abstract

We construct stable invariant manifolds for a large class of nonautonomous delay difference equations with infinite delay. This requires considering an appropriate class of phase spaces that are Banach spaces of sequences satisfying a certain axiom motivated by work of Hale and Kato for continuous time. We also give examples of these spaces. We consider the general cases when the linear part has a tempered exponential dichotomy and when the perturbation depends on a parameter. Moreover, we obtain the optimal regularity of the stable manifolds for Lipschitz and perturbations, jointly with respect to the base and to the parameter.