An application of KKM-map principle

Abstract

The following theorem is proved and several fixed point theorems and coincidence theorems are derived as corollaries. Let C be a nonempty convex subset of a normed linear space X, f:C→X a continuous function, g:C→C continuous, onto and almost quasi-convex. Assume that C has a nonempty compact convex subset D such that the setA={y∈C:‖g(x)−f(y)‖≥‖g(y)−f(y)‖   for   all   x∈D}is compact.Then there is a point y0∈C such that ‖g(y0)−f(y0)‖=d(f(y0),C).