Regular nonsmooth equations
type: Item
instance of: scholarly article
title: Regular nonsmooth equations
zbMATH Open document ID: 0826.65050
DOI: 10.1016/0304-4068(94)90003-5
author: Chris Shannon
published in: Journal of Mathematical Economics
publication date: 1994-06-16T00:00:00Z
review text: Notions of regular points and values for continuous (and not necessarily smooth) functions introduced by \textit{T. Rader} [Econometrica 41, 913-935 (1973; Zbl 0278.90007)] are used. It is shown that if \(y\) is a regular value of a continuous function \(f\), then the solutions to the equation \(f(x) = y\) are locally unique, and the degree \(d(f,A,y)\) of \(f\) on an open set \(A\) at \(y\) is given by \[ d(f,A,y) = \sum_{x \in f^{-1} (y)} \text{sgn det } Df(x). \] These results are used to extend \textit{E. Dierker's} results [Econometrica 40, 951-953 (1972; Zbl 0259.90005)] on the number of equilibria in regular exchange economies to nonsmooth exchange economies.
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 91B60
Mathematics Subject Classification ID: 91B50
zbMATH DE Number: 563444
zbMATH Keywords: regular economies
zbMATH Keywords: degree theory
zbMATH Keywords: nonsmooth equations
zbMATH Keywords: Lipschitz functions
zbMATH Keywords: regular points
zbMATH Keywords: exchange economies
reviewed by: Walter Zulehner
MaRDI profile type: Publication
full work available at URL: 0304-4068(94)90003-5
OpenAlex ID: W2086909774
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description: scientific article; zbMATH DE number 563444