On the iterative method for variational inequalities with nonsmooth unbounded operators in Banach space
type: Item
instance of: scholarly article
title: On the iterative method for variational inequalities with nonsmooth unbounded operators in Banach space
zbMATH Open document ID: 0815.41025
author: A. I. Notik
author: Yakov Alber
published in: Journal of Mathematical Analysis and Applications
publication date: 1995-06-11T00:00:00Z
review text: The authors continue their investigations concerning approximate methods for solving variational inequalities with unbounded and nonsmooth operators in Banach spaces. Their previous works were concerned with regularization and penality methods, and their convergence and stability. The first author [Sov. Math., Dokl. 20, 871-876 (1979; Zbl 0452.47079)] considered the iterative method \[ x^{n+1}= P_ \Omega (x^ n- \alpha_{n+\sigma} (z^ n- f)),\quad z^ n\in Ax^ n,\quad n=0,1,\dots, \quad x_ 0\in M, \tag{1} \] for the variational inequality \[ (Ax-f, y-x)\geq 0, \qquad \forall y\in \Omega, \tag{2} \] where \(A: H\to H\). (Here \((\cdot, \cdot)\) is the inner product in the Hilbert space \(H\), \(\alpha_{n+\sigma}\) are positive number parameters, \(C\geq \sigma\geq 0\), and \(P_ \Omega\) is a projection operator on the convex closed set \(\Omega \subset H\) in the sense of best approximation.) Here the authors consider the variational inequality (2) with operator \(A\) acting from uniformly convex Banach space \(B\) to \(B^*\), \((z,x)\) where \(z\in B^*\) and \(x\in B\) denotes a dual product pairing in \(B\) and \(\Omega\) is a closed convex set, \(\Omega \subset \text{int } D(A)\), \(\text{int } D(A) \neq \emptyset\). They study the iterative method \[ Ix^{n+1}= Ix^ n- \alpha_ n | z^ n |^{-1} z^ n, \quad n=0,1,\dots, x^ 0\in B, \quad x^ n= I^* Ix^ n, \tag{3} \] \(\forall z^ n\in Ax^ n -f\), prove convergence theorems for iterative processes and establish asymptotic estimates of their convergence rate.
Mathematics Subject Classification ID: 41A60
zbMATH DE Number: 720410
zbMATH Keywords: regularization method
zbMATH Keywords: penalty methods
zbMATH Keywords: variational inequalities
zbMATH Keywords: nonsmooth operators
reviewed by: T. D. Narang
MaRDI profile type: Publication
full work available at URL: jmaa.1994.1471
OpenAlex ID: W2060475625
DOI: 10.1006/JMAA.1994.1471
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description: scientific article; zbMATH DE number 720410