Perturbed iterative algorithms with errors for completely generalized strongly nonlinear implicit quasivariational inclusions
type: Item
instance of: scholarly article
title: Perturbed iterative algorithms with errors for completely generalized strongly nonlinear implicit quasivariational inclusions
zbMATH Open document ID: 1033.49020
DOI: 10.1155/S1025583400000205
published in: Journal of Inequalities and Applications
publication date: 2004-03-29T00:00:00Z
full work available at URL: https://eudml.org/doc/123360
review text: In this paper, the authors have considered the problem of finding \(u\in H\) such that \[ 0\in N(f(u),p(u))+M(g(u)-m(u),u)\tag{1} \] where \(f,p, g, m, N\) are single-valued mappings and \(M(.,.)\) is a maximal monotone mapping with respect to the first argument. Problem (1) is called the completely generalized strongly nonlinear implicit quasivariational inclusion. The authors have suggested Ishikawa and Mann type perturbed iterative algorithms for the problem (1) using the standard technique. In fact, all the results obtained in this paper are either incorrect or trivial generalizations of the known results. Even the problem (1) is \textit{not} well-defined from mathematical point of view. For example, if the mapping \(g = 0\), then problem (1) reduces to: Find \(u\in H\) such that \[ 0\in N(f(u),p(u)) + M(-m(u),u), \] which does not make any sense from mathematical point of view. In fact, this problem is not even a generalization of any type of the variational inclusion. Similarly, the problems (2.2), (2.3)[ 16], (2.5)[10], (2.6)[18] and (2.7), which can be obtained as special cases from (1) for \((g = 0)\) are no longer variational inclusions and variational inequalities.
reviewed by: Muhammad Aslam Noor
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 47H06
Mathematics Subject Classification ID: 90C33
zbMATH DE Number: 1526345
zbMATH Keywords: convergence
zbMATH Keywords: completely generalized strongly nonlinear implicit quasivariational inclusion
zbMATH Keywords: Ishikawa and Mann type perturbed iterative algorithms
zbMATH Keywords: variational inequalities
author: Soo Hak Shim
author: Shin Min Kang
author: Nan-Jing Huang
author: Yeol Je Cho
MaRDI profile type: Publication
OpenAlex ID: W2134436959
description: scientific article; zbMATH DE number 1526345