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On the Perron-Bellman theorem for \(C_0\)-semigroups and periodic evolutionary processes in Banach spaces. (English) Zbl 0981.47030
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 \(\mathbb{R}_+\).

47D06 One-parameter semigroups and linear evolution equations