On monotone trajectories

Janusz Mierczy{ński
1991 Proceedings of the American Mathematical Society  
In this paper C strongly monotone dynamical systems are investigated. It is proved that the set of points with precompact orbits which converge to a not unstable equilibrium but whose trajectories are not eventually strongly monotone is nowhere dense. This improves on and extends a recent result by P. Polácik [13] . For a metric space X with metric d, by a semiflow on X we mean a continuous mapping tp: [0, oo) x X -► X satisfying the following (we denote 0. We say a point x £ X (or its
more » ... y) is convergent (to an equilibrium e £ X) if d(cptx, e) -> 0 as t -> oo . It is easy to check that x £ X is convergent to í 6 I if and only if the orbit of x is precompact and co(x) = {e} , where the co-limit set co(x) := {y £ X : 3tk -► oo such that tpt x -> y as A: -» oo} . A real Banach space V with norm | • | is called strongly ordered if V is o endowed with a closed cone V+ having nonempty interior V+ . For v , w £ V, A semiflow cp on an open subset of a strongly ordered Banach space V is said to be strongly monotone if, for each t > 0, x, y g X, the inequality x < y implies tptx -C tpty . In a strongly monotone semiflow tp, the trajectory of x £ X is called eventually strongly decreasing (resp. eventually strongly increasing) when there is a r > 0 such that if T < tx < t2, then <pt x » tpt x (resp. 0r x < 0( x). A trajectory that is eventually strongly decreasing or eventually strongly increasing is called eventually strongly monotone. As proved in [6, Theorem 6.4], if the tra-
doi:10.1090/s0002-9939-1991-1056682-1 fatcat:vptcipwwsve4vn7qc3vfgz54gm