A representation formula for strongly continuous resolvent families
The resolvent of the linear Volterra equation \[ u(t)=f(t)+\int_0^ta(t-s)Au(s)ds \] in a Banach space with a closed operator \(A\) is related to \(a\) by a Laplace transform. This relation and a result on the inverse Laplace transform by \textit{B. Hennig} and \textit{F. Neubrander} [Appl. Anal. 49, No. 3-4, 151-170 (1993; Zbl 0791.44002)] leads straightforwardly to an involved formula for the resolvent. From this formula, criteria for a positive resolvent are gained.
linear Volterra equation in Banach space
scientific article; zbMATH DE number 1185926