The nonexpansive and mean nonexpansive fixed point properties are equivalent for affine mappings

Abstract

Let C be a convex subset of a Banach space X and let T be a mapping from C into C. Fix $$\alpha =(\alpha _1,\alpha _2,\ldots ,\alpha _n)$$ a multi-index in $${\mathbb {R}}^n$$ such that $$\alpha _i\ge 0$$ ( $$1\le i\le n$$ ), $$\sum _{i=1}^n\alpha _i=1$$ , $$\alpha _1,\alpha _n>0$$ , and consider the mapping $$T_\alpha :C\rightarrow C$$ given by $$T_\alpha =\sum _{i=1}^n \alpha _i T^i$$ . Every fixed point of T is a fixed point for $$T_\alpha $$ but the converse does not hold in general. In this paper we study necessary and sufficient conditions to assure the existence of fixed points for T in terms of the existence of fixed points of $$T_\alpha $$ and the behaviour of the T-orbits of the points in the domain of T. As a consequence, we prove that the fixed point property for nonexpansive mappings and the fixed point property for mean nonexpansive mappings are equivalent conditions when the involved mappings are affine. Some extensions for more general classes of mappings are also achieved.