On Perturbation of Convoluted C-Regularized Operator Families

It is well known that the cosine operator families (resp., theC 0 semigroups) and the fractionally integrated C-cosine operator families (resp., integrated C-semigroups) are important tools in studying incomplete second-order (resp., first-order) abstract Cauchy problems (cf., e.g., [1–17]). As an extension of the cosine operator families (resp., the C 0 semigroups) as well as the fractionally integrated C-cosine operator families (resp., integrated C-semigroups), the convoluted C-cosine operator families (resp., convoluted C-semigroups) (cf., e.g., [15, 18, 19]) are also good operator families in dealing with ill-posed incomplete second order (resp. first order) abstract Cauchy problems. In last two decades, there are many works on the perturbations on the C-regularized operator families (cf., e.g., [16, 20–24]). In the present paper, we will study the multiplicative and additive perturbation for two classes of convoluted Cregularized operator families: convoluted C-cosine operator families and convoluted C-semigroups, and our purpose is to obtain some new and general perturbation theorems for these convolutedC-regularized operator families and tomake the results new even for convoluted n-times integrated Ccosine operator families (resp., convoluted n-times integrated C-semigroups) (n ∈ N 0 , where N 0 denotes the nonnegative integers). Throughout this paper, N, R, C denote the set of positive integers, the real numbers, and the complex plane, respectively. X denotes a nontrivial complex Banach space, and L(X) denotes the space of bounded linear operators from X intoX. In the sequel, we assume that C ∈ L(X) is an injective operator. C([a, b], X) denotes the space of all continuous functions from [a, b] to X. For a closed linear operator A on X, its domain, range, resolvent set, and theC-resolvent set are denoted by D(A), R(A), ρ(A), and ρ c (A), respectively, where

In last two decades, there are many works on the perturbations on the -regularized operator families (cf., e.g., [16,[20][21][22][23][24]). In the present paper, we will study the multiplicative and additive perturbation for two classes of convoluted regularized operator families: convoluted -cosine operator families and convoluted -semigroups, and our purpose is to obtain some new and general perturbation theorems for these convoluted -regularized operator families and to make the results new even for convoluted -times integrated cosine operator families (resp., convoluted -times integrated -semigroups) ( ∈ N 0 , where N 0 denotes the nonnegative integers).
Throughout this paper, N, R, C denote the set of positive integers, the real numbers, and the complex plane, respectively. denotes a nontrivial complex Banach space, and () denotes the space of bounded linear operators from  into .In the sequel, we assume that  ∈ () is an injective operator.C([, ], ) denotes the space of all continuous functions from [, ] to .For a closed linear operator  on , its domain, range, resolvent set, and the -resolvent set are denoted by (), (), (), and   (), respectively, where   () is defined by ) is an exponentially bounded function, and for  ∈ R, where L[()]() is the Laplace transform of () as in the monograph [15].We define Next, we recall some notations and basic results from [15,19] about the convoluted -cosine operator families and convoluted -semigroups.
Proof.For any  0 ∈ () and  ∈ ( −1 ), let Then, Therefore, This means that  −1  ⊆ .Thus, by Remark 3, we see that Theorem 8. Let  be a closed linear operator on  and R ∈ ().Assume that there exists an injective operator  on  satisfying  ⊆ , R = R.Then, the following statements hold.
(1) If R subgenerates an exponentially bounded convoluted -cosine operator family on , then R subgenerates an exponentially bounded -convoluted -cosine operator family on .
In this case, it is easy to see that for any  ≥ 0, the operator is bounded, since where   () = ∫  0 C  ().Now, for each  ≥ 0, we define a bounded linear operator as follows: Clearly, the graph norms of R and  are equivalent.Therefore, noting that R subgenerates an exponentially bounded -convoluted -cosine operator family {  ()} ≥0 on , we obtain, for every  1 ,  2 ≥ 0, and  ∈ , that there exists a constant  1 such that Hence, Ĉ (⋅) is strongly continuous.Similarly, we can prove that Ĉ (⋅) is exponentially bounded; that is, there exists a constant M > 0 such that As in the monograph [15], we write for Re  > max (, ) ,  ∈ .
Then, by ( 16), we have Hence, Furthermore, On the other hand, for each  ∈ (R), Re  > max(, ), we obtain Therefore, It follows from (20) that Thus, by Definition 1, we know that R subgenerates an exponentially bounded -convoluted -cosine operator family on .
(2) Assume that R subgenerates an exponentially bounded -convoluted -cosine operator family on  and (R) ̸ = 0, and let It is not hard to see that  is closed operator on  and Since  = R subgenerates an exponentially bounded convoluted -cosine operator family on , we know from (1) that the operator  = R subgenerates an exponentially bounded -convoluted -cosine operator family on .

Remark 10. (1)
It is easy to see that if we take then (H1) is satisfied.
(2) In Theorem 9, if we take then we obtain the perturbations for -times integrated cosine operator families.

Remark 12.
(1) In Theorem 11, if we take then we obtain the perturbations for -times integrated semigroups.
(2) In Theorem 11, if we take and F := , then we have the multiplicative perturbations on the exponentially bounded -semigroups.
(i) Suppose that  is a subgenerator of an exponentially bounded -convoluted -cosine operator family {C  ()} ≥0 on .If there exists an operator F :  →  such that (ii) Suppose that  is a subgenerator of an exponentially bounded -convoluted -semigroup {T  ()} ≥0 on .
If there exists an operator F :  →  such that is Laplace transformable, and Proof.Replacing (37) with the following equality: and by the arguments similar to those in the proof of Theorem 9, we can prove (i).Point (ii) can also be deduced by a similar way.