• 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.

• Nonlinear Functional Analysis and Applications
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• v.26 no.1
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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.

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

• Nonlinear Functional Analysis and Applications
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• v.26 no.1
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• pp.165-175
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• 2021

In this article we derive some best approximation theorems for multimaps in abstract convex metric spaces. We are based on generalized KKM maps due to Kassay-Kolumbán, Chang-Zhang, and studied by Park, Kim-Park, Park-Lee, and Lee. Our main results are extensions of a recent work of Mitrović-Hussain-Sen-Radenović on G-convex metric spaces to partial KKM metric spaces. We also recall known works related to single-valued maps, and introduce new partial KKM metric spaces which can be applied our new results.

• Journal of the Korean Mathematical Society
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• v.44 no.2
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• pp.393-406
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• 2007

In the framework of generalized convex spaces, we show that generalized KKM maps can be regarded as ordinary KKM maps, and obtain some applications to equilibrium result, maximal element theorems, and almost fixed point theorems on multimaps of the Zima type.

• Journal of the Chungcheong Mathematical Society
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• v.34 no.4
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• pp.345-353
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• 2021

In this paper, first, we present new fixed point theorems for Mönch type multimaps on abstract convex uniform spaces and, also, a fixed point theorem for Mönch type multimaps in Hausdorff KKM L𝚪-spaces. Second, we show that Mönch type multimaps in the better admissible class defined on an L𝚪-space have fixed point properties whenever their ranges are Klee approximable. Finally, we obtain fixed point theorems on 𝔎ℭ-maps whose ranges are 𝚽-sets.

• Journal of the Korean Mathematical Society
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• v.32 no.2
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• pp.251-264
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• 1995

Applications of the classical Knaster-Kuratowski-Mazurkiewicz (si-mply, KKM) theorem 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 [L1]. In this direction, the first author [P5] found that certain coincidence theorems on a large class of composites of upper semicontinuous multifunctions imply many fundamental results in the KKM theory.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.305-320
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• 2013

Topological semilattices with path-connected intervals are special abstract convex spaces. In this paper, we obtain generalized KKM type theorems and their analytic formulations, maximal element theorems and collectively fixed point theorems on abstract convex spaces. We also apply them to topological semilattices with path-connected intervals, and obtain generalized forms of the results of Horvath and Ciscar, Luo, and Al-Homidan et al..

• Journal of applied mathematics & informatics
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• v.40 no.3_4
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• pp.577-585
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• 2022

In this work, we introduced and study vector equilibrium problems for trifunction in measurable space (for short, VEPMS). The existence of solutions of (VEPMS) are obtained by employing Aumann theorem and Fan KKM lemma. As an application, we prove an existence result for vector variational inequality problem for measurable space. Our results in this paper are new which can be considered as significant extension of previously known results in the literature.