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Generalized Knaster-Kuratowski-Mazurkiewicz type theorems and applications to minimax inequalities

Linh Manh Ha 1, *
  1. Department of Mathematics, Vietnam National University-HoChiMinh City, University of Information Technology, Thu Duc district, Saigon, Vietnam
Correspondence to: Linh Manh Ha, Department of Mathematics, Vietnam National University-HoChiMinh City, University of Information Technology, Thu Duc district, Saigon, Vietnam. Email: Nghiado@sci.edu.vn.
Volume & Issue: Vol. 20 No. K2 (2017) | Page No.: 131-140 | DOI: 10.32508/stdj.v20iK2.459
Published: 2017-06-30

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Copyright The Author(s) 2023. This article is published with open access by Vietnam National University, Ho Chi Minh city, Vietnam. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0) which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. 

Abstract

Knaster-Kuratowski-Mazurkiewicz type theorems play an important role in nonlinear analysis, optimization, and applied mathematics. Since the first well-known result, many international efforts have been made to develop sufficient conditions for the existence of points intersection (and their applications) in increasingly general settings: Gconvex spaces [21, 23], L-convex spaces [12], and FCspaces [8, 9]. Applications of Knaster-Kuratowski-Mazurkiewicz type theorems, especially in existence studies for variational inequalities, equilibrium problems and more general settings have been obtained by many authors, see e.g. recent papers [1, 2, 3, 8, 18, 24, 26] and the references therein. In this paper we propose a definition of generalized KnasterKuratowski-Mazurkiewicz mappings to encompass R-KKM mappings [5], L-KKM mappings [11], T-KKM mappings [18, 19], and many recent existing mappings. Knaster-KuratowskiMazurkiewicz type theorems are established in general topological spaces to generalize known results. As applications, we develop in detail general types of minimax theorems. Our results are shown to improve or include as special cases several recent ones in the literature.

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