An application of KKM-map principle

Abstract

The following theorem is proved and several fixed point theorems and coincidence theorems are derived as corollaries. LetCbe a nonempty convex subset of a normed linear spaceX,f:C→Xa continuous function,g:C→Ccontinuous, onto and almost quasi-convex. Assume thatChas a nonempty compact convex subsetDsuch that the setA={y∈C:‖g(x)−f(y)‖≥‖g(y)−f(y)‖   for   all   x∈D}is compact.Then there is a pointy0∈Csuch that‖g(y0)−f(y0)‖=d(f(y0),C).