Existence criteria for the solutions of two types of variational relation problems
type: Item
instance of: scholarly article
title: Existence criteria for the solutions of two types of variational relation problems
zbMATH Open document ID: 1272.49028
author: Mircea Balaj
author: Lai-Jiu Lin
published in: Journal of Optimization Theory and Applications
publication date: 2013-05-08T00:00:00Z
review text: Let \(X\) and \(Y\) be convex subsets of Hausdorff topological vector spaces and \(Z\) a topological space. Let \(S_1,S_2: X \multimap X\); \(T: X \multimap Y\) and \(P:X \multimap Z\) be multimaps and \(R(x,y,z)\) a ternary relation connecting elements \(x \in X,\) \(y \in Y\), and \(z \in Z\). The authors study the solvability of the following problems: [\((VRP1)\)] Find \((\overline{x},\overline{y}) \in X \times Y\) such that \(\overline{x} \in S_1(\overline{x}),\) \(\overline{y} \in T(\overline{x})\) and, for each \(u \in S_2(\overline{x}),\) there is \(z \in P(\overline{x})\) for which \(R(u,\overline{y},z)\) holds; [\((VRP2)\)] Find \((\overline{x},\overline{y}) \in X \times Y\) such that \(\overline{x} \in S_1(\overline{x}),\) \(\overline{y} \in T(\overline{x})\) and \(R(u,\overline{y},z)\) holds for each \(u \in S_2(\overline{x})\) and \(z \in P(\overline{x})\).
Mathematics Subject Classification ID: 49J53
Mathematics Subject Classification ID: 47J22
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 6161558
zbMATH Keywords: variational relation problem
zbMATH Keywords: set-valued map
zbMATH Keywords: fixed-point
reviewed by: Valerii V. Obukhovskij
MaRDI profile type: Publication
full work available at URL: s10957-012-0136-0
OpenAlex ID: W1985246712
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DOI: 10.1007/S10957-012-0136-0
description: scientific article; zbMATH DE number 6161558