A variant of a fixed point theorem of Browder-Fan and Reich (Q1071290)

Let S be a convex, weakly compact subset of a locally convex Hausdorff space (E,\(\tau)\) and \(f: S\to E\) be a continuous multifunction from its weak topology \(\omega\) to \(\tau\). Let p be a continuous seminorm on (E,\(\tau)\) and for subsets A, B of E, let \(p(A,B)=\inf \{p(x-y):\quad x\in A,\quad y\in B\}.\) In this paper, the author gives some sufficient conditions for the existence of an \(x\in S\) satisfying \(p(x,fx)=p(fx,S).\) By using this result the author presents several fixed point theorems. The result obtained in this paper generalizes some recent results of \textit{S. Reich} [Math. Z. 125, 17-31 (1972; Zbl 0216.173)] and \textit{Waters} [Ph. D. Thesis (1984), University of Wyoming].