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

• Journal of the Chungcheong Mathematical Society
• /
• v.27 no.1
• /
• pp.105-112
• /
• 2014

In this paper, we first introduce an R-E-KKM map in the E-convex settings, and next we prove an R-E-KKM theorem which generalizes the KKM theorem and the best proximity theorem simultaneously. As applications, a best proximity theorem and a fixed point theorem in E-convex sets are given.

• Journal of applied mathematics & informatics
• /
• v.7 no.1
• /
• pp.1-28
• /
• 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
• /
• v.47 no.5
• /
• pp.1037-1040
• /
• 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 the Korean Mathematical Society
• /
• v.42 no.1
• /
• pp.101-110
• /
• 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.

• Nonlinear Functional Analysis and Applications
• /
• v.28 no.1
• /
• pp.123-134
• /
• 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
• /
• v.53 no.5
• /
• pp.1363-1371
• /
• 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.

• Nonlinear Functional Analysis and Applications
• /
• v.26 no.1
• /
• pp.93-104
• /
• 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.

• Communications of the Korean Mathematical Society
• /
• v.25 no.2
• /
• pp.303-311
• /
• 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.

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