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

• 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.

• 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.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1363-1371
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• 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.

• 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 the Korean Mathematical Society
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• v.42 no.1
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• pp.101-110
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• 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
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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 applied mathematics & informatics
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• v.29 no.5_6
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• pp.1453-1462
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• 2011

In this paper, we study a new class of generalized nonlinear variational-like inequalities in reflexive Banach spaces. By using the KKM technique and the concept of the Hausdorff metric, we obtain some existence results for generalized nonlinear variational-like inequalities with generalized monotone multi-valued mappings in Banach spaces. These results improve and generalize many known results in recent literature.

• Communications of the Korean Mathematical Society
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• v.21 no.4
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• pp.689-700
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• 2006

In this paper, we introduce a new class of generalized nonlinear multivalued mixed quasi-variational-like inequalities and prove the existence and uniqueness of solutions for the class of generalized nonlinear multivalued mixed quasi-variational-like inequalities in reflexive Banach spaces using Fan-KKM Theorem.