An integrodifferential wave equation

This paper is devoted to the study of the integrodifferential equation $$u'(t) = Au(t) +\int_0^t a(t - s)A_1u(s)ds + f(t), t\geq 0, $$ where $A$ is a Hille-Yosida operator in a Banach space $X$, $A_1 \in \mathcal {L}(D(A);X)$ and $a$ has bounded variation. Existence, uniqueness and estimates of strict and weak solutions are proved by extrapolation methods and the Miller scheme. Applications are given to the Cauchy-Dirichlet problem for the integrodifferential wave equation $$ w_{tt}(t, x) = w_{xx}(t, x) +\int_0^ta(t -s)w_{xx}(s, x)ds + f(t, x), t\geq 0, x\in[0, l]. $$