# The Maximal Subspace for Generation of(a,k)-Regularized Families

Authors: Edgardo Alvarez-Pardo; Carlos Lizama
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

LetA be a linear operator in a Banach space X. We define a subspace of X and a norm such that the part of A in such subspace generates an (a, k)-regularized resolvent family. This space is maximal-unique in a suitable sense and nontrivial, under certain conditions on the kernels a and k.


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## Body

## 1. Introduction

Inspired by the well-known Hille-Yosida theorem, Kantorovitz defined in 1988 a linear subspace and a norm such that the restriction ofA to this subspace generates a strongly continuous semigroup of contractions (see [1]). This so-called Hille-Yosida space is maximal unique in a suitable sense. The same problem has been considered in the context of strongly continuous operator families of contractions by Cioranescu in [2]. In this case, the generation theorem of Sova and Fattorini was fundamental for her work. Later, Lizama in [3] used the generation theorem for resolvent families due to Da Prato and Iannelli (see [4]) as basis to generalize the results of Kantorovitz and Cioranescu to the context of resolvent families of bounded and linear operators. In this paper, also some applications to Volterra equations were given.It is remarkable that resolvent families do not includeα-times integrated semigroups, α-times integrated cosine functions, K-convoluted semigroups, K-convoluted cosine families, and integrated Volterra equations. For a historical account of these classes of operators, see ([5], page 234). Actually, these types of families are (a,k)-regularized. The concept of (a,k)-regularized resolvent families was introduced in [6]. The systematic treatment based on techniques of Laplace transforms was developed in several papers (see, e.g., [7–13]). The theory of (a,k)-regularized families has been developed in many directions and we refer to the recent monograph of Kostić [14] for further information. In this context, the problem to find maximal subspaces for generation of (a,k)-regularized families remained open in case k(t)≠1. In this paper, we are able to close this gap generalizing, in particular, [1, 2, 6].In this work, we will use the generation theorem for(a,k)-regularized resolvent family (see [6]) to show that there exists a linear subspace Za,k in X and a norm |·|a,k majorizing the given norm, such that (Za,k,|·|a,k) is a Banach space, and the part of A in Za,k generates a (a,k)-regularized resolvent family of contractions in Za,k. Moreover, the space (Za,k,|·|a,k) is a maximal-unique in a sense to be defined below. Concerning the non-triviality of Za,k, we prove that it contains the eigenvectors corresponding to non-positive eigenvalues of A. We close this paper with illustrative examples concerning the cases a(t)=tα-1 and k(t)=tβ in some region α>0 and β>0.This paper is organized as follows. In the first section, we recall the definition as well as basic results about(a,k)-regularized families.In Section2, we show the existence of the maximal subspace such that the part of A in this subspace generates an (a,k)-regularized family. We prove that such subspace is a Banach space with the norm defined below. The maximality is also proved and we show how this is used to obtain a relation with the Hille-Yosida space corresponding to the semigroup case.In Section3, we present some applications of the theory developed in the preceding section. Here we show the particular cases of generation corresponding to resolvent families, cosine operator families, semigroups, α-times semigroups and α-times cosine operator families. After that, we give concrete conditions on a given operator A to obtain the non-triviality of the maximal spaces and hence the well posedness on these spaces, for the abstract Cauchy problems of first and second order.


## 2. Preliminaries

In this section, we recall some useful results in the literature about(a,k)-regularized resolvent families. Let us fix some notations. From now on, we take X to be a Banach space with the norm ∥·∥. We denote by ℬ(X) the Banach algebra of all bounded linear operators on X endowed with the operator norm, which again is denoted by ∥·∥. The identity operator on X is denoted by I∈ℬ(X), and ℝ+ denotes the interval [0,∞). For a closed operator A, we denote by σ(A), σp(A), ρ(A) the spectrum, the point spectrum, and resolvent of A, respectively.Definition 2.1.
Letk∈C(ℝ+), k≠0, and a∈Lloc1(ℝ+) be given. Assume that A is a linear operator with domain D(A). A strongly continuous family {R(t)}t≥0⊂ℬ(X) is called (a,k)-regularized family on X having A as a generator if the following hold:(a)
R(0)=k(0)I;(b)
R(t)x∈D(A) and R(t)Ax=AR(t)x for all x∈D(A) and t≥0;(c)
R(t)x=k(t)x+∫0t‍a(t-s)AR(s)xds for all t≥0 and x∈D(A).In the case wherek(t)≡1, this definition corresponds to the resolvent family for the Volterra equation of convolution type in [6]. Moreover, if, in addition, a(t)≡1 then this family is a C0-semigroup on X or if a(t)≡t is a cosine family on X.We note that the study of(a,k)-regularized families is associated to a wide class of linear evolution equation, including, for example, fractional abstract differential equations (see [15]).Definition 2.2.
We say that(R(t))t≥0 is of type (M,ω) if there exist constants M≥0 and ω∈ℝ such that
(2.1)∥R(t)∥≤Meωt
for all t≥0.We will require the following theorem on generation of(a,k)-regularized families (see [6]).Theorem 2.3.
LetA be a closed and densely defined operator on a Banach space X. Then {R(t)}t≥0 is an (a,k)-regularized family of type (M,ω) if and only if the following hold:  (a)
a^(λ)≠0 and 1/a^(λ)∈ρ(A) for all λ>ω;(b)
H(λ):=k^(λ)(I-a^(λ)A)-1 satisfies the estimates
(2.2)∥H(n)(λ)∥≤Mn!(λ-ω)n+1,λ>ω,n∈ℕ0.In the case wherek(t)≡1, Theorem 2.3 is well known. In fact, if a(t)≡1 then it is just the Hille-Yosida theorem; if a(t)≡t, then it is the generation theorem due essentially to Da Prato and Iannelli in [4]. In the case where k(t)=tn/n! and a(t)≡1, it is the generation theorem for n-times integrated semigroups [16]; if k(t)=tn/n! and a(t) is arbitrary, it corresponds to the generation theorem for integrated solutions of Volterra equations due to Arendt and Kellerman [17].In order to give applications to our results we recall the following concepts of fractional calculus. The Mittag-Leffler function (see, e.g., [18–20]) is defined as follows:
(2.3)Eα,β(z):=∑n=0∞‍znΓ(αn+β)=12πi∫Haeμμα-βμα-zdμ,α,β>0,z∈ℂ,
where Ha is a Hankel path, that is, a contour which starts and ends at -∞ and encircles the disc |μ|≤|z|1/α counter clockwise. The function Eα,β is an entire function which provides a generalization of several usual functions. For a recent review, we refer to the monograph [21].An interesting property related with the Laplace transform of the Mittag-Leffler function is the following (cf. [18], (A.27) page 267):
(2.4)ℒ(tβ-1Eα,β(-ραtα))(λ)=λα-βλα+ρα,Reλ>|ρ|1/α;α>0,β>0,ρ∈ℝ.Remark 2.4 (see [22]).
If0<α<2 and β>0, then
(2.5)Eα,β(z)=1αz(1-β)/αexp(z1/α)+εα,β(z),|arg(z)|≤12απ;Eα,β(z)=εα,β(z),|arg(-z)|<(1-12α)π,
where
(2.6)εα,β(z):=-∑n=1N-1‍z-nΓ(β-αn)+𝒪(|z|-N),
as z→∞ with 2≤N∈ℕ. This implies that for each π(α/2)<ω<min{π,πα}, there is a constant C:=C(ω)>0 such that
(2.7)|Eα,β(z)|≤C1+|z|,ω≤|arg(z)|≤π.


## 3. The Maximal Subspace

In this section,X is a Banach space with norm ∥·∥. Let A be a linear operator and a∈Lloc1(ℝ+) and k∈C(ℝ+), k≠0. Assume that
(3.1)a^(λ)=∫0∞‍e-λta(t)dt≠0,∀λ>0,
and 1/a^(λ)∈ρ(A) for all λ>0. Observe that we are implicitly assuming that the inequality (2.2) holds with ω=0. Let Ha,k(λ):=(k^(λ)/a^(λ))((1/a^(λ))-A)-1 for λ>0, and define
(3.2)Ya,k:={x∈X:|x|a,k:=supλj>0,nj,l∈ℕ0∥∏j=1l‍(1nj!)λjnj+1Ha,k(nj)(λj)x∥<+∞},
where for l=0 the product is defined as x. It is clear that |·|a,k is a norm on Ya,k.Proposition 3.1.
(
Y
a
,
k
,
|
·
|
a
,
k
) is a Banach space.Proof.
Let{xi}⊂Ya,k be a Cauchy sequence. We observe that ∥x∥≤|x|a,k if x∈X. Then {xi} is a Cauchy sequence on X. Let x:=limi→∞xi.
First, we show thatx∈Ya,k. Indeed, let λj>0, nj,l∈ℕ0 be fixed. Then
(3.3)∥∏j=1l‍(1nj!)λjnj+1Ha,k(nj)(λj)x∥=limi→∞∥∏j=1l‍(1nj!)λjnj+1Ha,k(nj)(λj)xi∥≤limsupi→∞|xi|a,k<+∞.
Second, we prove that{xi} converges to x∈Ya,k. Let ε>0. There exists N:=N(ε)∈ℕ such that |xi-xm|a,k<ε/2 for i,m>N. Since x=limi→∞xi in the norm of X, we also have  ∥xm-x∥<ε/2M where M>0 is the constant given in (2.2). Hence for every λj>0, nj,l∈ℕ0, we have by inequality (2.2)
(3.4)∥∏j=1l‍(1nj!)λjnj+1Ha,k(nj)(λj)(xi-x)∥≤∥∏j=1l‍(1nj!)λjnj+1Ha,k(nj)(λj)(xi-xm)∥+∥∏j=1l‍(1nj!)λjnj+1Ha,k(nj)(λj)(xm-x)∥≤∥∏j=1l‍(1nj!)λjnj+1Ha,k(nj)(λj)(xi-xm)∥+∏j=1l‍(1nj!)λjnj+1∥Ha,k(nj)(λj)(xm-x)∥≤∥∏j=1l‍(1nj!)λjnj+1Ha,k(nj)(λj)(xi-xm)∥+M∥xm-x∥,i,m>N.
Taking supremum over allλj>0, nj,l∈ℕ0, we obtain
(3.5)|xi-x|a,k≤|xi-xm|a,k+M∥xm-x∥<ε2+ε2=ε,i>N.
Therefore the sequence{xi} converges to x∈Ya,k in the norm |·|a,k. Consequently (Ya,k,|·|a,k) is a Banach space.Definition 3.2.
LetAYa,k:D(AYa,k)⊆Ya,k→Ya,k be defined by AYa,kx:=Ax, where
(3.6)D(AYa,k):={x∈D(A):x,Ax∈Ya,k}.
This operator is sometimes called the part ofA in Ya,k.
We denote(3.7)Za,k:=D(AYa,k)¯,
where the closure is taken in the norm |·|a,k.Lemma 3.3.
With the preceding definitions and hypothesis, we have(a)
AYa,k is a closed linear operator on Ya,k,(b)
(1/a^(λ))-AYa,k is invertible on Ya,k for each λ>0,(c)
∥Ha,y(n)(λ)∥B(Ya,k)≤n!/λn+1 for each λ>0 and n∈ℕ0, where Ha,y(λ):=(k^(λ)/a^(λ))((1/a^(λ))-AYa,k)-1.Proof.
Letλ>0 be fixed. Since 1/a^(λ)∈ρ(A), then
(3.8)a^(λ)k^(λ)(1a^(λ)-A)Ha,k(λ)x=x,foreachx∈XHa,k(λ)a^(λ)k^(λ)(1a^(λ)-A)x=x,foreachx∈D(A).
Note thatAYa,k is closed because Ya,k is a Banach space and A is closed. So the first part is done. Now, let y∈Ya,k be fixed. Since Ha,k(λ):X→D(A), then Ha,k(λ)y∈D(A). Moreover,
(3.9)|Ha,k(λ)y|a,k∶=sup∥∏j=1l‍(1nj!)λjnj+1Ha,knj(λj)Ha,k(λ)y∥≤∥Ha,k(λ)∥sup∥∏j=1l‍(1nj!)λjnj+1Ha,knj(λj)y∥=∥Ha,k(λ)∥|y|a,k<∞,
therefore Ha,k(λ)y∈Ya,k. On the other hand, from identities above, we have
(3.10)AHa,k(λ)y=1a^(λ)Ha,k(λ)y-k^(λ)a^(λ)y;
thus AHa,k(λ)y∈Ya,k. Hence Ha,k(λ)y∈D(AYa,k), and we conclude that
(3.11)a^(λ)k^(λ)(1a^(λ)-AYa,k)Ha,k(λ)y=y,for  eachy∈Ya,k.
Now if y∈D(AYa,k) then, in particular, y∈D(A), and therefore,
(3.12)Ha,k(λ)a^(λ)k^(λ)(1a^(λ)-AYa,k)y=y.
This proves the second assertion. In particular, ρ(AYa,k)≠∅ and hence
(3.13)Ha,y(λ)=Ha,k(λ)∣Ya,k,∀λ>0.
Finally, lety∈Ya,k, λ>0, n∈ℕ0 be fixed. We have(3.14)|1n!λn+1Ha,y(λ)y|a,k∶=sup∥∏j=1l‍(1nj!)λjnj+1Ha,knj(λj)(1n!λn+1Ha,y(λ)y)∥≤sup∥∏j=1l+1‍(1nj!)λjnj+1Ha,y(nj)(λj)y∥≤|y|a,k,
where λj>0 are arbitrary for 1≤j≤l, λl+1:=λ and nl+1:=n. This proves the third part of the lemma.Lemma 3.4.
LetAa,k:D(Aa,k)⊆Za,k→Za,k be defined by Aa,kx:=AYa,kx, where
(3.15)D(Aa,k):={x∈D(AYa,k):x,AYa,kx∈Za,k}.
Then Aa,k is a closed operator such that D(Aa,k)¯=Za,k and  (a)
(1/a^(λ))-Aa,k is invertible on Za,k for each λ>0,(b)
∥Ha,k(n)(λ)∥B(Za,k)≤n!/λn+1 for each λ>0 and n∈ℕ0 where Ha,k(λ):=(k^(λ)/a^(λ))((1/a^(λ))-Aa,k)-1.Proof.
We observe thatHa,k(λ)=Ha,y(λ)∣Za,k. Then the result is a direct consequence of ([5], Lemma  3.10.2).As a consequence, we obtain the main result of this section on the existence of(a,k)-regularized families.Theorem 3.5.
LetA be a linear operator defined in a Banach space X and a∈Lloc1(ℝ+) and k∈C(ℝ+), k≠0. Assume that 1/a^(λ)∈ρ(A) for all λ>0. Then there exist a linear subspace Za,k and a norm |·|a,k such that (Za,k,|·|a,k) is a Banach space and Aa,k generates an (a,k)-regularized family of contractions in Za,k.Proof.
According to our hypothesis, we can apply the generation theorem for(a,k)-regularized family (see [6, Theorem  3.4]) and the result of Lemma 3.4.Concerning the non-triviality ofZa,k, we will prove that it contains the eigenvectors corresponding to nonpositive eigenvalues of A.Letμ∈ℂ be fixed. Let r(t,μ) be the unique solution to the scalar equation
(3.16)r(t,μ)=k(t)+μ∫0t‍a(t-s)r(s,μ)ds.Thus, provided the kernelsa(t) and k(t) are Laplace transformable, we have
(3.17)Ha,k(λ)=k^(λ)a^(λ)(1a^(λ)-μ)-1=∫0∞‍e-λtr(t,μ)dt.We define(3.18)Ca,k:={μ∈σ(A):themapt→r(t,μ),t≥0isbounded}.Proposition 3.6.
Letx be an eigenvector of A corresponding to the eigenvalue α∈Ca,k. Then x∈Za,k.Proof.
Letx be an eigenvector of A corresponding to the eigenvalue α such that the map t→r(t,α) is bounded. Let λ>0 and n∈ℕ be fixed. Then
(3.19)∥1n!λn+1Ha,k(n)(λ)x∥=∥1n!λn+1dndλn[k^(λ)a^(λ)(1a^(λ)-α)-1x]∥=∥1n!λn+1dndλn(∫0∞‍e-λtr(t,α)dt)x∥≤1n!λn+1∫0∞‍tne-λt|r(t,α)|dt∥x∥≤supt≥0|r(t,α)|∥x∥.
This implies that |x|a,k≤supt≥0|r(t,α)|∥x∥ and, consequently, x∈Za,k.The following result shows us that the spacesZa,k are maximal-unique in a certain sense.Theorem 3.7.
Under the same hypothesis of Theorem3.5, if (Wa,k,∥·∥a,k) is a Banach space such that Wa,k⊂X, ∥·∥≤∥·∥a,k and the operator Ba,k=A|D(Ba,k) with D(Ba,k):={x∈D(A):x,Ax∈Wa,k} generates an (a,k)-regularized family of contractions in Wa,k, then Wa,k⊂Za,k, |·|a,k≤∥·∥a,k and Ba,k⊂Aa,k.Proof.
Suppose that(Wa,k,∥·∥a,k), Ba,k are as in the statement of theorem. Since Ha,k(λ) is the Laplace transform of the (a,k)-regularized family Sa,k(t)  (t≥0), we have that for x∈Wa,k, λ>0, and n∈ℕ0(3.20)∥1n!λn+1Ha,k(n)(λ)x∥=∥1n!λn+1(ddλ)n∫0∞‍e-λtSa,k(t)xdt∥≤1n!λn+1∫0∞‍e-λt∥Sa,k(t)x∥dt≤1n!λn+1∫0∞‍e-λt∥Sa,k(t)x∥a,kdt≤1n!λn+1∫0∞‍e-λt∥x∥a,kdt=∥x∥a,k.
We conclude that for x∈Wa,k, |x|a,k≤∥x∥a,k, that is, Wa,k⊂Ya,k. It follows that
(3.21)D(Ba,k):={x∈D(A):x,Ax∈Wa,k}⊂{x∈D(A):x,Ax∈Ya,k}:=D(AYa,k).
Hence
(3.22)Wa,k=D(Ba,k)¯∥·∥a,k⊂D(AYa,k)¯∥·∥a,k⊂D(Ba,k)¯|·|a,k=Za,k.
Finally, this implies that D(Ba,k)⊂D(Aa,k) and Ba,k⊂Aa,k.The next result treats the “maximal property”. In order to obtain the analogous result to the resolvent families case, we need more information about the functionk(t).Theorem 3.8.
LetA be a linear operator defined in a Banach space X and k∈C1(ℝ+), k≠0 with absolutely convergent Laplace transform for λ>0. Assume that 1/a^(λ)∈ρ(A) for all λ>0. Suppose that 1≤k(0). Then Aa,k generates a strongly continuous semigroup of contractions on Za,k.Proof.
By Hille-Yosida theorem it is sufficient to have(α,∞)⊂ρ(Aa,k) and ∥λ(λ-Aa,k)∥B(Za,k)≤1 for all λ>α, for some real α.
In order to show this, we taken=0 in the second part of Lemma 3.4 and obtain
(3.23)∥k^(λ)a^(λ)(1a^(λ)-Aa,k)-1∥B(Za,k)≤1λ.
Let μ:=1/a^(λ). Then (3.23) gives
(3.24)∥μ(μ-Aa,k)-1∥≤1λk^(λ),(λ>0).
Since a^(λ)→0 as λ→∞, and limλ→∞λk^(λ)=k(0) by the initial value theorem, we obtain
(3.25)∥μ(μ-Aa,k)-1∥B(Za,k)≤1k(0),
for all μ sufficiently large. Since k(0)≥1, we get ∥μ(μ-Aa,k)∥B(Za,k)≤1 for μ sufficiently large, which concludes the proof.Remark 3.9.
From the maximal uniqueness ofZ1,1 and under the same hypothesis of preceding theorem we obtain that Za,k⊂Z1,1, |·|1,1≤|·|a,k and Aa,k⊂A1,1. Note that, in particular, Za,1⊂Z1,1 which include [3, Remark  2.10].


## 4.   Applications

Takingk(t)≡1 we obtain the main result in [3, Theorem  2.5].Corollary 4.1 (see [6], Theorem  2.5).
LetA be a linear operator defined in a Banach space X and a∈Lloc1(ℝ+), a≠0 with absolutely convergent Laplace transform for λ>0. Assume that 1/a^(λ)∈ρ(A) for all λ>0. Then there exist a linear subspace Za,1 and a norm |·|a,1 such that (Za,1,|·|a,1) is a Banach space and the equation
(4.1)u(t)=f(t)+∫0t‍a(s-t)Ak,1u(s)ds
admits resolvent family of contractions on Za,1, where Ak,1 is defined in Lemma 3.4.Takingk(t)≡1 or k(t)≡t, we obtain from the preceding corollary the following.Corollary 4.2 (see [1]).
LetA be a linear operator on X such that (0,∞)⊂ρ(A). Then there exist a linear subspace Z1⊂X and a norm |·|1 such that (Z1,|·|1) is a Banach space and the restriction A1 of A to Z1 is the infinitesimal generator of a C0-semigroup of contractions on Z1.Corollary 4.3 (see [2]).
LetA be a linear operator on X such that (0,∞)⊂ρ(A). Let At be the operator in Zt defined as above. Then At is the infinitesimal generator of a strongly continuous cosine family of contractions on Zt.Remark 4.4.
Applying Theorem3.5 with k(t)≡tβ/(Γ(β+1)) we obtain corresponding results for α-times integrated semigroups and β-times integrated cosine families taking a(t)≡1 and a(t)≡t, respectively.Suppose thatσp(A)≠∅ and μ∈σp(A). In the following examples we search conditions under which the function t→r(t,μ) is bounded. This ensure the non-triviality of the subspace Za,k.Example 4.5.
Let us considerk(t):=tβ/(Γ(β+1)) and a(t):=tα-1/Γ(α). Let A be a closed linear and densely defined operator on a Banach space X such that  (a)
(0,∞)⊂ρ(A),(b)
σp(A)∩ℝ-≠∅,(c)
0<α<2 and α≥β.
Letμ∈σp(A)∩ℝ-. Then applying Laplace transform, we have that
(4.2)r^(λ,μ)=λα-(β+1)λα-μ.
It follows from (2.4) that
(4.3)r(t,μ)=tβEα,β+1(μtα),(μ<0).
Takez=μtα with t>0, then arg(z)=π. By Remark 2.4, for each π(α/2)<ω<min{π,πα}, there is a constant C:=C(ω)>0 such that
(4.4)|Eα,β+1(μtα)|≤C1+|μ|tα,t>0.
Therefore
(4.5)|r(t,μ)|=|tβEα,β+1(μtα)|≤Ctβ1+|μ|tα,t>0.
From here and part (c), it follows that r(t,μ) is a bounded function for t>0. In particular, Za,k≡Zα,β≠{0} by Proposition 3.6. Since (0,∞)⊂ρ(A), we have that 1/a^(λ)=λ2∈ρ(A) for λ>0. It follows from Theorem 3.5 that A is the infinitesimal generator of a strongly continuous Laplace transformable (α,β)-resolvent family of contractions on Zα,β. Note in particular that if α=1 and β>0, then A is the generator of β-times integrated semigroup (Sβ(t))t≥0 on each subspace Z1,β for 0<β≤1. It means that the initial value problem
(4.6)u′(t)=Au(t),t≥0,u(0)=u0
is well posed in the sense that there exists a strongly continuous family of linear operators (Sβ(t))t≥0 on a subspace Z1,β of X such that for all initial values u0∈Z1,β there exists a unique classical solution of (4.6).Example 4.6.
Let us considerk(t):=tβ/(Γ(β+1)) and a(t):=tα-1/Γ(α) when α=2, that is, a(t):=t. Let A be a closed linear and densely defined operator on a Banach space X such that  (a)
(0,∞)⊂ρ(A),(b)
σp(A)∩ℝ-≠∅,(c)
0<β≤1.
Letμ∈σp(A)∩ℝ-. Then
(4.7)r^(λ,μ)=λ1-βλ2-μ=λλ2-μ1λβ.
Letω2:=-μ. First, we consider β=1. In this case, if we take inverse Laplace transform, we obtain
(4.8)r(t,μ)=1ωsin(ωt).
Obviously, this function is bounded fort≥0 and μ=-ω2<0.
Now, we consider0<β<1. In this case
(4.9)r(t,μ)=∫0t‍sβ-1Γ(β)cos(ω(t-s))ds=1Γ(β){cos(ωt)∫0t‍sβ-1cos(ωs)ds+sin(ωt)∫0t‍sβ-1sin(ωs)ds}.
Remember that the incomplete Gamma function is defined by(4.10)Γ(a,z):=∫z∞‍e-tta-1dt.
About the asymptotic behavior, we know that (see [23, formula (8.357)])
(4.11)Γ(a,z):=za-1e-z[1+𝒪(|z|-1)],
as |z|→∞ and -3π/2<arg(z)<3π/2.
It can be verified that the following formulas holds (see [23, formula (2.632)]):
(4.12)∫0t‍sβ-1sin(ωs)ds=-12ωβ[exp[iπ2(β-1)]Γ(β,-iωt)+exp[iπ2(1-β)]]Γ(β,iωt),(β<1),∫0t‍sβ-1cos(ωs)ds=-12ωβ[exp(iβπ2)Γ(β,-iωt)+exp(iβπ2)Γ(β,iωt)].
Note thatarg(iωt)=π/2 and arg(iωt)=-π/2, so we can apply (4.11). Then, for t sufficiently large, we obtain that
(4.13)|Γ(β,iωt)|=|Γ(β,-iωt)|≤ωβ-1tβ-1[1+𝒪(|ωt|-1)].
The boundedness follows from the fact thatβ<1. Therefore, for t sufficiently large
(4.14)|r(t,μ)|≤M[|Γ(β,iωt)|+|Γ(β,-iωt)|]≤K,
where K depends on ω and β. Since (0,∞)⊂ρ(A), we have that λ2∈ρ(A) for λ>0. It follows from Theorem 3.5 that A is the infinitesimal generator of β-times integrated cosine function (Cβ(t))t≥0 of contractions on Z2,β for 0<β≤1. As in the above example, it now means that the initial value problem
(4.15)u′′(t)=Au(t),t≥0,u(0)=u0,u'(0)=0
is well posed in the sense that there exists a strongly continuous family of linear operators (Cβ(t))t≥0 on a nontrivial subspace Z2,β of X such that for all initial values u0∈Z2,β there exists a unique classical solution of (4.15).



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*Source: 683021-2012-11-06.xml*