• 제목/요약/키워드: KKM-theorem

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ELEMENTS OF THE KKM THEORY FOR GENERALIZED CONVEX SPACE

  • Park, Se-Hei
    • Journal of applied mathematics & informatics
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    • 제7권1호
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    • pp.1-28
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    • 2000
  • In the present paper, we introduce fundamental results in the KKM theory for G-convex spaces which are equivalent to the Brouwer theorem, the Sperner lemma, and the KKM theorem. Those results are all abstract versions of known corresponding ones for convex subsets of topological vector spaces. Some earlier applications of those results are indicated. Finally, We give a new proof of the Himmelberg fixed point theorem and G-convex space versions of the von Neumann type minimax theorem and the Nash equilibrium theorem as typical examples of applications of our theory.

REMARKS ON THE KKM PROPERTY FOR OPEN-VALUED MULTIMAPS ON GENERALIZED CONVEX SPACES

  • KIM HOONJOO;PARK SEHIE
    • 대한수학회지
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    • 제42권1호
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    • pp.101-110
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    • 2005
  • Let (X, D; ${\Gamma}$) be a G-convex space and Y a Hausdorff space. Then $U^K_C$(X, Y) ${\subset}$ KD(X, Y), where $U^K_C$ is an admissible class (dup to Park) and KD denotes the class of multimaps having the KKM property for open-valued multimaps. This new result is used to obtain a KKM type theorem, matching theorems, a fixed point theorem, and a coincidence theorem.

REMARKS ON THE KKM STRUCTURES OF KHANH AND QUAN

  • Sehie Park
    • Nonlinear Functional Analysis and Applications
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    • 제28권1호
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    • pp.123-134
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    • 2023
  • Since Knaster, Kuratowski, and Mazurkiewicz established their KKM theorem in 1929, it was first applied to topological vector spaces mainly by Fan and Granas. Later it was extended to convex spaces by Lassonde and to extensions of c-spaces by Horvath. In 1992, such study was called the KKM theory by ourselves. Then the theory was extended to generalized convex spaces or G-convex spaces. Motivated by such spaces, there have appeared several particular types of artificial spaces. In 2006 we introduced abstract convex spaces which contain all existing spaces appeared in the KKM theory. Later in 2014-2020, Khahn and Quan introduced "topologically based existence theorems" and the so-called KKM structure. In the present paper, we show that their structure is a particular type of already known KKM spaces.

GENERALIZED KKM-TYPE THEOREMS FOR BEST PROXIMITY POINTS

  • Kim, Hoonjoo
    • 대한수학회보
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    • 제53권5호
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    • pp.1363-1371
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    • 2016
  • This paper is concerned with best proximity points for multimaps in normed spaces and in hyperconvex metric spaces. Using the generalized KKM theorem, we deduce new best proximity pair theorems for a family of multimaps with unionly open fibers in normed spaces. And we prove a new best proximity point theorem for quasi-lower semicontinuous multimaps in hyperconvex metric spaces.

COLLECTIVE FIXED POINTS FOR GENERALIZED CONDENSING MAPS IN ABSTRACT CONVEX UNIFORM SPACES

  • Kim, Hoonjoo
    • Nonlinear Functional Analysis and Applications
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    • 제26권1호
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    • pp.93-104
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    • 2021
  • In this paper, we present a fixed point theorem for a family of generalized condensing multimaps which have ranges of the Zima-Hadžić type in Hausdorff KKM uniform spaces. It extends Himmelberg et al. type fixed point theorem. As applications, we obtain some new collective fixed point theorems for various type generalized condensing multimaps in abstract convex uniform spaces.

COMMENTS ON GENERALIZED R-KKM TYPE THEOREMS

  • Park, Se-Hie
    • 대한수학회논문집
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    • 제25권2호
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    • pp.303-311
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    • 2010
  • Recently, some authors [3, 4, 11, 12, 15] adopted the concept of the so-called generalized R-KKM maps which are used to rewrite known results in the KKM theory. In the present paper, we show that those maps are simply KKM maps on G-convex spaces. Consequently, results on generalized R-KKM maps follow the corresponding previous ones on G-convex spaces.