THE KAKUTANI PROPERTY AND THE FIXED-POINT PROPERTY OF TOPOLOGICAL-SPACES WITH ABSTRACT CONVEXITY

The paper deals with the usual fixed point property and the following Kakutani property of a space X: for every upper semicontinuous function Φ from X to non-empty closed convex subsets of X, there exists x0 such that x0 ε{lunate} Φ(x0). We derive this property of X from various separation properties of convex subsets of X and a kind of local convexity of X. Convexity in our setup is given in an abstract axiomatic way. Special emphasis is given to the case where X has the form of a product. The obtained results cover several known fixed point theorems: Ky Fan-Glicksberg, Wallace, and special cases of Eilenberg-Montgomery. We also discuss an open problem concerning the fixed point property of finite posets and its role in proving more advanced theorems. © 1992.