在本篇論文中，我們證明一個在超凸度量空間度量空間上推廣型gkkm 定理。利用此定理，我們證得一些匹配定理、同質點定理及固定點定理。在應用方面，我們由此gkkm 定理，證明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 prove a generalized gkkm theorem in hyperconvex metric space. Use this theorem we get a matching theorem , coincidence theorem and fixed point theorem under some assumptions. As application, we use this theorem to establish some theorems about variational inequalities and minimax inequalities.

[1]A. Amini, M. Fakhar, and J. Zafarani, KKM mappings in metric spaces, Nonlinear Anal. 60(2005), 1045-1052.
[2]N. Aronszajn, and P. Panitchpakdi, Extensions of uniformly continuous
transformations and hyperconvex metric spaces, Pacific J. Math. 6(1956), 405-439.
[3]K. C. Border, Fixed point theorems with applications to economics and game theory, Cambridge University Press, 1989.
[4]S. S. Chang and Y. Zhang, Generalized KKM theorem and variational inequalities, J. Math. Anal. Appl. 159(1991), 208-233.
[5]T. H. Chang and C. L. Yen, KKM property and fixed point theorems, J. Math. Anal. Appl. 203(1996), 224-235.
[6]L. A. Dung and D. H. Tan, Some applications of the KKM-mapping principle in hyperconvex metric spaces, Nonlinear Ana, to appear.
[7]K. Fan, A generalization of Tychonoff’s fixed point theorem, Math. Ann. 142(1961),305-310.
[8]K. Fan, Some properties of convex sets related to fixed point theorem, Math. Ann. 266(1984), 519-537.
[9] J. C. Jeng, Hui-Chei Hsu, Young-Ye Huang, Fixed point theorem for multifuntions having KKM property on almost convex sets, J. Math. Anal. 319(2006), 187-198.
[10]B. Knaster, C. Kuratowski, and S. Mazurkiewicz, Ein Beweis des Fixpunksatzes fur n-dimensionale simplexe, Fund. Math. 14(1929),132-137.
[11] M. A. Khamsi, KKM and Ky Fan Theorems in Hyperconvex Metric Spaces, J. Math. Anal. Appl. 204(1996),298-306.
[12] J. H. Kim, and S. Park , Comments on some fixed point theorems in hyperconvex metric spaces, J.Math.Anal. 291 (2004),154-164 .
[13] 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.
[14] Y. L. Lee, G-S-KKM theorem and its applications, Graduate Institute of
Mathematics and Science, NHCTC, Hsin Chu, Taiwan. (2003).
[15] 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.
[16] J. T. Markin, A selection theorem for quasi-lower semicontinuous mappings in hyperconvex spaces, J. Math. Anal. Appl. 321(2006), 862-866.
[17] S. Park, Fixed point theorems in hyperconvex metric spaces, Nonlinear Anal. 37(1999), 467-472.
[18] N. Shioji, A further generalization of the Knaster-Kuratowski-Mazurkiewicz theorem, Proc. Amer. Math. Soc. 111(1991), 187-195.
[19] G. Q. Tina, Generalized KKM theorem, minimax qualities and their applications, J. Optim. Theory Appl. 83(1994), 375-389.
[20] X. Wu, B. Thompson, and G. X. Yuan, Fixed point theorems of upper semicontinuous multivalued mappings with applications in hyperconvex metric spaces, J. Math. Anal. Appl. 276(2002), 80-89.
[21] X. Wu, X.Yuan, Appoximate selections theorems on H-spaces with applications, J. Math. Anal. Appl. 231(1999)118-132.
[22] G. X. Z. Yuan, The Charactreization of Generalized Metric KKM Mappings with
Open Values in Hyperconvex Metric Spaces and Some Applications, J. Math.
Anal. Appl. 235(1999), 315-325.