Journal of the
Korean Mathematical Society JKMS

We obtain new fixed point theorems on maps defined on "locally $G$-convex" subsets of a generalized convex spaces. Our first theorem is a Schauder-Tychonoff type generalization of the Brouwer fixed point theorem for a $G$-convex space, and the second main result is a fixed point theorem for the Kakutani maps. Our results extend many known generalizations of the Brouwer theorem, and are based on the Knaster-Kuratowski-Mazurkiewicz theorem. From these results, we deduce new results on collectively fixed points, intersection theorems for sets with convex sections and quasi-equilibrium theorems.

Keywords: generalized convex space, $G$-convex space, fixed point, $H$-space, multifunction, upper semicontinuous (u.s.c.), star refinement, $Z$-type, type I, type II

MSC numbers: Primary 54H25, 54C60, 47H10