C^k invariant manifolds for infinite delay

Abstract

For a non-autonomous delay difference equation with infinite delay, we construct smooth stable and unstable invariant manifolds for any sufficiently small perturbation of an exponential dichotomy. We consider a general class of norms on the phase space satisfying an axiom considered by Matsunaga and Murakami that goes back to earlier work by Hale and Kato for continuous time. In addition, we show that the invariant manifolds are as regular as the perturbation. Finally, we consider briefly the case of center manifolds and we formulate corresponding results.