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On stabilization of discrete monotone dynamical systems. (English) Zbl 0652.47035

Let X be a strictly convex Banach lattice satisfying certain natural technical conditions, and let Y be a closed subset of X. For a mapping S:Y\(\to Y\), the positive orbit of x is \(\gamma\) \(+(x)=\{S\) n(x)\(\}\), while the positive limit set \(\omega\) (x) of x is the set of all subsequential limits of \(\{\) S n(x)\(\}\).
Theorem 1. Let S:Y\(\to Y\) be strongly monotone and nonexpansive. Let \(u_ 0\in Y\), and assume \(\gamma\) \(+(u_ 0)\) is relatively compact in X. Then \(\omega (u_ 0)=\{\nu \}\) for some \(\nu\in Y\) (in general depending on \(u_ 0).\)
The authors apply Theorem 1 to a semilinear parabolic initial-boundary value problem to infer convergence to periodic solutions.
Reviewer: G.Passty

MSC:

47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H20 Semigroups of nonlinear operators
35K55 Nonlinear parabolic equations
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