A singular limit problem for a linear Volterra equation

We study the dependence on ${c_1}$ and ${c_2}$ of the solution $u\left ( {t, {c_1}, {c_2}} \right )$ of the equation \[ u’\left ( t \right ) + \int _0^t A \left ( {t - s, {c_1}, {c_2}} \right )u\left ( s \right )ds = 0, \qquad u\left ( 0 \right ) = 1,\] where the conditions on $A$ are stated in terms of its Fourier transform. We obtain sufficient conditions and (weaker) necessary conditions for \[ \int _0^\infty {\sup \limits _{0 \le {c_i} \le 1} \left | {u\left ( {t, {c_1}, {c_2}} \right )} \right |dt} < \infty , i = 1,2\] and for \[ \int _0^\infty {\sup \limits _{0 \le {c_1},{c_2} \le 1} } \left | {u\left ( {t, {c_1}, {c_2}} \right )} \right |dt < \infty \] The kernel $A$ is a combination of nonnegative nonincreasing convex functions and arises in the linear theory of viscoelastic rods and plates.