Domains of Attraction of Generic $\omega$-Limit Sets for Strongly Monotone Semiflows

Abstract

Asymptotic behavior of a strongly increasing semiflow $\Omega$ in a strongly ordered metrizable topological space $X$ is investigated in terms of the $\omega$-limit set $\omega (x)$ of a generic point $x\in X$ whose positive semiorbit ${\mathcal{O}}^{+}(x)$ is assumed to be relatively compact. The domain of attraction of the $\omega$-limit set of a generic order $\omega$-stable point is determined. If $X$ is an open and order-convex subset of a separable strongly ordered Banach space $V$, it is proved that "almost all" points $x\in X$ are order $\omega$-stable, whereas the remaining $\omega$-unstable points are contained in the union of at most countably many Lipschitz manifolds of codimension one in $V$. If $\Omega$ admits a strongly positive, compact linearization about its equilibria, then $\omega (x)$ is a single equilibrium for every order $\omega$-stable point $x\in X$.