ON PERTURBATION OF α-TIMES INTEGRATED C-SEMIGROUPS
Let α ≥ 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 α-times integrated C-semigroup Sα(·) on X, B is a bounded
linear operator on D(A) such that BC = CB on D(A) and BA ⊂ AB, then
A + B generates an exponentially bounded nondegenerate α-times integrated
C-semigroup Tα(·) on X. Moreover, Tα(·) is also exponentially Lipschitz
continuous or norm continuous if Sα(·) is. We show that the exponential
boundedness of Tα(·) can be deleted and α-times integrated C-semigroups
can be extended to the context of local α-times integrated C-semigroups when
R(C) ⊂ D(A) and BSα(·) = Sα(·)B on D(A) both are added. Moreover,
Tα(·) is also locally Lipschitz continuous or norm continuous if Sα(·) is.
We show that A + B generates a nondegenerate local α-times integrated Csemigroup
Tα(·) on X if A generates a nondegenerate local α-times integrated
C-semigroup Sα(·) on X and B is a bounded linear operator on X such
that either BC = CB, BSα = SαB on X; or BC = CB on D(A) and
BA ⊂ AB. Moreover, Tα(·) is also locally Lipschitz continuous, (norm
continuous, exponentially bounded or exponentially Lipschitz continuous) if
Sα(·) is.