Topological degree for \((S)_{+}\)-mappings with maximal monotone perturbations and its applications to variational inequalities
type: Item
instance of: scholarly article
title: Topological degree for \((S)_{+}\)-mappings with maximal monotone perturbations and its applications to variational inequalities
zbMATH Open document ID: 1082.47050
author: Jun Kobayashi
author: Mitsuharu Ôtani
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-11-22T00:00:00Z
review text: This article deals with some generalization of the Browder degree theory for mappings of the type \(T + A: \;X \to X^*\), where \(T\) is a bounded demicontinuous \((S)_{(+)}\)-mapping and \(A\) a maximal monotone operator (\(X\) denoting a reflexive Banach space). The main part of the article is devoted to some revision of the original Browder theory. In particular, the authors study pseudo-monotone homotopies preserving the Browder degree, give a more general formulation for the normalization property, and show that the local degree for the generalized gradient of a nonsmooth functional at the local minimizer is equal to one. As applications, the variational inequalities \[ \int_\Omega | \nabla u| ^{p-2} \nabla u \cdot \nabla(v - u) \, dx \geq \int_\Omega f(x,u)(v - u) \, dx \;\;\forall v \in K, \] \[ K = \{u \in W_0^{1,p}(\Omega): \;| \nabla v| \leq 1 \;\text{ a.e. in } \Omega\}, \] and \[ \int_\Omega \nabla u \cdot \nabla(v - u) \, dx \geq \int_\Omega f(v - u) \, dx \;\;\forall \;v \in K, \qquad K = \{v \in H_0^1(\Omega: \;v \geq 0\;\text{ a.e. in } \Omega\}, \] are considered.
Mathematics Subject Classification ID: 47H11
Mathematics Subject Classification ID: 47H05
Mathematics Subject Classification ID: 47J20
zbMATH DE Number: 2117567
zbMATH Keywords: topological degree
zbMATH Keywords: mapping of class \((S)_+\)
zbMATH Keywords: maximal monotone operators
zbMATH Keywords: subdifferential operators
zbMATH Keywords: local minimizer of non-smooth functional
zbMATH Keywords: elliptic variational inequalities
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
full work available at URL: j.na.2004.07.007
OpenAlex ID: W2072300107
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DOI: 10.1016/J.NA.2004.07.007
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description: scientific article; zbMATH DE number 2117567