On Perturbation of Convoluted 𝐶-Regularized Operator Families

Abstract

Of concern are two classes of convoluted 𝐶-regularized operator families: convoluted 𝐶-cosine operator families and convoluted 𝐶-semigroups. We obtain new and general multiplicative and additive perturbation theorems for these convoluted 𝐶-regularized operator families. Two examples are given to illustrate our abstract results.

1. Introduction

It is well known that the cosine operator families (resp., the 𝐶₀ 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 𝐶₀ 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 𝐶-regularized 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 convoluted 𝐶-regularized operator families and to make the results new even for convoluted 𝑛-times integrated 𝐶-cosine operator families (resp., convoluted 𝑛-times integrated 𝐶-semigroups) (𝑛∈ℕ₀, where ℕ₀ 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 into 𝑋. 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 the 𝐶-resolvent set are denoted by 𝐷(𝐴), 𝑅(𝐴), 𝜌(𝐴), and 𝜌_𝑐(𝐴), respectively, where 𝜌_𝑐(𝐴) is defined by 𝜌_𝑐(𝐴)∶={𝜆∈ℂ∶𝑅(𝐶)⊂𝑅(𝜆−𝐴),𝜆−𝐴isinjective. (1)𝐾∈𝐂([0,∞),ℂ) is an exponentially bounded function, and for 𝛽∈ℝ, ℒ[𝐾(𝑡)](𝜆)≠0Re𝜆>𝛽, (2) where ℒ[𝐾(𝑡)](𝜆) is the Laplace transform of 𝐾(𝑡) as in the monograph [15]. We define Θ(𝑡)∶=^𝑡₀𝐾(𝑠)𝑑𝑠,𝑡≥0. (3)

Next, we recall some notations and basic results from [15, 19] about the convoluted 𝐶-cosine operator families and convoluted 𝐶-semigroups.

The following definition is the convoluted version of [15, Chapter 1, Definition 4.1].

Definition 1. Let 𝜔≥0 and (𝜔²,∞)⊂𝜌_𝑐(𝐴). Let {𝐶_𝐾(𝑡)}_𝑡≥0(𝐶_𝐾(𝑡)∈𝐿(𝑋),𝑡≥0) be a strongly continuous operator family such that C_𝐾(𝑡)≤𝑀𝑒^𝜔𝑡,𝑡≥0, (4) for some 𝑀>0, and 𝜆𝜆²−𝐴⁻¹1𝐶𝑥=ℒ[𝐾(𝑡)](𝜆)^∞₀𝑒^−𝜆𝑡C_𝐾(𝑡)𝑥𝑑𝑡,Re𝜆>max𝜔,𝛽,𝑥∈𝑋. (5) Then, 𝐴 is called a subgenerator of the exponentially bounded 𝐾-convoluted 𝐶-cosine operator family {C_𝐾(𝑡)}_𝑡≥0. Moreover, the operator 𝐴∶=𝐶⁻¹𝐴𝐶 is called the generator of the {C_𝐾(𝑡)}_𝑡≥0.

Proposition 2. Let 𝐴 be a closed operator and {𝐶_𝐾(𝑡)}_𝑡≥0 a strongly continuous, exponentially bounded operator family. Then 𝐴 is the subgenerator of a 𝐾-convoluted 𝐶-cosine operator family {𝐶_𝐾(𝑡)}_𝑡≥0 if and only if(1)C_𝐾(𝑡)𝐶=𝐶C_𝐾(𝑡), 𝑡≥0;(2)C_𝐾(𝑡)𝐴⊂𝐴C_𝐾(𝑡), 𝑡≥0, and 𝐴^𝑡₀^𝑠₀C_𝐾(𝜎)𝑥𝑑𝜎𝑑𝑠=C_𝐾(𝑡)𝑥−Θ(𝑡)𝐶𝑥,𝑡≥0,𝑥∈𝑋. (6)

Remark 3. If 𝐴 is the subgenerator of a 𝐾-convoluted 𝐶-cosine operator family, then 𝐶𝐴⊆𝐴𝐶.

Definition 4. Let 0≤𝜔<∞ and (𝜔,∞)⊂𝜌_𝑐(𝐴). Let {𝑇_𝐾(𝑡)}_𝑡≥0 be a strongly continuous operator family such that 𝑇_𝐾(𝑡)≤𝑀𝑒^𝜔𝑡,𝑡≥0, (7) for some 𝑀>0, and (𝜆−𝐴)⁻¹1𝐶𝑥=ℒ[𝐾(𝑡)](𝜆)^∞₀𝑒^−𝜆𝑡𝑇_𝐾(𝑡)𝑥𝑑𝑡,Re𝜆>max𝜔,𝛽,𝑥∈𝑋. (8) Then, 𝐴 is called a subgenerator of an exponentially bounded 𝐾-convoluted 𝐶-semigroup {𝑇_𝐾(𝑡)}_𝑡≥0. Moreover, the operator 𝐴∶=𝐶⁻¹𝐴𝐶 is called the generator of the {𝑇_𝐾(𝑡)}_𝑡≥0.

Proposition 5. Let 𝐴 be a closed operator, and {𝑇_𝐾(𝑡)}_𝑡≥0 a strongly continuous, exponentially bounded operator family. Then, 𝐴 is the subgenerator of a 𝐾-convoluted 𝐶-semigroup {𝑇_𝐾(𝑡)}_𝑡≥0 if and only if(1)T_𝐾(𝑡)𝐶=𝐶T_𝐾(𝑡), 𝑡≥0;(2)T_𝐾(𝑡)𝐴⊂𝐴T_𝐾(𝑡), 𝑡≥0, and 𝐴^𝑡₀T_𝐾(𝑠)𝑥𝑑𝑠=T_𝐾(𝑡)𝑥−Θ(𝑡)𝐶𝑥,𝑡≥0,𝑥∈𝑋. (9)

Remark 6. From [15], we know that the 𝐶-cosine operator families (resp., 𝐶-semigroups) are exactly the 0-times integrated 𝐶-cosine operator families (resp., the 0-times integrated 𝐶-semigroups). Let Γ(⋅) be the well-known Gamma function, and 𝐾(𝑡)=𝑡^𝛼−1Γ(𝛼). (10) Then, by Propositions 2 and 5, we get results for the 𝛼-times integrated 𝐶-cosine operator families (resp., 𝛼-times integrated 𝐶-semigroups) as well as 𝐶-cosine operator families (resp., 𝐶-semigroups). For more information on various 𝐶 operator families, we refer the reader to, for example, [3, 6–8, 14, 15, 17, 22] and references therein.

2. Multiplicative Perturbation Theorems

Lemma 7. Suppose that 𝐴 is a subgenerator of an exponentially bounded 𝐾-convoluted 𝐶-cosine operator family on 𝑋. If 𝜌(𝐴)≠∅, then 𝐶⁻¹𝐴𝐶=𝐴.

Proof. For any 𝜆₀∈𝜌(𝐴) and 𝑥∈𝐷(𝐶⁻¹𝐴𝐶), let 𝑦=𝜆₀𝑥−𝐶⁻¹𝐴𝐶𝑥. (11) Then, 𝜆₀−𝐴⁻¹𝐶=𝐶𝜆₀−𝐴⁻¹,𝐶𝑥=𝜆₀−𝐴⁻¹𝐶𝑦=𝐶𝜆₀−𝐴⁻¹𝑦. (12) Therefore, 𝑥=𝜆₀−𝐴⁻¹𝑦∈𝐷(𝐴). (13) This means that 𝐶⁻¹𝐴𝐶⊆𝐴. Thus, by Remark 3, we see that 𝐶⁻¹𝐴𝐶=𝐴.

Theorem 8. Let 𝐴 be a closed linear operator on 𝑋 and ℛ∈𝐿(𝑋). Assume that there exists an injective operator 𝐶 on 𝑋 satisfying 𝐶𝐴⊆𝐴𝐶, ℛ𝐶=𝐶ℛ. Then, the following statements hold.(1)If ℛ𝐴 subgenerates an exponentially bounded 𝐾-convoluted 𝐶-cosine operator family on 𝑋, then 𝐴ℛ subgenerates an exponentially bounded 𝐾-convoluted 𝐶-cosine operator family on 𝑋.(2)If 𝐴ℛ subgenerates an exponentially bounded 𝐾-convoluted 𝐶-cosine operator family on 𝑋 and 𝜌(ℛ𝐴)≠∅, then ℛ𝐴 generates an exponentially bounded 𝐾-convoluted 𝐶-cosine operator family on 𝑋.

Proof. (1) Assume that ℛ𝐴 subgenerates an exponentially bounded 𝐾-convoluted 𝐶-cosine operator family {C_𝐾(𝑡)}_𝑡≥0 on 𝑋.
In this case, it is easy to see that for any 𝑡≥0, the operator 𝑥↦𝐴^𝑡₀𝑆_𝐾(𝑠)ℛ𝑥𝑑𝑠 (14) is bounded, since ^𝑡₀𝑆_𝐾(𝑠)ℛ𝑥𝑑𝑠∈𝐷(ℛ𝐴), (15) where 𝑆_𝐾(𝑡)=∫^𝑡₀C_𝐾(𝑠)𝑑𝑠. Now, for each 𝑡≥0, we define a bounded linear operator as follows: C_𝐾(𝑡)𝑥=Θ(𝑡)𝐶𝑥+𝐴^𝑡₀𝑆_𝐾(𝑠)ℛ𝑥𝑑𝑠. (16) Clearly, the graph norms of ℛ𝐴 and 𝐴 are equivalent. Therefore, noting that ℛ𝐴 subgenerates an exponentially bounded 𝐾-convoluted 𝐶-cosine operator family {𝐶_𝐾(𝑡)}_𝑡≥0 on 𝑋, we obtain, for every 𝑡₁,𝑡₂≥0, and 𝑥∈𝑋, that there exists a constant 𝑀₁ such that 𝐶_𝐾𝑡₁𝑥−𝐶_𝐾𝑡₂𝑥≤Θ𝑡₁𝐶𝑥−Θ𝑡₂+𝐴𝐶𝑥_𝑡10𝑆_𝐾(𝑠)ℛ𝑥𝑑𝑠−^𝑡2₀𝑆_𝐾(𝑠)≤Θ𝑡ℛ𝑥𝑑𝑠₁𝐶𝑥−Θ𝑡₂𝐶𝑥+𝑀₁ℛ𝐴^𝑡1₀𝑆_𝐾(𝑠)ℛ𝑥𝑑𝑠−^𝑡2₀𝑆_𝐾(𝑠)+ℛ𝑥𝑑𝑠_𝑡10𝑆_𝐾(𝑠)ℛ𝑥𝑑𝑠−^𝑡2₀𝑆_𝐾(𝑠)=𝑀ℛ𝑥𝑑𝑠₁+1Θ𝑡₁𝐶𝑥−Θ𝑡₂𝐶𝑥+𝑀₁C_𝐾𝑡₁𝑥−C_𝐾𝑡₂𝑥+^𝑡1₀𝑆_𝐾(𝑠)−ℛ𝑥𝑑𝑠_𝑡20𝑆_𝐾(𝑠)ℛ𝑥𝑑𝑠⟶0,as𝑡₁⟶𝑡₂. (17) Hence, 𝐶_𝐾(⋅) is strongly continuous.
Similarly, we can prove that 𝐶_𝐾(⋅) is exponentially bounded; that is, there exists a constant 𝑀>0 such that 𝐶_𝐾(𝑡)≤𝑀𝑒^𝜔𝑡,𝑡≥0. (18)
As in the monograph [15], we write ℒ𝐶_𝐾(𝑡)(𝜆)𝑥=^∞₀𝑒^−𝜆𝑡𝐶_𝐾(𝑡)𝑥𝑑𝑡,forRe𝜆>max𝜔,𝛽,𝑥∈𝑋. (19) Then, by (16), we have ℒ𝐶_𝐾(𝑡𝜆)ℒ[𝐾(𝑡𝜆))(𝑥=)](𝜆ℒ[𝐾(𝑡𝜆)𝐶𝑥+𝐴)](𝜆𝜆²−ℛ𝐴⁻¹𝐶ℛ𝑥. (20) Hence, ℛℒ𝐶_𝐾(𝑡𝜆)𝑥=ℒ[𝐾(𝑡𝜆))()](𝜆×𝐶ℛ𝑥+ℛ𝐴𝜆²−ℛ𝐴⁻¹[𝐾(𝑡𝜆)𝜆𝐶ℛ𝑥=𝜆ℒ)](²−ℛ𝐴⁻¹(𝐴).𝐶ℛ𝑥∈𝐷 (21) Furthermore, 𝜆²−𝐴ℛℒ𝐶_𝐾(𝑡)(𝜆)𝑥=𝜆²ℒ𝐶_𝐾(𝑡𝜆)𝑥[𝐾(𝑡𝜆)𝐴𝜆)(−𝜆ℒ)](²−ℛ𝐴⁻¹[𝐾(𝑡𝜆)𝐶ℛ𝑥=𝜆ℒ)](𝐶𝑥. (22) On the other hand, for each 𝑥∈𝐷(𝐴ℛ), Re𝜆>max(𝜔,𝛽), we obtain ℒ[𝐾(𝑡)](𝜆)𝜆𝐶+𝐴𝜆²−ℛ𝐴⁻¹𝐶ℛ𝜆²𝑥[𝐾(𝑡𝜆)−𝐴ℛ=𝜆ℒ)](𝐶𝑥. (23) Therefore, 𝜆𝜆²−𝐴ℛ⁻¹1𝐶=𝜆𝐼+𝐴𝜆²−ℛ𝐴⁻¹ℛ𝐶. (24) It follows from (20) that ℒ𝐶_𝐾(𝑡)(𝜆)𝑥=𝜆ℒ[𝐾(𝑡)](𝜆)𝜆²−𝐴ℛ⁻¹𝐶𝑥. (25) Thus, by Definition 1, we know that 𝐴ℛ subgenerates an exponentially bounded 𝐾-convoluted 𝐶-cosine operator family on 𝑋.
(2) Assume that 𝐴ℛ subgenerates an exponentially bounded 𝐾-convoluted 𝐶-cosine operator family on 𝑋 and 𝜌(ℛ𝐴)≠∅, and let 𝜆₀(),𝜆∈𝜌ℛ𝐴𝐸=₀−ℛ𝐴ℛ,𝐹=𝐴𝜆₀−ℛ𝐴⁻¹. (26) It is not hard to see that 𝐸 is closed operator on 𝑋 and 𝐹∈𝐿(𝑋),𝐶𝐸⊆𝐸𝐶,𝐹𝐶=𝐶𝐹. (27) Since 𝐹𝐸=𝐴ℛ subgenerates an exponentially bounded 𝐾-convoluted 𝐶-cosine operator family on 𝑋, we know from (1) that the operator 𝐸𝐹=ℛ𝐴 subgenerates an exponentially bounded 𝐾-convoluted 𝐶-cosine operator family on 𝑋.
Noting that 𝜌(ℛ𝐴)≠∅ and in view of Lemma 7, we see that ℛ𝐴 generates an exponentially bounded 𝐾-convoluted 𝐶-cosine operator family on 𝑋.

Theorem 9. Let 𝐴 be a subgenerator of an exponentially bounded 𝐾-convoluted 𝐶-cosine operator family {𝐶_𝐾(𝑡)}_𝑡≥0 on 𝑋, S_𝐾(𝑡)=^𝑡₀C_𝐾(𝑠)𝑑𝑠,𝑡≥0, (28)𝐵∈𝐿(𝑋), and 𝑅(𝐵)⊂𝑅(𝐶). Suppose that(H1) there exists an operator ℱ∶𝑋→𝑋 such that ℱS_𝐾(𝑡)𝑥∶=𝐺_𝐾(𝑡)𝑥∈𝐂([0,∞),𝑋) (29) is Laplace transformable, and ℒ𝐺_𝐾(𝜆)=𝜆²−𝐴⁻¹𝐶𝑥,𝑥∈𝑋; (30)(H2) for any Φ∈𝐂([0,∞),𝑋), ∫^𝑡₀𝐺_𝐾(𝑡−𝑠)𝐶⁻¹𝐵Φ(𝑠)𝑑𝑠∈𝐷(𝐴), and 𝐴^𝑡₀𝐺_𝐾(𝑡−𝑠)𝐶⁻¹𝐵Φ(𝑠)𝑑𝑠≤𝑀^𝑡₀𝑒^{𝜔(𝑡−𝑠)}‖Φ(𝑠)‖𝑑𝑠,𝑡≥0, (31) where 𝑀 is a constant;(H3) there exists an injective operator 𝐶₁∈𝐿(𝑋) such that 𝑅(𝐶₁)⊂𝑅(𝐶) and 𝐶₁𝐴(𝐼+𝐵)⊂𝐴(𝐼+𝐵)𝐶₁. Then,(1)𝐴(𝐼+𝐵) subgenerates an exponentially bounded 𝐾-convoluted 𝐶₁-cosine operator family,(2)if 𝜌(𝐴)≠∅, then 𝐴(𝐼+𝐵) generates an exponentially bounded 𝐾-convoluted 𝐶₁-cosine operator family;(3)if 𝜌((𝐼+𝐵)𝐴)≠∅ and 𝐵𝐶₁=𝐶₁𝐵, 𝐶₁𝐴⊆𝐴𝐶₁, then (𝐼+𝐵)𝐴 generates an exponentially bounded 𝐾-convoluted 𝐶₁-cosine operator family on 𝑋.

Proof. (1) For each 𝑥∈𝑋, 𝑡≥0, define 𝐶₀(𝑡)𝑥=𝐶_𝐾(𝑡)𝑥,𝐶_𝑛(𝑡)𝑥=𝐴^𝑡₀𝐺_𝐾(𝑡−𝑠)𝐶⁻¹𝐵C_𝑛−1(𝑠)𝑥𝑑𝑠,𝑛=1,2,…. (32) Then, the operator family {𝐶_𝑛(𝑡)}_𝑡≥0 has the following properties:(i)for any 𝑥∈𝑋, 𝐶_𝑛(𝑡)𝑥∈𝐂([0,∞),𝑋); (ii)‖𝐶_𝑛(𝑡)‖≤(𝑀𝑀^𝑛𝑡^𝑛/𝑛!)𝑒^𝜔𝑡,𝑡≥0,∀𝑛∈ℕ₀. Therefore, the following series ^∞_𝑛=0𝐶_𝑛(𝑡)𝐶⁻¹𝐶₁,𝑡≥0, (33) is uniformly convergent on every compact interval in 𝑡, and we set ℎ(𝑡)=^∞_𝑛=0𝐶_𝑛(𝑡)𝐶⁻¹𝐶₁,𝑡≥0. (34) Clearly, ‖ℎ(𝑡)‖≤𝑀₁𝑒^(𝜔+𝑀_)𝑡,𝑡≥0, (35) where 𝑀₁=𝑀‖𝐶⁻¹𝐶₁‖, and 𝑡⟶ℎ(𝑡)𝑥iscontinuouson[0,∞)forany𝑥∈𝑋. (36) Moreover, ℎ(𝑡)𝑥=𝐶_𝐾(𝑡)𝐶⁻¹𝐶₁𝑥+𝐴^𝑡₀𝐺_𝐾(𝑡−𝑠)𝐶⁻¹𝐵ℎ(𝑠)𝑥𝑑𝑠,𝑥∈𝑋,𝑡≥0. (37) As in the monograph [15], we write, for sufficiently large 𝜆, ℒ[ℎ(𝑡)](𝜆)𝑥=^∞₀𝑒^−𝜆𝑡ℎ(𝑡)𝑥𝑑𝑡,𝑥∈𝑋. (38) Thus, by (5), we have ℒ[ℎ(𝑡)](𝜆)𝑥=𝜆ℒ[𝐾(𝑡)](𝜆)𝜆²−𝐴⁻¹𝐶₁𝑥+𝐴𝜆²−𝐴⁻¹𝐵ℒ[ℎ(𝑡)](𝜆)𝑥,𝑥∈𝑋. (39) This implies that 𝑅)ℒ[ℎ(𝑡𝜆(𝐴),𝜆((𝐼+𝐵)]())⊆𝐷₂−𝐴(𝐼+𝐵)ℒ[ℎ(𝑡)](𝜆)𝑥=𝜆²ℒ[ℎ(𝑡)](𝜆)𝑥−𝜆ℒ[𝐾(𝑡)](𝜆)𝐴𝜆²−𝐴⁻¹𝐶₁𝑥−𝜆²𝐴𝜆²−𝐴⁻¹[ℎ(𝑡𝜆)𝑥[𝐾(𝑡𝜆)𝐶𝐵ℒ)](=𝜆ℒ)](₁𝑥,𝑥∈𝑋. (40) Let 𝑈(𝑡)𝑥=𝐴^𝑡₀𝐺_𝐾(𝑡−𝑠)𝐶⁻¹𝐵𝑥𝑑𝑠,𝑥∈𝑋,𝑡≥0. (41) Then, for large 𝜆, we have ‖𝜆ℒ[𝑈(𝑡)](𝜆)‖=𝜆^∞₀𝑒^−𝜆𝑡𝑈(𝑡)≤𝑀𝑑𝑡.𝜆−𝜔 (42) So, for sufficiently large 𝜆, ‖𝜆ℒ[𝐾(𝑡)](𝜆)‖=𝐴𝜆²−𝐴⁻¹𝐵<1. (43) This means that the operator 𝐼−𝐴(𝜆²−𝐴)⁻¹𝐵 is invertible.
On the other hand, since 𝜆²−𝐴 and 𝐼−𝐴(𝜆²−𝐴)⁻¹𝐵 are injective, and 𝜆²−𝐴𝐼−𝐴𝜆²−𝐴⁻¹𝐵𝑥=𝜆²()(𝐴(,−𝐴𝐼+𝐵𝑥,𝑥∈𝐷𝐼+𝐵)) (44) we infer that 𝜆²−𝐴(𝐼+𝐵) is injective. This together with (40) implies that 𝜆𝜆²−𝐴(𝐼+𝐵)⁻¹𝐶₁1𝑥=ℒ[𝐾(𝑡)](𝜆)^∞₀𝑒^−𝜆𝑡ℎ(𝑡)𝑥𝑑𝑡. (45) By Definition 1, we know that 𝐴(𝐼+𝐵) subgenerates an exponentially bounded 𝐾-convoluted 𝐶₁-cosine operator family on 𝑋.
(2) By the proof of (1), we see that the operator 𝐼−𝐴(𝜆²−𝐴)⁻¹𝐵 is invertible, and 𝜌(𝐴)≠∅ implies that 𝜌(𝐴(𝐼+𝐵))≠∅. (46) In view of Lemma 7, we get 𝐶₁⁻¹𝐴(𝐼+𝐵)𝐶₁=𝐴(𝐼+𝐵). (47)
(3) By virtue of Theorem 8 (2), we have the conclusion.

Remark 10. (1) It is easy to see that if we take ℱ𝑆_𝐾(𝑡)𝑥∶=ℒ⁻¹1ℒ[𝐾(𝑡)](𝜆)∗𝑆_𝐾(𝑡)𝑥, (48) then (H1) is satisfied.
(2) In Theorem 9, if we take 𝐾(𝑡)=𝑡^𝑛−1Γ(𝑛)𝑑,ℱ∶=^𝑛𝑑𝑡^𝑛,𝑛∈ℕ, (49) then we obtain the perturbations for 𝑛-times integrated 𝐶-cosine operator families.
(3) In Theorem 9, if we take 𝐾(𝑡)≡1𝑡(𝑡≠0) (50) and ℱ∶=𝐼, then we have the multiplicative perturbations on the exponentially bounded 𝐶-cosine operator families.
By Theorem 9, we can immediately deduce the following theorem on 𝐾-convoluted 𝐶-semigroups.

Theorem 11. Let 𝐴 be a subgenerator of an exponentially bounded 𝐾-convoluted 𝐶-semigroup {𝑇_𝐾(𝑡)}_𝑡≥0 on 𝑋, 𝐵∈𝐿(𝑋) and 𝑅(𝐵)⊂𝑅(𝐶). Suppose that(H1) there exists an operator ℱ∶𝑋→𝑋 such that ℱT_𝐾(𝑡)𝑥∶=𝐻_𝐾(𝑡)𝑥∈𝐂([0,∞),𝑋) (51) is Laplace transformable, and ℒ𝐻_𝐾(𝜆)=(𝜆−𝐴)⁻¹𝐶𝑥,𝑥∈𝑋; (52)(H2) for any Φ∈𝐂([0,∞),𝑋), ∫^𝑡₀𝐻_𝐾(𝑡−𝑠)𝐶⁻¹𝐵Φ(𝑠)𝑑𝑠∈𝐷(𝐴), and 𝐴^𝑡₀𝐻_𝐾(𝑡−𝑠)𝐶⁻¹𝐵Φ(𝑠)𝑑𝑠≤𝑀^𝑡₀𝑒^{𝜔(𝑡−𝑠)}‖Φ(𝑠)‖𝑑𝑠,𝑡≥0, (53) where 𝑀 is a constant;(H3) there exists an injective operator 𝐶₁∈𝐿(𝑋) such that 𝑅(𝐶₁)⊂𝑅(𝐶) and 𝐶₁𝐴(𝐼+𝐵)⊂𝐴(𝐼+𝐵)𝐶₁.Then,(1)𝐴(𝐼+𝐵) subgenerates an exponentially bounded 𝐾-convoluted 𝐶₁-semigroup on 𝑋;(2)if 𝜌(𝐴)≠∅, then 𝐴(𝐼+𝐵) generates an exponentially bounded 𝐾-convoluted 𝐶₁-semigroup on 𝑋.(3)if 𝜌((𝐼+𝐵)𝐴)≠∅, then (𝐼+𝐵)𝐴 generates an exponentially bounded 𝐾-convoluted 𝐶₁-semigroup on 𝑋.

Remark 12. (1) In Theorem 11, if we take 𝐾(𝑡)𝑡∶=^𝑛−1Γ(𝑛)𝑑,ℱ∶=^𝑛𝑑𝑡^𝑛,𝑛∈ℕ, (54) then we obtain the perturbations for 𝑛-times integrated 𝐶-semigroups.
(2) In Theorem 11, if we take 𝐾(𝑡)1∶=𝑡(𝑡≠0) (55) and ℱ∶=𝐼, then we have the multiplicative perturbations on the exponentially bounded 𝐶-semigroups.

3. Additive Perturbation Theorem

Theorem 13. Let 𝐵∈𝐿(𝑋), 𝑅(𝐵)⊂𝑅(𝐶), and there exists an injective operator 𝐶₁∈𝐿(𝑋) such that 𝑅(𝐶₁)⊂𝑅(𝐶) and 𝐶₁(𝐴+𝐵)⊂(𝐴+𝐵)𝐶₁.(i) Suppose that 𝐴 is a subgenerator of an exponentially bounded 𝐾-convoluted 𝐶-cosine operator family {C_𝐾(𝑡)}_𝑡≥0 on 𝑋. If there exists an operator ℱ∶𝑋→𝑋 such that ℱC_𝐾(𝑡)𝑥∶=𝐺_𝐾(𝑡)𝑥∈𝐂([0,∞),𝑋) (56) is Laplace transformable, and ℒ𝐺_𝐾(𝜆)=𝜆²−𝐴⁻¹𝐶𝑥,𝑥∈𝑋, (57) then 𝐴+𝐵 subgenerates an exponentially bounded 𝐾-convoluted 𝐶₁-cosine operator family {C_𝐾(𝑡)}_𝑡≥0 on 𝑋, where C_𝐾(𝑡)𝑥=C_𝐾(𝑡)𝐶⁻¹𝐶₁𝑥+^𝑡₀S_𝐾(𝑡−𝑠)𝐶⁻¹𝐵C_𝐾(𝑡)𝑥𝑑𝑠,𝑡≥0,𝑥∈𝑋,S_𝐾(𝑡)𝑥=^𝑡₀C_𝐾(𝑠)𝑥𝑑𝑠,𝑡≥0,𝑥∈𝑋. (58)(ii) Suppose that 𝐴 is a subgenerator of an exponentially bounded 𝐾-convoluted 𝐶-semigroup {T_𝐾(𝑡)}_𝑡≥0 on 𝑋. If there exists an operator ℱ∶𝑋→𝑋 such that ℱT_𝐾(𝑡)𝑥∶=𝐻_𝐾(𝑡)𝑥∈𝐂([0,∞),𝑋) (59) is Laplace transformable, and ℒ𝐻_𝐾(𝜆)=(𝜆−𝐴)⁻¹𝐶𝑥,𝑥∈𝑋, (60) then 𝐴+𝐵 subgenerates an exponentially bounded 𝐾-convoluted 𝐶₁-semigroup {T_𝐾(𝑡)}_𝑡≥0 on 𝑋, where T_𝐾(𝑡)𝑥=T_𝐾(𝑡)𝐶⁻¹𝐶₁𝑥+^𝑡₀T_𝐾(𝑡−𝑠)𝐶⁻¹𝐵T_𝐾(𝑠)𝑥𝑑𝑠,𝑡≥0,𝑥∈𝑋. (61)

Proof. Replacing (37) with the following equality: ℎ(𝑡)𝑥=C_𝐾(𝑡)𝐶⁻¹𝐶₁𝑥+^𝑡₀𝐺_𝐾(𝑡−𝑠)𝐶⁻¹𝐵ℎ(𝑠)𝑥𝑑𝑠,𝑥∈𝑋,𝑡≥0, (62) 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.

Remark 14. In Theorem 13, if we take 𝐾(𝑡)=𝑡^𝑛−1Γ(𝑛)𝑑,ℱ∶=^𝑛𝑑𝑡^𝑛,𝑛∈ℕ₀, (63) then we obtain an additive perturbation theorem for the exponentially bounded 𝑛-times integrated 𝐶₁-cosine operator families (resp., 𝑛-times integrated 𝐶₁-semigroups) as well as 𝐶₁-cosine operator families (resp., 0-times integrated 𝐶₁-semigroup).

4. Examples

Example 1. Let 𝑋∶=𝐶₀(ℝ)⊕𝐶₀(ℝ)⊕𝐶₀(ℝ),𝐴𝑓,𝑔,ℎ(⋅)∶=𝑓^′,𝑔^′,𝜒_[0,∞)−𝜒_(−∞,0]ℎ, (64) where (𝐴)=𝑓,𝑔,ℎ∈𝐷𝑓,𝑔,ℎ∈𝑋∶𝑓^′∈𝐶₀(ℝ),𝑔^′∈𝐶₀(ℝ)(0),𝐶(⋅)ℎ(⋅,ℎ=0𝑓,𝑔,ℎ∶=𝑓,𝑔,sin),𝑓,𝑔,ℎ∈𝐶₀(ℝ). (65) As in [22, Examples 8.1 and 8.2], we can prove that 𝐴 is the generator of an exponentially bounded once integrated 𝐶-semigroup ([15]).
Define 𝐵(𝑡)=𝑒𝑓,𝑔,ℎ_−𝑡cos𝑡^𝑡₀𝑓(𝑠)𝑑𝑠,𝑒^−2𝑡×cos2𝑡_𝑡0𝑔(𝑠)𝑑𝑠,𝑡𝑒^−6𝑡sin𝑡⋅ℎ(𝑡), (66) for every 𝑡∈ℝ, and 𝑓,𝑔,ℎ∈𝐶₀(ℝ). Then, we can simply verify 𝐵∈𝐿(𝑋), 𝑅(𝐵)⊂𝐶(𝐷(𝐴)), and 𝐵𝐶𝑓,𝑔,ℎ=𝐶𝐵𝑓,𝑔,ℎ,𝑓,𝑔,ℎ∈𝑋. (67) Therefore, taking 𝐾(𝑡)𝑑≡1,ℱ∶=𝑑𝑡 (68) and using Remark 12 (1), we know that 𝐴(𝐼+𝐵) subgenerates an exponentially bounded once integrated 𝐶-semigroup on 𝑋.

Example 2. Let 𝑋₁=𝐿^∞(ℝ), 𝑋₂=𝐿²(ℝ), 𝐴₁=𝑑²𝑑𝜉²,𝐷𝐴₁=𝑊^2,∞(ℝ),𝐴₂=𝑑²𝑑𝜉²,𝐷𝐴₂=𝐻²(ℝ). (69) It follows from [15] that 𝐴₁ generates an exponentially bounded 𝐶₁-cosine operator family 𝐶₁(⋅) on 𝑋₁, where 𝐶₁=(1−𝑑²/𝑑𝜉²)⁻¹. Moreover, it is well known that 𝐴₂ generates a strongly continuous cosine operator family 𝐶₂(⋅) on 𝑋₂.
Let 𝑏₁(⋅)∈𝑊^4,∞(ℝ),𝑏₂(⋅)∈𝐻²(ℝ), (70) and define 𝐵₁∶𝑋₂→𝑋₁, 𝐵₂∶𝑋₁→𝑋₂ as follows: 𝐵₁𝜙(𝜉)=𝑏₁(𝜉)¹₀𝜙(𝜎)𝐵𝑑𝜎,₂𝜙(𝜉)=𝑏₂(𝜉)¹₀𝜙(𝜎)𝑑𝜎. (71) Set 𝑋=𝑋₁×𝑋₂, 𝐴𝐴=¹00𝐴₂,𝐷(𝐴)∶=𝐷𝐴₁×𝐷𝐴₂,𝐵=0𝐵₁𝐵₂0,𝐷(𝐵)∶=𝑋. (72) Clearly, 𝜌(𝐴)≠∅ and 𝐷(𝐴₁)=𝑅(𝐶₁). Take 𝜆₀∈𝜌(𝐴),𝐶=𝜆₀−𝐴⁻¹. (73) Then, 𝐴 generates an exponentially bounded 𝐶-cosine operator family 𝐶(⋅) on 𝑋, where 𝐶(𝑡)=𝐶₁(𝑡)𝐶₁⁻¹𝜆₀−𝐴₁⁻¹00𝐶₂(𝑡)𝜆₀−𝐴₂⁻¹. (74) Hence, 𝑆(𝑡)=𝑆₁(𝑡)𝐶₁⁻¹𝜆₀−𝐴₁⁻¹00𝑆₂(𝑡)𝜆₀−𝐴₂⁻¹, (75) where 𝑆(𝑡)∶=^𝑡₀𝐶(𝑠)𝑑𝑠,𝑆₁(𝑡)∶=^𝑡₀𝐶₁(𝑠)𝑑𝑠,𝑆₂(𝑡)∶=^𝑡₀𝐶₂(𝑠)𝑑𝑠. (76)(77) Therefore, we have, for each 𝑓=^𝑓1_𝑓2∈𝐂([0,∞),𝑋), ^𝑡₀𝑆(𝑡−𝑠)𝐶⁻¹(𝑠)=𝐴𝐵𝑓𝑑𝑠₁∫^𝑡₀𝑆₁(𝑡−𝑠)𝐶₁⁻¹𝐵₁𝑓₂(𝑠)𝐴𝑑𝑠₂∫^𝑡₀𝑆₂(𝑡−𝑠)𝐵₂𝑓₁(𝑠).𝑑𝑠 (78) Since 𝑅𝐵₁⊂𝐷𝐴₁𝐶₁⁻¹,𝑅𝐵₂⊂𝐷𝐴₂, (79) we see that there exist 𝑀, 𝜔>0 such that ^𝑡₀𝑆(𝑡−𝑠)𝐶⁻¹𝐵𝑓(𝑠)𝑑𝑠≤𝑀^𝑡₀𝑒^{𝜔(𝑡−𝑠)}𝑓(𝑠)𝑑𝑠,𝑡≥0. (80) Consequently, if there exists an injective operator 𝐶∈𝐿(𝑋) such that 𝑅(𝐶)⊂𝑅(𝐶) and 𝐶(𝐴+𝐵)⊂(𝐴+𝐵)𝐶, then taking 𝐾(𝑡)≡1𝑡,ℱ∶=𝐼 (81) and using Remark 14, we know that 𝐴+𝐵 subgenerates a 𝐶-cosine operator family on 𝑋.
Moreover, it is not hard to see that there exist 𝑀, 𝜔>0 such that 𝐴^𝑡₀𝑆(𝑡−𝑠)𝐶⁻¹𝐵𝑓(𝑠)𝑑𝑠≤𝑀^𝑡₀𝑒^{𝜔(𝑡−𝑠)}𝑓(𝑠)𝑑𝑠,𝑡≥0. (82) Hence, if there exists an injective operator 𝐶∈𝐿(𝑋) such that 𝑅(𝐶)⊂𝑅(𝐶) and 𝐶𝐴(𝐼+𝐵)⊂𝐴(𝐼+𝐵)𝐶, then by Remark 10 (3) (𝜌(𝐴)≠∅), we know that 𝐴(𝐼+𝐵) generates a 𝐶-cosine operator family on 𝑋.

Acknowledgments

The authors would like to thank the referees very much for helpful suggestions. The work was supported partly by the NSF of China (11201413, 11071042, and 11171210), the Educational Commission of Yunnan Province (2012Z010), and the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900).