Iterative algorithms for solving generalized nonlinear mixed variational inequalities
type: Item
instance of: scholarly article
title: Iterative algorithms for solving generalized nonlinear mixed variational inequalities
zbMATH Open document ID: 1071.65092
published in: Journal of Computational and Applied Mathematics
publication date: 2005-01-13T00:00:00Z
review text: The aim of this very important and useful paper is to investigate the iterative algorithms for solving generalized nonlinear mixed variational inequalities. Here the generalized nonlinear variational inequality problem: Find \(x\in H\), \(u\in T(x)\) and \(v\in A(x)\) such that \(\langle N(u, v),g(y)- g(x)\rangle+ \varphi(g(y))- \varphi(g(x))\geq 0\), \(\forall g(y)\in H\) is considered, where \(H\) is a real Hilbert space, \(C(H)\) are the families of all nonempty compact subsets of \(H\), \(T,A: H\to C(H)\), the set-valued mappings, \(N: H\times H\to H\) and \(g: H\to H\) are single-valued mappings, and \(\varphi: H\to(-\infty, +\infty]\) be a real function. The author introduces a concept of \(g\)-partially relaxed strong monotonicity for mappings. By applying the concept of auxiliary variational inequality technique some new predictor-corrector iterative schemes for solving a class of generalized nonlinear variational inequalities are suggested. Main result: The convergence of the iterative sequence generated by the suggested iterative algorithm is proved. The precise proof is proposed. We note that the convergence analysis of the predictor-corrector algorithm only requires (crucial point) that the underlying mappings are continuous and \(g\)-partially relaxed strongly monotone. Moreover, these algorithms and convergence results are new.
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 49J40
zbMATH DE Number: 2127103
zbMATH Keywords: Generalized nonlinear mixed variational inequality
zbMATH Keywords: convergence
zbMATH Keywords: Auxiliary variational principle
zbMATH Keywords: \(g\)-partially relaxed strongly monotone
zbMATH Keywords: Predictor-corrector iterative algorithm
author: Xieping Ding
reviewed by: Jan Lovíšek
MaRDI profile type: Publication
full work available at URL: j.cam.2004.02.016
OpenAlex ID: W2116131996
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DOI: 10.1016/J.CAM.2004.02.016
description: scientific article; zbMATH DE number 2127103