Set convergence for discretizations of the attractor

We consider the discretization of a dynamical system given by a C-o-semigroup S(t), defined on a Banach space X, possessing an attractor d. Under certain weak assumptions, Hale, Lin and Raugel showed that discretizations of S(t) possess local attractors, which may be considered as approximations to A. Without further assumptions, we show that these local attractors possess convergent subsequences in the Hausdorff or set metric, whose limit is a compact invariant subset of A. Using a new construction, we also consider the Kloeden and Lorenz concept of attracting sets in a Banach space, and show under mild assumptions that discretizations possess attracting sets converging to A in the Hausdorff metric.