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

Abstract

Let $T(·)$ be a (not necessarily exponentially bounded, not necessarily nondegenerate)$\alpha$-times integrated $C$-semigroup and let $-B$ be the generator of a $\left(C_0\right)\text{-}$ group $S(·)$ commuting with $T(·)$ and $C$. Under suitable conditions on $T(·)$ and $S(·)$ we prove the existence of an $\alpha$-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 $B$.