A Stieltjes type convolution for integrated semigroups of bounded strong variation and $L_p$-solutions to the abstract Cauchy problem

Abstract

A convolution of Stieltjes type is introduced for operator families of bounded strong variation and vector valued $L_1$-functions. Using this tool, perturbation theorems for integrated semigroups of bounded strong variation are derived, and improved results on $L_p$-solutions to the inhomogeneous Cauchy problem are obtained.