Nonautonomous dynamical systems provide a mathematical framework
for temporally changing phenomena, where the law of evolution
varies in time due to seasonal, modulation, controlling or even
random effects. Our goal is to provide an approach to the
corresponding geometric theory of nonautonomous discrete dynamical
systems in infinite-dimensional spaces by virtue of 2-parameter
semigroups (processes). These dynamical systems are generated by
implicit difference equations, which explicitly depend on time.
Compactness and dissipativity conditions are provided for such
problems in order to have attractors using the natural concept of
pullback convergence. Concerning a necessary linear theory, our
hyperbolicity concept is based on exponential dichotomies and
splittings. This concept is in turn used to construct nonautonomous
invariant manifolds, so-called fiber bundles, and deduce
linearization theorems. The results are illustrated using temporal
and full discretizations of evolutionary differential equations.
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