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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.

MSC:
 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators 47A25 Spectral sets 34G10 Linear ODE in abstract spaces 34C25 Periodic solutions of ODE
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References:
 [1] Daleckii, Ju.L.; Krein, M. G.: Stability of solutions of differential equations in Banach spaces. (1974) [2] Fattorini, H. O.: Second order linear differential equations in Banach spaces. (1985) · Zbl 0564.34063 [3] Katznelson, Y.: An introduction to harmonic analysis. (1976) · Zbl 0352.43001 [4] Nagel, R.: One-parameter semigroups of positive operators. (1986) · Zbl 0585.47030 [5] Neubrander, F.: Well-posedness of higher order abstract Cauchy problems. Trans. amer. Math. soc. 295, 257-290 (1986) · Zbl 0589.34004 [6] Pazy, A.: Semigroups of linear operators and applications to partial differential equations. (1983) · Zbl 0516.47023 [7] Gearhart, L.: Spectral theory for contraction semigroups on Hilbert space. Trans. amer. Math. soc. 236, 385-394 (1978) · Zbl 0326.47038 [8] Prüss, J.: On the spectrum ofc0. Trans. amer. Math. soc. 284, 847-857 (1984) · Zbl 0572.47030 [9] Vu, Ph.Q.: The operator equationaxxbcab. Math. Z. 208, 567-588 (1991) [10] Vu, Ph.Q.; Schüler, E.: The operator equationaxxbc. J. differential equations 2 (1998) [11] Goldstein, J.: Semigroups of linear operators and applications. (1985) · Zbl 0592.47034 [12] Nagy, B.: On cosine operator functions in Banach spaces. Acta sci. Math. 36, 281-293 (1974) · Zbl 0273.47008 [13] Lizama, C.: Mild almost periodic solutions of abstract differential equations. J. math. Anal. appl. 143, 560-571 (1989) · Zbl 0698.47035