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

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), ..., alpha(n)) a multi-index in R-n such that alpha(i) >= 0 (1 <= i <= n), 1 = Sigma(n)(i=1) alpha(i) = 1, alpha(1), alpha(n) > 0, and consider the mapping T-alpha : C -> C given by T-alpha = Sigma(n)(i-1) alpha T-i(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.