A General Fixed-Point Theorem for Correspondences

Abstract

A new local condition on correspondences called the "weak local connectedness property" (WLCP) is introduced. Working in ZFC, it is shown in our main theorem that - under mild restrictions - any correspondence from a connected subset X of a real TVS to itself will have a fixed point if the WLCP holds at each point of the domain. Since the WLCP generalizes several well - known conditions on correspondences, it is utilized in this paper to gather together a sampling of several disparate fixed-point and related results that are special cases of our main theorem.