KKM theorem with applications to lower and upper bounds equilibrium problem in \(G\)-convex spaces (Q1415180)

The aim of this paper is to provide conditions for the existence of solutions of the following lower and upper bound equilibrium problem, closely related to equilibrium problems: find \(\underline{x}\in K\) such that \[ \alpha\leq f(\underline{x},y)\leq \beta,\qquad \forall y\in K, \] where \(\alpha,\beta\in{\mathbb R},\) \(\alpha\leq \beta\), \(K\subset X\) and \(f:K\times K\to {\mathbb R};\) here \((X,D;\Gamma)\) is a \(G\)-convex space. To this purpose, the authors prove some refined versions of the KKM theorem, in the setting of \(G\)-convex spaces, and for transfer closed-valued maps, and, as a consequence, they obtain two existence results for the solution of the lower and upper bounds equilibrium problems, on \(G\)-convex spaces and on Hausdorff \(G\)-convex spaces. At the end, they give some applications of their existence results.