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The strong variant of a Barbashin theorem on stability of solutions for non-autonomous differential equations in Banach spaces. (English) Zbl 1160.47036
In the paper under review, the authors give a necessary and sufficient condition for the uniformly exponential stability of an evolution family of operators. More precisely, it is proved that an exponentially bounded evolution family \({\mathcal U}=\{U(t,s)\}_{t\geq s\geq 0}\) of bounded linear operators on a Banach space \(X\) is uniformly exponentially stable if and only if there exists a number \(q\in [1, \infty)\) such that \[ \sup_{t\geq 0}\bigg(\int_0^t\| U(t,\tau)^*x^*\| ^qd\tau\bigg)^{1/q}=M(x^*)<\infty \quad \text{ for all }x^*\in X^*. \]

47D06 One-parameter semigroups and linear evolution equations
35B35 Stability in context of PDEs
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