Existence of solutions of abstract volterra equations in a banach space and its subsets

Abstract

We consider a criterion and sufficient conditions for the existence of a solution of the equation $$Z_t x = \frac{{t^{n - 1} x}}{{\left( {n - 1} \right)!}} + \int\limits_0^t {a\left( {t - s} \right)AZ_s xds} $$ in a Banach space X. We determine a resolvent of the Volterra equation by differentiating the considered solution on subsets of X. We consider the notion of "incomplete" resolvent and its properties. We also weaken the Priiss conditions on the smoothness of the kernel a in the case where A generates a C ₀-semigroup and the resolvent is considered on D(A)