Positivity (2012) 16:653–684
Positive evolution families solving nonautonomous
Received: 5 April 2011 / Accepted: 11 July 2011 / Published online: 30 July 2011
© Springer Basel AG 2011
Abstract We study nonautonomous linear difference equations via the correspond-
ing abstract Cauchy problem. The domains of the associated operators are time depen-
dent and do not contain a common core, so we have to prove wellposedness by a direct
approach. In the periodic case we then investigate the asymptotics (in different topol-
ogies) of the corresponding evolution families using the Perron–Frobenius theory for
positive matrices combined with spectral decomposition methods. Finally, we show
how ﬂows in time-dependent networks can be treated within this framework.
Keywords Evolution familiy · Propagator · Abstract Cauchy problem ·
Jacobs–Glicksberg–DeLeeuw decomposition · Periodic process · Flows in networks
Mathematics Subject Classiﬁcation (2000) 47D06 · 34G10
Our starting point are systems of nonautonomous difference equations of the form
) = 0, t ≥ s ∈ R,
= g ∈ X.
The author expresses his gratitude both to Friedrich-Ebert-Stiftung and Wilhelm-Schuler-Stiftung.
F. Bayazit (
Arbeitsgruppe Funktionalanalysis, Mathematisch-Naturwissenschaftliche Fakultät,
Universität Tübingen, Auf der Morgenstelle 10, Tübingen, Germany