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Asymptotic stability of linear differential equations in Banach spaces. (English) Zbl 0639.34050
Let A be a generator of a strongly continuous bounded semigroup T(t), \(t\geq 0\). We prove that if the intersection of the spectrum of A and the imaginary axis is at most countable and \(A^*\) has no purely imaginary eigenvalues, then the Cauchy problem for the differential equation \(\dot x(t)=Ax(t)\), \(t\geq 0\), is asymptotically stable.

MSC:
34D05 Asymptotic properties of solutions to ordinary differential equations
34A30 Linear ordinary differential equations and systems, general
34G10 Linear differential equations in abstract spaces
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