Multiplicative Perturbations of Convoluted 𝐶-Cosine Functions and Convoluted 𝐶-Semigroups

Abstract

We obtain the multiplicative perturbation theorems for convoluted 𝐶-cosine functions (resp., convoluted 𝐶-semigroups) and 𝑛-times integrated 𝐶-cosine functions (resp., 𝑛-times integrated 𝐶-semigroups) for 𝑛∈ℕ. Moreover, we obtain some new results for perturbations on 𝐶-cosine functions (resp., 𝐶-semigroups). Some examples are presented.

1. Introduction and Preliminaries

The 𝛼-times integrated 𝐶-semigroups, 𝛼-times integrated 𝐶-cosine functions (𝛼>0) [1–6], 0-times integrated semigroups (i.e., 𝐶-semigroups), and 0-times integrated 𝐶-cosine functions (i.e., 𝐶-cosine functions) [5, 7–11] are powerful tools in studying ill-posed abstract Cauchy problems. The convoluted 𝐶-cosine functions (resp., convoluted 𝐶-semigroups) are the extension of 𝛼-times integrated 𝐶-cosine functions (resp., 𝛼-times integrated 𝐶-semigroups), they can be used to deal with more complicated ill-posed abstract Cauchy problems of evolution equations [5, 12–16].

Many researchers studied the perturbations on 𝐶-cosine functions and 𝐶-semigroups [17–22]. In [16], Kostić studied the additive perturbations of convoluted 𝐶-cosine functions and convoluted 𝐶-semigroups. However, to the authors’ knowledge, few papers can be found in the literature for the multiplicative perturbations on the convoluted 𝐶-cosine functions (resp., convoluted 𝐶-semigroups).

In this paper, based on the previously mentioned works we study the multiplicative perturbations on the convoluted 𝐶-cosine functions and convoluted 𝐶-semigroups. Moreover, we obtain the corresponding new results for 𝑛-times integrated 𝐶-semigroups (resp., 𝑛-times integrated 𝐶-cosine functions) (𝑛∈ℕ₀, ℕ₀ denotes the nonnegative integers).

Throughout this paper, ℕ, ℕ₀, ℝ, and ℂ denote the positive integers, the nonnegative integers, the real numbers, the complex plane, respectively. 𝑋 denotes a nontrivial complex Banach space, 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)and𝜆−𝐴isinjective. (1)𝐾∈𝐂([0,∞),ℂ) is an exponentially bounded function and for 𝛽∈ℝ, 𝐾(𝜆)≠0 (Re𝜆>𝛽), where 𝐾(𝜆) is the Laplace transform of 𝐾(𝑡). We define 𝜗(𝑡)∶=∫^𝑡₀𝐾(𝑠)𝑑𝑠.

The next definition is the convoluted version of Definition 4.1 in Chapter 1 of [5].

Definition 1 (see [5, 13, 15]). Let 𝜔≥0. If {𝜆²∶Re𝜆>max(𝜔,𝛽)}⊂𝜌_𝑐(𝐴) and there exists a strongly continuous operator family {C_𝐾(𝑡)}_𝑡≥0(C_𝐾(𝑡)∈𝐿(𝑋),𝑡≥0) such that ‖C_𝐾(𝑡)‖≤𝑀𝑒^𝜔𝑡,𝑡≥0 for some 𝑀>0, and𝜆𝜆²−𝐴⁻¹1𝐶𝑥=𝐾(𝜆)^∞0𝑒^−𝜆𝑡C_𝐾(𝑡)𝑥𝑑𝑡,Re𝜆>max𝜔,𝛽,𝑥∈𝑋, (2) then it is said that 𝐴 is a subgenerator of an exponentially bounded 𝐾-convoluted 𝐶-cosine function {C_𝐾(𝑡)}_𝑡≥0. The operator 𝐴∶=𝐶⁻¹𝐴𝐶 is called the generator of {C_𝐾(𝑡)}_𝑡≥0.

Theorem 2 (see [13–15]). Let {C_𝐾(𝑡)}_𝑡≥0 be a strongly continuous, exponentially bounded operator family, and let 𝐴 be a closed operator. Then the statements (i) and (ii) are equivalent, where(i)𝐴is the subgenerator of a 𝐾-convoluted 𝐶-cosine function {C_𝐾(𝑡)}_𝑡≥0,(ii)  (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. (3)

Definition 3. Let 0≤𝜔<∞. If {𝜆∶Re𝜆>max(𝜔,𝛽)}⊂𝜌_𝑐(𝐴) and there exists a strongly continuous operator family {T_𝐾(𝑡)}_𝑡≥0 such that ‖T_𝐾(𝑡)‖≤𝑀𝑒^𝜔𝑡,𝑡≥0 for some 𝑀>0, and (𝜆−𝐴)⁻¹1𝐶𝑥=𝐾(𝜆)^∞0𝑒^−𝜆𝑡T_𝐾(𝑡)𝑥𝑑𝑡,Re𝜆>max𝜔,𝛽,𝑥∈𝑋, (4) then it is said that 𝐴 is a subgenerator of an exponentially bounded 𝐾-convoluted 𝐶-semigroup {T_𝐾(𝑡)}_𝑡≥0. The operator 𝐴∶=𝐶⁻¹𝐴𝐶 is called the generator of {T_𝐾(𝑡)}_𝑡≥0.

Theorem 4. Let {T_𝐾(𝑡)}_𝑡≥0 be a strongly continuous, exponentially bounded operator family, and let 𝐴 be a closed operator. Then the assertions (i) and (ii) are equivalent, where(i)𝐴 is the subgenerator of a 𝐾-convoluted 𝐶-semigroup {T_𝐾(𝑡)}_𝑡≥0,(ii)(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. (5)

Remark 5 (see [16]). In Theorems 2 and 4, putting 𝐾(𝑡)=𝑡^𝑟−1/Γ(𝑟), where Γ(⋅) denotes the Gamma function, one obtains the classes of 𝑟-times integrated 𝐶-cosine functions and 𝑟-times integrated 𝐶-semigroups; a 0-times integrated 𝐶-cosine function (resp., 0-times integrated 𝐶-semigroup) is defined to be a 𝐶-cosine function (resp., 𝐶-semigroup). More knowledge for them, we refer the reader to, for example, [1–3, 5, 7–11, 18] and references there in.

Next, we recall the definitions of 𝑟-times integrated 𝐶-semigroup and 𝑟-times integrated 𝐶-cosine functions (𝑟≥0).

Definition 6 (see [5]). Let 0≤𝜔<∞ and let 𝑟∈[0,∞). If (𝜔²,∞)⊂𝜌_𝑐(𝐴) (resp., (𝜔,∞)⊂𝜌_𝑐(𝐴)) and there exists a strongly continuous operator family {C_𝑟(𝑡)}_𝑡≥0 (resp., {T_𝑟(𝑡)}_𝑡≥0) such that ‖C_𝑟(𝑡)‖≤𝑀𝑒^𝜔𝑡,𝑡≥0 (resp., ‖T_𝑟(𝑡)‖≤𝑀𝑒^𝜔𝑡,𝑡≥0) for some 𝑀>0, and 𝜆𝜆²−𝐴⁻¹𝐶𝑥=𝜆^𝑟∞₀𝑒^−𝜆𝑡C_𝑟(𝑡)𝑥𝑑𝑡,𝜆>𝜔,𝑥∈𝑋,resp.(𝜆−𝐴)⁻¹𝐶𝑥=𝜆^𝑟∞₀𝑒^−𝜆𝑡T_𝑟(𝑡)𝑥𝑑𝑡,𝜆>𝜔,𝑥∈𝑋,, (6) then it is said that 𝐴 is a subgenerator of an exponentially bounded 𝑟-times integrated 𝐶-cosine function {C_𝑟(𝑡)}_𝑡≥0 (resp., 𝑟-times integrated 𝐶-semigroup {T_𝑟(𝑡)}_𝑡≥0) on 𝑋. If 𝑟=0, then {C_𝑟(𝑡)}_𝑡≥0 (resp., {T_𝑟(𝑡)}_𝑡≥0) is called an exponentially bounded 0-times integrated 𝐶-cosine function (resp., 0-times integrated 𝐶-semigroup).

We present the definition of 𝐶-cosine functions which will be used in the proof of Theorem 12.

Definition 7 (see [1, 5]). A strongly continuous family {C(𝑡)}_𝑡≥0 of bounded linear operators on 𝑋 is called a 𝐶-cosine function on 𝑋, if 𝐶C(⋅)=C(⋅)𝐶, C(0)=𝐶 and C(𝑡+𝑠)𝐶+C(|𝑡−𝑠|)𝐶=2C(𝑡)C(𝑠), for all 𝑡,𝑠≥0.

2. Main Results

Suppose that 𝐴 is a subgenerator of an exponentially bounded 𝐾-convoluted 𝐶-cosine function {C_𝐾(𝑡)}_𝑡≥0 on 𝑋, S_𝐾(𝑡)=∫^𝑡₀C_𝐾(𝑠)𝑑𝑠, for any Ψ∈𝐂([0,∞),𝐿(𝑋)) with ‖Ψ(𝑡)‖=𝑂(𝑒^𝜔𝑡), we set 𝐿(𝜆)∶=sup^𝑎₀𝑒^{−𝜆𝑡𝑡}₀𝛿(𝜆)Ψ(𝑠)𝐶⁻¹𝑃𝐴C_𝐾()(𝐴),‖𝑥‖𝑡−𝑠𝑥𝑑𝑠𝑑𝑡,𝑥∈𝐷≤1<∞, (7) for some 𝑎∈(0,+∞] and 𝜆>max(𝜔,𝛽), where 𝛿(𝜆) is some function and 𝑃=𝐵/𝐾(𝜆), 𝐵∈𝐿(𝑋) with 𝑅(𝐵)⊂𝑅(𝐶).

We have the following multiplicative perturbation theorem.

Theorem 8. Suppose that 𝐴 is a subgenerator of an exponentially bounded 𝐾-convoluted 𝐶-cosine function {C_𝐾(𝑡)}_𝑡≥0 on 𝑋. Let 𝐵𝐶=𝐶𝐵, and 𝐷(𝐴) is dense in 𝑋, 𝜆²∶𝜆>max𝜔,𝛽⊂𝜌((𝐼+𝛿(𝜆)𝐵)𝐴). (8) If lim_𝜆→∞𝐿(𝜆)𝑒^𝜆𝑡=0 for all 𝑡≥0, then (𝐼+𝛿(𝜆)𝐵)𝐴 subgenerates an exponentially bounded 𝐾-convoluted 𝐶-cosine function on 𝑋.

Proof. For all 𝑥∈𝐷(𝐴), ‖𝑥‖≤1, 𝜆 is large enough and 𝜀 is small enough, we have ^𝑡₀𝛿(𝜆)Ψ(𝑠)𝐶⁻¹𝑃𝐴S_𝐾()=𝑡−𝑠𝑥𝑑𝑠_𝑡0𝑠0𝛿(𝜆)Ψ(𝜎)𝐶⁻¹𝑃𝐴C_𝐾(𝑠−𝜎)𝑥𝑑𝜎𝑑𝑠≤𝑒^𝜆𝑡𝑡₀𝑒^{−𝜆𝑠𝑠}₀𝛿(𝜆)Ψ(𝜎)𝐶⁻¹𝑃𝐴C_𝐾(𝑠−𝜎)𝑥𝑑𝜎𝑑𝑠≤𝑒^𝜆𝑡𝐿(𝜆)<𝜀<1,𝑡≥0. (9)
Let 𝒱∶[0,∞)→𝐿(𝑋) be any strongly continuous function with ‖𝒱(𝑡)‖=𝑂(𝑒^𝜔𝑡); we define (ℳ𝒱)(𝑡)𝑥=^𝑡₀𝛿(𝜆)𝒱(𝑠)𝐶⁻¹𝑃𝐴S_𝐾()(𝐴)𝑡−𝑠𝑥𝑑𝑠,𝑥∈𝐷,𝑡≥0. (10) Obviously, (ℳ𝒱)(𝑡)𝑥 is continuous on 𝑡≥0, from (9) and the denseness of 𝐷(𝐴), ℳ maps 𝐂([0,∞),𝐿(𝑋)) into 𝐂([0,∞),𝐿(𝑋)).
It follows from (9) that (𝐼−ℳ)⁻¹ is bounded. For each 𝑡≥0, set C_𝐾(𝑡)𝑥∶=(𝐼−ℳ)⁻¹C_𝐾(⋅)𝑥(𝑡),𝑥∈𝑋. (11) Then, C_𝐾(𝑡)𝐶=𝐶C_𝐾(𝑡), and there exists a constant 𝑀 such that ‖C_𝐾(𝑡)‖≤𝑀𝑒^𝜔𝑡, C_𝐾(𝑡)𝑥=C_𝐾(𝑡)𝑥+𝛿(𝜆)^𝑡₀C_𝐾(𝑠)𝐶⁻¹𝑃𝐴S_𝐾(𝑡−𝑠)𝑥𝑑𝑠. (12) For sufficiently large 𝜆, we set ℒ(𝜆)𝑥=^∞₀𝑒^−𝜆𝑡C_𝐾(𝑡)𝑥𝑑𝑡,𝑥∈𝑋. (13) Taking Laplace transform of (12), we have ℒ(𝜆)𝑥=𝜆𝐾(𝜆)𝜆²−𝐴⁻¹(𝜆)ℒ(𝜆)𝐶𝐶𝑥+𝛿⁻¹𝐵𝐴𝜆²−𝐴⁻¹𝐶𝑥,𝑥∈𝑋. (14) Therefore for 𝑥∈𝐷(𝐴), ℒ(𝜆)𝜆²−(𝐼+𝛿(𝜆)𝐵)𝐴𝑥=𝜆𝐾(𝜆)𝐶𝑥. (15) Noting (8), for 𝑥∈𝑋, we have ℒ(𝜆)𝜆²−(𝐼+𝛿(𝜆)𝐵)𝐴𝜆²−(𝐼+𝛿(𝜆)𝐵)𝐴⁻¹𝑥=𝜆𝐾(𝜆)𝜆²−(𝐼+𝛿(𝜆)𝐵)𝐴⁻¹𝐶𝑥, (16) that is 1𝐾(𝜆)^∞0𝑒^−𝜆𝑡C_𝐾(𝑡)1𝑥𝑑𝑡=𝐾(𝜆)ℒ(𝜆)𝑥=𝜆𝜆²−(𝐼+𝛿(𝜆)𝐵)𝐴⁻¹𝐶𝑥. (17) Then from Definition 1, (𝐼+𝛿(𝜆)𝐵)𝐴 subgenerates an exponentially bounded 𝐾-convoluted 𝐶-cosine function {C_𝐾(𝑡)}_𝑡≥0.

Theorem 9. Suppose 𝐴 is a subgenerator of an exponentially bounded 𝐾-convoluted 𝐶-cosine function {C_𝐾(𝑡)}_𝑡≥0 on 𝑋, S_𝐾(𝑡)=∫^𝑡₀C_𝐾(𝑠)𝑑𝑠. Let 𝐵∈𝐿(𝑋) with 𝐵𝐶=𝐶𝐵 and let 𝑅(𝐵)⊂𝑅(𝐶), and 𝐷(𝐴) is dense in 𝑋. If for any Φ∈𝐂([0,∞),𝐿(𝑋)), ^𝑡₀Φ(𝑠)𝐶⁻¹𝐵𝐴S_𝐾()≤𝑀𝑡−𝑠𝑥𝑑𝑠_𝑡0𝑒^{𝜔(𝑡−𝑠)}‖Φ(𝑠‖𝑥‖,(𝐴))‖𝑑𝑠⋅𝑥∈𝐷,𝑡≥0, (18) where 𝑀 is a constant, then for some (and all) 𝜆, Re𝜆>max(𝜔,𝛽), (𝐼+𝐾(𝜆)𝐵)𝐴 subgenerates an exponentially bounded 𝐾-convoluted 𝐶-cosine function on 𝑋.

Proof. Define the operator functions {C_𝑛(𝑡)}_𝑡≥0 as follows: C₀(𝑡)𝑥=C_𝐾(𝑡)𝑥,C_𝑛(𝑡)𝑥=^𝑡₀C_𝑛−1(𝑠)𝐶⁻¹𝐵𝐴S_𝐾()(𝐴)𝑡−𝑠𝑥𝑑𝑠,𝑥∈𝐷,𝑡≥0,𝑛=1,2,…. (19)
By induction, we obtain (i)for any 𝑥∈𝑋, C_𝑛(𝑡)𝑥∈𝐂([0,∞),𝑋), (ii)‖C_𝑛(𝑡)𝑥‖≤(𝑀𝑀^𝑛𝑡^𝑛/𝑛!)𝑒^𝜔𝑡‖𝑥‖, 𝑡≥0, 𝑥∈𝑋, for all 𝑛≥0.
Define the operator function ℎ(𝑡)=^∞_𝑛=0C_𝑛(𝑡),𝑡≥0. (20)
Noting that the series ∑^∞_𝑛=0(𝑀𝑀^𝑛𝑡^𝑛/𝑛!)𝑒^𝜔𝑡 is uniformly converge on every compact interval in 𝑡, we can see that the series (20) is uniformly converge on every compact interval in 𝑡, so does the operator ℎ(𝑡). It is obvious that ‖ℎ(𝑡)‖≤𝑀𝑒^(𝜔+𝑀^)𝑡 and 𝑡→ℎ(𝑡)𝑥 is continuous on [0,∞) for any 𝑥∈𝑋. Moreover, ℎ(𝑡)𝑥=C_𝐾(𝑡)𝑥+^𝑡₀ℎ(𝑠)𝐶⁻¹𝐵𝐴S_𝐾(𝑡−𝑠)𝑥𝑑𝑠,𝑥∈𝑋,𝑡≥0. (21) For Re𝜆 sufficiently large, we set ℒ(𝜆)𝑥=^∞₀𝑒^−𝜆𝑡ℎ(𝑡)𝑥𝑑𝑡,𝑥∈𝑋. (22) Next, we show that the following equalities hold: ℒ(𝜆)𝜆²−𝐼+𝐾(𝜆)𝐵𝐴𝑥=𝜆𝐾(𝜆)𝐶𝑥,𝑥∈𝐷(𝐴),𝜆²−𝐼+𝐾(𝜆)𝐵𝐴ℒ(𝜆)𝑥=𝜆𝐾(𝜆)𝐶𝑥,𝑥∈𝑋. (23)(24) By induction, it is not difficult to see that ^∞₀𝑒^−𝜆𝑡C_𝑛(𝑡)𝐾(𝜆)𝜆𝑥𝑑𝑡=𝜆²−𝐴⁻¹𝐾(𝜆)𝐵𝐴𝜆²−𝐴^−1𝑛𝐶𝑥,𝑥∈𝑋,𝑛≥0. (25) Let 𝑄(𝑡)𝑥=^𝑡₀𝐶⁻¹𝐵𝐴S_𝐾(𝑡−𝑠)𝑥𝑑𝑠,𝑥∈𝐷(𝐴). (26) By hypothesis, 𝑄(𝑡) can be extended to 𝑋 and satisfies ‖𝑄(𝑡≤𝑀)‖𝜔𝑒^𝜔𝑡−1,𝑡≥0. (27) Set 𝑄(𝜆)𝑥=^∞₀𝑒^−𝜆𝑡𝑄(𝑡)𝑥𝑑𝑡,𝑥∈𝑋. (28) Then from (25) and (27), ‖𝜆𝑄(𝜆)‖=‖𝐶⁻¹𝐾(𝜆)𝐵𝐴(𝜆²−𝐴)⁻¹𝐶‖<1 for |𝜆| sufficiently large. Therefore, the series ^∞_𝑛=0𝐾(𝜆)𝐵𝐴𝜆²−𝐴^−1𝑛𝐶=^∞_𝑛=0𝐶𝐶⁻¹𝐾(𝜆)𝐵𝐴𝜆²−𝐴⁻¹𝐶^𝑛 (29) converges.
For 𝑥∈𝐷(𝐴) and Re𝜆>max(𝜔,𝛽), from (25), we have ℒ(𝜆)𝜆²−𝐾(𝜆)𝐵𝐴𝑥=𝐼+_∞0𝑒^{∞−𝜆𝑡}_𝑛=0C_𝑛(𝑡)𝜆²−𝐾(𝜆)𝐵𝐴=𝐾(𝜆)𝐼+𝑥𝑑𝑡_∞𝑛=0𝜆𝜆²−𝐴⁻¹𝐾(𝜆)𝐵𝐴𝜆²−𝐴^−1𝑛×𝐶𝜆²−𝐾(𝜆)𝐵𝐴𝑥𝐾(𝜆)𝐾(𝜆)𝐼+=𝜆𝐶𝑥−𝜆²𝜆²−𝐴⁻¹+𝐶𝐵𝐴𝑥_∞𝑛=1𝜆𝐾(𝜆)𝜆²−𝐴⁻¹𝐾(𝜆)𝐵𝐴𝜆²−𝐴^−1𝑛×𝐶𝜆²𝑥−−𝐴_∞𝑛=1𝜆𝐾(𝜆)𝜆²−𝐴⁻¹𝐾(𝜆)𝐵𝐴𝜆²−𝐴^−1𝑛𝐾(𝜆)𝐾(𝜆)×𝐶𝐵𝐴𝑥=𝜆𝐶𝑥+^∞_𝑛=2𝜆𝐾(𝜆)𝜆²−𝐴⁻¹×𝐾(𝜆)𝐵𝐴𝜆²−𝐴^−1𝑛𝐶𝜆²𝑥−−𝐴_∞𝑛=1𝜆𝐾(𝜆)𝜆²−𝐴⁻¹𝐾(𝜆)𝐵𝐴𝜆²−𝐴^−1𝑛𝐾(𝜆)𝐾(𝜆)×𝐶𝐵𝐴𝑥=𝜆𝐶𝑥. (30) Similarly, we can prove (24). Now, from Definition 1, we conclude that (𝐼+𝐾(𝜆)𝐵)𝐴 subgenerates an exponentially bounded 𝐾-convoluted 𝐶-cosine function on 𝑋.

By the proof of Theorems 8 and 9, we immediately obtain the following results for 𝐾-convoluted 𝐶-semigroups.

Theorem 10. Suppose that 𝐴 is a subgenerator of an exponentially bounded 𝐾-convoluted 𝐶-semigroup {T_𝐾(𝑡)}_𝑡≥0 on 𝑋. 𝐷(𝐴) is dense in 𝑋. Let 𝐵∈𝐿(𝑋) with 𝐵𝐶=𝐶𝐵 and let 𝑅(𝐵)⊂𝑅(𝐶). (i)One sets 𝐿(𝜆)∶=sup^𝑎₀𝑒^−𝜆𝑠𝛿(𝜆)𝐶⁻¹𝑃𝐴T_𝐾(𝑠)𝑥(𝐴),‖𝑥‖𝑑𝑠,𝑥∈𝐷≤1<∞, (31) for some 𝑎∈(0,+∞] and 𝜆>max(𝜔,𝛽), where 𝛿(𝜆) is a function and 𝑃=𝐵/𝐾(𝜆). If {𝜆∶𝜆>max(𝜔,𝛽)}⊂𝜌((𝐼+𝛿(𝜆)𝐵)𝐴), then (𝐼+𝛿(𝜆)𝐵)𝐴 subgenerates an exponentially bounded 𝐾-convoluted 𝐶-semigroup on 𝑋 provided that lim_𝜆→∞𝐿(𝜆)𝑒^𝜆𝑡=0 for all 𝑡≥0. (ii)If for any Φ∈𝐂([0,∞),𝐿(𝑋)), ^𝑡₀Φ(𝑠)𝐶⁻¹𝐵𝐴T_𝐾()≤𝑀𝑡−𝑠𝑥𝑑𝑠_𝑡0𝑒^{𝜔(𝑡−𝑠)}‖Φ(𝑠‖𝑥‖,(𝐴))‖𝑑𝑠⋅𝑥∈𝐷,𝑡≥0, (32) where 𝑀 is a constant, then for some (and all) 𝜆, Re𝜆>max(𝜔,𝛽), (𝐼+𝐾(𝜆)𝐵)𝐴 subgenerates an exponentially bounded 𝐾-convoluted 𝐶-semigroup on 𝑋.

Proof. (i) For any Ψ∈𝐂([0,∞),𝐿(𝑋)) with ‖Ψ(𝑡)‖=𝑂(𝑒^𝜔𝑡), sufficiently large 𝜆 and sufficiently small 𝜀, we have ^𝑡₀𝛿(𝜆)Ψ(𝑠)𝐶⁻¹𝑃𝐴T_𝐾(𝑡−𝑠)𝑥𝑑𝑠≤𝑀^∗𝑒^{(𝜆+𝜔)𝑡𝑡}₀𝑒^−𝜆𝑠𝛿(𝜆)𝐶⁻¹𝑃𝐴T_𝐾(𝑠)𝑥𝑑𝑠≤𝑀^∗𝑒^{(𝜆+𝜔)𝑡}𝐿(𝜆)(𝐴),‖𝑥‖<𝜀<1,𝑡≥0,𝑥∈𝐷≤1, (33) where 𝑀^∗ is a constant. The rest part of the proof is exactly the same as the corresponding part of the proof of Theorem 8.
The proof of (ii) is similar to the one of Theorem 9.

In Theorems 8–10, take 𝐾(𝑡)=𝑡^𝑛−1/Γ(𝑛), we have the following result for 𝑛-times integrated 𝐶-cosine function (resp., 𝑛-times integrated 𝐶-semigroup).

Theorem 11. Suppose 𝐴 is a subgenerator of an exponentially bounded 𝑛-times integrated 𝐶-cosine function {C_𝑛(𝑡)}_𝑡≥0 (resp., 𝑛-times integrated 𝐶-semigroup {T_𝑛(𝑡)}_𝑡≥0) on 𝑋. Let 𝐵∈𝐿(𝑋) with 𝐵𝐶=𝐶𝐵 and let 𝑅(𝐵)⊂𝑅(𝐶), and 𝐷(𝐴^𝑛+1) is dense in 𝑋. (i)One sets 𝐿(𝜆)∶=sup^𝑎₀𝑒^{−𝜆𝑡𝑡}₀𝛿(𝜆)Ψ(𝑠)𝐶⁻¹×𝑑𝐵𝐴_𝑛𝑑𝑡^𝑛C_𝑛()𝑥𝐴𝑡−𝑠𝑑𝑠𝑑𝑡,𝑥∈𝐷^𝑛+1,‖𝑥‖≤1<∞, (34) for any Ψ∈𝐂([0,∞),𝐿(𝑋)) with ‖Ψ(𝑡)‖=𝑂(𝑒^𝜔𝑡), resp.𝐿(𝜆)∶=sup^𝑎₀𝑒^−𝜆𝑠𝛿(𝜆)𝐶⁻¹𝑑𝐵𝐴^𝑛𝑑𝑠^𝑛T_𝑛(𝑠)𝑥𝐴𝑑𝑠,𝑥∈𝐷^𝑛+1,‖𝑥‖≤1<∞ (35) for some 𝑎∈(0,+∞] and 𝜆>𝜔, where 𝛿(𝜆) is a function. If (𝜔²,∞)⊂𝜌((𝐼+𝛿(𝜆)𝐵)𝐴) (resp., (𝜔,∞)⊂𝜌((𝐼+𝛿(𝜆)𝐵)𝐴)), then (𝐼+𝛿(𝜆)𝐵)𝐴 subgenerates an exponentially bounded 𝑛-times integrated 𝐶-cosine function (resp., 𝑛-times integrated 𝐶-semigroup) on 𝑋 provided that lim_𝜆→∞𝐿(𝜆)𝑒^𝜆𝑡=0 for all 𝑡≥0.(ii)If for any Φ∈𝐂([0,∞),𝐿(𝑋)), ^𝑡₀Φ(𝑠)𝐶⁻¹𝑑𝐵𝐴^𝑛𝑑𝑡^𝑛S_𝑛()𝑥≤𝑀𝑡−𝑠𝑑𝑠_𝑡0𝑒^{𝜔(𝑡−𝑠)}‖Φ(𝑠‖𝑥‖,𝐴)‖𝑑𝑠⋅𝑥∈𝐷^𝑛+1,𝑡≥0, (36) where S_𝑛(𝑡)=∫^𝑡₀C_𝑛(𝑠)𝑑𝑠 and 𝑀 is a constant, resp.^𝑡₀Φ(𝑠)𝐶⁻¹𝑑𝐵𝐴^𝑛𝑑𝑡^𝑛T_𝑛()𝑥≤𝑀𝑡−𝑠𝑑𝑠_𝑡0𝑒^{𝜔(𝑡−𝑠)}‖Φ(𝑠‖𝑥‖,𝐴)‖𝑑𝑠⋅𝑥∈𝐷^𝑛+1,𝑡≥0 (37) then for some (and all) 𝜆, 𝜆>𝜔, (𝐼+𝐾(𝜆)𝐵)𝐴 subgenerates an exponentially bounded 𝑛-times integrated 𝐶-cosine function (resp., 𝑛-times integrated 𝐶-semigroup) on 𝑋.

When 𝑛=0, from Theorem 11(ii), we immediately obtain the result of 0-times integrated 𝐶-cosine function (resp., 0-times integrated 𝐶-semigroup).

Theorem 12. Let 𝐵∈𝐿(𝑋) with 𝐵𝐶=𝐶𝐵 and let 𝑅(𝐵)⊂𝑅(𝐶), and 𝐷(𝐴) is dense in 𝑋. Suppose that 𝐴 is an exponentially bounded generator of a 𝐶-cosine function {C(𝑡)}_𝑡≥0 (resp., 𝐶-semigroup {T(𝑡)}_𝑡≥0) on 𝑋. If for any Φ∈𝐂([0,∞),𝐿(𝑋)), ^𝑡₀Φ(𝑠)𝐶⁻¹𝐵𝐴S(𝑡−𝑠)𝑥𝑑𝑠≤𝑀^𝑡₀𝑒^{𝜔(𝑡−𝑠)}‖Φ(𝑠‖𝑥‖,(𝐴))‖𝑑𝑠⋅𝑥∈𝐷,𝑡≥0, (38) where S(𝑡)=∫^𝑡₀C(𝑠)𝑑𝑠. resp.^𝑡₀Φ(𝑠)𝐶⁻¹𝐵𝐴T(𝑡−𝑠)𝑥𝑑𝑠≤𝑀^𝑡₀𝑒^{𝜔(𝑡−𝑠)}‖Φ(𝑠‖𝑥‖,(𝐴))‖𝑑𝑠⋅𝑥∈𝐷,𝑡≥0 (39) for some 𝑎∈(0,+∞] and 𝜆>𝜔, then (𝐼+𝐵)𝐴 subgenerates an exponentially bounded 𝐶-cosine function (resp., 𝐶-semigroup) on 𝑋.

Noting the Definition 7 and the special properties of 𝐶-cosine functions (resp., 𝐶-semigroups), we obtain a different result from Theorem 11(i) (when 𝑛=0).

Theorem 13. Let 𝐵∈𝐿(𝑋) with 𝐵𝐶=𝐶𝐵 and let 𝑅(𝐵)⊂𝑅(𝐶), and 𝐷(𝐴) is dense in 𝑋, (𝜔²,∞)⊂𝜌((𝐼+𝐵)𝐴) (resp., (𝜔,∞)⊂𝜌((𝐼+𝐵)𝐴)). Suppose that 𝐴 is an exponentially bounded generator of a 𝐶-cosine function {C(𝑡)}_𝑡≥0 (resp., 𝐶-semigroup {T(𝑡)}_𝑡≥0) on 𝑋. If 𝐿(𝜆)∶=sup^𝑎₀𝑒^{−𝜆𝑡𝑡}₀𝐶⁻¹𝐵𝐴C()(𝐴),‖𝑥‖𝑡−𝑠𝑥𝑑𝑠𝑑𝑡,𝑥∈𝐷≤1<∞,resp.𝐿(𝜆)∶=sup^𝑎₀𝑒^−𝜆𝑠𝐶⁻¹𝐵𝐴T(𝑠)𝑥(𝐴),‖𝑥‖𝑑𝑠,𝑥∈𝐷≤1<∞ (40)(41) for some 𝑎∈(0,+∞] and 𝜆>𝜔, letting 𝐿(∞)=lim_𝜆→∞𝐿(𝜆), then for any 𝜀<(𝐿(∞))⁻¹, (𝐼+𝜀𝐵)𝐴 subgenerates an exponentially bounded 𝐶-cosine function (resp., 𝐶-semigroup) on 𝑋.

Proof. We prove only for 𝐶-cosine functions. Choose 0<𝜇<𝜇₁<𝜇₂<1 such that |𝜀|=𝜇(𝐿(∞))⁻¹. For any Ψ∈𝐂([0,𝑡],𝐿(𝑋)), pick a 𝜆 large enough such that 𝐿(𝜆)/𝐿(∞)<𝜇₁/𝜇, and then pick a 𝜏∈(0,𝑎) small enough such that 𝑒^𝜆𝜏sup_{𝑠∈[0,𝜏]}‖Ψ(𝑠)‖≤𝜇₂/𝜇₁, then for all 𝑥∈𝐷(𝐴), ‖𝑥‖≤1, we have 𝜀Ψ(^𝑡₀𝑠)𝐶⁻¹𝐵𝐴S()=𝑡−𝑠𝑥𝑑𝑠_𝑡0𝑠0𝜀Ψ(𝜎)𝐶⁻¹𝐵𝐴C(𝑠−𝜎)𝑥𝑑𝜎𝑑𝑠≤𝑒^𝜆𝑡𝑡₀𝑒^{−𝜆𝑠𝑠}₀𝜀𝐶⁻¹𝐵𝐴C(𝑠−𝜎)𝑥𝑑𝜎𝑑𝑠⋅sup_{𝑠∈[0,𝜏]}‖Ψ(𝑠)‖≤𝑒^𝜆𝜏|𝜀|𝐿(𝜆)⋅sup_{𝑠∈[0,𝜏]}‖Ψ(𝑠)‖<𝜇₂<1,𝑡∈[0,𝜏], (42) where S(𝑡)=∫^𝑡₀C(𝑠)𝑑𝑠.
Let 𝒱∶[0,𝜏]→𝐿(𝑋) be any strongly continuous function; we define (𝑡)𝑥=ℳ𝒱)(_𝑡0𝜀𝒱(𝑠)𝐶⁻¹𝐵𝐴S(𝑡−𝑠)𝑥𝑑𝑠,𝑥∈𝐷(𝐴),𝑡∈[0,𝜏]. (43) Obviously, (ℳ𝒱)(𝑡)𝑥 is continuous on 𝑡≥0, from (42) and the denseness of 𝐷(𝐴), ℳ maps 𝐂([0,𝜏],𝐿(𝑋)) into 𝐂([0,𝜏],𝐿(𝑋)).
It follows from (42) that (𝐼−ℳ)⁻¹ is bounded. For each 𝑡∈[0,𝜏], set 𝑉(𝑡)𝑥∶=(𝐼−ℳ)⁻¹[C(⋅)𝑥](𝑡),𝑥∈𝑋. (44) Then, 𝑉(𝑡)𝐶=𝐶𝑉(𝑡), and there exists a constant 𝑀 such that ‖𝑉(𝑡)‖≤𝑀𝑒^𝜔𝑡: 𝑉(𝑡)𝑥=C(𝑡)𝑥+^𝑡₀𝜀𝑉(𝑠)𝐶⁻¹𝐵𝐴S(𝑡−𝑠)𝑥𝑑𝑠,𝑡∈[0,𝜏]. (45) For 𝑡∈((𝑛−1)𝜏,𝑛𝜏], 𝑛=2,3,…, we define inductively 𝑉(𝑡)∶=−𝑉(2(𝑛−1)𝜏−𝑡)+2𝐶⁻¹𝑉(𝑡−(𝑛−1)𝜏)𝑉((𝑛−1)𝜏). (46) Next, we will prove by induction that for any 𝑛∈ℕ, 𝑅(𝑉(𝜎)𝑉((𝑛−1)𝜏))⊂𝑅(𝐶), for 𝜎∈[0,𝜏], and that for every 𝑛∈ℕ, 𝑉(⋅) is strongly continuous in [0,𝑛𝜏] and 𝑉(𝑡)𝑥=C(𝑡)𝑥+^𝑡₀𝜀𝑉(𝑠)𝐶⁻¹𝐵𝐴S()[].𝑡−𝑠𝑥𝑑𝑠,𝑥∈𝑋,𝑡∈0,𝑛𝜏 (47) Indeed for 𝑛=1, this is true. Assume that (47) holds for 𝑛. Then for 𝑥∈𝑋, 𝜎∈[0,𝜏] we get 2𝑉(𝜎)𝑉(𝑛𝜏)𝑥=2C(𝜎)C(𝑛𝜏)𝑥+2₀^𝑛𝜏𝜀𝑉(𝜎)𝑉(𝑠)𝐶⁻¹𝐵𝐴S(𝑛𝜏−𝑠)𝑥𝑑𝑠+2^𝜎₀𝜀𝑉(𝑠)𝐶⁻¹𝐵𝐴S(𝜎−𝑠)C()[𝐶()𝑛𝜏𝑥𝑑𝑠=𝐶𝜎+𝑛𝜏𝑥+C(𝑛𝜏−𝜎)𝑥]+2₀^𝑛𝜏𝜀𝑉(𝜎)𝑉(𝑠)𝐶⁻¹𝐵𝐴S(𝑛𝜏−𝑠)𝑥𝑑𝑠+2^𝜎₀𝜀𝑉(𝑠)𝐶⁻¹𝐵𝐴S(𝜎−𝑠)C()[𝑉(𝜎)𝑉(⋅)[𝐶()𝑛𝜏𝑥𝑑𝑠=2ℳ)](𝑛𝜏𝑥+𝐶𝜎+𝑛𝜏𝑥+C(𝑛𝜏−𝜎)𝑥]+𝐶^𝜎₀𝜀𝑉(𝑠)𝐶⁻¹×[𝐵𝐴S(𝑛𝜏+𝜎−𝑠)𝑥−S(𝑛𝜏−𝜎+𝑠)𝑥]𝑑𝑠. (48) Then for 𝑥∈𝑋, 𝜎∈[0,𝜏], (𝜎)𝑉()𝑥()2𝑉𝑛𝜏=𝐶𝐼−ℳ⁻¹{C(𝜎+𝑛𝜏)𝑥+C()𝑥+𝑛𝜏−𝜎_𝜎0𝜀𝑉(𝑠)𝐶⁻¹𝐵𝐴[S()𝑥−𝑛𝜏+𝜎−𝑠S(𝑛𝜏−𝜎+𝑠)𝑥]𝑑𝑠}. (49) Hence, 𝑉(𝜎)𝑉(𝑛𝜏)⊂𝑅(𝐶), 𝜎∈[0,𝜏], and 𝜎→𝐶⁻¹𝑉(𝜎)𝑉(𝑛𝜏)𝑥 is continuous in [0,𝜏] for each 𝑥∈𝑋. From (48), we have (𝜎)𝑉()𝑥[2𝑉𝑛𝜏=𝐶C(𝜎+𝑛𝜏)𝑥+C(𝑛𝜏−𝜎)𝑥]+2₀^𝑛𝜏𝜀𝑉(𝜎)𝑉(𝑠)𝐶⁻¹𝐵𝐴S(𝑛𝜏−𝑠)𝑥𝑑𝑠+𝐶^𝜎₀𝜀𝑉(𝑠)𝐶⁻¹𝐵𝐴[S()𝑥−𝑛𝜏+𝜎−𝑠S()𝑥][𝑛𝜏−𝜎+𝑠𝑑𝑠=𝐶C(𝜎+𝑛𝜏)𝑥+C(𝑛𝜏−𝜎)𝑥]+𝐶₀^𝑛𝜏𝜀[𝑉()𝐶𝜎+𝑠+𝑉(|𝑠−𝜎|)]⁻¹𝐵𝐴S(𝑛𝜏−𝑠)𝑥𝑑𝑠+𝐶^𝜎₀𝜀𝑉(𝑠)𝐶⁻¹𝐵𝐴[S()𝑥−𝑛𝜏+𝜎−𝑠S()𝑥][𝑛𝜏−𝜎+𝑠𝑑𝑠=𝐶C(𝜎+𝑛𝜏)𝑥+C(𝑛𝜏−𝜎)𝑥]+𝐶_𝜎^{𝜎+𝑛𝜏}𝜀𝑉(𝑠)𝐶⁻¹𝐵𝐴S(𝑛𝜏+𝜎−𝑠)𝑥𝑑𝑠+𝐶^𝜎₀𝜀𝑉(𝑠)𝐶⁻¹𝐵𝐴S(𝑛𝜏−𝜎+𝑠)𝑥𝑑𝑠+𝐶₀^{𝑛𝜏−𝜎}𝜀𝑉(𝑠)𝐶⁻¹𝐵𝐴S(𝑛𝜏−𝜎−𝑠)𝑥𝑑𝑠+𝐶^𝜎₀𝜀𝑉(𝑠)𝐶⁻¹𝐵𝐴[S()𝑥−𝑛𝜏+𝜎−𝑠S()𝑥][𝑛𝜏−𝜎+𝑠𝑑𝑠=𝐶C(𝜎+𝑛𝜏)𝑥+C(𝑛𝜏−𝜎)𝑥]+𝐶₀^{𝜎+𝑛𝜏}𝜀𝑉(𝑠)𝐶⁻¹𝐵𝐴S(𝑛𝜏+𝜎−𝑠)𝑥𝑑𝑠+𝐶₀^{𝑛𝜏−𝜎}𝜀𝑉(𝑠)𝐶⁻¹𝐵𝐴S()()()𝑛𝜏−𝜎−𝑠𝑥𝑑𝑠=𝐶𝑉𝜎+𝑛𝜏𝑥+𝐶𝑉𝑛𝜏−𝜎𝑥. (50) Therefore, 𝑉(⋅) is strongly continuous in [0,∞) and (47) holds for all 𝑡≥0. Taking Laplace transform of (47), then the conclusion can be proved in a similar way in the proof of Theorem 8.
We can prove the case of 𝐶-semigroups in a similar way.

3. Examples

Example 14. Let 𝑋∶=𝑓∈𝐶^∞[0,1]∶𝑓∶=sup_𝑝≥0‖𝑓^(𝑝)‖_∞𝑝!²,𝑑<∞𝐴∶=−(𝐴)𝑑𝑥,𝐷∶=𝑓∈𝑋∶𝑓^′∈𝑋,𝑓(0)=0. (51)
It is well known that there exist positive real numbers 𝑚 and 𝑀 such that {𝜆∈ℂ∶Re𝜆≥0}⊂𝜌(𝐴),‖𝑅(𝜆,𝐴)‖≤𝑀𝑒^𝑚|𝜆|,Re𝜆≥0. (52)
Moreover, 𝐴 generates an exponentially bounded 𝐾_𝑎-convoluted semigroup {T_𝐾(𝑡)}_𝑡≥0 for some 𝑎>√2𝑚, where 𝐾(𝑡)=(𝑎/(2√𝜋𝑡³))𝑒^{−𝑎2/(4𝑡)},𝑡≥0, then 𝐾_𝑎(𝜆)=𝑒^{√−𝑎𝜆},Re𝜆>0 [14, 23]. We set 𝐵𝑓(𝑥)∶=^𝑗_𝑛=1^𝑥₀(𝑥−𝑠)^𝑛−1()!𝑓(𝑠)[],𝑛−1𝑑𝑠,𝑗∈ℕ,𝑥∈0,1𝑓∈𝑋. (53) Obviously, 𝐵∈𝐿(𝑋) and 𝐵𝐴⊂𝐴𝐵. Then from Theorem 10(ii) (𝐶=𝐼), (𝐼+𝑒^{√−𝑎𝜆}𝐵)𝐴 subgenerates an exponentially bounded 𝐾_𝑎-convoluted semigroup {T_𝐾(𝑡)}_𝑡≥0 on 𝑋.

Example 15. Let 𝑋∶=𝐶₀(ℝ)⊕𝐶₀(ℝ)⊕𝐶₀(ℝ), 𝐴𝑓,𝑔,ℎ(⋅)∶=𝑓^′,𝑔^′,𝜒_[0,∞)−𝜒_(−∞,0]ℎ,(𝐴)=𝑓,𝑔,ℎ∈𝐷𝑓,𝑔,ℎ∈𝑋∶𝑓^′∈𝐶₀(ℝ),𝑔^′∈𝐶₀(ℝ),ℎ(0)=0 (54) and 𝐶(𝑓,𝑔,ℎ)∶=(𝑓,𝑔,sin(⋅)ℎ(⋅)),𝑓,𝑔,ℎ∈𝐶₀(ℝ). Arguing as in [3, Examples 8.1 and 8.2], one gets that 𝐴 is a generator of an exponentially bounded once integrated 𝐶-semigroup [16].
For 𝑓,𝑔,ℎ∈𝐶₀(ℝ) and 𝑡∈ℝ, we set 𝐵(𝑡)=𝑒𝑓,𝑔,ℎ_−𝑡𝑡0𝑓(𝑠)𝑑𝑠,𝑒^−2𝑡𝑡₀𝑔(𝑠)𝑑𝑠,𝑡𝑒^−3𝑡sin𝑡⋅ℎ(𝑡). (55) Then one can simply verify that 𝐵∈𝐿(𝑋),𝑅(𝐵)⊂𝐶(𝐷(𝐴)), and 𝐵𝐶(𝑓,𝑔,ℎ)=𝐶𝐵(𝑓,𝑔,ℎ), (𝑓,𝑔,ℎ)∈𝑋. Then from Theorem 11(i), (𝐼+𝑒^−𝜆2𝐵)𝐴 subgenerates an exponentially bounded once integrated 𝐶-semigroup on 𝑋.

Example 16. Let 𝑋₁=𝐿²(ℝ³), 𝑋₂=𝐿^𝑝(ℝ³) (1≤𝑝≤∞), 𝐴₁=Δ,𝐷𝐴₁=𝐻²ℝ³,𝐴₂=𝑎Δ+³_𝑖=1𝑐_𝑖𝜕𝜕𝑥₁+𝑐₄𝑎>0,𝑐_𝑖,𝐷𝐴∈ℝ,𝑖=1,2,3,4₂=𝑊^2,𝑝ℝ³. (56) Then 𝐴₁ generates a strongly continuous cosine function C₁(⋅) on 𝑋₁. It follows from [5] that 𝐴₂ generates an exponentially bounded 𝐶₂-cosine function C₂(⋅) on 𝑋₂, where 𝐶₂=(1−Δ)⁻¹.
Set 𝑟₁(⋅)∈𝐻²(ℝ³), 𝑟₂(⋅)∈𝑊^2,𝑝(ℝ³), 𝑞₁(⋅)∈𝐶²_𝑐(ℝ³), 𝑞₂(⋅)∈𝐶_𝑐(ℝ³). Define bounded linear operators 𝐵₁∶𝑋₂→𝑋₁, 𝐵₂∶𝑋₁→𝑋₂ as follows: 𝐵₁𝜙(𝜉)=𝑟₁(𝜉)_ℝ3𝑞₁(𝜎)𝜙(𝜎)𝐵𝑑𝜎,₂𝜙(𝜉)=𝑟₂(𝜉)_ℝ3𝑞₂(𝜎)𝜙(𝜎)𝑑𝜎. (57) Let 𝑋=𝑋₁×𝑋₂, 𝐴𝐴=¹00𝐴₂,𝐷(𝐴)∶=𝐷𝐴₁×𝐷𝐴₂,𝐵=0𝐵₁𝐵₂0,𝐷(𝐵)∶=𝑋. (58) Taking 𝜆₀∈𝜌(𝐴) and putting 𝐶=(𝜆₀−𝐴)⁻¹, then 𝐴 generates an exponentially bounded 𝐶-cosine function C(⋅) on 𝑋, where C(𝑡)=C₁(𝑡)𝜆₀−𝐴₁⁻¹00C₂(𝑡)𝐶₂⁻¹𝜆₀−𝐴₂⁻¹. (59) We denote S₁(𝑡)∶=∫^𝑡₀C₁(𝑠)𝑑𝑠, S₂(𝑡)∶=∫^𝑡₀C₂(𝑠)𝑑𝑠, S(𝑡)∶=∫^𝑡₀C(𝑠)𝑑𝑠, then S(𝑡)=S₁(𝑡)𝜆₀−𝐴₁⁻¹00S₂(𝑡)𝐶₂⁻¹𝜆₀−𝐴₂⁻¹, (60) and for any 𝑥=^𝑥1_𝑥2∈𝐷(𝐴), 0≤𝑠≤𝑡<∞, 𝐶⁻¹𝐵𝐴S()𝑥=𝜆𝑡−𝑠₀−𝐴₁𝐵₁𝐴₂S₂(𝑡−𝑠)𝐶₂⁻¹𝜆₀−𝐴₂⁻¹𝑥₂𝜆₀−𝐴₂𝐵₂𝐴₁S₁(𝑡−𝑠)𝜆₀−𝐴₁⁻¹𝑥₁. (61) It follows from 𝑅(𝐵₁)⊂𝐷(𝐴₁) and 𝑅(𝐵₂)⊂𝐷(𝐴₂) that there exist 𝑀, 𝜔>0 such that 𝑒^{−𝜆𝑡𝑡}₀𝐶⁻¹𝐵𝐴C()≤𝑀𝑡−𝑠𝑥𝑑𝑠𝜔𝑒^−𝜆𝑡𝑒^𝜔𝑡−1‖𝑥‖,𝑥∈𝐷(𝐴), (62) then 𝐿(𝜆)∶=sup^𝑎₀𝑒^{−𝜆𝑡𝑡}₀𝐶⁻¹𝐵𝐴C()(𝐴),‖𝑥‖𝑡−𝑠𝑥𝑑𝑠𝑑𝑡,𝑥∈𝐷≤1<∞, (63) and then (40) is satisfied.

Acknowledgments

The authors are grateful to the referee for his/her valuable suggestions. This work was partly supported by the NSF of China (11201413), the NSF of Yunnan Province (2009ZC054M), the Educational Commission of Yunnan Province (2012Z010), and the Foundation of Key Program of Yunnan Normal University.