Nonautonomous integro-differential equations of hyperbolic type

Abstract

This paper is devoted to two problems for the nonautonomous integrodifferential equation in a Banach space $X$ of hyperbolic type $$ \text{$u'(t) = A(t)u(t)+ \int_0^t B(t,s)u(s) ds + f(t)$ for $t\in[0,T]$, and $u(0) = u_0$.} $$ One is the problem of existence and uniqueness of classical solutions without assuming that the common domain of $A(t)$ is dense in $X$, and the other is the regularity problem in the case where the common domain of $A(t)$ is dense in $X$. The regularity result will play a role in developing an abstract theory which can be applied to the second-order integrodifferential equations with the third kind boundary conditions.