Virtually nonexpansive maps and their convergence sets

We introduce the notion of a virtually nonexpansive selfmap of a metric space and show that the fixed point set of such a map is generally a retract of its convergence set. We also show that the class of virtually nonexpansive maps properly contains the class of (continuous) asymptotically quasi-nonexpansive maps. When the domain is complete and the fixed point set is totally bounded, we give another description of the convergence set of a virtually nonexpansive map and use it to show that the convergence set is always a Gδ-set. We also discuss some criteria to obtain an explicit retraction from the domain onto the fixed point set in Banach space settings.