Variational inequalities for \((\eta,\theta)\)-pseudomonotone operators in nonreflexive Banach spaces
type: Item
instance of: scholarly article
title: Variational inequalities for \((\eta,\theta)\)-pseudomonotone operators in nonreflexive Banach spaces
zbMATH Open document ID: 0954.47052
DOI: 10.1016/S0893-9659(99)00050-6
published in: Applied Mathematics Letters
publication date: 2001-01-29T00:00:00Z
review text: The paper concerns a variational problem: Find \(x_0\in K\) such that, for all \(x\in K\), \(\langle Tx_0,\eta(x,x_0)\rangle\geq 0\). Here \(X\) is a nonreflexive Banach space, \(K\subseteq X^{**}\) is a convex set, \(T:K\to X^{*}\) is a hemicontinuous operator, and \(\eta:K\times K\to X^{**}\) is an appropriate substitute for the standard operator \((x,x_0)\to x-x_0\). The main result asserts solvability in case \(T\) is \((\eta,\theta)\)-pseudomonotone for some \(\theta:K\times K\to X^{**}\), which resembles \(\eta\). This generalized monotonicity is a property which, for some \(r>0\), renders the problem above equivalent to the problem below: find \(x_0\in K\) such that, for all \(x\in K\), \(\langle Tx,\eta(x,x_0)\rangle\geq r\|\theta(x,x_0)\|^2\).
Mathematics Subject Classification ID: 47J20
Mathematics Subject Classification ID: 49J40
zbMATH DE Number: 1369320
zbMATH Keywords: variational problem
zbMATH Keywords: hemicontinuous operator
zbMATH Keywords: generalized monotonicity
author: Byung-Soo Lee
author: Gue Myung Lee
reviewed by: Corneliu Ursescu
MaRDI profile type: Publication
cites work: Nonlinear Elliptic Boundary Value Problems. II
cites work: Sensitivity Analysis in Variational Inequalities
cites work: Variational inequalities with nonmonotone operators
cites work: On some non-linear elliptic differential functional equations
cites work: Variational inequalities for monotone operators in nonreflexive Banach spaces
cites work: Variational inequalities involving strongly pseudomonotone hemicontinuous mappings in nonreflexive Banach spaces
cites work: Variational inequalities in nonreflexive Banach spaces
cites work: Variational inequalities, complementarity problems, and duality theorems
cites work: A generalization of Tychonoff's fixed point theorem
full work available at URL: s0893-9659(99)00050-6
OpenAlex ID: W2984746387
description: scientific article; zbMATH DE number 1369320