The rise and fall of L-spaces

Abstract

For a quite long period, the so-called L-structure or L-spaces have been studied by some authors. They have several trivial misconceptions such as their L-spaces extend the well-known generalized convex (G-convex) spaces. In order to clarify this matter and others, we show that our KKM theory on abstract convex spaces improves typical results in L-spaces. Main topics in this paper are related to extensions of the Himmelberg fixed point theorem. Since such studies are beyond of L-spaces, we cordially claim that now is the proper time to give up the useless study on L-spaces and their variants FC-spaces.

The rise and fall of L-spaces, II

“…In our previous work [34] in 2013, we showed that better forms of theorems in [5,7,8,9,11] can be deduced from a KKM theorem on abstract convex spaces in our sense [2327]. Moreover, in our recent work [38], we showed that our KKM theory on abstract convex spaces improves and extends all of typical results on L-spaces. Since such studies are beyond of L-spaces, we cordially claimed that now is the proper time to…”

The rise and fall of MC-spaces

“…The present article is a continuation of our previous works [34], [35] on the rise and fall of the L-spaces due to Ben-El-Mechaiekh, Chebbi, Florenzano, and Llinares [1] in 1998. Since then there have appeared some people concentrating mainly on L-spaces and they can be adequately called L-space theorists.…”

“…In [34], we showed that our KKM theory on abstract convex spaces improved typical results on L-spaces. Main topics there were related to extensions of the Himmelberg fixed point theorem.…”

“…The present article is a continuation of our previous [34], [35] and concerns with the rise and fall of mc-spaces. This will be an important record for the history of KKM theory.…”

“…In [35], various types of coercing families initiated by Ben-El-Mechaiekh, Chebbi, and Florenzano [2] are unified by a single coercivity condition. We showed that better forms of results using several coercing families can be deduced from a general KKM theorem on abstract convex spaces in our previous works.…”