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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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