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We apply the contraction mapping theorem to establish some bounded and unbounded perturbation theorems concerning nondegenerate local α-times integrated semigroups. Some unbounded perturbation results of Wang et al. [Studia Math. 170 (2005)] are also generalized. We also establish some growth properties of perturbations of local α-times integrated semigroups.

This paper is concerned with α-times integrated C-semigroups for α > 0 and the associated abstract Cauchy problem: $u^{\text{'}}\left(t\right)=Au\left(t\right)+\frac{t^{\alpha -1}}{\Gamma \left(\alpha \right)}x$, t >0; u(0) = 0. We first investigate basic properties of an α-times integrated C-semigroup which may not be exponentially bounded. We then characterize the generator A of an exponentially bounded α-times integrated C-semigroup, either in terms of its Laplace transforms or in terms of existence of a unique solution of the above abstract Cauchy problem for every x in ${(\lambda -A)}^{-1}C\left(X\right)$.