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Remarks on the KKM property for open-valued multimaps on generalized convex spaces. (English) Zbl 1083.47041

If \((X,D;\Gamma)\) is a G-convex space, \(Y\) a Hausdorff space, then denote by \(\mathcal{U}_{C}^{\mathcal{K}}(X,Y) \) the admissible class of Park and by \(k\mathcal{D}(X,Y)\) the class of multimaps having the KKM property for open-valued multifunctions. The main result of the present paper says that \(\mathcal{U}_{C}^{\mathcal{K}}(X,Y)=k \mathcal{D}(X,Y)\). Some applications to KKM theorems, matching theorems and coincidence results are given.

MSC:

47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
55M20 Fixed points and coincidences in algebraic topology
47H04 Set-valued operators
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