設X是一個非空G-凸空間，Y是一個拓樸空間，F屬於G-KKM(X,Y)，Q是一個從Y映到2的X次方的Φ-函數。在某些假設條件之下，我們證得F與Q的一些同質點定理。我們也證明了一些推廣型G-KKM定理，並利用這些推廣型G-KKM定理證明一些變分不等式的存在性定理。本文的結果推廣了許多學者的研究結果。

Let X be a nonempty G-convex space, let Y be a topological space, let F in G-KKM(X,Y) , and let Q is a set-valued mapping from Y into X be a Φ-mapping. In this paper, we establish some coincidence theorems of F and Q under some assumptions. We also establish some generalized G-KKM theorems and apply these generalized G-KKM theorems to establish the existence theorems concerning variational inequalities. Our results generalize many other authors’ results (for example, see, [7,13,20,23]).

[1] Q. H. Ansari, A. Idzik, and J. C. Yao, Coincidence and fixed point theorems with applications, Topol. Methods Nonlinear Anal. 15(2000), 191-202.
[2] 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.
[3] K. C. Border, Fixed point theorems with applications to economics and game theory, Cambridge University Press, 1989.
[4] Mircea Balaj, Weakly G-KKM mappings, G-KKM property, and minimax inequalities, J. Math. Anal. Appl. 294(2004), 237-245.
[5] S. S. Chang and Y. Zhang, Generalized KKM theorem and variational inequalities, J. Math. Anal. Appl. 159(1991), 208-233.
[6] S. S. Chang, B. S. Lee, X. Wu, Y. J. Cho, and G. M. Lee, On the generalized quasivariational inequality problems, J. Math. Anal. Appl. 203(1996), 686-711.
[7] T. H. Chang and C. L. Yen, KKM property and fixed point theorems, J. Math. Anal. Appl. 203(1996), 224-235.
[8] P. Deguire and M. Lassonde, Familles selectantes, Topol. Methods Nonlinear Anal. 5(1995), 261-269.
[9] 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.
[10] X. P. Ding, Best approximation and coincidence theorems, J. Sichuan Normal Univ. Nat. Sci. 18(1995), 21-29.
[11] X. P. Ding, Coincidence theorems in topological spaces and their applications, applied. Math. Lett. 12(1999), 99-105.
[12] X. P. Ding, Existence of solutions for quasi-equilibrium problems in noncompact topological spaces, Comput. Math. Appl. 39(2000), 13-21.
[13] 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.
[14] K. Fan, A generalization of Tychonoff’s fixed point theorem, Math. Ann. 142(1961),305-310.
[15] A. Granas and F. C. Liu, Coincidence for set valued maps and inequalities, J. Math. Anal. Appl. 165(1986), 119-148.
[16] B. Knaster, C. Kuratowski, and S. Mazurkiewicz, Ein Beweis des Fixpunksatzes fur n-dimensionale simplexe, Fund. Math. 14(1929),132-137.
[17] 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.
[18] L. J. Lin, A KKM type theorem and its applications, Bull. Austral. Math. Soc. 59(1999), 481-493.
[19] L. J. Lin, Applications of a fixed point theorem in G-convex space, Nonlinear Anal. 46(2001), 601-608.
[20] L. J. Lin, Q. H. Ansari, and J. Y. Wu, Geometric properties and coincidence theorems with applications to generalized vector equilibrium problems, J. Optim. Theory Appl. 117(1)(2003), 121-137.
[21] L. J. Lin and H. I. Chen, Coincidence theorems for family of multimaps and their applications to equilibrium problems, J. Abstr. Anal. 5(2003), 295-305.
[22] L. J. Lin, System of coincidence theorems with applications, J. Math. Anal. Appl. 285(2003), 408-418.
[23] 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.
[24] M. Lassonde, On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl. 97(1983), 151-201.
[25] Y. J. Lin and G. Tina, Minimax inequalities equivalent to the Fan-Knaster- Kuratowski-Mazurkiewicz theorem, Appl. Math. Optim. 28(1993), 173-179.
[26] J. von Neumann, Uber ein okonomsiches Gleichungssystem und eine Verallgemeinering des Browerschen Fixpunktsatzes, Ergeb. Math. Kolloq. 8(1937), 73-83.
[27] S. Park, Foundations of the KKM theory via coincidences of composites of upper semi-continuous maps, J. Korean Math. Soc. 31(1994), 164-176.
[28] S. Park and H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal. Appl. 197(1996), 173-187.
[29] S. Park and H. Kim, Functions of the KKM theory on generalized convex spaces, J. Math. Anal. Appl. 209(1997), 551-571.
[30] N. Shioji, A further generalization of the Knaster-Kuratowski-Mazurkiewicz theorem, Proc. Amer. Math. Soc. 111(1991), 187-195.
[31] G. Q. Tina, Generalized KKM theorem, minimax inequalities and their applications, J. Optim. Theory Appl. 83(1994), 375-389.
[32] G. Tina and J. Zhou, Transfer continuities, generalizations of the Weierstrass and maximum theorems: a full characterization, J. Math. Econom. 24(1995), 281-303.
[33] Z. T. Yu and L. J. Lin, Continuous selection and fixed point theorems, Nonlinear Anal. 52(2003), 445-453.