Continuum of Zero Points of a Mapping on a Compact, Convex Set

Abstract

Let X be a nonempty, compact and convex set in Rn and $\phi$ be an outer semicontinuous mapping from X to the collection of nonempty, compact convex subsets of Rn. We show that for any nonzero vector c in Rn there exists a set of stationary points of $\phi$ on X with respect to c connecting a point in the boundary of X at which $c^{\top} x$ is minimized on X to another point in the boundary of X at which $c^{\top}x$ is maximized on X. We provide several conditions on $\phi$ under which there exists a continuum of zero points of $\phi$ connecting two such points in the boundary of X, and an intersection result on a convex, compact set. An application to constrained equilibria is also given.