Perturbations of local C-cosine functions

We show that A+B is a closed subgenerator of a local C-cosine function T (.) on a complex Banach space X defined by T(t)x = Sigma(infinity)(N=0) B-n integral(t)(o) j(n-1) (s)j(n) (t - s) C (vertical bar t - s vertical bar)xds for all x is an element of X and 0 <= t <= T-0, if A is a closed subgenerator of a local C-cosine function C(.) on X and one of the following cases holds: (i) C(.) is exponentially bounded, and B is a bounded linear operator on D((A) over bar) so that BC = CB on D((A) over bar) and BA subset of AB; (ii) B is a bounded linear operator on D((A) over bar) which commutes with C(.) on D((A) over bar) and BA subset of AB; (iii) B is a bounded linear operator on X which commutes with C(.) on X. Here j(n)(t) = t(n)/n! for all t is an element of R, and integral(t)(0) j(-1)(s)j(0)(t - s)C (vertical bar t - 2s vertical bar)xds = C(t)x for all x is an element of X and 0 <= t < T-0.