Reghiş, Mircea; Buşe, Constantin On the Perron-Bellman theorem for \(C_0\)-semigroups and periodic evolutionary processes in Banach spaces. (English) Zbl 0981.47030 Ital. J. Pure Appl. Math. 4, 155-166 (1998). Let \(T= \{T(t)\}_{t\geq 0}\) be a \(C_0\)-semigroup on \(X\) and \(A\) be its infinitesimal generator. First, the authors present the example of \(T\) such that \(A\) is not bounded on all \(X\), \[ \sup_{t\geq 0} \Biggl\|\int^t_0 e^{i\rho s}T(s) x ds\Biggr\|< \infty,\quad\forall\rho\in \mathbb{R},\quad\forall x\in X, \] but \(T\) is not uniformly exponentially stable (counterexample to Balint’s 1983’s result).Motivated by this counterexample, the authors proved that a periodic evolutionary process \(U\) of linear operators on a Banach space \(X\) is uniformly exponentially stable if and only if \[ \sup_{t\geq 0} \Biggl\|\int^t_0 u(t,s) f(s) ds\Biggr\|< \infty,\quad\forall f\in C_0(\mathbb{R}_+, X) \] (of all continuous functions \(f: \mathbb{R}_+\to X\) with \(\lim_{t\to\infty} f(t)= 0\)). As application, the authors show that \(U\) is uniformly exponentially stable if and only if the above inequality holds for all \(X\)-valued almost periodic functions \(f\) on \(\mathbb{R}_+\). Reviewer: Nguyêñ Hôǹg Thái (Minsk) Cited in 5 Documents MSC: 47D06 One-parameter semigroups and linear evolution equations Keywords:\(C_0\)-semigroup; infinitesimal generator; periodic evolutionary process; uniformly exponentially stable PDFBibTeX XMLCite \textit{M. Reghiş} and \textit{C. Buşe}, Ital. J. Pure Appl. Math. 4, 155--166 (1998; Zbl 0981.47030)