On the Exponential Convergence to a Limit of Solutions of Perturbed Linear Volterra Equations

We consider a system of perturbed Volterra integrodifferential equations for which the solution approaches a nontrivial limit and the difference between the solution and its limit is integrable. Under the condition that the second moment of the kernel is integrable we show that the solution decays exponentially to its limit if and only if the kernel is exponentially integrable and the tail of the perturbation decays exponentially.