Abstract
Abstract The purpose of this informal paper is three-fold: First, filling a gap in the literature, we provide a (necessary and sufficient) principle of linearized stability for nonautonomous difference equations in Banach spaces based on the dichotomy spectrum. Second, complementing the above, we survey and exemplify an ambient nonautonomous and infinite-dimensional center manifold reduction, that is Pliss’s reduction principle suitable for critical stability situations. Third, these results are applied to integrodifference equations of Hammerstein- and Urysohn-type both in C- and Lp -spaces. Specific features of the nonautonomous case are underlined. Yet, for the simpler situation of periodic time-dependence even explicit computations are feasible.
Highlights
The purpose of this informal paper is three-fold: First, lling a gap in the literature, we provide a principle of linearized stability for nonautonomous di erence equations in Banach spaces based on the dichotomy spectrum
By virtue of the dichotomy spectrum, the above statements canonically generalize to the time-dependent situation. This dynamical spectrum is a subset of the positive real line. It proved to be an e cient tool in the geometric theory of nonautonomous dynamical systems when it comes to the construction of invariant manifolds and foliations, as well as for topological or smooth linearization questions
It is rather surprising that the crucial role of the dichotomy spectrum in stability criteria based on linearization is not explicitly present in the literature to our best knowledge
Summary
The purpose of this informal paper is three-fold: First, lling a gap in the literature, we provide a (necessary and su cient) principle of linearized stability for nonautonomous di erence equations in Banach spaces based on the dichotomy spectrum. Given a permanent solution (φ*t )t∈I of (∆) we assume that the stability boundary is contained in the dominant spectral interval of the variational eqn (Vφ* ).
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