An extension to a theorem of Himmelberg
Let \(E=(E,\tau)\) be a topological vector space with topology \(\tau\), \(\widehat E=(E, \tau)^\wedge\) the completion of \(E\), \(E^*\) the dual of \(E\) and \((E^*, \tau_w)\) the weak topology of \(E\). We denote by \(co\) and \(cl\) the convex hull and the closure. The following theorem from this paper is an extension of a fixed-point theorem of \textit{C. J. Himmelberg} [J. Math. Anal. Appl. 38, 205--207 (1972; Zbl 0204.23104)]. Theorem: Let \(E\) be a topological vector space, \(C\) a convex subset of \(E\) and \(K\) a compact subset of \(C\) such that \(cl(co(K))\) is \(\widehat E\)-compact. Let \(f:C\to K\) be an usc multifunction such that for each \(x\in X\), \(f(x)\) is a nonempty, convex and closed subset of \(K\). If \((\widehat E,\tau_w)\) separates the points of \(\widehat E\), then \(f\) has a fixed-point in \(K\).
scientific article; zbMATH DE number 5083277