Almost-periodic solutions for a second order abstract Cauchy problem (Q812800)

In the paper under review, the authors establish an existence result for almost-periodic solutions of a second order abstract Cauchy problem, that is, almost-periodic solutions of the equation \(x''(t)=Ax(t)+f(t)\), \(t\geq0\), with the conditions \(x(0)=x_0\), \(x'(0)=x_1\), where \(A\) is a densely defined closed operator acting in a Banach space \(X\) and all functions are \(X\)-valued. They show that if \(f\) is almost periodic and satisfies certain additional conditions, then the ``mild'' solution \(x\), given by a known formula, is almost-periodic. The authors often use the compactness of the sine function in Banach spaces, with important consequences for the spectral properties of the operator \(A\).