Let y(t, x, f) denote the solution of y'(t) + ∫₀ᵗ [d + a(t - s)] Ly(s)ds = f(t),t 0, y(0) = x, where d 0 and L is a selfadjoins densely defined operator on a Hilbert space H with L Λ > 0. Let U(t) x = y(t, x, 0). By analyzing a related scalar equation with parameter, we find sufficient conditions on the kernel $a(t)$ for \| U(t) \| → 0(t → ∞ ) and ∫₀^∞ \| U(t) \|dt < ∞. These results and a resolvent formula can be combined to reveal the behavior of y(t, x, f) as t → ∞.
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Carr et al. (1979) studied this question.
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