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.
@article{bwmeta1.element.bwnjournal-article-div19i1-2n2bwm,
author = {George Isac and George Xian-Zhi Yuan},
title = {The dual form of Knaster-Kuratowski-Mazurkiewicz principle in hyperconvex metric spaces and some applications},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
volume = {19},
year = {1999},
pages = {17-33},
zbl = {0962.47022},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-div19i1-2n2bwm}
}
George Isac; George Xian-Zhi Yuan. The dual form of Knaster-Kuratowski-Mazurkiewicz principle in hyperconvex metric spaces and some applications. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 19 (1999) pp. 17-33. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-div19i1-2n2bwm/
[000] [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.
[001] [2] N. Aronszajn and P. Panitchpakdi, Extensions of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405-439. | Zbl 0074.17802
[002] [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. | Zbl 0667.49016
[003] [4] F.E. Browder, The fixed point theory of multivalued mappings in topological vector spaces, Math. Ann. 177 (1968), 283-301. | Zbl 0176.45204
[004] [5] S.S. Chang and Y. Zhang, Generalized KKM theorem and variational inequalities, J. Math. Anal. Appl. 159 (1993), 208-223. | Zbl 0739.47026
[005] [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. | Zbl 0833.49009
[006] [7] J. Dugundji and A. Granas, Fixed Point Theory 1, PWN Warszawa 1982.
[007] [8] K. Fan, A generalization of Tychonoff's fixed point theorem, Math. Ann. 142 (1961), 305-310. | Zbl 0093.36701
[008] [9] K. Fan, Extensions of two fixed point theorems of F.E. Browder, Math. Z. 112 (1969), 234-240. | Zbl 0185.39503
[009] [10] K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), 519-537. | Zbl 0515.47029
[010] [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.
[011] [12] C. Horvath, Contractibility and generalized convexity, J. Math. Anal. Appl. 156 (1991) 341-357. | Zbl 0733.54011
[012] [13] J.R. Isbell, Six theorems about injective metric spaces, Comm. Math. Helvetici 39 (1964), 65-76. | Zbl 0151.30205
[013] [14] J.L. Kelley, Banach spaces with the extension property, Trans. Amer. Math. Soc. 72 (1952), 323-326. | Zbl 0046.12002
[014] [15] M.A. Khamsi, KKM and Ky Fan theorems in hyperconvex metric spaces, J. Math. Anal. Appl. 204 (1996), 298-306. | Zbl 0869.54045
[015] [16] B. Knaster, C. Kuratowski and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes für n-dimensional simplexe, Fund. Math. 14 (1929), 132-137. | Zbl 55.0972.01
[016] [17] W.K. Kim, Some applications of the Kakutani fixed point theorems, J. Math. Anal. Appl. 121 (1987), 119-122. | Zbl 0612.54055
[017] [18] W.A. Kirk and S.S. Shin, Fixed point theorems in hyperconvex spaces, Houston J. Math. 23 (1997), 175-187. | Zbl 0957.46033
[018] [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). | Zbl 1068.47072
[019] [20] V.L. Klee, On certain intersection properties of convex sets, Canad. J. Math. 3 (1951), 272-275. | Zbl 0042.40701
[020] [21] H.E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer Verlag, New York 208 (1974). | Zbl 0285.46024
[021] [22] M. Lassonde, On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl. 97 (1983), 151-201. | Zbl 0527.47037
[022] [23] M. Lassonde, Sur le principle KKM, C.R. Acad. Sci. Paris. Série I. 310 (1990), 573-576. | Zbl 0715.47038
[023] [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. | Zbl 0747.47045
[024] [25] L. Nachbin, A theorem of Hahn-Banach type for linear transformations, Trans. Amer. Math. Soc. 68 (1960), 28-54.
[025] [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.
[026] [27] S. Park, Fixed point theorems in hyperconvex metric spaces, Nonlinear Anal., T.M.A. 37 (1999), 467-472. | Zbl 0930.47023
[027] [28] M.H. Shih, Covering properties of convex sets, Bull. London Math. Soc. 18 (1986), 57-59. | Zbl 0579.52004
[028] [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.
[029] [30] R.C. Sine, On nonlinear contraction semigroups in Sup-norm spaces, Nonlinear Anal., T.M.A. 3 (1979), 885-890. | Zbl 0423.47035
[030] [31] R.C. Sine, Hyperconvexity and nonexpansive multifunctions, Trans. Amer. Math. Soc. 315 (1989), 755-767. | Zbl 0682.47029
[031] [32] R.C. Sine Hyperconvexity and approximate fixed points, Nonlinear Anal., T.M.A. 13 (1989), 863-869. | Zbl 0694.54033
[032] [33] P.M. Soardi, Existence of fixed points of nonexpansinve mappings in certain Banach lattices, Proc. Amer. Math. Soc. 73 (1979), 25-29. | Zbl 0371.47048
[033] [34] E. Tarafdar, A fixed point theorem equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz theorem, J. Math. Anal. Appl. 128 (1987), 475-479. | Zbl 0644.47050
[034] [35] G.X.Z. Yuan, Fixed points of upper semicontinuous mappings in locally G-convex spaces, Bull. Austral. Math. Soc. 58 (1998), 469-478. | Zbl 0927.47034
[035] [36] G.X.Z. Yuan, KKM Theory and Applications in Nonlinear Analysis, Marcel Dekker, New York 1999.
[036] [37] E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed Point Theorems, Springer Verlag, New York 1986. | Zbl 0583.47050