Strongly nonlinear variational inequalities and generalized pseudocontractions
type: Item
instance of: scholarly article
title: Strongly nonlinear variational inequalities and generalized pseudocontractions
zbMATH Open document ID: 0953.49020
DOI: 10.1007/BF02857297
author: Ram U. Verma
publication date: 2001-01-29T00:00:00Z
review text: Let \(K\) be a closed convex set in a Hilbert space \(H\). For given nonlinear operators \(T,S:H\to H\), find \(u\in K\) such that \[ \langle Su- Tu,v-u\rangle\geq 0,\quad\text{for all }v\in K.\tag{1} \] (1) is called the strongly nonlinear variational inequality, which was considered and studied by \textit{M. A. Noor} [C. R. Math. Acad. Sci., Soc. R. Can. 4, 213-218 (1982; Zbl 0502.49008)]. In this paper, the author has suggested and analyzed an iterative method. It is worth mentioning that Theorem 2.1 was proved by Noor in 1982. Proofs of the main results, Theorem 2.2 and its Corollary 2.1, are wrong and incorrect. In fact, all the results are misleading. It can be proved easily that Lipschitz continuity implies the so-called generalized pseudocontractivity. Consequently, this paper does not contain any new and original result.
reviewed by: Muhammad Aslam Noor
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 90C33
Mathematics Subject Classification ID: 47J20
zbMATH DE Number: 1339479
zbMATH Keywords: convergence analysis
zbMATH Keywords: strongly nonlinear variational inequality
zbMATH Keywords: iterative method
zbMATH Keywords: Lipschitz continuity
zbMATH Keywords: pseudocontractivity
MaRDI profile type: Publication
cites work: Construction of fixed points of nonlinear mappings in Hilbert space
cites work: Q3321366
cites work: Q3914338
cites work: Q3905599
cites work: Remarks on Pseudo-Contractive Mappings
cites work: A fixed point theorem for demicontinuous pseudo-contractions in Hilbert apace
cites work: Q4881785
cites work: Q5640302
cites work: Strongly nonlinear quasivariational inequalities
cites work: General approximation-solvability of nonlinear equations involving \({\mathcal A}\)-regular operators
cites work: Iterative algorithms for variational inequalities and associated nonlinear equations involving relaxed Lipschitz operators
cites work: Nonlinear Variational and Constrained Hemivariational Inequalities Involving Relaxed Operators
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 1339479