Variational inequalities and fixed point theorems for PM maps
type: Item
instance of: scholarly article
title: Variational inequalities and fixed point theorems for PM maps
zbMATH Open document ID: 0942.49011
author: K. Q. Lan
author: J. Webb
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-08-21T00:00:00Z
review text: Let \(X\) be a Banach space, \(X^*\) be the dual space of \(X\), \(K\) be a closed convex subset in \(X\), \(J:X\to X^*\) be a duality map and \(T: D\subset K\to X^*\) a PM-map. The authors consider the problem of finding \(y\in D\) such that \[ \langle Jy- Ty,y-x\rangle\leq 0\quad\text{for all }x\in K. \] They used Brezis results on minimax principle to obtain results on variational inequalities for \(J- T\) with \(T\) being a PM-map and establish an acute angle principle for a PM-map. They use the theory of variational inequalities for \(J-T\) where \(T\) is a PM-map, to give new applications on the existence of fixed points for generalized inward PM-maps in Hilbert spaces. They establish a new equivalence between variational inequalities and nearest points of maps. For generalized inward maps the equivalence between variational inequalities and fixed points of these maps is established. They used this relationship not only to study properties of PM-maps but also to derive a fixed point principle for non-self maps from variational inequalities. The results generalize and unify many earlier results with different and new methods. This paper also has applications of fixed point theory to homogeneous integral equations.
reviewed by: Lai-Jiu Lin
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 1215757
zbMATH Keywords: demicontinuous
zbMATH Keywords: pseudo-monotone
zbMATH Keywords: duality map
zbMATH Keywords: PM-map
zbMATH Keywords: variational inequalities
zbMATH Keywords: fixed points
zbMATH Keywords: nearest points of maps
zbMATH Keywords: non-self map
MaRDI profile type: Publication
full work available at URL: jmaa.1998.5988
OpenAlex ID: W1980491218
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DOI: 10.1006/JMAA.1998.5988
description: scientific article; zbMATH DE number 1215757