A fixed point theorem for conical shells (Q721202)

The authors prove a fixed point theorem for mappings \(f\) defined on conical shells in \(\mathbb{R}^n\) that extends Krasnosel'skiĭ fixed point theorem on cones in Banach spaces. A conical shell \(F\) is the subset of a cone \(C\) defined by the inequalities \(r \leq \|x\| \leq R\) for some \(0 < r < R\). The top, bottom and lateral faces of \(F\) are defined by \(F_R = F \cap \{\|x\| = R\}\), \(F_r = F \cap \{\|x\| = r\}\) and \(L = \mathrm{bd}\,C \cap F\), respectively. Theorem. Let \(C, K\) be closed, convex cones in \(\mathbb{R}^n\) with nonempty interior and such that \(C \cap K = \{0\}\), \(F \subset C\) be a conical shell and \(f : F \to \mathbb{R}^n \setminus K\). Suppose the following two conditions are satisfied: {\parindent=6mm \begin{itemize}\item[(i)] \(f\) maps \(L\) outward and around \(C\) or \(f\) maps \(L\) inward into \(C\). \item[(ii)] \(f\) maps the top and bottom faces of \(F\) in opposite directions. \end{itemize}} Then \(f\) has a fixed point. Roughly speaking, \(f\) maps \(F_R\) inward (outward) if \(f(F_R)\) is included in (disjoint to) \(\{\|x\|\leq R\}\) and \(f\) maps \(F_r\) inward (outward) if \(f(F_r)\) is disjoint to (included in) \(\{\|x\| \leq r\}\). Similarly, \(f\) maps \(L\) inward (outward) if \(f(L) \subset C\) (\(f(L) \cap C = \emptyset\)) and \(f\) maps \(L\) around \(C\) if \(f : L \cap \{\|x\|=t\} \to \mathbb{R}^n \setminus (C \cup K)\) is essential for some \(r \leq t \leq R\). The proof of the theorem uses Lefschetz theory (and its generalization to compact ANRs) and is elementary. The article contains several remarks that show that previous results in the literature did not address the cases considered by this theorem. In the last section, it is shown that the theorem partly generalizes to infinite-dimensional Banach spaces.