Generic quasi-convergence for strongly order preserving semiflows: a new approach
The authors consider strongly order-preserving semiflows \(\Phi\) on partially ordered metric spaces \(X\), with \(\Phi\) having compact orbit closures. Let \(E\) denote the set of equilibria of \(\Phi\). Under the condition (L) Either every omega limit set has an infimum in \(X\) and the set of points that are doubly accessible from below (with respect to the order) has dense interior, or every omega limit set has a supremum in \(X\) and the set of points that are doubly accessible from above has dense interior, the authors prove that the set \(Q\) of points with limit set in \(E\) and the set \(C\subseteq Q\) of points with single point limit set satisfy \[ X\setminus Q\subseteq\overline{\text{Int\,}C}, \] and \(\text{Int\,}Q\) is dense in \(X\). An application to delay differential equations is given.
scientific article; zbMATH DE number 2145897
scientific article