Nonsmooth variational problems in the limit case and duality
type: Item
instance of: scholarly article
title: Nonsmooth variational problems in the limit case and duality
zbMATH Open document ID: 1104.49010
DOI: 10.1023/B:JOGO.0000047913.06908.37
author: V. V. Motreanu
author: Dumitru Motreanu
published in: Journal of Global Optimization
publication date: 2005-04-07T00:00:00Z
review text: Given an extended real-valued function \(f\) defined on a reflexive Banach space (typically of the form \(f=\Phi+\alpha,\) where \(\Phi\) is a locally Lipschitz function and \(\alpha\) a lower semicontinuous convex function), and a nonempty subset \(S\) of \(X,\) the authors give conditions guaranteeing that the value \(a=\inf_{S} f\) is critical and attained in \(S\), that is, there exists \(u\in S\) with \(f(u)=a\) and such that for all \(v\in X,\) \(\Phi ^{0}(u,v-u)+\alpha(v)\geq\alpha(u)\), where \(\Phi^{0}(u;.)\) denotes the Clarke directional derivative of \(\Phi\). The paper uses duality (minimax principles) and results from [\textit{S. Marano} and \textit{D. Motreanu}, Topol. Methods Nonlinear Anal. 22, 139--158 (2003; Zbl 1213.58010)]. The last part discusses applications to boundary value problems.
reviewed by: A. Daniilidis
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35J85
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 49N15
Mathematics Subject Classification ID: 90C46
zbMATH DE Number: 2153473
zbMATH Keywords: variational inequality
zbMATH Keywords: critical point
zbMATH Keywords: nonsmooth analysis
zbMATH Keywords: minimax principle
zbMATH Keywords: duality
MaRDI profile type: Publication
full work available at URL: b:jogo.0000047913.06908.37
OpenAlex ID: W1972830866
description: scientific article; zbMATH DE number 2153473