Fixed points of nonexpansive mappings in spaces of continuous functions

Abstract

Let K be a compact metrizable space and C(K) the Banach space of all real continuous functions defined on K with the maximum norm. It is known that C(K) fails to have the weak fixed point property for nonexpansive mappings (w-FPP) when K contains a perfect set. However the space C(ω n + 1), where n ∈ N and ω is the first infinite ordinal number, enjoys the w-FPP and so C(K) also satisfies this property if K(ω) = ∅. It is unknown if C(K) has the w-FPP when K is a scattered set such that K(ω) 6= ∅. In this paper we prove that certain subspaces of C(K), with K(ω) 6= ∅, satisfy the w-FPP. To prove this result we introduce the notion of ω-almost weak orthogonality and we prove that an ω-almost weakly orthogonal closed subspace of C(K) enjoys the w-FPP. We show an example of an ω-almost weakly orthogonal subspace of C(ω ω + 1) which is not contained in C(ω n + 1) for any n ∈ N.