Generalized KKM theorems and common fixed point theorems
Let \(Y\) be an arbitrary nonempty set and \(X\) be a nonempty convex set in a vector space. A multivalued mapping \(T:Y\multimap X\) is said to be a generalized \(KKM\)-mapping if for any nonempty finite subset \(\{y_1,\dots,y_n\}\) of \(Y\), there exists a subset \(\{x_1,\dots,x_n\}\) in \(X\) such that \(\text{co}\; \{x_i:i\in I\} \subseteq \bigcup\limits_{i\in I}T(y_i)\) for each nonempty subset \(I\) of \(\{1,\dots,n\}\). If \(\mathcal{T}\) is a family of multivalued mappings from \(Y\) into \(X\), then \(\mathcal{T}\) is said to be equi-KKM if for any nonempty finite subset \(\{y_1,\dots,y_n\}\) of \(Y\), there exists a subset \(\{x_1,\dots,x_n\}\) in \(X\) such that \(\text{co}\; \{x_i:i\in I\} \subseteq \bigcup\limits_{i\in I}T(y_i)\) for each nonempty subset \(I\) of \(\{1,\dots,n\}\) and each \(T\in \mathcal{T}\). The authors show first that, if \(T\) is a generalized \(KKM\) multivalued mapping defined on an arbitrary set and with closed values into a closed subset of a topological vector space, then the family of its values has the finite intersection property. Relying on this result, the authors establish several intersection theorems for the family of values of a generalized \(KKM\) mapping, and common fixed point theorems for families of equi-\(KKM\) mappings.
scientific article; zbMATH DE number 5930952