Uniform L¹ Behavior for an Integrodifferential Equation with Parameter

For a family of real integrodifferential equations with nonnegative, nonincreasing, convex, strongly positive convolution kernel λ [c + a(t)], depending on the parameter λ, 0 < Λ λ < ∞, we show that the solutions u(t,λ ), normalized by the initial condition u(0,λ ) = 1, satisfy _λ | u(t,λ ) | = u⁰ (t) where u⁰ ∈ 1(0,∞ ) and u⁰ (∞ ) = 0. A result of the same type holds for uₜ. 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.