For a continuous, increasing function ω : R + → R + \{0} of finite exponential type, this paper introduces the set Z(A, ω) of all x in a Banach space X for which the second order abstract differential equation (2) has a mild solution such that [ω(t)] −1 u(t, x) is uniformly continues on R + , and show that Z(A, ω) is a maximal Banach subspace continuously embedded in X, where A ∈ B(X) is closed. Moreover, A| Z(A,ω) generates an O(ω(t)) strongly continuous cosine operator function family.