On multivaled fixed-point free maps on R^n

Abstract

To formulate our results let $f$ be a continuous map from ${\mathbb{R}}^n$ to $2^{{\mathbb{R}}^n}$ and $k$ a natural number such that $|f(x)|\le k$ for all $x$. We prove that $f$ is fixed-point free if and only if its continuous extension $\tilde{f}:\beta {\mathbb{R}}^n\to 2^{\beta {\mathbb{R}}^n}$ is fixed-point free. If one wishes to stay within metric terms, the result can be formulated as follows: $f$ is fixed-point free if and only if there exists a continuous fixed-point free extension $\bar{f}:b{\mathbb{R}}^n\to 2^{b{\mathbb{R}}^n}$ for some metric compactificaton $b{\mathbb{R}}^n$ of ${\mathbb{R}}^n$. Using the classical notion of colorablity, we prove that such an $f$ is always colorable. Moreover, a number of colors sufficient to paint the graph can be expressed as a function of $n$ and $k$ only. The mentioned results also hold if the domain is replaced by any closed subspace of ${\mathbb{R}}^n$ without any changes in the range.