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

Authors: Fang Li; Huiwen Wang; Jun Zhang
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We obtain the multiplicative perturbation theorems for convolutedC-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.


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## Body

## 1. Introduction and Preliminaries

Theα-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 onC-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 convolutedC-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 ρc(A), respectively, where ρc(A) is defined by
(1)ρc(A)∶={λ∈ℂ:R(C)⊂R(λ-A)andλ-Ais injective}.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)∶=∫0t‍K(s)ds.The next definition is the convoluted version of Definition 4.1 in Chapter 1 of [5].Definition 1 (see [5, 13, 15]).
Letω≥0. If {λ2:Reλ>max(ω,β)}⊂ρc(A) and there exists a strongly continuous operator family {CK(t)}t≥0(CK(t)∈L(X),t≥0) such that ∥CK(t)∥≤Meωt,t≥0 for some M>0, and(2)λ(λ2-A)-1Cx=1K^(λ)∫0∞‍e-λtCK(t)xdt,Reλ>max(ω,β),x∈X,
then it is said that A is a subgenerator of an exponentially bounded K-convoluted C-cosine function {CK(t)}t≥0. The operator A¯∶=C-1AC is called the generator of {CK(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=CCK(t), t≥0,(2)CK(t)A⊂ACK(t), t≥0 and
(3)A∫0t‍∫0s‍CK(σ)xdσds=CK(t)x-ϑ(t)Cx,t≥0,x∈X.Definition 3.
Let0≤ω<∞. If {λ:Reλ>max(ω,β)}⊂ρc(A) and there exists a strongly continuous operator family {TK(t)}t≥0 such that ∥TK(t)∥≤Meωt,t≥0 for some M>0, and
(4)(λ-A)-1Cx=1K^(λ)∫0∞‍e-λtTK(t)xdt,Reλ>max(ω,β),x∈X,
then it is said that A is a subgenerator of an exponentially bounded K-convoluted C-semigroup {TK(t)}t≥0. The operator A¯∶=C-1AC is called the generator of {TK(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=CTK(t), t≥0,(2)
TK(t)A⊂ATK(t), t≥0 and
(5)A∫0t‍TK(s)xds=TK(t)x-ϑ(t)Cx,t≥0,x∈X.Remark 5 (see [16]).
In Theorems2 and 4, putting K(t)=tr-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 ofr-times integrated C-semigroup and r-times integrated C-cosine functions (r≥0).Definition 6 (see [5]).
Let0≤ω<∞ and let r∈[0,∞). If (ω2,∞)⊂ρc(A) (resp., (ω,∞)⊂ρc(A)) and there exists a strongly continuous operator family {Cr(t)}t≥0 (resp., {Tr(t)}t≥0) such that ∥Cr(t)∥≤Meωt,t≥0 (resp., ∥Tr(t)∥≤Meωt,t≥0) for some M>0, and
(6)λ(λ2-A)-1Cx=λr∫0∞‍e-λtCr(t)xdt,λ>ω,x∈X,(∫0∞‍e-λtresp.(λ-A)-1Cx=λr∫0∞‍e-λtTr(t)xdt,λ>ω,x∈X,),
then it is said that A is a subgenerator of an exponentially bounded r-times integrated C-cosine function {Cr(t)}t≥0 (resp., r-times integrated C-semigroup {Tr(t)}t≥0) on X. If r=0, then {Cr(t)}t≥0 (resp., {Tr(t)}t≥0) is called an exponentially bounded 0-times integrated C-cosine function (resp., 0-times integrated C-semigroup).We present the definition ofC-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 thatA is a subgenerator of an exponentially bounded K-convoluted C-cosine function {CK(t)}t≥0 on X, SK(t)=∫0t‍CK(s)ds, for any Ψ∈C([0,∞),L(X)) with ∥Ψ(t)∥=O(eωt), we set
(7)L(λ)∶=sup{∫0a‍e-λt∥∫0t‍δ(λ)Ψ(s)C-1PACK(t-s)xds∥dt,x∈D(A),∥x∥≤1∫0t‍δ(λ)}<∞,
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 thatA 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,
(8){λ2:λ>max(ω,β)}⊂ρ((I+δ(λ)B)A).
If limλ→∞L(λ)eλt=0 for all t≥0, then (I+δ(λ)B)A subgenerates an exponentially bounded K-convoluted C-cosine function on X.Proof.
For allx∈D(A), ∥x∥≤1, λ is large enough and ε is small enough, we have
(9)∥∫0t‍δ(λ)Ψ(s)C-1PASK(t-s)xds∥=∥∫0t‍∫0s‍δ(λ)Ψ(σ)C-1PACK(s-σ)xdσds∥≤eλt∫0t‍e-λs∥∫0s‍δ(λ)Ψ(σ)C-1PACK(s-σ)xdσ∥ds≤eλtL(λ)<ε<1,t≥0.
Let𝒱:[0,∞)→L(X) be any strongly continuous function with ∥𝒱(t)∥=O(eωt); we define
(10)(ℳ𝒱)(t)x=∫0t‍δ(λ)𝒱(s)C-1PASK(t-s)xds,x∈D(A),t≥0.
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
(11)C^K(t)x∶=(I-ℳ)-1[CK(·)x](t),x∈X.
Then, C^K(t)C=CC^K(t), and there exists a constant M^ such that ∥C^K(t)∥≤M^eωt,
(12)C^K(t)x=CK(t)x+δ(λ)∫0t‍C^K(s)C-1PASK(t-s)xds.
For sufficiently large λ, we set
(13)ℒ(λ)x=∫0∞‍e-λtC^K(t)xdt,x∈X.
Taking Laplace transform of (12), we have
(14)ℒ(λ)x=λK^(λ)(λ2-A)-1Cx+δ(λ)ℒ(λ)C-1BA(λ2-A)-1Cx,x∈X.
Therefore for x∈D(A),
(15)ℒ(λ)(λ2-(I+δ(λ)B)A)x=λK^(λ)Cx.
Noting (8), for x∈X, we have
(16)ℒ(λ)(λ2-(I+δ(λ)B)A)(λ2-(I+δ(λ)B)A)-1x=λK^(λ)(λ2-(I+δ(λ)B)A)-1Cx,
that is
(17)1K^(λ)∫0∞‍e-λtC^K(t)xdt=1K^(λ)ℒ(λ)x=λ(λ2-(I+δ(λ)B)A)-1Cx.
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.
SupposeA is a subgenerator of an exponentially bounded K-convoluted C-cosine function {CK(t)}t≥0 on X, SK(t)=∫0t‍C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)),
(18)∥∫0t‍Φ(s)C-1BASK(t-s)xds∥≤M~∫0t‍eω(t-s)∥Φ(s)∥ds·∥x∥,x∈D(A),t≥0,
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:
(19)C¯0(t)x=CK(t)x,C¯n(t)x=∫0t‍C¯n-1(s)C-1BASK(t-s)xds,x∈D(A),t≥0,n=1,2,….
By induction, we obtain(i)
for anyx∈X, C¯n(t)x∈C([0,∞),X),(ii)
∥C¯n(t)x∥≤(MM~ntn/n!)eωt∥x∥, t≥0, x∈X, for all n≥0.
Define the operator function(20)h(t)=∑n=0∞‍C¯n(t),t≥0.
Noting that the series∑n=0∞(MM~ntn/n!)eωt 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,
(21)h(t)x=CK(t)x+∫0t‍h(s)C-1BASK(t-s)xds,x∈X,t≥0.
For Reλ sufficiently large, we set
(22)ℒ(λ)x=∫0∞‍e-λth(t)xdt,x∈X.
Next, we show that the following equalities hold:
(23)ℒ(λ)[λ2-(I+K^(λ)B)A]x=λK^(λ)Cx,x∈D(A),(24)[λ2-(I+K^(λ)B)A]ℒ(λ)x=λK^(λ)Cx,x∈X.
By induction, it is not difficult to see that
(25)∫0∞‍e-λtC¯n(t)xdt=λK^(λ)(λ2-A)-1[K^(λ)BA(λ2-A)-1]nCx,x∈X,n≥0.
Let
(26)Q(t)x=∫0t‍C-1BASK(t-s)xds,x∈D(A).
By hypothesis, Q(t) can be extended to X and satisfies
(27)∥Q(t)∥≤M~ω(eωt-1),t≥0.
Set
(28)Q^(λ)x=∫0∞‍e-λtQ(t)xdt,x∈X.
Then from (25) and (27), ∥λQ^(λ)∥=∥C-1K^(λ)BA(λ2-A)-1C∥<1 for |λ| sufficiently large. Therefore, the series
(29)∑n=0∞‍[K^(λ)BA(λ2-A)-1]nC=∑n=0∞‍C[C-1K^(λ)BA(λ2-A)-1C]n
converges.
Forx∈D(A) and Reλ>max(ω,β), from (25), we have
(30)ℒ(λ)[λ2-(I+K^(λ)B)A]x=∫0∞‍e-λt∑n=0∞‍C¯n(t)[λ2-(I+K^(λ)B)A]xdt=K^(λ)∑n=0∞‍λ(λ2-A)-1[K^(λ)BA(λ2-A)-1]n×C[λ2-(I+K^(λ)B)A]x=λK^(λ)Cx-λ(K^(λ))2(λ2-A)-1CBAx+∑n=1∞‍λK^(λ)(λ2-A)-1[K^(λ)BA(λ2-A)-1]n×C(λ2-A)x-∑n=1∞‍λK^(λ)(λ2-A)-1[K^(λ)BA(λ2-A)-1]n×CK^(λ)BAx=λK^(λ)Cx+∑n=2∞‍λK^(λ)(λ2-A)-1×[K^(λ)BA(λ2-A)-1]nC(λ2-A)x-∑n=1∞‍λK^(λ)(λ2-A)-1[K^(λ)BA(λ2-A)-1]n×CK^(λ)BAx=λK^(λ)Cx.
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 Theorems8 and 9, we immediately obtain the following results for K-convoluted C-semigroups.Theorem 10.
Suppose thatA 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(31)L(λ)∶=sup{∫0a‍e-λs∥δ(λ)C-1PATK(s)x∥ds,x∈D(A),∥x∥≤1∫0a‍}<∞,
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 limλ→∞L(λ)eλt=0 for all t≥0.(ii)
If for anyΦ∈C([0,∞),L(X)),
(32)∥∫0t‍Φ(s)C-1BATK(t-s)xds∥≤M~∫0t‍eω(t-s)∥Φ(s)∥ds·∥x∥,x∈D(A),t≥0,
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 anyΨ∈C([0,∞),L(X)) with ∥Ψ(t)∥=O(eωt), sufficiently large λ and sufficiently small ε, we have
(33)∥∫0t‍δ(λ)Ψ(s)C-1PATK(t-s)xds∥≤M*e(λ+ω)t∫0t‍e-λs∥δ(λ)C-1PATK(s)x∥ds≤M*e(λ+ω)tL(λ)<ε<1,t≥0,x∈D(A),≤M*e(λ+ω)tL(λ)<ε<1,t≥0,∥x∥≤1,
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 Theorem9.In Theorems8–10, take K(t)=tn-1/Γ(n), we have the following result for n-times integrated C-cosine function (resp., n-times integrated C-semigroup).Theorem 11.
SupposeA 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(An+1) is dense in X. (i)
One sets(34)L(λ)∶=sup{∫0a‍e-λt∥∫0t‍δ(λ)Ψ(s)C-1BAL(λ)∶=sup{∫0a‍e-λt×(dndtnCn(t-s)x)ds∥dt,x∈D(An+1),∥x∥≤1∫0t‍}<∞,
for any Ψ∈C([0,∞),L(X)) with ∥Ψ(t)∥=O(eωt),
(35)((dndsnTn(s)x)resp.L(λ)∶=sup{∫0a‍e-λs∥δ(λ)C-1BA(dndsnTn(s)x)∥ds,x∈D(An+1),∥x∥≤1(dndsnTn(s)x)}<∞)
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 limλ→∞L(λ)eλt=0 for all t≥0.(ii)
If for anyΦ∈C([0,∞),L(X)),
(36)∥∫0t‍Φ(s)C-1BA(dndtnSn(t-s)x)ds∥≤M~∫0t‍eω(t-s)∥Φ(s)∥ds·∥x∥,x∈D(An+1),t≥0,
where Sn(t)=∫0t‍Cn(s)ds and M~ is a constant,
(37)(resp.∥∫0t‍Φ(s)C-1BA(dndtnTn(t-s)x)ds∥≤M~∫0t‍eω(t-s)∥Φ(s)∥ds·∥x∥,x∈D(An+1),t≥0∫0t‍)
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.Whenn=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.
LetB∈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)),
(38)∥∫0t‍Φ(s)C-1BAS(t-s)xds∥≤M∫0t‍eω(t-s)∥Φ(s)∥ds·∥x∥,x∈D(A),t≥0,
where S(t)=∫0t‍C(s)ds.
(39)(resp.∥∫0t‍Φ(s)C-1BAT(t-s)xds∥≤M∫0t‍eω(t-s)∥Φ(s)∥ds·∥x∥,x∈D(A),t≥0∫0t‍)
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 Definition7 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.
LetB∈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
(40)L(λ):=sup{∫0a‍e-λt∥∫0t‍C-1BAC(t-s)xds∥dt,x∈D(A),∥x∥≤1∥∫0t‍C-1BAC(t-s)xds∥}<∞,(41)(resp.L(λ):=sup{∫0a‍e-λs∥C-1BAT(s)x∥ds,x∈D(A),∥x∥≤1∫0a‍}<∞∫0a‍)
for some a∈(0,+∞] and λ>ω, letting L(∞)=limλ→∞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 forC-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 eλτsups∈[0,τ]∥Ψ(s)∥≤μ2/μ1, then for all x∈D(A), ∥x∥≤1, we have
(42)∥∫0t‍εΨ(s)C-1BAS(t-s)xds∥=∥∫0t‍∫0s‍εΨ(σ)C-1BAC(s-σ)xdσds∥≤eλt∫0t‍e-λs∥∫0s‍εC-1BAC(s-σ)xdσ∥ds·sups∈[0,τ]∥Ψ(s)∥≤eλτ|ε|L(λ)·sups∈[0,τ]∥Ψ(s)∥<μ2<1,t∈[0,τ],
where S(t)=∫0t‍C(s)ds.
Let𝒱:[0,τ]→L(X) be any strongly continuous function; we define
(43)(ℳ𝒱)(t)x=∫0t‍ε𝒱(s)C-1BAS(t-s)xds,x∈D(A),t∈[0,τ].
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
(44)V(t)x∶=(I-ℳ)-1[C(·)x](t),x∈X.
Then, V(t)C=CV(t), and there exists a constant M¯ such that ∥V(t)∥≤M¯eωt:
(45)V(t)x=C(t)x+∫0t‍εV(s)C-1BAS(t-s)xds,t∈[0,τ].
For t∈((n-1)τ,nτ], n=2,3,…, we define inductively
(46)V(t)∶=-V(2(n-1)τ-t)+2C-1V(t-(n-1)τ)V((n-1)τ).
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
(47)V(t)x=C(t)x+∫0t‍εV(s)C-1BAS(t-s)xds,x∈X,t∈[0,nτ].
Indeed for n=1, this is true. Assume that (47) holds for n. Then for x∈X, σ∈[0,τ] we get
(48)2V(σ)V(nτ)x=2C(σ)C(nτ)x+2∫0nτ‍εV(σ)V(s)C-1BAS(nτ-s)xds+2∫0σ‍εV(s)C-1BAS(σ-s)C(nτ)xds=C[C(σ+nτ)x+C(nτ-σ)x]+2∫0nτ‍εV(σ)V(s)C-1BAS(nτ-s)xds+2∫0σ‍εV(s)C-1BAS(σ-s)C(nτ)xds=2ℳ[V(σ)V(·)](nτ)x+C[C(σ+nτ)x+C(nτ-σ)x]+C∫0σ‍εV(s)C-1BA×[S(nτ+σ-s)x-S(nτ-σ+s)x]ds.
Then for x∈X, σ∈[0,τ],
(49)2V(σ)V(nτ)x=C(I-ℳ)-1{C(σ+nτ)x+C(nτ-σ)x=C(I-ℳ)-1+∫0σ‍εV(s)C-1BA[S(nτ+σ-s)x=C(I-ℳ)-1-S(nτ-σ+s)x]ds}.
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
(50)2V(σ)V(nτ)x=C[C(σ+nτ)x+C(nτ-σ)x]+2∫0nτ‍εV(σ)V(s)C-1BAS(nτ-s)xds+C∫0σ‍εV(s)C-1BA[S(nτ+σ-s)x-S(nτ-σ+s)x]ds=C[C(σ+nτ)x+C(nτ-σ)x]+C∫0nτ‍ε[V(σ+s)+V(|s-σ|)]C-1BAS(nτ-s)xds+C∫0σ‍εV(s)C-1BA[S(nτ+σ-s)x-S(nτ-σ+s)x]ds=C[C(σ+nτ)x+C(nτ-σ)x]+C∫σσ+nτ‍εV(s)C-1BAS(nτ+σ-s)xds+C∫0σ‍εV(s)C-1BAS(nτ-σ+s)xds+C∫0nτ-σ‍εV(s)C-1BAS(nτ-σ-s)xds+C∫0σ‍εV(s)C-1BA[S(nτ+σ-s)x-S(nτ-σ+s)x]ds=C[C(σ+nτ)x+C(nτ-σ)x]+C∫0σ+nτ‍εV(s)C-1BAS(nτ+σ-s)xds+C∫0nτ-σ‍εV(s)C-1BAS(nτ-σ-s)xds=CV(σ+nτ)x+CV(nτ-σ)x.
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 ofC-semigroups in a similar way.


## 3. Examples

Example 14.
Let(51)X∶={f∈C∞[0,1]:∥f∥∶=supp≥0∥f(p)∥∞p!2<∞},A∶=-ddx,D(A)∶={f∈X:f′∈X,f(0)=0}.
It is well known that there exist positive real numbersm and M such that
(52){λ∈ℂ:Reλ≥0}⊂ρ(A),∥R(λ,A)∥≤Mem|λ|,Reλ≥0.
Moreover,A generates an exponentially bounded Ka-convoluted semigroup {TK(t)}t≥0 for some a>2m, where K(t)=(a/(2πt3))e-a2/(4t),t≥0, then K^a(λ)=e-aλ,Reλ>0 [14, 23]. We set
(53)Bf(x)∶=∑n=1j‍∫0x‍(x-s)n-1(n-1)!f(s)ds,j∈ℕ,x∈[0,1],f∈X.
Obviously, B∈L(X) and BA⊂AB. Then from Theorem 10(ii) (C=I), (I+e-aλB)A subgenerates an exponentially bounded Ka-convoluted semigroup {T~K(t)}t≥0 on X.Example 15.
LetX∶=C0(ℝ)⊕C0(ℝ)⊕C0(ℝ),
(54)A(f,g,h)(·)∶=(f′,g′,(χ[0,∞)-χ(-∞,0])h),(f,g,h)∈D(A)={(f,g,h)∈X:f′∈C0(ℝ),g′∈C0(ℝ),h(0)=0}
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].
Forf,g,h∈C0(ℝ) and t∈ℝ, we set
(55)B(f,g,h)(t)=(e-t∫0t‍f(s)ds,e-2t∫0t‍g(s)ds,te-3tsint·h(t)).
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.
LetX1=L2(ℝ3), X2=Lp(ℝ3) (1≤p≤∞),
(56)A1=Δ,D(A1)=H2(ℝ3),A2=aΔ+∑i=13‍ci∂∂x1+c4(a>0,ci∈ℝ,i=1,2,3,4),D(A2)=W2,p(ℝ3).
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.
Setr1(·)∈H2(ℝ3), r2(·)∈W2,p(ℝ3), q1(·)∈Cc2(ℝ3), q2(·)∈Cc(ℝ3). Define bounded linear operators B1:X2→X1, B2:X1→X2 as follows:
(57)(B1ϕ)(ξ)=r1(ξ)∫ℝ3‍q1(σ)ϕ(σ)dσ,(B2ϕ)(ξ)=r2(ξ)∫ℝ3‍q2(σ)ϕ(σ)dσ.
Let X=X1×X2,
(58)A=(A100A2),D(A)∶=D(A1)×D(A2),B=(0B1B20),D(B)∶=X.
Taking λ0∈ρ(A) and putting C=(λ0-A)-1, then A generates an exponentially bounded C-cosine function C(·) on X, where
(59)C(t)=(C1(t)(λ0-A1)-100C2(t)C2-1(λ0-A2)-1).
We denote S1(t)∶=∫0t‍C1(s)ds, S2(t)∶=∫0t‍C2(s)ds, S(t)∶=∫0t‍C(s)ds, then
(60)S(t)=(S1(t)(λ0-A1)-100S2(t)C2-1(λ0-A2)-1),
and for any x=(x1x2)∈D(A), 0≤s≤t<∞,
(61)C-1BAS(t-s)x=((λ0-A1)B1A2S2(t-s)C2-1(λ0-A2)-1x2(λ0-A2)B2A1S1(t-s)(λ0-A1)-1x1).
It follows from R(B1)⊂D(A1) and R(B2)⊂D(A2) that there exist M, ω>0 such that
(62)e-λt∥∫0t‍C-1BAC(t-s)xds∥≤Mωe-λt(eωt-1)∥x∥,x∈D(A),
then
(63)L(λ)∶=sup{∫0a‍e-λt∥∫0t‍C-1BAC(t-s)xds∥dt,x∈D(A),∥x∥≤1∫0a‍}<∞,
and then (40) is satisfied.



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*Source: 426459-2013-03-20.xml*