Weak topology and Browder--Kirk's theorem on hyperspace (Q996864)

Let \(WCS(X)\) denote the metric space of all nonempty compact convex subsets of a Banach space \(X\) with the Hausdorff distance and \(K\) be a nonempty \(\tau\)-compact convex subset of \(WCS(X)\), where the topology \(\tau\) is defined such that the closed balls are closed. The authors prove that if \(T\) is a nonexpansive mapping which maps \(K\) into \(K\), then \(T\) has a fixed point. One should note that once the authors have proved that the closed balls are closed, this result can be deduced directly from a result of \textit{W.\,A.\thinspace Kirk} [Proc.\ Am.\ Math.\ Soc.\ 82, 640--642 (1981; Zbl 0471.54027)].