The authors investigate the existence of mild solutions to second-order initial value problems for a class of damped differential inclusions with nonlocal conditions of the form \(y''-Ay \in By'+F(t,y)\), \(t \in J=[0,b]\) and \(y(0)+f(y)=y_0\), \(y'(0)=\eta\), where \(F:J \times E \to 2^E\) is a multifunction, \(f:C(J,E) \to E\) is a continuous function, \(A\) is the infinitesimal generator of a strongly continuous cosine family \(\{C(t): t\in \mathbb{R}\}\) in the Banach space \(E\), \(B\) is a bounded linear operator on \(E\) and \(y_0,\eta \in E\). Using an approach based on fixed-point theory, the authors prove sufficient conditions for the existence of mild solutions in the cases in which \(F\) has convex or nonconvex values.