Resolvent Formulas for a Volterra Equation in Hilbert Space

Let y(t, x, f) denote the solution of the Cauchy problem \[ { y}'(t) + ∫_0^t {[d + a(t - s)]{ Ly}(s)ds = { f}(t), t 0, { y}(0) = { x},} \] where d 0 and L is a self-adjoint densely defined linear operator on a Hilbert space H with L λ ₁ I. Let U(t) x = y(t, x, 0), V = U'. By analyzing a related scalar equation with parameter, we find sufficient conditions on the kernel a in order that ∫₀^∞ \| V(t)L- γ \| dt < ∞ (γ > 0). These results and certain resolvent formulas can be used to study the asymptotic behavior of the solution y(t, x, f) as t → ∞. An application to a semilinear integro-partial differential equation is presented.