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The operator equation \(AX-XB=C\), admissibility, and asymptotic behavior of differential equations. (English) Zbl 0918.34059
For abstract evolution equations of the form \(u'(t)=Au(t)+f(t)\) one knows the notion of mild solutions. These are solutions which are representable in the form \[ u(t)=T(t-s)u(s)+\int^t_sT(t-\tau)f(\tau)d\tau\quad\text{for all }t\geq s, \] where \(A\) is the generator of a \(C_0\)-semigroup \(T(t)\) on a Banach space \(E\). A subspace \({\mathcal M}\) of the Banach space of all uniformly continuous bounded functions from \(\mathbb{R}\) to \(E\) is regularly admissible if for every \(f\in{\mathcal M}\) there exists a unique mild solution \(u\in{\mathcal M}\) of the abstract evolution equation.
The authors show that the problem is connected with the study of operator equations of the form \(AX-XB=C\) if a subspace \(\mathcal M\) is regularly admissible. As applications the authors study the exponential dichotomy, stability, the existence of periodic and almost-periodic solutions to the starting equation \(u'(t)=Au(t)+f(t)\).

MSC:
34G10 Linear differential equations in abstract spaces
47A62 Equations involving linear operators, with operator unknowns
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