The Resolvent Kernel of an Integrodifferential Equation in Hilbert Space

Let y(t, x, f) denote the solution of y'(t) + ∫ ₀ᵗ a(t - s) Ly(s)ds = f(t), y(0) = x, t 0, where L is a self adjoint densely defined operator on a Hilbert space H, with L μ > 0. Let U(t) x = y(t, x, 0). It is shown that if $a(t)$ is continuous (t 0) and completely monotonic $(t > 0)$, but not constant, then U(t) is a bounded operator on H with \| U(t) \| √ 2, \| U(t) \| → 0(t → ∞ ), and ∫ ₀^∞ \| U(t) \|dt < ∞. This result is useful when the representation y(t, x, f) = U(t) x + ∫ ₀ᵗ U(t - s) f(s)ds holds. The proof starts with the inequality \| U(t) \| λ μ | u(t,λ ) |, where uₜ (t,λ ) + λ ∫ ₀ᵗ a(t - s)u(s,λ )ds = 0, u(0,λ ) = 1.