• 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.

• Bulletin of the Korean Mathematical Society
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• v.27 no.2
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• pp.151-155
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• 1990

Recently, H.Z.Ming [7] obtained a necessary and sufficient condition for the existence of a solution to a general operator equation. In the present paper, we obtain such conditions in general forms and give some examples. We begin with the well-known Fan-Browder fixed point theorem, from which we deduce two general theorems on such necessary and sufficient conditiions. We give some examples of such conditions, which are improved versions of fixed point theorems of Halpern-Bergman [5], Ky Fan [3], [4], Kaczynski [6], Reich [9], Schauder [10], Tychonoff [11], and Ming [7]. In fact, we restate Ming's result in its correct form. The following is known as the Fan-Browder fixed point theorem [1], [2].

• 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.

• Bulletin of the Korean Mathematical Society
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• v.30 no.1
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• pp.83-89
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• 1993

Applications of the classical Knaster-Kuratowski-Mazurkiewicz theorem [KKM] and the fixed point theory of multifunctions defined on convex subsets of topological vector spaces have been greatly improved by adopting the concept of convex spaces due to Lassonde[L]. Recently, this concept has been extended to pseudo-convex spaces, contractible spaces, or spaces having certain families of contractible subsets by Horvath[H1-4]. In the present paper we give a far-reaching generalization of the best approximation theorem of Ky Fan[F1, 2] to pseudo-metric spaces and improved versions of the well-known fixed point theorems due to Brouwer [B] and Schauder [S] for spaces having certain families of contractible subsets. Our basic tool is a generalized Fan-Browder type fixed point theorem in our previous works [P3, 4].

• Journal of the Korean Mathematical Society
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• v.37 no.6
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• pp.885-899
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• 2000

We obtain generalized versions of the Fan-Browder fixed point theorem for G-convex spaces. We define the class B of better admissible multimaps on G-convex spaces and show that any closed compact map in b fro ma locally G-convex uniform space into itself has a fixed point.

• Journal of the Korean Mathematical Society
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• v.45 no.1
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• pp.1-27
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• 2008

We introduce a new concept of convex spaces and a multimap class K having certain KKM property. From a basic KKM type theorem for a K-map defined on an convex space without any topology, we deduce ten equivalent formulations of the theorem. As applications of the equivalents, in the frame of convex topological spaces, we obtain Fan-Browder type fixed point theorems, almost fixed point theorems for multimaps, mutual relations between the map classes K and B, variational inequalities, the von Neumann type minimax theorems, and the Nash equilibrium theorems.

• Communications of the Korean Mathematical Society
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• v.21 no.4
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• pp.665-673
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• 2006

In a recent paper, Domokos and $Kolumb\'{a}}n$ [2] gave an interesting interpretation of variational inequalities (VI) and vector variational inequalities (VVI) in Banach space settings in terms of variational inequalities with operator solutions (in short, OVVI). Inspired by their work, in a former paper [4], we proposed the scheme of generalized vector variational inequality with operator solutions (in short, GOVVI) which extends (OVVI) into a multivalued case. In this note, we further develop the previous work [4]. A more general pseudomonotone operator is treated. We present a result on the existence of solutions of (GVVI) under the weak pseudomonotonicity introduced in Yu and Yao [8] within the framework of (GOVVI) by exploiting some techniques on (GOVVI) or (GVVI) in [4].