Perturbation of non-exponentially-bounded α-times integrated C-semigroups

Abstract

Let T(•) be a (not necessarily exponentially bounded, not necessarily nondegenerate) α-times integrated C-semigroup and let -B be the generator of a (C₀)- group S(•) commuting with T(•) and C. Under suitable conditions on T(•) and S(•) we prove the existence of an α-times integrated C-semigroup V(•), which has generator overline{A+B} provided that T(•) is nondegenerate and has generator A. Explicit expressions of V(•) in terms of T(•) and S(•) are obtained. In particular, when B is bounded, V(•) can be constructed by means of a series in terms of T(•) and powers of B.