Regularized Solutions for Abstract Volterra Equations

We introduce a two-kernel dependent family of strong continuous operators defined in a Banach space, which allows us to consider in an unified treatment the notions of, among others, C0-semigroups of operators, cosine families, n-times integrated semigroups, resolvent families and k-generalized solutions. The results are applied to the study of existence and uniqueness of solutions for the Volterra equation of convolution type usfqa) Au, in the case A is not necessarily densely defined. Examples for equations defined in L p spaces are also given.