本文首先證明一個在FC空間上的 型函數的固定點定理，也利用這個定理證明了一個推廣型變分不等式的存在性定理。我們證得一些關於三個多值函數的循環點存在性定理。

In this paper, we establish a fixed theorem of a in FC-space. Apply this fixed theorem, we get an existence theorem concerning generalized variational inequalities. We also establish some cycle point theorems for three set-valued mappings.

[1] H. Ben-El-Mechaiekh, P. Deguire, and A. Granas, Points fixes et coincidences pour les functions multivoques II, C. R. Acad. Sci. Paris Ser. I 295(1982), 381-388.
[2] H. Ben-El-Mechaiekh, P. Deguire, Approachability and fixed points for
non-convex set-valued maps, J. Math. Anal. Appl. 170 (1992) 477-500.
[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, Generalized KKM property and fixed point theorems, J. Math. Anal. Appl. 203(1996), 224-235.
[6] T. H. Chang, Y. Y. Huang, J. C. Jeng, K. W. Kuo, On S-KKM property and related topics, J. Math. Anal. Appl. 229(1999), 212 –227.
[7] T. H. Chang, Y. Y. Huang, J. C. Jeng, Fixed-point theorems for multifunctions in S-KKM class, Nonl. Anal. 44(2001), 1007-1017.
[8]C. M. Chen, Fixed point theorems, KKM theorems and its application. Department of Mathematics, NTNU, Taipei, Taiwan, ROC.
[9] P. Deguire and M. Lassonde, Familles selectantes, Topol. Methods Nonlinear Anal. 5(1995), 261-269.
[10] P. Deguire, K. K. Tan, and G. X. Z. Yuan, The study of maximal elements, fixed point for -majorized mappings and their applications to minimax and variational inequalities in product topological spaces, Nonlinear Anal. 37(1999), 933-951.
[11] X. P. Ding, Best approximation and coincidence theorems, J. Sichuan Normal Univ. Nat. Sci. 18(1995), 21-29.
[12] X. P. Ding, Coincidence theorems in topological spaces and their applications, applied. Math. Lett. 12(1999), 99-105.
[13] X. P. Ding, Existence of solutions for quasi-equilibrium problems in noncompact topological spaces, Comput. Math. Appl. 39(2000), 13-21.
[14] X. P. Ding and J. Y. Park, Fixed points and generalized vector equilibrium problems in generalized convex spaces, Indian J. Pure Appl. Math. 34(6)(2003), 973-990.
[15]X.P. Ding, System of generalized vector quasi-equilibrium problem in locally FC-space, Acta Math. Sinica, in press.
[16] X. P. Ding, Maximal element theorems in product and generalized games, J. Math. Anal. Appl. 305(2005), 29-42.
[17] X. P. Ding, The Generalized game and system of generalized vector quasi-equilibrium problems in locally -umiform spaces.Nonl. Anal., in press.
[18] K. Fan, A generalization of Tychonoff’s fixed point theorem, Math. Ann. 142(1961),305-310.
[19] A. Granas and F. C. Liu, Coincidence for set valued maps and inequalities, J.
Math. Anal. Appl. 165(1986), 119-148.
[20] B. Knaster, C. Kuratowski, and S. Mazurkiewicz, Ein Beweis des Fixpunksatzes fur n-dimensionale simplexe, Fund. Math. 14(1929),132-137.
[21] J. L. Kelly, General Topology, Van Nostrand, Princeton, NJ, 1955.
[22] 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.
[23] Y. L. Lee, G-S-KKM theorem and its applications, graduate Institute of Mathematics and Science, NHCTC, Hsin Chu, Taiwan.(2003).
[24] L. J. Lin, System of coincidence theorems with applications, J. Math. Anal. Appl. 285(2003), 408-418.
[25] M. Lassonde, On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl. 97(1983), 151-201.
[26] Y. J. Lin and G. Tina, Minimax inequalities equivalent to the Fan-Knaster- Kuratowski-Mazurkiewicz theorem, Appl. Math. Optim. 28(1993), 173-179.
[27] J. von Neumann, Uber ein okonomsiches Gleichungssystem und eine Verallgemeinering des Browerschen Fixpunktsatzes, Ergeb. Math. Kolloq. 8(1937), 73-83.
[28] S. Park, Foundations of the KKM theory via coincidences of composites of upper semi-continuous maps, J. Korean Math. Soc. 31(1994), 164-176.
[29] S. Park and H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal. Appl. 197(1996), 173-187.
[30] S. Park and H. Kim, Functions of the KKM theory on generalized convex spaces, J. Math. Anal. Appl. 209(1997), 551-571.
[31] N. Shioji, A further generalization of the Knaster-Kuratowski-Mazurkiewicz theorem, Proc. Amer. Math. Soc. 111(1991), 187-195.
[32] G. Q. Tina, Generalized KKM theorem, minimax inequalities and their applications, J. Optim. Theory Appl. 83(1994), 375-389.
[33] G. Tina and J. Zhou, Transfer continuities, generalizations of the Weierstrass and maximum theorems: a full characterization, J. Math. Econom. 24(1995), 281-303.
[34] X. Z. Yuan, KKM Theorem and application in nonlinear analysis, Marcel Dekker, New York, 1999, 189-192.