Periodic and almost periodic cosine operator functions

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.
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