Metric regularity and subdifferential calculus in Banach spaces
type: Item
instance of: scholarly article
title: Metric regularity and subdifferential calculus in Banach spaces
zbMATH Open document ID: 0854.49015
DOI: 10.1007/BF01033643
author: Abderrahim Jourani
author: L. Thibault
published in: Set-Valued Analysis
publication date: 1997-01-05T00:00:00Z
review text: Throughout the paper it is assumed that \(X\) and \(Y\) are reflexive Banach spaces. The authors consider the special multivalued mapping \(F(x)=- g(x)+D\), where \(g:X \to Y\) need not satisfy any differentiability assumptions and \(D \subset Y\). Let \(C \subset X\) and \(x_0\in C\). \(F\) is called metrically regular at \((x_0, y_0)\in \text{graph} F\) w.r.t. the set \(C\) iff there exist \(k \geq 0\) and \(\varepsilon > 0\) such that \(d(x,C \cap F^{-1} (y)) \leq k d(x, F(x))\) for all \(x\in (x_0+\varepsilon B_X) \cap C\) and \(y\in y_0+\varepsilon B_Y\), where \(d(x, F(x))\) is the distance of \(x\) to the set \(F(x)\) \((d (x, \emptyset) :=\infty)\). At first they give sufficient conditions for the above reglarity. Using the subdifferential calculus for limiting Fréchet subdifferentials \(\partial_F\) (sum of subdifferentials, chain rule) and the Fréchet coderivative \(D^*_F g(x_0)\) defined by \[ D^*_F g(x_0) (y^*) :=\bigl\{ x^*\in X^* \mid (x^*,-y^*)\in \mathbb{R}_+\partial_F d \bigl(\bigl( x_0, g(x_0) \bigr), \text{graph } g \bigr) \bigr\} \] they prove (Theorem 2.3) the metrical regularity of \(F\) in the above sense under the assumptions that \(C,D\) are closed, \(g\) is Lipschitzian around \(x_0\) with constant \(K_g\), \(D\) is epi-Lipschitz at \(g (x_0)\) and that the regularity condition \[ y^*\in \partial_Fd \bigl( g(x_0), D \bigr) \wedge 0\in D^*_F g \bigl( x_0 (y^*) \bigr)+K_g \partial_F d(x_0, C) \Rightarrow y^*=0 \] is satisfied. In their work they do not need the epi-Lipschitz property of the set \(C\). As application of this theorem they give conditions for the metrical regularity in the special cases \(F(x)=- x+D\) (Cor. 2.4), \(D=X \times \text{epi} f\) (Cor. 2.5), \(g\) is strictly differentiable at \(x_0\) (Cor. 2.6). Using the above results they can prove the more general chain rule for the limiting Fréchet subdifferential \[ \partial_F (f \circ g) (x_0) \subset \bigcup_{y^*\in \partial_F f(y_0)} D^*_F g(x_0) (y^*), \] where \(g:X \to Y\)…
reviewed by: Armin Hoffmann
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 49J27
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 26E25
zbMATH DE Number: 773791
zbMATH Keywords: subdifferential calculus
zbMATH Keywords: limiting Fréchet subdifferentials
zbMATH Keywords: metrical regularity
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 773791