Variational-like inequalities with generalized monotone mappings in Banach spaces
type: Item
instance of: scholarly article
title: Variational-like inequalities with generalized monotone mappings in Banach spaces
zbMATH Open document ID: 1041.49006
published in: Journal of Optimization Theory and Applications
publication date: 2004-02-12T00:00:00Z
review text: Let a Banach space \(E\) with dual \(E^{\ast}\) and a convex set \(K\subseteq E\) be given. Let further \(T:K\rightarrow E^{\ast}\), \(\eta:K\times K\rightarrow E\) be maps and \(f:K\rightarrow\mathbb{R}\cup\{+\infty\}\) be a proper convex function. This paper investigates the existence of solutions for the following ``variational-like'' inequality problem: find \(x\in K\) such that \(\left\langle Tx,\eta(y,x)\right\rangle +f(y)-f(x)\geq0\), for all \(y\in K\). It is assumed that \(T\) is relaxed \(\eta\)-\(\alpha\) monotone in the sense that for some function \(\alpha:E\rightarrow\mathbb{R}\), the inequality \(\left\langle Tx-Ty,\eta(x,y)\right\rangle \geq\alpha(x-y)\) holds for all \(x,y\in K\). The proof makes use of the Ky Fan Lemma. A similar result is established for another class of maps introduced in the paper, the relaxed \(\eta\)-\(\alpha\) semimonotone maps.
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 47J20
zbMATH DE Number: 2041231
zbMATH Keywords: variational-like inequalities
zbMATH Keywords: generalized monotone operators
zbMATH Keywords: KKM maps
zbMATH Keywords: hemicontinuity
author: Ya Ping Fang
author: Nan-Jing Huang
reviewed by: Nicolas Hadjisavvas
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2041231