Periodic and fixed points of multivalued maps on Euclidean spaces

Abstract

We show, in particular, that a multivalued map $f$ from a closed subspace $X$ of ${\mathbb{R}}^n$ to ${\exp }_k({\mathbb{R}}^n)$ has a point of period exactly $M$ if and only if its continuous extension $\tilde{f}:\beta X\to {\exp }_k(\beta {\mathbb{R}}^n)$ has such a point. The result also holds if one repace ${\mathbb{R}}^n$ by a locally compact Lindelöf space of finite dimension. We also show that if $f$ is a colorable map froma normal space $X$ to the space $\mathcal{K}(X)$ of all compact subsets of $X$ then its extension $\tilde{f}:\beta X\to \mathcal{K}(\beta X)$ is fixed-point free.

13 pages