Maximal subspaces for solutions of the second order abstract cauchy problem

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 showthat 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.