Time dependent Desch-Schappacher type perturbations of Volterra integral equations (Q1865908)

The authors study perturbations of the Volterra integral equation \[ u(t)=f(t)+ \int_0^t a(t-s)Au(s) ds, \quad t\geq 0, \] where \(A\) is a closed linear operator in a Banach space and where it is assumed that there is a strongly continuous resolvent for the equation. The perturbations are either multiplicative, i.e., \(u(t)=f(t)+\int_0^t a(t-s)B(s)Au(s) ds\) or additive, i.e., \(u(t)=f(t)+\int_0^t a(t-s)(B(s)+A)u(s) ds\). In both cases the authors first prove a more general result with complicated assumptions and then in a series of corollaries give simple and sufficient conditions for the assumptions of the general theorem to hold.