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Comments on generalized R-KKM type theorems. (English) Zbl 1213.49008

Summary: Recently, some authors [L. Deng and X. Xia, J. Math. Anal. Appl. 285, No. 2, 679–690 (2003; Zbl 1039.54011); L. Deng and M. G. Yang, Acta Math. Sin., Engl. Ser. 22, No. 6, 1809–1818 (2006; Zbl 1129.54024); X. P. Ding, Acta Math. Sin., Engl. Ser. 23, No. 10, 1869–1880 (2007; Zbl 1138.49006); X. P. Ding, Y. C. Liou and J. C. Yao, Appl. Math. Lett. 18, No. 12, 1345–1350 (2005; Zbl 1083.54532); J. Huang, J. Math. Anal. Appl. 312, No. 1, 374–382 (2005; Zbl 1090.54038)] adopted the concept of the so-called generalized R-KKM maps which are used to rewrite known results in the KKM theory. In the present paper, we show that those maps are simply KKM maps on \(G\)-convex spaces. Consequently, results on generalized R-KKM maps follow the corresponding previous ones on \(G\)-convex spaces.

MSC:

49J35 Existence of solutions for minimax problems
49J27 Existence theories for problems in abstract spaces
91B50 General equilibrium theory
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