Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 394584, 9 pages
doi:10.1155/2011/394584
Research Article
Some Results on n-Times Integrated C-Regularized
Semigroups
Fang Li, Huiwen Wang, and Zihai Qu
School of Mathematics, Yunnan Normal University, Kunming 650092, China
Correspondence should be addressed to Huiwen Wang, hwwang114@gmail.com
Received 21 October 2010; Accepted 13 December 2010
Academic Editor: Toka Diagana
Copyright q 2011 Fang Li et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We present a generation theorem of n-times integrated C-regularized semigroups and clarify the
relation between differentiable n 1-times integrated C-regularized semigroups and singular
n-times integrated C-regularized semigroups.
1. Introduction and Preliminaries
In 1987, Arendt 1 studied the n-times integrated semigroups, which are more general than
C0 semigroups there exist many operators that generate n-times integrated semigroups but
not C0 semigroups.
In recent years, the n-times integrated C-regularized semigroups have received much
attention because they can be used to deal with ill-posed abstract Cauchy problems and
characterize the “weak” well-posedness of many important differential equations cf., e.g.,
2–18.
Stimulated by the works in 2, 5–7, 9, 12–18, in this paper, we present a generation
theorem of the n-times integrated C-regularized semigroups for the case that the domain
of generator and the range of regularizing operator C are not necessarily dense, and prove
that the subgenerator of an exponentially bounded, differentiable n 1-times integrated Cregularized semigroup is also a subgenerator of a singular n-times integrated C-regularized
semigroup.
Throughout this paper, X is a Banach space; X ∗ denotes the dual space of X; LX, X
denotes the space of all linear and bounded operators from X to X, it will be abbreviated
to LX; LX∗ denotes the dual space of LX. By C1 0, ∞, X we denote the space of all
continuously differentiable X-valued functions on 0, ∞. C0, ∞, X is the space of all
continuous X-valued functions on 0, ∞.
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All operators are linear. For a closed linear operator A, we write DA, RA, ρA for
the domain, the range, the resolvent set of A in a Banach space X, respectively.
We denote by A0 A|DA the part of A in DA, that is,
DA0 : x ∈ DA; Ax ∈ DA ,
A0 x Ax, for x ∈ DA0 .
1.1
The C-resolvent set of A is defined as:
ρC A λ ≥ 0; λ − A is injective, RC ⊂ Rλ − A and λ − A−1 C ∈ LX .
1.2
We abbreviate n-times integrated C-regularized semigroup to n-times integrated
C-semigroup.
Definition 1.1. Let n be a nonnegative integer. Then A is the subgenerator of an exponentially
bounded n-times integrated C-semigroup {St}t≥0 if ω, ∞ ⊂ ρC A for some ω ≥ 0 and
there exists a strongly continuous family S· : 0, ∞ → LX with St ≤ Meω t for some
M > 0 such that
−1
λ − A Cx λ
n
∞
e−λt Stx dt
λ > ω, x ∈ X.
1.3
0
In this case, {St}t≥0 is called the exponentially bounded n-times integrated
: C−1 AC.
C-semigroup generated by A
If C I resp., n 0, then A is called a generator of an exponentially bounded n-times
integrated semigroup resp., C-semigroup.
We recall some properties of n-times integrated C-semigroup.
Lemma 1.2 see 10, Lemma 3.2. Assume that A is a subgenerator of an n-times integrated
C-semigroup {St}t≥0 . Then
i StC CSt t ≥ 0,
ii Stx ∈ DA, and AStx StAx t ≥ 0, x ∈ DA,
t
iii Stx tn /n!Cx A 0 Ssx ds t ≥ 0, x ∈ X.
In particular, S0 0.
Definition 1.3. Let ω ≥ 0. If ω, ∞ ⊂ ρC A and there exists {St}t≥0 ⊂ LX such that
i S0 0 and S· : 0, ∞ → LX is strongly continuous,
∞
ii for λ > ω, 0 e−λt Stdt < ∞,
∞
iii λ − A−1 Cx λn 0 e−λt Stx dt, λ > ω, x ∈ X,
then we say that {St}t≥0 is a singular n-times integrated C-semigroup with subgenerator A.
Remark 1.4. Clearly, an exponentially bounded n-times integrated C-semigroup is a singular
n-times integrated C-semigroup. But the converse is not true.
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2. The Main Results
Theorem 2.1. Let M > 0, ω ≥ 0 be constants, and let A be a closed operator satisfying ω, ∞ ⊂
ρC A. Assume that ϕt is the nonnegative measurable function on 0, ∞. A necessary and sufficient
condition for A is the subgenerator of an n 1-times integrated C-semigroup {St}t≥0 satisfying
∞
A1 lim supλ → ∞ λn2 0 e−λt Stdt ≤ M,
s
A2 St − Ss ≤ t ϕueωu du, 0 ≤ t ≤ s, is that for λ > ω,
i lim supλ → ∞ λλ − A−1 C ≤ M,
∞
m
ii λ − A−1 C/λn ≤ 0 e−λ−ωt tm ϕtdt, m 1, 2, . . ..
Proof. Sufficiency. Let ψt eωt ϕt. Set
fλ ∞
e
−λt
ψtdt ∞
0
e−λ−ωt ϕtdt,
2.1
λ > ω.
0
For x∗ ∈ X ∗ , we have
m ∞
−1
C
−
A
λ
∗ ∗
x
, x ≤ x · x e−λt tm ϕtdt
λn
0
m ≤ x · x∗ · fλ
, m 1, 2, . . . .
2.2
Using this fact together with Widder’s classical theorem, it is not difficult to see that
the existence of a measurable function h·, x, x∗ with |ht, x, x∗ | ≤ x∗ xψt, a.e., t ≥ 0
such that
λ − A−1 C
x, x∗
λn
∞
e−λt ht, x, x∗ dt,
2.3
λ > ω.
0
t
Let Ht, x, x∗ 0 hs, x, x∗ ds, t ≥ 0, x∗ ∈ X ∗ . In view of the convolution theorem for
Laplace transforms and from 2.3, we have
λ − A−1 C
x, x∗
λn
λ
∞
e−λt Ht, x, x∗ dt,
λ > ω, x∗ ∈ X ∗ .
2.4
0
Using the uniqueness of Laplace transforms and the linearity of h·, x, x∗ for each
x ∈ X , x ∈ X, we can see that for each t ≥ 0, Ht, x, x∗ is linear and
∗
∗
∗
∗
|Ht h, x, x − Ht, x, x | ≤
th
t
∗
∗
|hs, x, x |ds ≤ x · x th
ψsds.
t
2.5
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Hence for all t ≥ 0, there exists St ∈ LX∗∗ such that
Ht, x, x∗ Stx, x∗ ,
St h − St ≤
x ∈ X, x∗ ∈ X ∗ ,
th
ψsds,
t
λ − A−1 C
λ
λn
∞
2.6
t ≥ 0, h ≥ 0,
2.7
e−λt Stdt.
2.8
0
Denote by q : Lx∗∗ → Lx∗∗ /LX the quotient mapping. Since λ − A−1 C ∈ LX,
we deduce
0q
λ − A−1 C
λn
λ
∞
e−λt qStdt.
2.9
0
It follows from the uniqueness theorem for Laplace transforms that qSt 0, that is, St ∈
LX.
Combining 2.7 and 2.8 yields that St : 0, ∞ → LX is strongly continuous and
∞
e−λt Stdt ≤
0
∞
e−λt
0
t
ψsds dt 0
1
λ
∞
e−λt ψtdt < ∞.
2.10
0
Now, we conclude that {St}t≥0 is an n 1-times integrated C-semigroup satisfying
A2. Assertion A1 is immediate, by 2.8 and i.
Necessity. Let ψt eωt ϕt. Since {St}t≥0 is an n 1-times integrated C-semigroup on X,
we have
λ − A−1 C λn1
∞
e−λt Stdt
2.11
0
for λ > ω. Noting that St h − St ≤
th
t
St ≤
ψs ds h ≥ 0 and S0 0, we find
t
2.12
ψsds.
0
Then for any y∗ ∈ LX∗ and λ > ω, we obtain
λ − A−1 C ∗
,y
λn
∞
λ
e−λt Stdt, y∗
0
≤λ
∞
0
e−λt St · y∗ dt ≤ y∗ ∞
0
2.13
e−λt ψtdt.
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Therefore, there exists a measurable function ηt on 0, ∞ with |ηt| ≤ ψt a.e.
such that
λ − A−1 C ∞
e−λt ηtdt.
λn
0
2.14
Furthermore, by calculation, we have
m ∞
∞
λ − A−1 C
−λt m
≤
e t ψtdt e−λ−ωt tm ϕtdt,
λn
0
0
m 1, 2, . . . .
2.15
Assertion i is an immediate consequence of 2.11 and A1.
Remark 2.2. If n 0 and C I, then {St}t≥0 is an integrated semigroup in the sense of
Bobrowski 2.
Theorem 2.3. Let M > 0, ω ≥ 0 be constants, and let A be a closed operator satisfying ω, ∞ ⊂
ρA. Assume that A is a subgenerator of an n 1-times integrated C-semigroup {St}t≥0 and
satisfies (ii) of Theorem 2.1 and lim supλ → ∞ λλ − A−1 ≤ M. If A0 A|DA is a subgenerator of
an n-times integrated C-semigroup {S0 t}t≥0 on DA, then for μ ∈ ρA, x ∈ X,
Stx μ − A0
Stx lim μ
μ→∞
t
t
−1
S0 s μ − A x ds,
2.16
0
−1
S0 s μ − A x ds.
2.17
0
Proof. For μ ∈ ρA, x ∈ X, set {St}
t≥0 as follows:
Stx
μ
t
−1
−1
−1
tn S0 s μ − A x ds − S0 t μ − A x μ − A Cx.
n!
0
2.18
is strongly continuous on X.
Since S0 t is strongly continuous on DA, St
Fixing λ > ω, we have
λ
n1
∞
e
0
−λt n
Stx dt λ μ − λ
∞
−1
−1
e−λt S0 t μ − A x dt μ − A Cx
0
−1
−1
μ − λ λ − A−1 C μ − A x μ − A Cx
2.19
λ − A−1 Cx.
It follows from the uniqueness of Laplace transforms that Stx Stx,
x ∈ X. So we get
−1
2.16. By the hypothesis lim supλ → ∞ λλ − A ≤ M, we see
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t
−1
−1
−1
tn μ − A Cx
Stx lim μ S0 s μ − A x ds − S0 t μ − A x μ→∞
n!
0
2.20
t
−1
lim μ S0 s μ − A Cx ds,
μ→∞
0
and the proof is completed.
Now, we study the relation between differentiable n 1-times integrated Csemigroups and singular n-times integrated C-semigroups.
Theorem 2.4. Let ω ≥ 0, and let A be a closed operator satisfying ω, ∞ ⊂ ρC A. Assume that
ϕt is the nonnegative measurable function on 0, ∞. The following two assertions are equivalent:
1 A is the subgenerator of a singular n-times integrated C-semigroup {Ut}t≥0 satisfying
Ut ≤ ϕteωt .
2 A is the subgenerator of an exponentially bounded n 1-times integrated C-semigroup
{St}t≥0 satisfying
St − Ss ≤
s
0 ≤ t ≤ s,
ϕτeωτ dτ,
t
Stx ∈ C 0, ∞, X,
1
2.21
for x ∈ X.
Proof. 1⇒2: we set
t
Stx :
Usx ds,
2.22
t ≥ 0.
0
Since Utx is locally integrable on 0, ∞, Stx is well-defined for any x ∈ X. It is easy to
check that Stx belongs to C1 0, ∞, X.
For every λ > ω, since
t
t
−λs λs
Stx e e Usx ds ≤ eλt e−λs Usxds ≤ Meλt x,
0
0
2.23
we deduce that St is exponentially bounded.
Moreover, for λ > ω, we have
−1
λ − A Cx λ
n
∞
e
−λt
Utx dt λ
0
n1
∞
e−λt Stx dt,
0
s
s
≤
Uτdτ
ϕτeωτ dτ,
St − Ss t
Thus {St}t≥0 is the desired semigroup in 2.
t
2.24
0 ≤ t ≤ s.
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2⇒1: for any x ∈ X, we set
Utx :
d
Stx,
dt
U0x : 0,
for t > 0,
2.25
for t 0.
Then Utx ∈ C0, ∞, X and U0 0.
Noting that
St h − St ≤
th
2.26
ϕseωs ds,
t
we find
St h − St 1 th
≤
ϕseωs ds.
h
h
t
2.27
Since Stx is continuously differentiable for t > 0, we get
Ut ≤ ϕteωt
2.28
a.e..
Moreover, for λ > ω, we have
∞
e
−λt
Utdt ≤
0
−1
λ − A Cx λ
n1
∞
e−λ−ωt ϕtdt < ∞,
0
∞
e
−λt
Stx dt λ
n
2.29
∞
0
e
−λt
Utx dt.
0
Thus, {Ut}t≥0 is a singular n-times integrated C-semigroup with subgenerator A.
Theorem 2.5. Let M > 0, ω ≥ 0 be constants, and let A be a closed operator satisfying ω, ∞ ⊂
ρA. Let ϕt be the function in Theorem 2.4. If A is the subgenerator of a singular n-times integrated
C-semigroup {Ut}t≥0 , satisfying Ut ≤ ϕteωt , and satisfies
lim supλλ − A−1 ≤ M
λ→∞
λ > ω,
then
1 for λ > ω, x ∈ X, Utx λ − A0 S0 tλ − A−1 x,
2 for x ∈ DA, limt → 0 Utx 0,
3 for λ > ω, x ∈ X, Utx limλ → ∞ λS0 tλ − A−1 x,
∞
4 for λ > ω, x ∈ DA if and only if limλ → ∞ λn1 0 e−λt Utx dt Cx,
where A0 and S0 t are the symbols mentioned in Theorem 2.3.
2.30
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Proof. It follows from Theorems 2.3 and 2.4 that A subgenerates an n 1-times integrated
C-semigroup {St}t≥0 , which is continuously differentiable for t > 0 and satisfies 2.16 and
2.17.
Differentiating 2.16 with respect to t, we obtain
Utx d
Stx λ − A0 S0 tλ − A−1 x,
dt
x ∈ X, λ > ω.
2.31
This completes the proof of 1.
To show 2, for x ∈ DA, we have
Utx λ − A0 S0 tλ − A−1 x S0 tx.
2.32
Letting t → 0 , we get
lim Utx 0,
x ∈ DA.
t → 0
2.33
To show 3, for x ∈ X, since Stx ∈ C1 0, ∞, X, it follows from 2.17 that
limλ → ∞ λS0 tλ − A−1 x is continuous for t > 0, thus, we have
Utx d
Stx lim λS0 tλ − A−1 x,
λ→∞
dt
t > 0.
2.34
Obviously, the equality above is true for t 0.
Noting that
lim supλλ − A−1 ≤ M
λ→∞
λ > ω,
2.35
we can deduce that x ∈ DA implies limλ → ∞ λλ − A−1 Cx Cx, and from
λ − A−1 Cx λn
∞
e−λt Utx dt,
2.36
0
assertion 4 is immediate if we note that limλ → ∞ λλ − A−1 Cx Cx implies x ∈ DA.
Acknowledgments
The authors are grateful to the referees for their valuable suggestions. This work is supported
by the NSF of Yunnan Province 2009ZC054M.
References
1 W. Arendt, “Vector-valued Laplace transforms and Cauchy problems,” Israel Journal of Mathematics,
vol. 59, no. 3, pp. 327–352, 1987.
Advances in Difference Equations
9
2 A. Bobrowski, “On the generation of non-continuous semigroups,” Semigroup Forum, vol. 54, no. 2,
pp. 237–252, 1997.
3 Y.-C. Li and S.-Y. Shaw, “On local α-times integrated C-semigroups,” Abstract and Applied Analysis,
vol. 2007, Article ID 34890, 18 pages, 2007.
4 Y.-C. Li and S.-Y. Shaw, “On characterization and perturbation of local C-semigroups,” Proceedings of
the American Mathematical Society, vol. 135, no. 4, pp. 1097–1106, 2007.
5 J. Liang and T.-J Xiao, “Integrated semigroups and higher order abstract equations,” Journal of
Mathematical Analysis and Applications, vol. 222, no. 1, pp. 110–125, 1998.
6 J. Liang and T.-J Xiao, “Wellposedness results for certain classes of higher order abstract Cauchy
problems connected with integrated semigroups,” Semigroup Forum, vol. 56, no. 1, pp. 84–103, 1998.
7 J. Liang and T.-J Xiao, “Norm continuity for t > 0 of linear operator families,” Chinese Science Bulletin,
vol. 43, no. 9, pp. 719–723, 1998.
8 K. Nagaoka, “Generation of the integrated semigroups by superelliptic differential operators,” Journal
of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 1143–1154, 2008.
9 N. Tanaka, “Locally Lipschitz continuous integrated semigroups,” Studia Mathematica, vol. 167, no. 1,
pp. 1–16, 2005.
10 H. R. Thieme, ““Integrated semigroups” and integrated solutions to abstract Cauchy problems,”
Journal of Mathematical Analysis and Applications, vol. 152, no. 2, pp. 416–447, 1990.
11 H. R. Thieme, “Differentiability of convolutions, integrated semigroups of bounded semi-variation,
and the inhomogeneous Cauchy problem,” Journal of Evolution Equations, vol. 8, no. 2, pp. 283–305,
2008.
12 T.-J Xiao and J. Liang, “Integrated semigroups, cosine families and higher order abstract Cauchy
problems,” in Functional Analysis in China, vol. 356 of Mathematics and Its Applications, pp. 351–365,
Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
13 T.-J Xiao and J. Liang, “Widder-Arendt theorem and integrated semigroups in locally convex space,”
Science in China. Series A, vol. 39, no. 11, pp. 1121–1130, 1996.
14 T.-J Xiao and J. Liang, “Laplace transforms and integrated, regularized semigroups in locally convex
spaces,” Journal of Functional Analysis, vol. 148, no. 2, pp. 448–479, 1997.
15 T.-J. Xiao and J. Liang, The Cauchy Problem for Higher-Order Abstract Differential Equations, vol. 1701 of
Lecture Notes in Mathematics, Springer, Berlin, Germany, 1998.
16 T.-J. Xiao and J. Liang, “Approximations of Laplace transforms and integrated semigroups,” Journal
of Functional Analysis, vol. 172, no. 1, pp. 202–220, 2000.
17 T.-J. Xiao and J. Liang, “Higher order abstract Cauchy problems: their existence and uniqueness
families,” Journal of the London Mathematical Society, vol. 67, no. 1, pp. 149–164, 2003.
18 T.-J. Xiao and J. Liang, “Second order differential operators with Feller-Wentzell type boundary
conditions,” Journal of Functional Analysis, vol. 254, no. 6, pp. 1467–1486, 2008.