scholarly journals Equivalence between the Existence of an Approximate Equilibrium in a Competitive Economy and Sperner's Lemma: A Constructive Analysis

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Yasuhito Tanaka
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Constructive Mathematics ◽  
Approximate Equilibrium ◽  
Sperner’S Lemma ◽  
Brouwer’S Fixed Point Theorem ◽  
Brouwer's Fixed Point Theorem ◽  
Competitive Economy ◽  
Approximate Version

Brouwer's fixed point theorem cannot be constructively proved, so the existence of an equilibrium in a competitive economy also cannot be constructively proved. On the other hand, Sperner's lemma which is used to prove Brouwer's theorem is constructively proved. Some authors have presented a constructive (or an approximate) version of Brouwer's fixed point theorem using Sperner's lemma. In this paper, I prove the existence of an approximate equilibrium in a competitive economy directly by Sperner's lemma. Also I show that the existence of an approximate equilibrium leads to Sperner's lemma. I follow the Bishop style constructive mathematics according to Bishop and Bridges (1985), Bridges and Richman (1987), and Bridges and Vîţă (2006).

scholarly journals A Proof of Constructive Version of Brouwer's Fixed Point Theorem with Uniform Sequential Continuity

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
Yasuhito Tanaka
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Uniform Continuity ◽  
Constructive Mathematics ◽  
Sperner’S Lemma ◽  
Brouwer’S Fixed Point Theorem ◽  
Constructive Version ◽  
Sequential Continuity ◽  
Brouwer's Fixed Point Theorem

It is often said that Brouwer's fixed point theorem cannot be constructively proved. On the other hand, Sperner's lemma, which is used to prove Brouwer's theorem, can be constructively proved. Some authors have presented a constructive (or an approximate) version of Brouwer's fixed point theorem using Sperner's lemma. They, however, assume uniform continuity of functions. We consider uniform sequential continuity of functions. In classical mathematics, uniform continuity and uniform sequential continuity are equivalent. In constructive mathematics a la Bishop, however, uniform sequential continuity is weaker than uniform continuity. We will prove a constructive version of Brouwer's fixed point theorem in an n-dimensional simplex for uniformly sequentially continuous functions. We follow the Bishop style constructive mathematics.

Brauer's Fixed Point Theorem

2021 ◽  
Vol 8 (1) ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Simple Proof ◽  
Exchange Model ◽  
Existence Of Equilibrium ◽  
Economic Applications ◽  
Brouwer’S Fixed Point Theorem ◽  
Brouwer's Fixed Point Theorem

The primary goal of the paper is to deliver a simple proof of equivalence between Brouwer’s fixed point theorem and the existence of equilibrium in a simple exchange model with monotonic consumers. To achieve this end, we discuss some equivalent formulations of Brouwer’s theorem and prove additional ones, that are ’approximating’ in character or seem to be better suited for economic applications than the standard results.

scholarly journals Brouwer's Fixed Point Theorem with Isolated Fixed Points and His Fan Theorem

2012 ◽  
Vol 2012 ◽  
pp. 1-3
Author(s):  
Yasuhito Tanaka
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Point Theorem ◽  
Brouwer’S Fixed Point Theorem ◽  
Fan Theorem ◽  
Brouwer's Fixed Point Theorem

We show that Brouwer’s fixed point theorem with isolated fixed points is equivalent to Brouwer’s fan theorem.

scholarly journals On Some Applications of Kakutani’s Fixed Point Theorem in Game Theory

2021 ◽  
Vol 66 (1) ◽  
pp. 49-69
Author(s):  
Md. Alamgir Hossain ◽  
◽  
Md. Zulfikar Ali ◽  
Md. Asaduzzaman ◽  
Md. Sazzad Hossain ◽  
...  
Keyword(s):  
Game Theory ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Research Work ◽  
Topological Properties ◽  
Set Functions ◽  
Brouwer’S Fixed Point Theorem ◽  
Effective Role ◽  
Brouwer's Fixed Point Theorem

In this paper, we discuss some major applications of Kakutani’s fixed point theorem in game theory. In the course of research work we mostly use the idea of mathematical set, functions, topological properties and Brouwer’s fixed point theorem to make the Kakutani’s fixed point theorem more conspicuous. In the key point of idea, we include how this theory can play the effective role to highlight new fixed point results and their applications in different fields of game theory.

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