The KKM principle in abstract convex spaces: equivalent formulations and applications. (English) Zbl 1214.47042
This paper is a valuable contribution to KKM theory. Precisely, the author shows that a sequence of a dozen statements characterize the KKM spaces and are equivalent formulations of the partial KKM principle. As applications, the author adds more than a dozen statements including generalized formulations of the von Neumann minimax theorem, the von Neumann intersection lemma, the Nash equilibrium theorem, and the Fan type minimax inequalities for any KKM spaces. Consequently, this paper unifies and enlarges previously known several proper examples of such statements for particular types of KKM spaces.
Reviewer: Long Wei (Jiangxi)
MSC:
| 47H04 | Set-valued operators |
| 47H10 | Fixed-point theorems |
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 46A16 | Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.) |
| 46A55 | Convex sets in topological linear spaces; Choquet theory |
| 49J27 | Existence theories for problems in abstract spaces |
| 49J35 | Existence of solutions for minimax problems |
| 52A07 | Convex sets in topological vector spaces (aspects of convex geometry) |
| 54C60 | Set-valued maps in general topology |
Keywords:
abstract convex space; \(G\)-convex spaces; KKM space; (partial) KKM principle; minimax inequality; minimax theorem; Nash equilibrium point; variational inequalityReferences:
| [1] | Knaster, B.; Kuratowski, K.; Mazurkiewicz, S., Ein Beweis des Fixpunktsatzes für \(n\)-Dimensionale Simplexe, Fund. Math., 14, 132-137 (1929) · JFM 55.0972.01 |
| [2] | Park, S., Some coincidence theorems on acyclic multifunctions and applications to KKM theory, (Tan, K.-K., Fixed Point Theory and Applications (1992), World Scientific Publ.: World Scientific Publ. River Edge, NJ), 248-277 · Zbl 1426.47005 |
| [3] | Park, S., Ninety years of the Brouwer fixed point theorem, Vietnam J. Math., 27, 193-232 (1999) |
| [4] | Sion, M., On general minimax theorems, Pacific J. Math., 8, 171-176 (1958) · Zbl 0081.11502 |
| [5] | Granas, A., Quelques Méthodes topologique èn analyse convexe, (Méthdes topologiques en analyse convexe, Sém. Math. Supér., vol. 110 (1990), Press. Univ. Montréal), 11-77 · Zbl 0745.49015 |
| [6] | Lassonde, M., On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl., 97, 151-201 (1983) · Zbl 0527.47037 |
| [7] | Horvath, C. D., Contractibility and generalized convexity, J. Math. Anal. Appl., 156, 341-357 (1991) · Zbl 0733.54011 |
| [8] | Horvath, C. D., Extension and selection theorems in topological spaces with a generalized convexity structure, Ann. Fac. Sci. Toulouse, 2, 253-269 (1993) · Zbl 0799.54013 |
| [9] | Park, S., Elements of the KKM theory for generalized convex spaces, Korean J. Comput. Appl. Math., 7, 1-28 (2000) · Zbl 0959.47035 |
| [10] | Park, S., On generalizations of the KKM principle on abstract convex spaces, Nonlinear Anal. Forum, 11, 67-77 (2006) · Zbl 1120.47038 |
| [11] | Park, S., Various subclasses of abstract convex spaces for the KKM theory, Proc. Nat. Inst. Math. Sci., 2, 4, 35-47 (2007) |
| [12] | Park, S., Elements of the KKM theory on abstract convex spaces, J. Korean Math. Soc., 45, 1, 1-27 (2008) · Zbl 1149.47040 |
| [13] | Park, S., Equilibrium existence theorems in KKM spaces, Nonlinear Anal., 69, 4352-4364 (2008) · Zbl 1163.47044 |
| [14] | Park, S., New foundations of the KKM theory, J. Nonlinear Convex Anal., 9, 3, 331-350 (2008) · Zbl 1167.47041 |
| [15] | Park, S., Generalized convex spaces, \(L\)-spaces, and \(F C\)-spaces, J. Global Optim., 45, 2, 203-210 (2009) · Zbl 1225.47059 |
| [16] | Park, S., Fixed point theory of multimaps in abstract convex uniform spaces, Nonlinear Anal., 71, 2468-2480 (2009) · Zbl 1222.47078 |
| [17] | Park, S., From the KKM principle to the Nash equilibria, Int. J. Math. Stat., 6, S10, 77-88 (2010) |
| [18] | Park, S., Remarks on the partial KKM principle, Nonlinear Anal. Forum, 14, 1, 1-12 (2009) · Zbl 1484.54049 |
| [19] | Fan, K., A generalization of Tychonoff’s fixed point theorem, Math. Ann., 142, 305-310 (1961) · Zbl 0093.36701 |
| [20] | Reich, S.; Shafrir, I., Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal., 15, 537-558 (1990) · Zbl 0728.47043 |
| [21] | Horvath, C. D.; Llinares-Ciscar, J. V., Maximal elements and fixed points for binary relations on topological ordered spaces, J. Math. Econom., 25, 291-306 (1996) · Zbl 0852.90006 |
| [22] | Park, S., Five episodes related to generalized convex spaces, Nonlinear Funct. Anal. Appl., 2, 49-61 (1997) |
| [23] | Kulpa, W.; Szymanski, A., Applications of general infimum principles to fixed-point theory and game theory, Set-valued Anal., 16, 375-398 (2008) · Zbl 1158.91423 |
| [24] | Khanh, P. Q.; Quan, N. H.; Yao, J. C., Generalized KKM type theorems in GFC-spaces and applications, Nonlinear Anal., 71, 1227-1234 (2009) · Zbl 1176.47041 |
| [25] | Kirk, W. A.; Panyanak, B., Best approximations in \(R\)-trees, Numer. Funct. Anal. Optimiz., 28, 5-6, 681-690 (2007) · Zbl 1132.54025 |
| [26] | Horvath, C. D., Topological convexities, selections and fixed points, Topology Appl., 155, 830-850 (2008) · Zbl 1144.54030 |
| [27] | Briec, W.; Horvath, C., Nash points, Ky Fan inequality and equilibria of abstract economies in Max-Plus and \(B\)-convexity, J. Math. Anal. Appl., 341, 1, 188-199 (2008) · Zbl 1151.91017 |
| [28] | von Neumann, J., Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes, Ergeb. eines Math. Kolloq., 8, 73-83 (1937), [Rev. Econom. Stud. XIII, No.33 (1945-46), 1-9.] · JFM 63.1167.03 |
| [29] | J.P. Torres-Martínez, Fixed points as Nash equilibria, Fixed Point Theory Appl. 2006, 1-4. Article ID 36135.; J.P. Torres-Martínez, Fixed points as Nash equilibria, Fixed Point Theory Appl. 2006, 1-4. Article ID 36135. · Zbl 1150.91303 |
| [30] | Nash, J. F., Equilibrium points in N-person games, Proc. Natl. Acad. Sci. USA, 36, 48-49 (1950) · Zbl 0036.01104 |
| [31] | Nash, J., Non-cooperative games, Ann. of Math., 54, 286-295 (1951) · Zbl 0045.08202 |
| [32] | Gwinner, J., On fixed points and variational inequalities — A circular tour, Nonlinear Anal., TMA, 5, 565-583 (1981) · Zbl 0461.47037 |
| [33] | Horvath, C. D.; Lassonde, M., Intersection of sets with \(n\)-connected unions, Proc. Amer. Math. Soc., 125, 1209-1214 (1997) · Zbl 0861.54020 |
| [34] | Park, S.; Jeong, K. S., Fixed point and non-retract theorems — Classical circular tours, Taiwan. J. Math., 5, 97-108 (2001) · Zbl 0991.47043 |
| [35] | Fan, K., Applications of a theorem concerning sets with convex sections, Math. Ann., 163, 189-203 (1966) · Zbl 0138.37401 |
| [36] | Park, S., Acyclic versions of the von Neumann and Nash equilibrium theorems, J. Comput. Appl. Math., 113, 83-91 (2000) · Zbl 0947.47039 |
| [37] | Park, S., Remarks on acyclic versions of generalized von Neumann and Nash equilibrium theorems, Appl. Math. Lett., 15, 641-647 (2002) · Zbl 1019.46039 |
| [38] | Park, S., A generalized minimax inequality related to admissible multimaps and its applications, J. Korean Math. Soc., 34, 719-730 (1997) · Zbl 0904.49004 |
| [39] | Park, S., Further generalizations of the Gale-Nikaido-Debreu theorem, J. Appl. Math. Comput., 32, 1, 171-176 (2010) · Zbl 1194.47066 |
| [40] | Park, S., A unified fixed point theory in generalized convex spaces, Acta Math. Sin., Engl. Ser., 23, 8, 1509-1536 (2007) · Zbl 1129.47041 |
| [41] | S. Park, Applications of the KKM theory to fixed point theory, J. Nat. Acad. Sci., ROK (in press).; S. Park, Applications of the KKM theory to fixed point theory, J. Nat. Acad. Sci., ROK (in press). |
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