Abstract
Recently we established the Knaster, Kuratowski, and Mazurkiewicz (KKM) theory on abstract convex spaces. In our research, we noticed that there are quite a few generalizations and applications of the 1984 KKM theorem of Ky Fan compared with his celebrated 1961 KKM lemma. In a certain sense, the relationship between the 1984 theorem and hundreds of known generalizations of the original KKM theorem has not been recognized for a long period. There would be some reasons to explain this fact. Instead, in this paper, we give several generalizations of the 1984 theorem and some known applications in order to reveal the close relationship among them.MSC: 47H04, 47H10, 47J20, 47N10, 49J53, 52A99, 54C60, 54H25, 58E35, 90C47, 91A13, 91B50.
Highlights
One of the earliest equivalent formulations of the Brouwer fixed point theorem of is the theorem of Knaster, Kuratowski, and Mazurkiewicz of [ ], which was concerned with a particular type of multimaps, later called KKM maps
In our another review [ ], we recalled our versions of general KKM type theorems for abstract convex spaces and introduced relatively recent applications of various generalized KKM type theorems due to other authors in the twenty-first century
In Section of this paper, we introduce the recent concepts of abstract convex spaces and partial KKM spaces and the modern versions of some most general KKM type theorems
Summary
One of the earliest equivalent formulations of the Brouwer fixed point theorem of is the theorem of Knaster, Kuratowski, and Mazurkiewicz (the KKM theorem for short) of [ ], which was concerned with a particular type of multimaps, later called KKM maps. Definition The partial KKM principle for an abstract convex space (E, D; ) is the statement E ∈ KC(E, D, E); that is, for any closed-valued KKM map G : D E, the family {G(y)}y∈D has the finite intersection property. Theorem A Let (E, D; ) be an abstract convex space, the identity map E ∈ KC(E, D, E) (resp., E ∈ KO(E, D, E)), and G : D E be a multimap satisfying ( ) G has closed (resp., open) values; and ( ) N ⊂ G(N) for any N ∈ D (that is, G is a KKM map). D ⊂ D such that N ⊂ D and LN ∩ G(y) ⊂ K
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