Periodic and fixed points of multivalued maps on Euclidean spaces

We show, in particular, that a multivalued map $f$ from a closed subspace $X$ of $\mathbb R^n$ to ${\rm exp}_k(\mathbb R^n)$ has a point of period exactly $M$ if and only if its continuous extension $\tilde f: \beta X\to {\rm exp}_k(\beta \mathbb R^n)$ has such a point. The result also holds if one repace $\mathbb R^n$ by a locally compact Lindel\"of 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.