Brauer'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.

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

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.

Be grateful when a solution exists, because of 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.

Stability Analysis of Stochastic Generalized Equation via Brouwer’s Fixed Point Theorem

The stochastic generalized equation provides a unifying methodology to study several important stochastic programming problems in engineering and economics. Under some metric regularity conditions, the quantitative stability analysis of solutions of a stochastic generalized equation with the variation of the probability measure is investigated via Brouwer’s fixed point theorem. In particular, the error bounds described by Hausdorff distance between the solution sets are established against the variation of the probability measure. The stability results obtained are finally applied to a stochastic conic programming.