Let A be a closed linear operator such that the abstract Cauchy problem u''(t) = Au(t),t epsilon R; u(0) = x, u'(0) = y, is well-posed. We present some multiplicative perturbation theorems which give conditions on an operator C so that the abstract Cauchy problems for differential equations u''(t) = ACu(t) and u''(t) = C Au(t) also are well-posed. Some new or known additive perturbation theorems and mixed-type perturbation theorems are deduced as corollaries. Applications to characterization of the infinitesimal comparison of two cosine operator functions are also discussed.
