Attractive sets of periodic integrodifference equations under discretization

We consider periodic difference equations in infinite-dimensional Banach spaces possessing compact asymptotically stable set A. It is established that such A persists under spatial discretizations by means of projection methods as nearby closed and bounded uniformly asymptotically stable sets, which moreover converge to A in the Hausdorff metric for increasingly more accurate schemes. The proof is based on a Lyapunov function guaranteed by an ambient converse theorem and a pullback construction. As application serves integrodifference equations on spaces of continuous and p-integrable functions over a compact habitat.