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Domains of Attraction of Generic u-Limit Sets for Strongly Monotone Semiflows

Asymptotic behavior of a strongly increasing semiftow 'Z' in a strongly ordered metrizable topological space X is investigated in terms of the w-limit set w(x) of a generic point z E X whose positive semiorbit 0+ ( X) is assumed to be relatively compact. The domain of attraction of the w-limit set of a generic order w-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 E X are order t'-stable, whereas the remaining w-unstable points are contained in the union of at most countably many Lipschitz manifolds of codimension one in V. If 41' admits a strongly positive, compact linearization about its equilibria, then w(x) is a single equilibrium for every order t&-stable point x E X.