A generalization of antipodal point theorems for set-valued mappings

Abstract

Let $U$ be a bounded symmetric open neighborhood of the origin of $\mathbf{R}^{m+k} \ (k\geqq 1)$. We shall prove a generalization of the Borsuk's antipodal theorem for an admissible mapping $\varphi:\partial\overline{U}\to \mathbf{R}^m$ and the related topic. We shall generalize the theorem for the case of a bounded symmetric open neighborhood $U$ of the origin of an infinite dimensional normed space $\mathbf{E}$. The Borsuk-Ulam theorem shall be studied for the case of a bounded symmetric open neighborhood $U$ of the origin of an infinite dimensional normed space $\mathbf{E}$.