A characterization of well-posedness for the second order abstract Cauchy problems with finite delay

In this work, we introduce a strongly continuous one-parameter family of bounded linear operators that completely describes the well-posedness of a second order abstract differential delay equation in the initial history space $L^p([-r,0];X)$, $r>0$. This family, which satisfies a specific functional equation is applied to characterize the mild solution of the considered second order delay equation.