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Spectral properties of cosine operator functions. (English) Zbl 0675.47029
Summary: Let A be the generator of a cosine function ${C}_{t}$, $t\in ℝ$ in a Banach space X; we shall connect the existence and uniqueness of a T- periodic mild solution of the equation ${u}^{\text{'}\text{'}}=Au+f$ with the spectral property $1\in \rho \left({C}_{T}\right)$ and, in case X is a Hilbert space, also with spectral properties of A.

##### MSC:
 47D03 (Semi)groups of linear operators 34G10 Linear ODE in abstract spaces 47D99 Algebraic systems of linear operators
##### References:
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