Fakhar, M.; Zafarani, J. KKM theorem with applications to lower and upper bounds equilibrium problem in \(G\)-convex spaces. (English) Zbl 1044.47044 Int. J. Math. Math. Sci. 2003, No. 51, 3267-3276 (2003). 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 Keywords:lower and upper bounds equilibrium problem; KKM map; \(G\)-convex space PDFBibTeX XMLCite \textit{M. Fakhar} and \textit{J. Zafarani}, Int. J. Math. Math. Sci. 2003, No. 51, 3267--3276 (2003; Zbl 1044.47044) Full Text: DOI EuDML