Theorem on an implicit function in quasidifferential calculus
type: Item
instance of: scholarly article
title: Theorem on an implicit function in quasidifferential calculus
zbMATH Open document ID: 0859.49019
DOI: 10.1007/BF02363856
author: V. F. Demyanov
published in: Journal of Mathematical Sciences (New York)
publication date: 1997-04-06T00:00:00Z
review text: The author proves two generalizations of the implicit function theorem for quasidifferentiable functions. Therefore let \(f:\mathbb{R}^m\times\mathbb{R}^n\to\mathbb{R}^n\) be componentwise quasidifferentiable in the second argument with \(f(x_0,y_0)=0_n\). The first result is a restatement of the implicit function theorem of classical analysis in terms of quasidifferentiable calculus, giving necessary conditions for the existence and uniqueness of a continuous \(y:\mathbb{R}^m\to\mathbb{R}^n\) with \(y(x_0)=y_0\) such that \[ f(x,y(x))=0_n \] for \(x\) near \(x_0\). Given in addition \(g\in\mathbb{R}^m\) and \(q_0\in\mathbb{R}^n\), the second theorem gives necessary conditions for the existence and uniqueness of a continuous \(q:[0,\alpha_0]\to\mathbb{R}^n\) with \(q(0)=q_0\) such that \[ f(x_0+\alpha g,y_0+\alpha q(\alpha))=0_n \] for \(\alpha\in [0,\alpha_0]\), that is, the implicit function exists only in the direction \((g,q_0)\). The differences between both theorems are exhibited in a lot of examples.
reviewed by: Sebastian Goette
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 26B10
Mathematics Subject Classification ID: 26B05
zbMATH DE Number: 903431
zbMATH Keywords: quasidifferential calculus
zbMATH Keywords: nonsmooth analysis
zbMATH Keywords: implicit function theorem
zbMATH Keywords: quasidifferentiable functions
MaRDI profile type: Publication
cites work: Q4692280
cites work: Q5558820
cites work: Q3041848
cites work: Q3828596
cites work: On inverse function theorems for set-valued maps
cites work: On the inverse function theorem
cites work: On implicit function theorems for set-valued maps and their application to mathematical programming under inclusion constraints
cites work: Nonsmooth Analysis: Differential Calculus of Nondifferentiable Mappings
cites work: A generalization of Brouwer's fixed point theorem
cites work: General constraint qualifications in nondifferentiable programming
cites work: A constraint qualification in quasidifferentiable programming
cites work: Q4126592
cites work: Q3793253
full work available at URL: https://doi.org/10.1007/bf02363856
OpenAlex ID: W2066658748
Recommended article: Q3974403
Recommended article: Q3788758
Recommended article: A directional implicit function theorem for quasidifferentiable functions
Recommended article: Q4971909
Recommended article: Q3072609
Recommended article: The implicit function theorem revisited
Recommended article: Implicit function theorems for nondifferentiable mappings
Recommended article: An implicit function theorem
description: scientific article; zbMATH DE number 903431