Fixed‐point and coincidence theorems for set‐valued maps with nonconvex or noncompact domains in topological vector spaces

Abstract

A technique, based on the investigations of the images of maps, for obtaining fixed‐point and coincidence results in a new class of maps and domains is described. In particular, we show that the problem concerning the existence of fixed points of expansive set‐valued maps and inner set‐valued maps on not necessarily convex or compact sets in Hausdorff topological vector spaces has a solution. As a consequence, we prove a new intersection theorem concerning not necessarily convex or compact sets and its applications. We also give new coincidence and section theorems for maps defined on not necessarily convex sets in Hausdorff topological vector spaces. Examples and counterexamples show a fundamental difference between our results and the well‐known ones.