• Journal of the Chungcheong Mathematical Society
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• v.26 no.3
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• pp.533-539
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• 2013

In this paper, using the E-convexity, we introduce the generalized E-KKM map, and next prove the generalized E-KKM theorem which generalizes the KKM theorem due to Fan [4]. As an application, a fixed point theorem in E-convex sets is given.

• Bulletin of the Korean Mathematical Society
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• v.47 no.5
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• pp.1037-1040
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• 2010

For convex subsets X of a topological vector space E, we show that a KKM principle implies a Fan-Browder type fixed point theorem and that this theorem implies generalized forms of the Sion minimax theorem.

• 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.

• 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.

• East Asian mathematical journal
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• v.25 no.4
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• pp.505-516
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• 2009

In this paper, we consider a new class of generalized mixed vector equilibrium-like problems in Banach spaces. By using Fan-KKM Theorem and Nadler's Theorem, we prove the existence theorem of solution for this class of generalized mixed vector equilibrium-like problems.

• 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.

• 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.

• 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.

• Honam Mathematical Journal
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• v.29 no.3
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• pp.455-464
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• 2007

We consider existence theorems for a new class of mixed vector F-implicit complementarity problems and the corresponding mixed vector F-implicit variational inequality problems.

• 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.