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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 $$\mathbb{R}_+$$.

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations