The Operator EquationAX−XB=C, Admissibility, and Asymptotic Behavior of Differential Equations

It is shown that there exists a natural relationship between the regular admissibility of a translation-invariant subspace M ofBUC(R, E) (the space of uniformly continuous bounded functions onRwith values in a Banach spaceE), w.r.t. the differential equationu′(t)=Au(t)+f(t) (∗), and the unique solva...

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Bibliographic Details
Published in:Journal of Differential Equations 1998-05, Vol.145 (2), p.394-419
Main Authors: Phong, Vu Quoc, Schüler, E.
Format: Article
Language:English
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Summary:It is shown that there exists a natural relationship between the regular admissibility of a translation-invariant subspace M ofBUC(R, E) (the space of uniformly continuous bounded functions onRwith values in a Banach spaceE), w.r.t. the differential equationu′(t)=Au(t)+f(t) (∗), and the unique solvability of a special operator equation of Lyapunov's typeAX−XDM=−δ0, where DMis the restriction of the operator D≡d/dtonto M andδ0(f)=f(0). This leads to some spectral conditions for the regular admissibility. Applications to questions of exponential dichotomy, exponential stability, the existence of periodic and almost periodic solutions of equation (∗), as well as analogous questions for some nonlinear equations and functional differential equations are presented.
ISSN:0022-0396
1090-2732
DOI:10.1006/jdeq.1998.3418