Geodesic
Periodic and almost periodic cosine operator functions
Sbornik. Mathematics, Tome 46 (1983) no. 3, pp. 391-402

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Necessary and sufficient conditions are established for a well-posed Cauchy problem \begin{equation} \label{1} u''(t)=Au(t),\qquad u(0)=u^0,\quad u'(0)=u^1, \end{equation} to have an almost periodic (periodic) solution in a Banach space. The influence of how “scattered” the spectrum $\sigma(A)$ is on almost-periodicity of a cosine operator function giving the solution of (1) is determined. Results relating to the Cauchy problem for the nonhomogeneous equation are presented. Bibliography: 29 titles. 
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     author = {S. I. Piskarev},
     title = {Periodic and almost periodic cosine operator functions},
     journal = {Sbornik. Mathematics},
     pages = {391--402},
     publisher = {mathdoc},
     volume = {46},
     number = {3},
     year = {1983},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1983_46_3_a4/}
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S. I. Piskarev. Periodic and almost periodic cosine operator functions. Sbornik. Mathematics, Tome 46 (1983) no. 3, pp. 391-402. http://geodesic.mathdoc.fr/item/SM_1983_46_3_a4/