Minimax principles for convex eigenvalue problems
type: Item
instance of: scholarly article
title: Minimax principles for convex eigenvalue problems
zbMATH Open document ID: 0591.47045
DOI: 10.1016/0022-247X(84)90174-4
author: Theodore Laetsch
published in: Journal of Mathematical Analysis and Applications
publication date: 1984-01-01T00:00:00Z
review text: Let A be a compact, isotone operator on the positive cone K of a partially ordered locally convex topological vector space E satisfying \(A0>0\). The following problem is considered in the work: To find all numbers \(\lambda >0\) and all \(u\in K\) satisfying \(u=\lambda Au.\) Let \(K^*\) be the dual cone of K and \(\sigma^*=\inf \{1/\lambda:\exists u\in K,u=\lambda Au\}\). It is proved that \(\sigma^*=\inf_{u\in K}\sup_{\phi \in K^*}\phi (Au)/\phi (u)\), and under an additional condition on A (A is convex and maps K into the interior of K) it follows \(\sigma^*=\sup_{\phi \in K^*}\inf_{u\in K}\phi (Au)/\phi (u)\).
Mathematics Subject Classification ID: 47J10
Mathematics Subject Classification ID: 47H07
zbMATH DE Number: 3949105
zbMATH Keywords: convex eigenvalue problems
zbMATH Keywords: minimax principle
zbMATH Keywords: saddle point
zbMATH Keywords: characterization
zbMATH Keywords: compact, isotone operator
zbMATH Keywords: partially ordered locally convex topological vector space
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3949105