International Journal of Mathematics and Mathematical Sciences
Volume 15 (1992), Issue 4, Pages 659-661
doi:10.1155/S0161171292000875
An application of KKM-map principle
Dipartimento di Matematica, Universitá degli studi della Calabria, Arcavacata di Rende (Cosenza) 87036, Italy
Received 4 February 1991; Revised 6 June 1991
Copyright © 1992 A. Carbone. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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).