On the existence of zeros of monotone operators in reflexive Banach spaces
type: Item
instance of: scholarly article
title: On the existence of zeros of monotone operators in reflexive Banach spaces
zbMATH Open document ID: 1110.47053
author: N. E. Zubov
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-12-07T00:00:00Z
review text: Let \(E\) be a reflexive, strictly convex and smooth Banach space with dual \(E^*\), \(C\) be a nonempty bounded closed convex subset of \(E\) and \(T: E\to E^*\) be a hemicontinuous monotone mapping. The authors study the existence of zeros \(x\in C\) of the nonlinear equation \(T(x)= 0\). For this, they introduce a new sufficient condition of the following kind: \[ \text{inf}_{y\in C} V(y,(J- T)x)< V(x,(J- T)x)\quad \forall x\in C: J^{-1}(J- T)x\not\in C, \] where \(J: E\to E^*\) is the duality mapping of \(E\) and \(V:E\times E^*\to\mathbb{R}_+\) is the (generalized) quadratic distance function defined by \(V(x, x^*)= \| x\|^2- 2\langle x,x^*\rangle+\| x^*\|^2\). In the first part of the paper, this condition is characterized and compared with other known conditions. Moreover, examples of mappings which satisfy this new condition are provided. In the second part, the authors derive the existence theorem for zeros by means of this condition, using a known fixed point theorem of Takahashi. The result is used for the derivation of a surjectivity assertion for the mapping \(T\) and for a fixed point theorem for nonexpansive mappings in Hilbert spaces. In the last part, the authors present a sufficient and necessary condition for the existence of zeros of maximal monotone set-valued mappings.
reviewed by: Jörg Thierfelder
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 47H05
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 5077912
zbMATH Keywords: monotone operator
zbMATH Keywords: existence of zeros
zbMATH Keywords: duality mapping
zbMATH Keywords: generalized projection
zbMATH Keywords: resolvent
zbMATH Keywords: Yosida approximation
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.11.054
OpenAlex ID: W2082708933
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DOI: 10.1016/J.JMAA.2005.11.054
description: scientific article; zbMATH DE number 5077912