Virtually stable maps and their convergence sets

Abstract

We generalize the concept of a virtually nonexpansive selfmap of a metric space by introducing the notion of a virtually stable selfmap of a Hausdorff space. We also prove that every virtually nonexpansive selfmap is virtually stable, and the fixed point set of a virtually stable selfmap is a retract of the convergence set. Some properties of the convergence set and the fixed point set of a virtually stable map are also investigated.