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On the spectrum of cosine functions. (English) Zbl 0921.34073
The author gives some characterizations of the resolvent set $\rho(A)$ of the generator $A$ of a strongly continuous cosine function $C(t)$ with the aid of the equation $$u''(t)=Au(t)+f(t).$$ One of these theorems shows that $1\in\rho(C(1))$ if and only if, for every $1$-periodic function $f\in C([0,1],X)$ ($X$ is a Banach space), the above equation has a unique $1$-periodic mild solution of class $C^1$. In case of a Hilbert space $X$, the above equation has a unique $1$-periodic mild solution for any $1$-periodic function $f\in C([0,1],X)$, if and only if, the associated sine function $S(t)$ has the property that $S(1)$ is invertible.

34L15Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators
47A25Spectral sets
34G10Linear ODE in abstract spaces
34C25Periodic solutions of ODE
Full Text: DOI
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