The equivalence of several basic theorems for subdifferentials
type: Item
instance of: scholarly article
title: The equivalence of several basic theorems for subdifferentials
zbMATH Open document ID: 0917.49017
DOI: 10.1023/A:1008656207314
author: Qiji J. Zhu
published in: Set-Valued Analysis
publication date: 1999-05-09T00:00:00Z
review text: In the paper it is shown that special basic tools for \(\beta\)-subdifferentials introduced by different authors are equivalent. Let \(X\) be a Banach space and \(\beta\) be a bornology on \(X\). Then the lower-semicontinuous function \(g:X\to \mathbb{R}\) is called to be \(\beta\)-differentiable at \(x\) with \(\beta\)-derivative \(\nabla g(x)\) if the limit \[ \lim_{x\to 0} {g(x+ tu)- g(x)- t\cdot\langle g(x),u\rangle\over t}= 0 \] is uniform in \(u\in V\) for every set \(V\in\beta\). It is \(\beta\)-smooth at \(x\) if the mapping \(\nabla g:X\to X^*_\beta\) is continuous around \(x\) (where \(X^*_\beta\) is the dual space of \(X\) provided with the topology of uniform convergence on \(\beta\)-sets). Finally, the function \(f: X\to\mathbb{R}\) is called to be \(\beta\)-subdifferentiable and \(x^*\) is a \(\beta\)-subgradient at \(x\) if there exists a \(\beta\)-smooth function \(g\) such that \(f\)-\(g\) admits a minimum at \(x\). Using special bornologies for \(\beta\) (e.g., the family of singleton sets, bounded sets,\dots) then of course we get special differentiability notions. In detail, regarding this general differentiability concept, the author compares the following sum rules for \(N\) functions which are assumed to be lower-semicontinuous: -- the nonlocal fuzzy sum rule of Borwein, Treiman and Zhu; -- the local fuzzy sum rule of Ioffe; -- the Lipschitz local fuzzy sum rule (which is the same, but all functions are locally Lipschitz \phantom{-- }at \(x\)); -- the multidirectional mean-value theorem of Clarke, Ledyaev, Stern and Wolenski; -- the extremal principle of Mordukhovich. In the main theorem of the paper, the author proves the equivalence of these conditions -- without additional assumptions. Beside this, further interesting consequences of this result are given, e.g., a sufficient condition for these tools in form of the existence of a \(\beta\)-smooth Lipschitz bump-function and a new characterization of Asplund spaces by means of the mentioned …
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 26E15
zbMATH DE Number: 1230453
zbMATH Keywords: extremal principles
zbMATH Keywords: \(\beta\)-subdifferentials
zbMATH Keywords: \(\beta\)-subgradient
zbMATH Keywords: sum rules
zbMATH Keywords: Asplund spaces
zbMATH Keywords: Fréchet subdifferentiability
reviewed by: Jörg Thierfelder
MaRDI profile type: Publication
full work available at URL: a:1008656207314
OpenAlex ID: W20559199
description: scientific article; zbMATH DE number 1230453