Linear Volterra equations and integrated solution families

We consider the linear Volterra equation (VE;A,a) u(t)=x+integral(0)(t) a(t-x)Au(s)ds for t greater than or equal to 0. Here A is an unbounded closed linear operator in a Banach space X and a is a scaler valued function. We study the theory of solution families which are not necessarily exponentially bounded and also, as their generalizations, consider the notion of n-times integrated solution families for (VE;A,a). These families are characterized in terms of the associated Volterra integral equation (VE;A,a)(n) u(t)=t(n)/n!x+A integral(0)(t)a(t-x)u(s)ds for t greater than or equal to 0. The results are applied to additive and multiplicative perturbation theorems and adjoint problems.