LAUSR

The paper deals with the functional differential equation \begin{document}$\begin{equation*} y'(t)+∈t_0^{∞}μ_0(ds)\, y(t-s)+\sum_{k=1}^∞ e^{iω_kt}∈t_0^{∞}μ_k(ds)\, y(t-s)=f(t), \end{equation*}$ \end{document} where the functions \begin{document} $y$ \end{document} and \begin{document} $f$ \end{document} take their values in a Hilbert space, \begin{document} $ω_k∈\mathbb{R}$ \end{document} , \begin{document} $μ_k$ \end{document} are bounded operator-valued measures concentrated on \begin{document} $[0, +∞)$ \end{document} , and \begin{document} $\sum_{k=1}^∞\Vertμ_k\Vert . It is shown that the equation is stable provided the unperturbed equation \begin{document} $y'(t)+\int{{_0^{∞}}}μ_0(ds)\, y(t-s)=f(t)$ \end{document} is at least strictly passive (and consequently stable) and a special estimate holds; this estimate is certainly true if \begin{document} $|ω_k|$ \end{document} are sufficiently large.