Linear Volterra equations and integrated solution families

Abstract

We consider the linear Volterra equation

$${\text{(VE;}}A{\text{,}}a{\text{)}}u{\text{(}}t{\text{) = }}x {\text{ + }}\int_{\text{0}}^{\text{t}} { a{\text{(}}t{\text{ - }}s{\text{)}}Au{\text{(}}s{\text{)}}ds {\text{for }}t \geqslant {\text{0}}{\text{.}}} $$

HereA is an unbounded closed linear operator in a Banach spaceX anda is a scalar valued function. We study the theory of solution families which are not necessarily exponentially bounded and also, as their generalizations, consider the notion ofn-times integrated solution families for (VE;A, a). These families are characterized in terms of the associated Volterra integral equation

$${\text{(VE;}}A{\text{,}}a{\text{)}}_n u{\text{(}}t{\text{) = }}\frac{{t^n }}{{n!}}x {\text{ + }}A{\text{ }}\int_{\text{0}}^{\text{t}} { a{\text{(}}t{\text{ - }}s{\text{)}}u{\text{(}}s{\text{)}}ds {\text{for }}t \geqslant {\text{0}}{\text{.}}} $$

The results are applied to additive and multiplicative perturbation theorems and adjoint problems.