Be grateful when a solution exists, because of Brouwer’s Fixed Point Theorem

2020 ◽  
pp. 227-232
Author(s):  
Susan D'Agostino
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Economic Model ◽  
Existence Of Solutions ◽  
Mathematics Students ◽  
Existence Of A Solution ◽  
Mathematical Theorem ◽  
Brouwer’S Fixed Point Theorem ◽  
Brouwer's Fixed Point Theorem

“Be grateful when solutions exist, because of Brouwer’s Fixed Point Theorem” offers a introduction to a mathematical theorem asserting the existence of solutions to problems in engineering, medicine, economics, and other fields, as long as certain criteria are satisfied. The discussion is illustrated with numerous hand-drawn sketches. While this theorem assures the existence of a solution—a “fixed point”—it does not provide insight on how to find it. Still, it can be reassuring to know that a solution exists. For example, economist John von Newmann used this theorem in his 1937 economic model establishing that there exist prices at which supply equals demand. Mathematics students and enthusiasts are encouraged to be grateful for knowledge that a solution exists in mathematical and life pursuits, even when the solution itself remains elusive. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

scholarly journals Existence of solutions of inverted variational inequalities

2012 ◽  
Vol 28 (2) ◽  
pp. 271-278
Author(s):  
SZILARD LASZLO ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Variational Inequalities ◽  
Point Theorem ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Generalized Variational Inequalities ◽  
Easy Consequence ◽  
Brouwer’S Fixed Point Theorem ◽  
Brouwer's Fixed Point Theorem

In this paper we introduce two new generalized variational inequalities and we give some existence results of the solutions for these variational inequalities involving operators belonging to a recently introduced class of operators. We show by examples, that our results fail outside of this class. Further, we establish a result that may be viewed as a generalization of Minty’s theorem, that is, we show that under some circumstances the set of solutions of these variational inequalities coincide. We also show, the condition that the operators, involved in these variational inequalities, belong to the above mentioned class, is essential in obtaining this result. As application, we show that Brouwer’s fixed point theorem is an easy consequence of our results.

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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