Local Lipschitz-constant functions and maximal subdifferentials
type: Item
instance of: scholarly article
title: Local Lipschitz-constant functions and maximal subdifferentials
zbMATH Open document ID: 1029.49016
DOI: 10.1023/A:1021975622768
published in: Set-Valued Analysis
publication date: 2003-04-27T00:00:00Z
review text: Let \(X\) be a Banach space. For the study of generalized subdifferentials, the authors introduce the notion of topologically robust upper semicontinuity for real-valued functions. So the function \(k(x)\) is called to be topologically robust usc if it is upper semicontinuous and quasi lower semicontinuous. After the discussion of basic properties of such functions it is shown that the set-valued mapping \(k(x) B_{X^*}\) (where \(B_{X^*}\) is the dual unit ball) is a Clarke subdifferential map and (if \(X\) is separable) also an approximate subdifferential map. This result is extended showing that even the mapping \[ T(x):= CSC(\Omega+ kB_{X^*})(x) \] (where \(k(x)\) is an lsc function, \(\Omega(x)\) is the union of Clarke subdifferentials of a countable family of equi-locally Lipschitz functions, and \(CSC\) means the cusco hull, i.e., the convex upper semi-continuous hull of a map) is a Clarke subdifferential map. In the second part the authors regard the associated local Lipschitz-constant function \(\text{Lip}_f\) to a given locally Lipschitz function \(f\) according to \[ \text{Lip}_f(x):= \limsup_{\delta\downarrow 0} \Biggl\{{|f(y)- f(z)|\over\|y- z\|}:\|y\|,\|z\|\leq \delta, y\neq z\Biggr\}. \] Topological properties but also special chain rules for this notion are presented. Moreover, it is shown that every nonnegative topologically robust usc function is a local Lipschitz-constant function. Additionally assumptions are presented which ensure that conversely every local Lipschitz-constant function is also topologically robust usc and hence that the mappings \(\text{Lip}_f(x) B_{X^*}\) are Clarke subdifferential maps. In the last part of the paper it is shown that one can always find a maximal \(\beta\)-subderivative whenever the one-sided Dini directional derivatives equal the local Lipschitz-constant.
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 54E52
Mathematics Subject Classification ID: 49J53
zbMATH DE Number: 1901530
zbMATH Keywords: Lipschitz function
zbMATH Keywords: Baire category
zbMATH Keywords: Clarke subdifferential
zbMATH Keywords: approximate subdifferential
zbMATH Keywords: local Lipschitz-constant function
zbMATH Keywords: topologically robust upper semicontinuity
zbMATH Keywords: lsc function
zbMATH Keywords: cusco hull
zbMATH Keywords: Dini directional derivatives
author: Jonathan M. Borwein
author: J. D. Vanderwerff
author: Xianfu Wang
reviewed by: Jörg Thierfelder
MaRDI profile type: Publication
Wikidata QID: Q56935748
description: scientific article; zbMATH DE number 1901530