A new topological degree theory for densely defined \((S_+)_L\)-perturbations of multivalued maximal monotone operators in reflexive separable Banach spaces
type: Item
instance of: scholarly article
title: A new topological degree theory for densely defined \((S_+)_L\)-perturbations of multivalued maximal monotone operators in reflexive separable Banach spaces
zbMATH Open document ID: 1158.47045
author: Athanassios G. Kartsatos
author: Joseph Quarcoo
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2008-10-21T00:00:00Z
review text: The authors introduce a topological degree theory for densely defined perturbations of multivalued maximal monotone operators in reflexive separable Banach spaces, which is based on the degree theory of \textit{A.\,G.\thinspace Kartsatos} and \textit{I.\,V.\thinspace Skrypnik} [Adv.\ Differ.\ Equ.\ 4, No.\,3, 413--456 (1999; Zbl 0959.47037)]. More precisely, let \(X\) be a real reflexive separable Banach space with its dual \(X^*\). Suppose that \(T: D(T)\subset X\to 2^{X^*}\) is maximal monotone such that \(0\) is an interior point of \(D(T)\) and \(0\in T(0)\), and that \(C: D(C)\subset X\to X^*\) is of class \((S_{+})_L\) such that \(\langle Cx,x\rangle \geq - \psi(| \!| x | \!| )\) for \(x\in D(C)\), where \(L\subset D(C)\) is a dense linear subspace of \(X\) and \(\psi: [0,\infty) \to [0,\infty)\) is nondecreasing. The authors define a topological degree for the sum \(T+C\) as the limit of the Kartsatos--Skrypnik degree of single-valued mappings \(T_t+C\), where each \(T_t=(T^{-1}+tJ^{-1})^{-1}: X\to X^*\) is maximal monotone, \(J\) being the normalized duality mapping. This degree theory extends the theory by the first author and \textit{I.\,V.\thinspace Skrypnik} [Abstr.\ Appl.\ Anal.\ 2005, No.\,2, 121--158 (2005; Zbl 1110.47049)] in some sense. Moreover, this degree is applied to establish an abstract result on the existence of zeros for inclusions.
reviewed by: In-Sook Kim
Mathematics Subject Classification ID: 47H11
Mathematics Subject Classification ID: 47H05
Mathematics Subject Classification ID: 47H14
Mathematics Subject Classification ID: 47H04
zbMATH DE Number: 5354984
zbMATH Keywords: topological degree
zbMATH Keywords: maximal monotone operators
zbMATH Keywords: multivalued operators
zbMATH Keywords: densely defined \((S_{+})_L\)-perturbations
zbMATH Keywords: existence of zeros
zbMATH Keywords: Kartsatos-Skrypnik degree theory
MaRDI profile type: Publication
full work available at URL: j.na.2007.08.017
OpenAlex ID: W1549419072
cites work: On the degree theory for densely defined mappings of class \((S_+)_L\)
cites work: Perturbations of nonlinear maximal monotone sets in banach space
cites work: Q4092518
cites work: Fixed point theory and nonlinear problems
cites work: The degree of mapping, and its generalizations
cites work: Degree of mapping for nonlinear mappings of monotone type
cites work: Nonlinear mappings of monotone type in Banach spaces
cites work: Q3141921
cites work: Topological degree theory for multi-valued mappings of class \((S_+)_L\)
cites work: Q3999276
cites work: Ranges of densely defined generalized pseudomonotone perturbations of maximal monotone operators.
cites work: New results in the perturbation theory of maximal monotone and $M$-accretive operators in Banach spaces
cites work: Normalized eigenvectors for nonlinear abstract and elliptic operators
cites work: Topological degree theories for densely defined mappings involving operators of type \((S_+)\)
cites work: A new topological degree theory for densely defined quasibounded \(({\widetilde{S}}_{+})\)-perturbations of multivalued maximal monotone operators in reflexive Banach spaces
cites work: Degree theories and invariance of domain for perturbed maximal monotone operators in Banach spaces
cites work: On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces
cites work: Nonlinear operators of monotone type in reflexive Banach spaces and nonlinear perturbations
cites work: Q4143788
cites work: Q4173851
cites work: Remarks on the approximation-solvability of nonlinear functional equations
cites work: Q3730503
cites work: Minimax monotonicity
cites work: Q4049449
cites work: Q4312115
cites work: Q3992492
DOI: 10.1016/J.NA.2007.08.017
Recommended article: Ranges of densely defined generalized pseudomonotone perturbations of maximal monotone operators.
Recommended article: Q5192134
Recommended article: Q3451281
Recommended article: The Leray-Schauder approach to the degree theory for \((S_+)\)-perturbations of maximal monotone operators in separable reflexive Banach spaces
Recommended article: Strongly quasibounded maximal monotone perturbations for the Berkovits--Mustonen topological degree theory
Recommended article: Topological degree and application to a parabolic variational inequality problem
Recommended article: Q4789582
Recommended article: Q3388762
Recommended article: A new topological degree theory for perturbations of demicontinuous operators and applications to nonlinear equations with nonmonotone nonlinearities
Recommended article: A new topological degree theory for perturbations of the sum of two maximal monotone operators
description: scientific article; zbMATH DE number 5354984