Minimax principles for convex eigenvalue problems (Q1074847)

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 \(\sigma^*=\sup_{\phi \in K^*}\inf_{u\in K}\phi (Au)/\phi (u)\).