A topological intersection theorem with applications in \(H\)-spaces
Let \(X\) be a topological space and let \(F(X)\) be a family of non-empty finite subsets of \(X\). If \(\{\Gamma_A\}\) denotes a family of non-empty subsets of \(X\) such that \(\Gamma_A\subset\Gamma_B\) for \(A,B\in F(X)\) and \(A\subset B\), then the pair \((X,\{\Gamma_A\})\) is said to be an \(H\)-space. It is clear that the Cartesian product of \(H\)-spaces is still an \(H\)-space. A subset \(B\subset X\) is called \(H\)-convex in \((X,\{\Gamma_A\})\) if \(\Gamma_A\subset B\) for every non-empty finite subset \(A\subset B\). Denote \[ X= \sqcap \{X_s: s\in I\},\quad X^s= \sqcap\{X_k: k\in I,\;k\neq s\}. \] The following topological intersection theorem of a family of sets with \(H\)-convex sections has been proved. Theorem. Let \(\{(X_s,\{\Gamma_{A_s}\}):s\in I\}\) be a family of compact \(H\)-spaces and let \(\{E_s\subset X\}\), \(s\in I\) be a family of subsets. Suppose that the following conditions are satisfied: 1) for each fixed \(x^s\in X\) the subset of \(E_s\) defined by \[ E_s(x^s)= \{x_s\in X_s: (x_s, x^s)\in E_s\} \] is non-empty \(H\)-convex; 2) for each fixed \(x_s\in X_s\), the subset of \(E_s\) defined by \[ E_s(x_s)= \{x^s\in X:(x_s, x^s)\in E_s\} \] is an open subset of \(X^s\). Then \(\bigcap\{E_s: s\in I\}\neq\emptyset\). A special case of this theorem is the Ky-Fan minimax theorem. The application of this topological intersection theorem to study the Nash equilibrium of an abstract economy as well as the existence of solutions for quasi-variational inequalities are presented in the paper.
scientific article; zbMATH DE number 1507471