Uniform $L^1$ behavior in a class of linear Volterra equations

We find sufficient conditions for the solution of the equation $u’\left ( t \right ) + \\ \int _0^t {\sum \nolimits _{i = 1}^n {{\lambda _i}{a_i}\left ( {t - s} \right )u\left ( s \right )ds = 0, u\left ( 0 \right ) = 1} }$, to satisfy $\int _0^\infty {{{\sup }_{{\lambda _1},...,{\lambda _n} \ge 1}}{{\left ( {{\lambda _1} + \cdot \cdot \cdot + {\lambda _n}} \right )}^{ - 1/2}} \times \\ u’\left ( {t, {\lambda _1},...,{\lambda _n}} \right )dt < \infty }$. Our results generalize the case $n = 1$. Applications to a related equation in Hilbert space are given.