ON PERTURBATION OF α-TIMES INTEGRATED C-SEMIGROUPS

Let alpha >= 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(.) on X, B is a bounded linear operator on <(D(A))over bar> such that BC = CB on <(D(A))over bar> and BA subset of AB, then A + B generates an exponentially bounded nondegenerate alpha-times integrated C-semigroup T-alpha(.) on X. Moreover, T-alpha(.) is also exponentially Lipschitz continuous or norm continuous if S-alpha(.) is. We show that the exponential boundedness of T-alpha(.) 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 of <(D(A))over bar> and BS alpha(.) = S-alpha(.)B on <(D(A))over bar> both are added. Moreover, T-alpha(.) is also locally Lipschitz continuous or norm continuous if S-alpha(.) is. We show that A + B generates a nondegenerate local alpha-times integrated C-semigroup T-alpha(.) on X if A generates a nondegenerate local alpha-times integrated C-semigroup S-alpha(.) 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 <(D(A))over bar> and BA subset of AB. Moreover, T-alpha(.) is also locally Lipschitz continuous, (norm continuous, exponentially bounded or exponentially Lipschitz continuous) if S-alpha(.) is.