ABSTRACT CONVEXITY AND FIXED-POINTS

The purpose of this paper is to extend Himmelberg's fixed-point theore m, replacing the usual convexity in topological vector spaces with an abstract topological notion of convexity that generalizes classical co nvexity as well as several metric convexity structures found in the li terature. We prove the existence, under weak hypotheses, of a fixed po int for a compact approachable map, and we provide sufficient conditio ns under which this result applies to maps whose values are convex in the abstract sense mentioned above. (C) 1998 Academic Press.