Fixed points of nonexpansive mappings in spaces of continuous functions

Abstract

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