This problem is Theorem 2 from an old paper "ASYMPTOTIC BEHAVIOR OP SOLUTIONS OF NONLINEAR VOLTERRA EQUATIONS" by R. K. Miller. Consider a scalar equation \begin{equation} x(t)=f(t)-\int_0^ta(t-s)x(s)ds. \end{equation} The theorem says
Suppose $a(t)\in L_1(0,\infty)$, $f(t)$ is bounded, measurable and tends to a limit $f_0$ as $t\rightarrow\infty$. For each such $f$ the solution of the scalar equation is bounded and tend to the limit \begin{equation} x(t)\rightarrow x_0=f_0/(1+\int_0^{\infty}a(s)ds) ~\text{as} ~t\rightarrow\infty \end{equation} if and only if when $Re(u)\geq 0$ one has \begin{equation} \int_0^{\infty}a(t)e^{-ut}dt\neq-1 \end{equation}
The author gave reference "Soc. 68 (1962), 323-329. 4. R. E. A. C. Paley and N. Wiener, Fourier transforms in the complex domain, Amer. Math.Soc.Colloq.Publ. Vol.XIX,Amer. Math. Soc, Providence, R. I., 1934", which is not available to me.
I do not understand how this theorem comes from, any kind of help is appreciated, THANK YOU!