Uniform L¹ Behavior in Classes of Integrodifferential Equations with Completely Monotonic Kernels

We find conditions on a family a of completely monotone, locally integrable, nonconstant functions a which enable us to write the solution $u(t;a)$ of u'(t) + ∫₀ᵗ a(t - s)u(s)ds = 0, $u(0) = 1$, as u(t;a) = - ∫- ε⁰ eσ t dμ (σ ;a) + u₁ (t;a), where μ ( · ;a) is a finite nonnegative measure on [ - ε ,0] and |u₁ (t;a)| Qe- ε t with Q, ε positive constants independent of a ∈ a. This formula is then utilized to give conditions on the collection a which ensure that ρ (t) a ∈ a |u(t;a)| → 0(t → ∞ ) and ∫₀^∞ ρ (t) a ∈ a |u(t;a)|dt < ∞ , where ρ is a given weight function. These results can be combined with a resolvent formula to investigate the asymptotic behavior as t → ∞ of solutions of certain integrodifferential equations in Hilbert space.