A note on the convergence of the solutions of a linear functional differential equation (Q908474)

The paper deals with the equation \((1)\quad x'(t)=-\mu_ 0x(t)+\int^{0}_{-\infty}x(t+s)d\mu (s),\) where \(\mu_ 0\in R\) and \(\mu\) : (-\(\infty,\infty)\to R\) is a nondecreasing function and \(\mu_ 0=\int^{0}_{-\infty}d\mu\). The author proves that any solution to (1) belonging to a bounded continuous initial function converges to a finite limit as \(t\to \infty\) if and only if \(\int^{\infty}_{-\infty}| s| d\mu (s)<\infty\) holds.