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On weak exponential stability of evolution operators in Banach spaces. (English) Zbl 1218.47065
According to the authors, an evolution operator is a family of (not necessarily linear) operators $$\{ U(t, s)\mid t \geq s \geq 0 \}$$ on a Banach space $$E$$ satisfying $$U(t, t) = I$$, $$U(t, s)U(s, r) = U(t, r)$$ $$(t \geq s \geq r)$$ plus a continuity assumption and the exponential growth condition
$\|U(t, t_0)u\| \leq Me^{\omega(t - t_0)} \|u\| \quad (t \geq t_0 \geq 0 , \;u \in E) \, . \tag{1}$ If (1) is reinforced to
$\|U(t, t_0)u\| \leq Ne^{-\nu(t - s)} \|U(s, t_0) u\| \quad (t \geq s \geq t_0 \geq 0, \;u \in E) , \tag{2}$ then $$\{U(t, s)\}$$ is uniformly exponentially stable. Finally, if (2) holds for $$t_0$$ depending on $$u$$, the evolution operator is weakly exponentially stable. The results are on various properties of $$\{U(t, s)\}$$ that are equivalent to weak exponential stability.

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 34D05 Asymptotic properties of solutions to ordinary differential equations 34D23 Global stability of solutions to ordinary differential equations 34G20 Nonlinear differential equations in abstract spaces
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