The authors find sufficient conditions to assure the existence of at least one mild solution to the initial value problem for the second-order differential inclusion \[ y''-Ay\in F(t,y_t),\quad t\in J=[0,b];\quad y_0=\phi,\;y'(0)=\eta, \] where \(F:J\times C(J_0,E)\to 2^{E}\) is a bounded, closed, convex multivalued map, \(\phi\in C(J_0,E)\), \(J_0=[-r,0]\), \(A\) is the infinitesimal generator of a strongly continuous cosine family in a Banach space \(E\), and \(\eta \in E\). The main tool used to prove the existence result is a fixed point theorem for condensing maps.