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KKM theorem with applications to lower and upper bounds equilibrium problem in \(G\)-convex spaces. (English) Zbl 1044.47044

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.
Reviewer: Rita Pini (Milano)

MSC:

47J20 Variational and other types of inequalities involving nonlinear operators (general)
47H10 Fixed-point theorems
54C60 Set-valued maps in general topology
49J35 Existence of solutions for minimax problems
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