The Kakutani property and the fixed point property of topological spaces with abstract convexity
1992; Elsevier BV; Volume: 168; Issue: 2 Linguagem: Inglês
10.1016/0022-247x(92)90174-c
ISSN1096-0813
Autores Tópico(s)Optimization and Variational Analysis
ResumoThe 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 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.
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