A hierarchy of nonexpansivity and a natural transfer of fixed point properties

Abstract This study provides an analysis of nonexpansive-related operators resulted through affine displacements of conventional nonexpansive mappings. In particular, a unitary and simultaneous approach on both averaged (as nonexpansivity subclass) and enriched nonexpansive mappings (as nonexpansivity superclass) is enabled, after unveiling some structural analogy and a certain duality of theses two classes. Moreover, we emphasize that the displacement technique creates a natural transfer-return dynamics between operators, which preserves important operator properties and converts conventional iterative procedures related to nonexpansivity to adequate inertial procedures for displaced nonexpansive operators. The resulting equivalences allow us to take fundamental results on fixed points of conventional nonexpansive mappings and transfer them to results related to fixed points of averaged or enriched nonexpansive operators.