Asymptotically almost periodic solutions of abstract retarded functional differential equations of second order (Q1041107)

The paper deals with the existence of asymptotically almost periodic mild solutions of the delay Cauchy problem \[ x''(t)=Ax(t)+F(t,x_t, x'_t),\quad t\geq 0, \;x_0=\varphi,\quad x'_0=\psi \] and the existence of asymptotically almost periodic mild solutions of the semi-linear abstract retarded functional differential equation \[ x''(t)=Ax(t)+F(t,x_t, x'_t),\quad t\in{\mathbb{R}}, \] in a complex Banach space \(X,\) with infinite delay. Here, the operator \(A\) is the infinitesimal generator of a strongly continuous cosine function of linear operators on \(X.\) To establish his existence results the author assumes that the expression \(F(t,x_t, x'_t)\) can be written in the form \(F_1(t,x_t)+F_2(t,x_t),\) with \(F_1, F_2\) satisfying appropriate conditions. An application to the non-linear wave equation with infinite delay illustrates the results.