The stability of the set of KKM points
Let \(X\) be a bounded complete convex subset of a normed linear space. A multivalued function \(G\) on \(X\) such that each \(G(x)\) is a compact nonempty subset of \(X\) is called a KKM mapping if, for every finite subset \(\{x_1,x_2,\dots, x_n\}\) of \(X\), its convex hull is contained in \(\bigcup^n_{i=1} G(x_i)\). Let \(M\) be a the set of KKM mappings on \(X\) and define a metric on \(M\) by setting \(\rho(G_1, G_2)= \sup_{x\in X} h(G_1(x), G_2(x))\) where \(h(G_1(x), G_2(x))\) denotes the Hausdorff distance between the compact sets \(G_1(x)\) and \(G_2(x)\). A theorem of Fan implies that the set \(F(G)= \bigcap_{x\in X} G(x)\), called the set of KKM points, is nonempty. A point \(x\in F(G)\) is essential if, given \(\varepsilon> 0\), there exists \(\delta> 0\) such that \(G'\in M\) and \(\rho(G, G')< \delta\), then there exists \(x'\in F(G')\) whose distance from \(x\) is less than \(\varepsilon\). A mapping \(G\in M\) is essential if all its KKM points are essential. The authors prove that there is a subset \(Q\) of \(M\) that is the countable intersection of open subsets of \(M\), and hence is dense in \(M\), that has the property that every mapping in \(Q\) is essential. Moreover, they show that for every \(G\in M\) there is component \(C(G)\) of \(F(G)\) that is essential. This means that, given \(\varepsilon> 0\), there exists \(\delta> 0\) such that if \(G'\in M\) with \(\rho(G, G')\), then \(F(G')\) intersects the \(\varepsilon\)-neighborhood of \(C(G)\) in \(X\).
scientific article; zbMATH DE number 1963665