On a second-order delay integro-differential equation in Banach spaces

Consider in a Banach space $ (X, \Vert \cdot \Vert) $ the following delay integro-differential equation$ \begin{align} u^{\prime \prime}(t) = Au(t)+\int_{-\infty}^0g(s,u_t(s))\, ds +f(t), \ \ 0\le t\le T, \end{align} $where $ A:D(A)\subset X \to X $ is a densely defined, closed linear operator generating a strongly continuous cosine family, while $ g $ is an unbounded nonlinear function which is dominated in some sense by $ A $. Global existence and uniqueness are derived for the initial value problem associated with the above equation by using a fixed point approach combined with properties of the cosine and sine families. Continuity of solutions with respect to the initial data and some stability properties will be also addressed. We illustrate our results with an application to the wave equation with nonlinear memory.