Exponential stability and uniform boundedness of solutions for nonautonomous periodic abstract Cauchy problems. An evolution semigroup approach
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.\ [\textit{C. Corduneanu}, Almost periodic oscillations and waves. New York, NY: Springer (2009; Zbl 1163.34002)]).
scientific article; zbMATH DE number 6153933
scientific article