A generalization of antipodal point theorems for set-valued mappings (Q985050)

The authors obtain interesting generalizations of the Borsuk-Ulam theorem on antipodes and the Schauder fixed point theorem for multivalued mappings. Namely, let \(U\) be an open bounded and symmetric neighbourhood of the origin in a normed space \(E\). Assume that \(\varphi:\overline U\multimap E\) is a compact admissible multivalued map (in the sense of \textit{L.\,Górniewicz} [Dissertationes Math., Warszawa 129 (1976; Zbl 0324.55002)]) which is equivariant on the border of \(U\). Then \(\varphi\) has a fixed point. Moreover, if \(\varphi\) is a compact admissible vector field on \(\overline U\) into \(E_k\), then a Borsuk-Ulam type theorem is proved, where \(E_k\) is a closed subspace of \(E\) such that \(\text{codim\,}E_k= k\). Note that, in the case when \(U\) is the unit open ball in \(E\), the above results were proved by \textit{K.\,Geba} and \textit{L.\,Górniewicz} in [Bull.\ Pol.\ Acad.\ Sci., Math.\ 34, 315--322 (1986; Zbl 0613.55001) and ibid.\ 34, 323--328 (1986; Zbl 0613.55002)].