Intermediate value theorem for simplices for simplicial approximation of fixed points and zeros

Two short proofs of a recently proposed intermediate value theorem for simplices are given. The obtained proofs are based on Sperner covering principles. Furthermore, this intermediate value theorem is applied for the localization and approximation of fixed points and zeros of continuous mappings us...

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Bibliographic Details
Published in:Topology and its applications 2020-04, Vol.275, p.107036, Article 107036
Main Author: Vrahatis, Michael N.
Format: Article
Language:English
Subjects:
Bolzano theorem
Bolzano-Poincaré-Miranda theorem
Brouwer fixed point theorem
Equilibria
Intermediate value theorems
Knaster-Kuratowski-Mazurkiewicz lemma
Mathematical economics
Simplicial approximation
Sperner covering
Sperner lemma
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Summary:Two short proofs of a recently proposed intermediate value theorem for simplices are given. The obtained proofs are based on Sperner covering principles. Furthermore, this intermediate value theorem is applied for the localization and approximation of fixed points and zeros of continuous mappings using a simplicial subdivision of a simplex. Also, a theorem for the existence of a Sperner simplex (panchromatic simplex) in the considered simplicial subdivision is proved. In addition, an error estimate is presented.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2019.107036