Uniform $L^1 $ Behavior for an Integrodifferential Equation with Parameter

For a family of real integrodifferential equations with nonnegative, nonincreasing, convex, strongly positive convolution kernel $\lambda [c + a(t)]$, depending on the parameter $\lambda $, $0 < \Lambda \leqq \lambda < \infty $, we show that the solutions $u(t,\lambda )$, normalized by the initial condition $u(0,\lambda ) = 1$, satisfy $\sup _\lambda | {u(t,\lambda )} | = u^0 (t)$ where $u^0 \in 1(0,\infty )$ and $u^0 (\infty ) = 0$. A result of the same type holds for $u_t $. The proof uses a Fourier integral representation for the solution. Applications to equations in Hilbert space and to scalar equations with a complex parameter are given.