Approximate Rolle's theorems for the proximal subgradient and the generalized gradient.
type: Item
instance of: scholarly article
title: Approximate Rolle's theorems for the proximal subgradient and the generalized gradient.
zbMATH Open document ID: 1034.46040
DOI: 10.1016/S0022-247X(03)00267-1
author: Daniel Azagra
author: J. Ferrera
author: Fernando López-Mesas
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-08-25T00:00:00Z
review text: The authors obtain an approximative Rolle's theorem for proximinal subgradients in real Hilbert spaces and for the generalized gradients \(\partial f\): Theorem. Let \(U\) be a bounded connected open subset of a real Banach space \(X\) and \(f:\overline U\to {\mathbb R}\) be a bounded locally Lipschitz function satisfying \(\sup f(u) > \sup f (\partial U)\). Then, for every \(\alpha>0\), there exist \(x\in U\) and \(\xi\in \partial f(x)\) such that \(\| \xi\| <\alpha\). It is also proved that an exact Rolle's theorem for the generalized gradients \(\partial f\) is false in every infinite-dimensional Banach spaces \(X\): Theorem. For every~\(X\) there exists a bounded Lipschitz function \(f\) defined on a bounded convex body~\(U\) such that \(f\) vanishes at~\(\partial U\) and yet \(y\notin \partial f(x)\). Here, given a locally Lipschitz \(f:X\to {\mathbb R}\), \(\partial f(x)=\{\xi \in X^*\mid f^0(x,v)\geq \langle \xi,v\rangle\) \(\forall v\in X\}\), \(f^0(x,v)=\limsup (f(y+tv)-f(y))/t\) as \((y,t)\to (x,0)\).
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 26A24
zbMATH DE Number: 1970517
zbMATH Keywords: Rolle theorem
zbMATH Keywords: proximinal subspace
zbMATH Keywords: generalized gradient
MaRDI profile type: Publication
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full work available at URL: s0022-247x(03)00267-1
OpenAlex ID: W2019055612
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description: scientific article; zbMATH DE number 1970517