Fixed point property of direct sums

Abstract

For a uniformly convex space Z, we show that Z-direct sums (X1⊕⋯⊕XN)Z of Banach spaces X1,...,XN with R(a,Xi)<1+a for some a ∈(0,1] have the fixed point property for nonexpansive mappings. As a direct consequence, the result holds for all ψ-direct sums with ψ being strictly convex. The same result is extended to all ψ-direct sums X⊕ψY of spaces X and Y with property (M), whenever ψ≠ψ1. The permanence of properties that are sufficient for the fixed point property are obtained for Z-direct sums (and then for ψ-direct sums). Such properties include the properties R(X)<2, WNUS, CNJ(a,X)<2, UKK, and NUC. © 2005 Elsevier Ltd. All rights reserved.