ON PERTURBATION OF $\alpha$-TIMES INTEGRATED $C$-SEMIGROUPS

Abstract

Let $\alpha \geq 0$, and $C$ be a bounded linear injection on a complex Banach space $X$. We first show that if $A$ generates an exponentially bounded nondegenerate $\alpha$-times integrated $C$-semigroup $S_{\alpha}(\cdot)$ on $X$, $B$ is a bounded linear operator on $\overline{D(A)}$ such that $BC = CB$ on $\overline{D(A)}$ and $BA \subset AB$, then $A+B$ generates an exponentially bounded nondegenerate $\alpha$-times integrated $C$-semigroup $T_{\alpha}(\cdot)$ on $X$. Moreover, $T_{\alpha}(\cdot)$ is also exponentially Lipschitz continuous or norm continuous if $S_{\alpha}(\cdot)$ is. We show that the exponential boundedness of $T_{\alpha}(\cdot)$ can be deleted and $\alpha$-times integrated $C$-semigroups can be extended to the context of local $\alpha$-times integrated $C$-semigroups when $R(C) \subset \overline{D(A)}$ and $BS_{\alpha}(\cdot) = S_{\alpha}(\cdot)B$ on $\overline{D(A)}$ both are added. Moreover, $T_{\alpha}(\cdot)$ is also locally Lipschitz continuous or norm continuous if $S_{\alpha}(\cdot)$ is. We show that $A+B$ generates a nondegenerate local $\alpha$-times integrated $C$-semigroup $T_{\alpha}(\cdot)$ on $X$ if $A$ generates a nondegenerate local $\alpha$-times integrated $C$-semigroup $S_{\alpha}(\cdot)$ on $X$ and $B$ is a bounded linear operator on $X$ such that either $BC = CB$, $BS_{\alpha} = S_{\alpha}B$ on $X$; or $BC = CB$ on $\overline{D(A)}$ and $BA \subset AB$. Moreover, $T_{\alpha}(\cdot)$ is also locally Lipschitz continuous, (norm continuous, exponentially bounded or exponentially Lipschitz continuous) if $S_{\alpha}(\cdot)$ is.