An open mapping theorem using unbounded generalized Jacobians
type: Item
instance of: scholarly article
title: An open mapping theorem using unbounded generalized Jacobians
zbMATH Open document ID: 1035.49016
DOI: 10.1016/S0362-546X(01)00774-X
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-08-25T00:00:00Z
review text: The authors present a new open mapping theorem for nonsmooth functions by means of (not necessary bounded) approximate Jacobians. Let \(f: \mathbb{R}^n\to \mathbb{R}^m\) be a continuous function and \(x\in\mathbb{R}^n\). A closed set of \((m\times n)\)-matrices \(Jf(x)\subset L(\mathbb{R}^n, \mathbb{R}^m)\) is called to be an approximate Jacobian of \(f\) at \(x\), iff for every \(u\in \mathbb{R}^n\) and \(v\in \mathbb{R}^m\), one has \[ \limsup_{t\downarrow 0} {(vf)(x+ tu)- (vf)(x)\over t}\leq \sup_{M\in Jf(x)}\langle v, Mu\rangle. \] This notion was introduced by the authors some years ago. If \(f\) is locally Lipschitzian then the Clarke generalized Jacobian is an example of a locally bounded approximate Jacobian. Moreover, a continuous mapping is locally Lipschitzian if and only if it admits a locally bounded approximate Jacobian. A basic tool for the discussion of not Lipschitz functions is the limiting singular approximate Jacobian \[ J^\infty f(x):= \limsup\{t_k Jf(x_k)\mid t_k\downarrow 0\text{ and }x_k\to x\} \] which is nonempty in this case and (if \(Jf\) is assumed to be upper semicontinuous) can be described by the recession cone of the set \(Jf(x)\). After the discussion of properties of recession cones and using a special meanvalue theorem, the authors establish a chain rule for functions with unbounded, but upper semicontinuous approximate Jacobians. On the base of these results they present their main theorem. It is shown that -- assuming a continuous function \(f: \mathbb{R}^n\to \mathbb{R}^m\) with an upper semicontinuous approximate Jacobian mapping \(Jf\) and invertible matrices \(A\in \overline{\text{conv\,}} Jf(x_0)\cup\text{conv}(J^\infty f(x_0)\setminus\{0\})\) -- there exist numbers \(\delta> 0\) and \(\varepsilon> 0\) such that \[ \| f(x_0+ h)- f(x_0)\|\geq \varepsilon\| h\|\quad\text{for all }h\neq 0,\;\| h\|< \delta \] and \[ f(x_0)+ {\varepsilon\delta\over 2}\text{\,int\,} B(0,1)\subseteq f(x_0+ \delta\text{\,int\,}B(0, 1)). \] As corol…
reviewed by: Jörg Thierfelder
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 46A30
Mathematics Subject Classification ID: 26B10
zbMATH DE Number: 1788033
zbMATH Keywords: approximate Jacobian
zbMATH Keywords: open mapping theorem
zbMATH Keywords: implicit function theorem
zbMATH Keywords: inverse function theorem
author: V. Jeyakumar
author: Dinh The Luc
MaRDI profile type: Publication
cites work: A survey of subdifferential calculus with applications
cites work: On the inverse function theorem
cites work: Q4000274
cites work: Q4875353
cites work: Approximate Jacobian Matrices for Nonsmooth Continuous Maps and C1-Optimization
cites work: Nonsmooth calculus, minimality, and monotonicity of convexificators
cites work: Q4216243
cites work: Approximate Hessian matrices and second-order optimality conditions for nonlinear programming problems with C1-data
cites work: Lipschitzian inverse functions, directional derivatives, and applications in \(C^{1,1}\) optimization
cites work: Recession maps and applications
cites work: Convergence of asymptotic directions
cites work: Generalized differential calculus for nonsmooth and set-valued mappings
cites work: Mixed Coderivatives of Set–Valued Mappings in Variational Analysis
cites work: Extensions of subgradient calculus with applications to optimization
cites work: Q4218971
cites work: An Implicit Function Theorem without Differentiability
cites work: Fat homeomorphisms and unbounded derivate containers
full work available at URL: s0362-546x(01)00774-x
OpenAlex ID: W2076959604
description: scientific article; zbMATH DE number 1788033