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Exponential stability and uniform boundedness of solutions for nonautonomous periodic abstract Cauchy problems. An evolution semigroup approach. (English) Zbl 1279.47063
Let $$\mathcal{U}=\{U(t,s)\}_{t\geq s}$$ be a strongly continuous and $$q$$-periodic evolution family on a Banach space $$X$$. The authors show that, if $\sup_{\mu\in \mathbb R}\sup_{t\geq \tau\geq 0}\left\|\int^t_\tau e^{i\mu s}U(t,s)x\, ds\right\|<\infty \text{ for all } x\in X,$ then $\sup_{\mu\in \mathbb R}\sup_{t\geq 0}\left\|\int^t_\tau e^{-i\mu s}\mathcal{T}(s)f\, ds\right\|<\infty \text{ for all } f\in \mathrm{span}\{\cup_{t\geq 0}\mathcal{A}_t\},$ where $\mathcal{T}(t)f(s):=\left\{ \begin{matrix} U(s,s-t)f(s-t), & s\geq t \\ 0, & s<t, \\ \end{matrix} \right. \;\;t\geq 0, \;s\in \mathbb R, \;f\in \overline{\mathrm{span}}\{\cup_{t\geq 0}\mathcal{A}_t\},$ and $$\mathcal{A}_t$$ is the set of all $$X$$-valued functions $$f$$ on $$\mathbb R$$ for which there exists a function $$F\in P_q(R,X)\cap AP_1(R,X)$$ such that $$F(t)=0$$, $$f=F|_{[t,\infty)}$$ and $$f(s)=0$$ for $$s<t$$. Here, $$P_q(\mathbb R,X)$$ denotes the $$q$$-periodic, $$X$$-valued, uniformly continuous functions on $$\mathbb R$$ and $$AP_1(\mathbb R,X)$$ is a certain space of almost periodic functions (cf. [C. Corduneanu, Almost periodic oscillations and waves. New York, NY: Springer (2009; Zbl 1163.34002)]).

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47A30 Norms (inequalities, more than one norm, etc.) of linear operators 47A10 Spectrum, resolvent 35B15 Almost and pseudo-almost periodic solutions to PDEs 35B10 Periodic solutions to PDEs
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