The rise and fall of L-spaces, II

Year 2021, Volume: 5 Issue: 1, 7 - 24, 31.03.2021

Sehie Park

https://doi.org/10.31197/atnaa.847835

Abstract

In 2005, Ben-El-Mechaiekh, Chebbi, and Florenzano obtained a generalization of Ky Fan's 1984 KKM theorem on the intersection of a
family of closed sets on non-compact convex sets in a topological vector space. They also extended the Fan-Browder fixed point theorem
to multimaps on non-compact convex sets. Since then several groups of the L-space theorists introduced coercivity families and applied
them to L-spaces, H-spaces, etc. In this article, we show that better forms of such works can be deduced from a general KKM theorem
on abstract convex spaces in our previous works. Consequently, all of the known KKM theoretic results on L-spaces related coercivity
families are extended to corresponding better forms on abstract convex spaces.

This article is a continuation of our \cite{38} and a revised and extended version of \cite{34}.

Keywords

KKM theorem, Fan's 1961, 1984 KKM theorem, Fan-Browder fixed point theorem, minimax inequality, abstract convex space, (partial) KKM space

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