Remarks on simplicial spaces and \(L^*\)-spaces of Kulpa and Szymanski (Q2910134)
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scientific article; zbMATH DE number 6079058
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Remarks on simplicial spaces and \(L^*\)-spaces of Kulpa and Szymanski |
scientific article; zbMATH DE number 6079058 |
Statements
7 September 2012
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abstract convex space
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KKM principle
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matching property
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fixed point theorems (of Schauder, Tychonoff, Kakutani, and Fan-Browder)
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minimax theorem
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Nash equilibrium theorem
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Gale-Nikaido-Debreu theorem
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Fan minimax inequality
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Remarks on simplicial spaces and \(L^*\)-spaces of Kulpa and Szymanski (English)
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Starting from the recent paper [\textit{W. Kulpa} and \textit{A. Szymanski}, ``Applications of general infimum principles to fixed-point theory and game theory'', Set-Valued Anal. 16, No. 4, 375--398 (2008; Zbl 1158.91423)], where a series of theorems called Infimum Principles were introduced and applied to some classical results (like fixed point theorems, minimax theorems, Nash equilibrium theorem), in this paper the author recalls that, for any abstract convex space satisfying an abstract KKM principle, some classical theorems can be obtained without using any Infimum Principle. In addition, it is noted that the \(L^*\)-spaces defined in [loc. cit.] are particular types of abstract convex spaes satisfying the abstract KKM principle.
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