Fixed points of nonexpansive mappings in spaces of continuous functions

Let $K$ be a compact metrizable space and let $C(K)$ be 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 $C(\omega ^{n}+1)$, where $n\in \mathbb{N}$ and $\omega$ is the first infinite ordinal number, enjoys the w-FPP, and so $C(K)$ also satisfies this property if $K^{(\omega )}=\emptyset$. It is unknown if $C(K)$ has the w-FPP when $K$ is a scattered set such that $K^{(\omega )}\not =\emptyset$. In this paper we prove that certain subspaces of $C(K)$, with $K^{(\omega )}\not = \emptyset$, satisfy the w-FPP. To prove this result we introduce the notion of $\omega$-almost weak orthogonality and we prove that an $\omega$-almost weakly orthogonal closed subspace of $C(K)$ enjoys the w-FPP. We show an example of an $\omega$-almost weakly orthogonal subspace of $C(\omega ^{\omega }+1)$ which is not contained in $C(\omega ^{n}+1)$ for any $n\in \mathbb{N}$.