Convergence for Strongly Order-Preserving Semiflows

Convergence of almost all trajectories of strongly order-preserving semiflows is derived under suitable additional assumptions. These essentially consist in slightly sharpening the strongly order-preserving property and in the continuous differentiability of the flow with respect to the state variable. Required spectral properties of the linearizations of the flow around equilibria usually follow in the same way as the compactness and monotonicity assumptions for the flow itself. The proofs are based on sharpened versions of the limit set dichotomy and the sequential limit set trichotomy.