Nonlinear Functional Analysis and Applications

Kyungnam University, Department of Mathematics Eduaction (경남대학교 수학교육과)
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REMARKS ON THE KKM STRUCTURES OF KHANH AND QUAN

  • Sehie Park (The National Academy of Sciences, Department of Mathematical Sciences, Seoul National University)
  • Received : 2022.03.07
  • Accepted : 2022.08.05
  • Published : 2023.03.03
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Abstract

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.

Keywords

Acknowledgement

In this occasion, the author would like to express his sincere gratitude to Professor Jong Kyu Kim for his almost thirty years contribution to the journal Nonlinear Functional Analysis and Applications(NFAA) as Chief of Editors.

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