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1999 | 19 | 1-2 | 17-33

Tytuł artykułu

The dual form of Knaster-Kuratowski-Mazurkiewicz principle in hyperconvex metric spaces and some applications

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In this paper, we first establish the dual form of Knaster- Kuratowski-Mazurkiewicz principle which is a hyperconvex version of corresponding result due to Shih. Then Ky Fan type matching theorems for finitely closed and open covers are given. As applications, we establish some intersection theorems which are hyperconvex versions of corresponding results due to Alexandroff and Pasynkoff, Fan, Klee, Horvath and Lassonde. Then Ky Fan type best approximation theorem and Schauder-Tychonoff fixed point theorem for set-valued mappings (i.e., Fan-Glicksberg fixed point theorem) in hyperconvex spaces are also developed, and finally one unified form of Browder-Fan fixed point theorem for set-valued mappings in hyperconvex spaces is given. These results include corresponding results in the literature due to Khamsi, Kirk and Shin, Kirk et al. as special cases.


  • Department of Mathematics and Computer Sciences, Royal Military College of Canada, Kingston, Ont. Canada K7K 5L0
  • Department of Mathematics, The University of Queensland, Brisbane, Australia 4072


  • [1] P. Alexandroff and B. Pasynkoff, Elementary proof of the essentiality of the identical mapping of a simplex (in Russian), Uspehi Mat. Nauk (N.S.) 12 (1957), 175-179.
  • [2] N. Aronszajn and P. Panitchpakdi, Extensions of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405-439.
  • [3] C. Bardaro and R. Ceppitelli, Some further generalizations of Knaster-Kuratowski-Mazurkiewicz theorem and minimax inequalities, J. Math. Anal. Appl. 132 (1988), 484-490.
  • [4] F.E. Browder, The fixed point theory of multivalued mappings in topological vector spaces, Math. Ann. 177 (1968), 283-301.
  • [5] S.S. Chang and Y. Zhang, Generalized KKM theorem and variational inequalities, J. Math. Anal. Appl. 159 (1993), 208-223.
  • [6] X.P. Ding and K.K. Tan, A minimax inequality with applications to existence of equilibrium points and fixed point theorems, Coll. Math. 68 (1992), 233-247.
  • [7] J. Dugundji and A. Granas, Fixed Point Theory 1, PWN Warszawa 1982.
  • [8] K. Fan, A generalization of Tychonoff's fixed point theorem, Math. Ann. 142 (1961), 305-310.
  • [9] K. Fan, Extensions of two fixed point theorems of F.E. Browder, Math. Z. 112 (1969), 234-240.
  • [10] K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), 519-537.
  • [11] A. Granas, KKM-maps and their applications to nonlinear problems, The Scottish Book: Mathematics from the Scottish Cafe ed., R. Daniel Mauldin, Birkhäuser, Boston (1982) 45-61.
  • [12] C. Horvath, Contractibility and generalized convexity, J. Math. Anal. Appl. 156 (1991) 341-357.
  • [13] J.R. Isbell, Six theorems about injective metric spaces, Comm. Math. Helvetici 39 (1964), 65-76.
  • [14] J.L. Kelley, Banach spaces with the extension property, Trans. Amer. Math. Soc. 72 (1952), 323-326.
  • [15] M.A. Khamsi, KKM and Ky Fan theorems in hyperconvex metric spaces, J. Math. Anal. Appl. 204 (1996), 298-306.
  • [16] B. Knaster, C. Kuratowski and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für n-dimensional simplexe, Fund. Math. 14 (1929), 132-137.
  • [17] W.K. Kim, Some applications of the Kakutani fixed point theorems, J. Math. Anal. Appl. 121 (1987), 119-122.
  • [18] W.A. Kirk and S.S. Shin, Fixed point theorems in hyperconvex spaces, Houston J. Math. 23 (1997), 175-187.
  • [19] W.A. Kirk, B. Sims and X.Z. Yuan, The Knaster-Kuratowski and Mazurkiewicz theory in hyperconvex metric spaces and some of its applications, Nonlinear Anal., T.M.A. (in press) (1999).
  • [20] V.L. Klee, On certain intersection properties of convex sets, Canad. J. Math. 3 (1951), 272-275.
  • [21] H.E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer Verlag, New York 208 (1974).
  • [22] M. Lassonde, On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl. 97 (1983), 151-201.
  • [23] M. Lassonde, Sur le principle KKM, C.R. Acad. Sci. Paris. Série I. 310 (1990), 573-576.
  • [24] M. Lin and R.C. Sine, Retractions on the fixed point set of semigroup of nonexpansive maps in hyperconvex spaces, Nonlinear Anal., T.M.A. 15 (1990), 943-954.
  • [25] L. Nachbin, A theorem of Hahn-Banach type for linear transformations, Trans. Amer. Math. Soc. 68 (1960), 28-54.
  • [26] S. Park, Some coincidence theorems on acyclic multifunctions and applications to KKM theory, Fixed Point Theory and Applications, Ed. K.K. Tan, World Scientific, Singapore (1992), 248-278.
  • [27] S. Park, Fixed point theorems in hyperconvex metric spaces, Nonlinear Anal., T.M.A. 37 (1999), 467-472.
  • [28] M.H. Shih, Covering properties of convex sets, Bull. London Math. Soc. 18 (1986), 57-59.
  • [29] M.H. Shih and K.K. Tan, Covering theorems of convex sets related to fixed-point theorems, Nonlinear Analysis and Convex Analysis, Eds. B.L. Lin and S. Simons, Marcel Dekker Inc., New York and Basel (1987) 235-244.
  • [30] R.C. Sine, On nonlinear contraction semigroups in Sup-norm spaces, Nonlinear Anal., T.M.A. 3 (1979), 885-890.
  • [31] R.C. Sine, Hyperconvexity and nonexpansive multifunctions, Trans. Amer. Math. Soc. 315 (1989), 755-767.
  • [32] R.C. Sine Hyperconvexity and approximate fixed points, Nonlinear Anal., T.M.A. 13 (1989), 863-869.
  • [33] P.M. Soardi, Existence of fixed points of nonexpansinve mappings in certain Banach lattices, Proc. Amer. Math. Soc. 73 (1979), 25-29.
  • [34] E. Tarafdar, A fixed point theorem equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz theorem, J. Math. Anal. Appl. 128 (1987), 475-479.
  • [35] G.X.Z. Yuan, Fixed points of upper semicontinuous mappings in locally G-convex spaces, Bull. Austral. Math. Soc. 58 (1998), 469-478.
  • [36] G.X.Z. Yuan, KKM Theory and Applications in Nonlinear Analysis, Marcel Dekker, New York 1999.
  • [37] E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed Point Theorems, Springer Verlag, New York 1986.

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