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The Brouwer Fixed Point Theorem and Related Theorems

The Brouwer Fixed Point Theorem and Related Theorems
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On the equivalence of the Arrow impossibility theorem and the Brouwer fixed point theorem

Applied Mathematics and Computation, 2006
We will show that in the case where there are two individuals and three alternatives (or under the assumption of free-triple property) the Arrow impossibility theorem [K.J. Arrow, Social Choice and Individual Values, second ed., Yale University Press, 1963] for social welfare functions that there exists no social welfare function which satisfies ...
Yasuhito Tanaka
exaly   +2 more sources

On the L-fuzzy Brouwer fixed point theorem

Fuzzy Sets and Systems, 1999
Let \(L\) be a fuzzy lattice, i.e. a completely distributive lattice with an order-reversing involution. In the paper, the authors prove successfully that (1) the \(L\)-cube has the fixed point property (under the Hutton product \(L\)-topology); (2) the stratified \(L\)-cube is formed by the product of \(I^*(L)\) which is the stratification of the \(L\)
Tomasz Kubiak, Dexue Zhang
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On Brouwer's fixed point theorem

Topology Proceedings, 2022
Summary: It is shown by Klaas Pieter Hart, Jan van Mill, and Roman Pol [\textit{K. P. Hart} et al., Topol. Proc. 25(Summer), 179--206 (2000; Zbl 1027.54051)] that Brouwer's fixed point theorem can be reduced to its 3-dimensional case by using the hyperspace of a 2-dimensional hereditarily indecomposable continuum.
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A Remark on the Caristi’s Fixed Point Theorem and the Brouwer Fixed Point Theorem

2020
It is well-known that a partial order induced from a lower semi-continuous map gives us a clear picture of a proof of the Caristi’s fixed point theorem. The proof utilized Zorn’s lemma to guarantee the existence of a minimal element which turns out to be a desired fixed point.
S. Dhompongsa, P. Kumam
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Brouwer’s Fixed-Point Theorem

2018
It is more than a century since Brouwer [4] proved a fixed- point theorem of great consequence, in the setting of finite-dimensional Euclidean spaces. It was subsequently extended to normed linear spaces by Schauder [25], and later to locally convex linear topological spaces by Tychonoff [31].
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Some complements to Brouwer’s fixed point theorem

Israel Journal of Mathematics, 1967
The sets which can be the fixed points of a continuous function or a homeomorphism ofB n are investigated.
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Brouwer's fixed point theorem (abstract only)

ACM Communications in Computer Algebra, 2008
"Brouwer's Fixed Point Theorem" was named after the Dutch mathematician "L. E. J. Brouwer". This theorem states that for every continuous function "f" mapped onto itself at a given interval, has at least one fixed point such that "f(x0) = x0". The proof of this theorem uses what is called "Sperner's Lemma" (by Emanuel Sperner), which would be the one ...
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Brouwer’s Fixed-Point Theorem: An Alternative Proof

SIAM Journal on Mathematical Analysis, 1974
This paper gives an alternative proof of Brouwer’s fixed-point theorem. The expected preknowledge on the part of the reader in following the proof is the continuity of the roots of polynomial equations with respect to the coefficients, and the standard compactness argument.
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History of the Brouwer Fixed Point Theorem

2020
The history of the Brouwer fixed point theorem, closely linked to the history of the Brouwer degree, is particularly intricated and is a case story showing the ‘nonlinear’ character of the evolution of mathematics. After describing and commenting the fundamental contribution of Brouwer, it is shown how it has been in one way or another anticipated by ...
George Dinca, Jean Mawhin
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