• Title/Summary/Keyword: Fan-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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    • v.7 no.1
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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.

ON THE EXISTENCE OF SOLUTIONS FOR SOME VARIATIONAL-LIKE INEQUALITIES

  • Lee, Byung-Soo;Jung, Jong-Soo
    • Communications of the Korean Mathematical Society
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    • v.14 no.2
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    • pp.319-328
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    • 1999
  • In this paper we consider a kind of Minty`s lemma for multifunctions in Banach spaces, and apply it to obtain existence theorems for two kinds of variational-like inequalities using the KKM-Fan theorem.

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HYBRID-TYPE SET-VALUED VARIATIONAL-LIKE INEQUALITIES IN REFLEXIVE BANACH SPACES

  • Lee, Byung-Soo;Khan, Mohd. Firdosh;Salahuddin, Salahuddin
    • Journal of applied mathematics & informatics
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    • v.27 no.5_6
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    • pp.1371-1379
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    • 2009
  • In this paper, we introduce a relaxed hybrid-type$\eta$-f-${\alpha}$-pseudomo-notonicity. By using the KKM-technique, we establish some existence results for set-valued variational-like inequalities with $\eta-f-\alpha$-pseudomonotone, relaxed $\eta-f-\alpha$-pseudomonotone, Fan-KKM Theorem.

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REMARKS ON THE KKM STRUCTURES OF KHANH AND QUAN

  • Sehie Park
    • Nonlinear Functional Analysis and Applications
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    • v.28 no.1
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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.

MATCHING THEOREMS AND SIMULTANEOUS RELATION PROBLEMS

  • Balaj, Mircea;Coroianu, Lucian
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.5
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    • pp.939-949
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    • 2011
  • In this paper we give two matching theorems of Ky Fan type concerning open or closed coverings of nonempty convex sets in a topological vector space. One of them will permit us to put in evidence, when X and Y are convex sets in topological vector spaces, a new subclass of KKM(X, Y) different by any admissible class $\mathfrak{u}_c$(X, Y). For this class of set-valued mappings we establish a KKM-type theorem which will be then used for obtaining existence theorems for the solutions of two types of simultaneous relation problems.

MULTIVALUED MIXED QUASI-VARIATIONAL-LIKE INEQUALITIES

  • Lee Byung-Soo
    • The Pure and Applied Mathematics
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    • v.13 no.3 s.33
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    • pp.197-206
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    • 2006
  • This paper introduces a class of multivalued mixed quasi-variational-like ineqcalities and shows the existence of solutions to the class of quasi-variational-like inequalities in reflexive Banach spaces.

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