A note on the lower semicontinuity of efficient solutions for parametric vector equilibrium problems
type: Item
instance of: scholarly article
title: A note on the lower semicontinuity of efficient solutions for parametric vector equilibrium problems
zbMATH Open document ID: 1264.90156
author: Yanyan Li
published in: Applied Mathematics Letters
publication date: 2013-03-07T00:00:00Z
review text: Assume \(B\), \(\Lambda\) are metric spaces and \(Y\) is a metric vector space with closed convex cone \(C\) such that \(\text{int} C\neq \emptyset\). The authors consider for \(F:B\times B\times \Lambda \rightarrow Y\) and a set valued map \(A:\Lambda \rightarrow 2^B\) the mapping \(S:\Lambda\rightarrow B\) of efficient solutions of a parametric vector equilibrium problem given by \[ S(\mu):=\{x\in A(\mu) \mid F(x,y,\mu)\notin -C\setminus\{0\}, \forall y\in A(\mu)\}. \] Theorem 3.1: If \(A\) and \(F\) are continuous mappings, \(A\) is compact valued and the metric regularity condition (the authors use more restrictively \(D=1\)) \[ \exists D>0~ \forall \mu\in \Lambda~ \forall x\in A(\mu)\setminus S(\mu) ~\exists y\in S(\mu): d_B(x,y)\leq D \inf_{w\in Y\setminus (-\text{int}\,C)}d_Y(F(x,y,\mu),w) \] holds, where \(d_B\) resp. \(d_Y\) are the metrics on \(B\) and \(Y\), then the set valued mapping \(S\) is lower semicontinuous in the sense of Kuratowski. Under the same assumptions (Theorem 3.2) the lower semicontinuity is valid for the weak efficient solution set mapping [replace in \(S\) the set \(-C\setminus\{0\}\) by \(-\text{int}\,C\)]. They illustrate their results by two finite dimensional examples.
reviewed by: Armin Hoffmann
Mathematics Subject Classification ID: 90C29
Mathematics Subject Classification ID: 58C06
Mathematics Subject Classification ID: 90C31
Mathematics Subject Classification ID: 90C33
zbMATH DE Number: 6142818
zbMATH Keywords: parametric vector equilibrium problems
zbMATH Keywords: lower semicontinuity
zbMATH Keywords: efficient solutions
zbMATH Keywords: weak efficient solutions
MaRDI profile type: Publication
full work available at URL: j.aml.2012.11.010
OpenAlex ID: W2046021185
DOI: 10.1016/J.AML.2012.11.010
description: scientific article; zbMATH DE number 6142818