An ultrafilter approach to locally almost nonexpansive maps

Let K be a bounded closed convex subset of a Banach space X, and suppose f:K→K is 'locally almost nonexpansive' in the sense of Nussbaum. It is shown that the mapping Id-f is demiclosed if X is either uniformly convex or satisfies the Opial property. These facts are known, but the ultrapower approach given here is new. In fact, we give ultrapower characterizations of locally almost nonexpansive maps and of the Opial property. Finally, we obtain a new demiclosedness result for the class of 'locally almost pointwise contractive mappings'