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

Abstract

Let S be a convex, weakly compact subset of a locally convex Hausdorff space (E,τ) and f:S→E be a continuous multifunction from its weak topology ω to τ. let ρ be a continuous seminorm on (E,τ) and for subsets A, B of E let p(A,B)=inf{p(x−y):x ϵ A, y ϵ B}. In this paper, sufficient conditions are developed for the existence of an x ϵ S satisfying p(x,fx)=p(fx,S). The result is then used to prove several fixed point theorems.