在本文中，我們先定義2-KKM函數及推廣型2-KKM函數，再利用超凸度量空間的特性，我們在不需任何緊緻性的條件下，證得了一個KKM定理及固定點定理。利用這個KKM定理，我們證得一些變分不等式及大中取小不等式的存在性定理。

In this paper, we first define 2-KKM mapping and generalized 2-KKM mapping. Then we apply the property of hyperconvex metric space to get a KKM theorem and a fixed point theorem without compact assumption. By using this KKM theorem we get some theorems about variational inequalities and minimax inequalities.

[1]N. Aronszajn, and P. Panitchpakdi, Extensions of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6(1956) 405-439.
[2]K. C. Border, Fixed point theorems with applications to economics and game theory, Cambridge University Press, 1989.
[3]S. S. Chang and Y. Zhang, Generalized KKM theorem and variational inequalities, J. Math. Anal. Appl. 159(1991), 208-233.
[4]T. H. Chang and C. L. Yen, KKM property and fixed point theorems, J. Math. Anal. Appl. 203(1996), 224-235.
[5]L. A. Dung and D. H. Tan, Some applications of the KKM-mapping principle in hyperconvex metric spaces, Nonlinear Anal. In press.
[6]K. Fan, A generalization of Tychonoff’s fixed point theorem, Math. Ann. 142(1961),305-310.
[7]A. Granas and F. C. Liu, Coincidence for set valued maps and inequalities, J. Math. Anal. Appl. 165(1986), 119-148.
[8]B. Knaster, C. Kuratowski, and S. Mazurkiewicz, Ein Beweis des Fixpunksatzes fur n-dimensionale simplexe, Fund. Math. 14(1929),132-137.
[9]M. A. Khamsi, KKM and Ky Fan Theorems in Hyperconvex Metric Spaces, J. Math. Anal. Appl. 204(1996),298-306.
[10]W. A. Kirk, B. Sims, and G. X .Z. Yuan, The Knaster-Kuratowski and Mazurkiewicz theory in hyperconvex metric spaces and some of its applications, Nonlinear Anal. 39(2000), 611-627.
[11]F. J. Liu, On a form of KKM principle and supinfsup inequalities of von Neumann and Ky Fan type, J. Math. Anal. Appl. 155(1991), 420-436.
[12]L. J. Lin and W. P. Wan, KKM type theorems and coincidence theorems with applications to the existence of equilibria, J. Optim. Theory Appl. 123(1)(2004), 105-122.
[13]M. Lassonde, On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl. 97(1983), 151-201.
[14]N. Shioji, A further generalization of the Knaster-Kuratowski-Mazurkiewicz theorem, Proc. Amer. Math. Soc. 111(1991), 187-195.
[15]G. Q. Tina, Generalized KKM theorem, minimax inequalities and their applications, J. Optim. Theory Appl. 83(1994), 375-389.