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
Trang 1Volume 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
Copyrightq 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, Arendt1 studied the n-times integrated semigroups, which are more general than
C0semigroupsthere exist many operators that generate n-times integrated semigroups but not C0semigroups
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 in2,5 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
C-regularized semigroup is also a subgenerator of a singular n-times integrated C-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 C10, ∞, 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, ∞.
Trang 2All 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,
D A0 :x ∈ D A; Ax ∈ DA, A0x Ax, for x ∈ D A0. 1.1
The C-resolvent set of A is defined as:
ρ C A λ ≥ 0; λ − A is injective, RC ⊂ Rλ − A and λ − A−1C ∈ L X. 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 ifω, ∞ ⊂ ρ C A for some ω ≥ 0 and there exists a strongly continuous family S· : 0, ∞ → LX with St ≤ Me ω tfor some
M > 0 such that
λ − A−1Cx λ n
∞
0
e −λt S tx dt λ > ω, x ∈ X. 1.3
In this case, {St} t≥0 is called the exponentially bounded n-times integrated
C-semigroup generated by A : C−1AC.
If C I resp., n 0, then A is called a generator of an exponentially bounded n-times
integrated semigroupresp., 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,
iii Stx t n /n!Cx At
0Ssx ds t ≥ 0, x ∈ X.
In particular, S0 0.
Definition 1.3 Let ω ≥ 0 If ω, ∞ ⊂ ρ C 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−1Cx λ n∞
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.
Trang 32 The Main Results
Theorem 2.1 Let M > 0, ω ≥ 0 be constants, and let A be a closed operator satisfying ω, ∞ ⊂
ρ C 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 → ∞ λ n2∞
0 e −λt Stdt ≤ M,
A2 St − Ss ≤s
t ϕue ωu du, 0 ≤ t ≤ s, is that for λ > ω,
i lim supλ → ∞ λλ − A−1C ≤ M,
ii λ − A−1C/λ nm ≤0∞e −λ−ωt t m ϕtdt, m 1, 2,
Proof Sufficiency Let ψt e ωt ϕt Set
f λ
∞
0
e −λt ψ tdt
∞
0
e −λ−ωt ϕ tdt, λ > ω. 2.1
For x∗∈ X∗, we have
λ − A−1C
m
, x∗
≤ x · x∗
∞
0
e −λt t m ϕ tdt
≤
x · x∗ · fλ m, 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−1C
λ n x, x∗
∞
0
e −λt h t, x, x∗dt, λ > ω. 2.3
Let Ht, x, x∗ t
0hs, x, x∗ds, t ≥ 0, x∗∈ X∗ In view of the convolution theorem for Laplace transforms and from2.3, we have
λ − A−1C
λ n x, x∗
λ
∞
0
e −λt H t, x, x∗dt, λ > ω, x∗∈ X∗. 2.4
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
t
ψ sds. 2.5
Trang 4Hence for all t ≥ 0, there exists St ∈ LX∗∗such that
H t, x, x∗ ∗ , x ∈ X, x∗∈ X∗, 2.6
St h − St ≤
th
t
ψ sds, t ≥ 0, h ≥ 0, 2.7
λ − A−1C
λ n λ
∞
0
e −λt S tdt. 2.8
Denote by q : Lx∗∗ → Lx∗∗/LX the quotient mapping Since λ − A−1C ∈ LX,
we deduce
0 q
λ − A−1C
λ n
λ
∞
0
e −λt q Stdt. 2.9
It follows from the uniqueness theorem for Laplace transforms that qSt 0, that is, St ∈
LX.
Combining2.7 and 2.8 yields that St : 0, ∞ → LX is strongly continuous and
∞
0
e −λt Stdt ≤
∞
0
e −λt
t
0
ψ sds dt 1
λ
∞
0
e −λt ψ tdt < ∞. 2.10
Now, we conclude that{St} t≥0is ann 1-times integrated C-semigroup satisfying
A2 Assertion A1 is immediate, by 2.8 and i
Necessity Let ψt e ωt ϕt Since {St} t≥0is ann 1-times integrated C-semigroup on X,
we have
λ − A−1C λ n1
∞
0
e −λt S tdt 2.11
for λ > ω Noting that St h − St ≤th
t ψs ds h ≥ 0 and S0 0, we find
St ≤
t
0
Then for any y∗ ∈ LX∗and λ > ω, we obtain
λ − A−1C
λ n , y∗
λ
∞
0
e −λt S tdt, y∗
≤ λ
∞
0
e −λt St ·y∗dt ≤y∗∞
0
e −λt ψ tdt.
2.13
Trang 5Therefore, there exists a measurable function ηt on 0, ∞ with |ηt| ≤ ψt a.e.
such that
λ − A
−1C
λ n
∞
0
e −λt η tdt. 2.14
Furthermore, by calculation, we have
λ − A−1C
λ n
m
≤
∞
0
e −λt t m ψ tdt
∞
0
e −λ−ωt t m ϕ tdt, m 1, 2, 2.15
Assertioni 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 Bobrowski2
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 sup λ → ∞ 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,
S tx μ − A0
t
0
S0 s μ − A −1x ds, 2.16
S tx lim
μ → ∞ μ
t
0
Proof For μ ∈ ρA, x ∈ X, set { St} t≥0as follows:
Stx μt
0S0 s μ − A −1x ds − S0 t μ − A −1x t
n n!
μ − A −1Cx. 2.18
Since S0t is strongly continuous on DA, St is strongly continuous on X.
Fixing λ > ω, we have
λ n1
∞
0
e −λt Stx dt λ n
μ − λ
∞ 0
e −λt S0 t μ − A −1x dt
μ − A −1Cx
μ − λ λ − A−1C
μ − A −1x
μ − A −1Cx
λ − A−1Cx.
2.19
It follows from the uniqueness of Laplace transforms that Stx Stx, x ∈ X So we get
2.16 By the hypothesis lim supλ → ∞ λλ − A−1 ≤ M, we see
Trang 6S tx lim
μ → ∞
μ
t
0
S0 s μ − A −1x ds − S0 t μ − A −1x t
n n!
μ − A −1Cx
lim
μ → ∞ μ
t
0
S0 s μ − A −1Cx ds,
2.20
and the proof is completed
Now, we study the relation between differentiable n 1-times integrated
C-semigroups and singular n-times integrated C-C-semigroups.
Theorem 2.4 Let ω ≥ 0, and let A be a closed operator satisfying ω, ∞ ⊂ ρ C 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 ≤ ϕte ωt
2 A is the subgenerator of an exponentially bounded n 1-times integrated C-semigroup {St} t≥0 satisfying
St − Ss ≤
s
t
ϕ τe ωτ dτ, 0≤ t ≤ s,
S tx ∈ C10, ∞, X, for x ∈ X.
2.21
Proof. 1⇒2: we set
S tx :
t
0
U sx ds, t ≥ 0. 2.22
Since Utx is locally integrable on 0, ∞, Stx is well-defined for any x ∈ X It is easy to check that Stx belongs to C10, ∞, X.
For every λ > ω, since
Stx
t
0
e −λs e λs U sx ds
≤ e λt
t
0
e −λs Usxds ≤ Me λt x, 2.23
we deduce that St is exponentially bounded.
Moreover, for λ > ω, we have
λ − A−1Cx λ n
∞
0
e −λt U tx dt λ n1
∞
0
e −λt S tx dt,
St − Ss
s
t
U τdτ
≤s
t
ϕ τe ωτ dτ, 0≤ t ≤ s.
2.24
Thus{St} t≥0is the desired semigroup in2
Trang 72⇒1: for any x ∈ X, we set
U tx : d
dt S tx, for t > 0,
U 0x : 0, for t 0.
2.25
Then Utx ∈ C0, ∞, X and U0 0.
Noting that
St h − St ≤
th
t
ϕ se ωs ds, 2.26
we find
S t h − St h ≤ h1th
t
ϕ se ωs ds. 2.27
Since Stx is continuously differentiable for t > 0, we get
Ut ≤ ϕte ωt a.e.. 2.28
Moreover, for λ > ω, we have
∞
0
e −λt Utdt ≤
∞
0
e −λ−ωt ϕ tdt < ∞,
λ − A−1Cx λ n1
∞
0
e −λt S tx dt λ n
∞
0
e −λt U tx dt.
2.29
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 ≤ ϕte ωt , and satisfies
lim sup
λ → ∞
λλ − A−1 ≤ M λ > ω,
2.30
then
1 for λ > ω, x ∈ X, Utx λ − A0S0tλ − A−1x,
2 for x ∈ DA, lim t → 0Utx 0,
3 for λ > ω, x ∈ X, Utx lim λ → ∞ λS0 tλ − A−1x,
4 for λ > ω, x ∈ DA if and only if lim λ → ∞ λ n1∞
0 e −λt Utx dt Cx, where A0 and S0 t are the symbols mentioned in Theorem 2.3
Trang 8Proof It follows from Theorems2.3and2.4that 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
U tx d
dt S tx λ − A0S0tλ − A−1x, x ∈ X, λ > ω. 2.31 This completes the proof of1
To show2, for x ∈ DA, we have
U tx λ − A0S0tλ − A−1x S0 tx. 2.32
Letting t → 0, we get
lim
t → 0U tx 0, x ∈ DA. 2.33
To show 3, for x ∈ X, since Stx ∈ C10, ∞, X, it follows from 2.17 that limλ → ∞ λS0 tλ − A−1x is continuous for t > 0, thus, we have
U tx d
dt S tx lim
λ → ∞ λS0 tλ − A−1x, t > 0. 2.34
Obviously, the equality above is true for t 0.
Noting that
lim sup
λ → ∞
λλ − A−1 ≤ M λ > ω,
2.35
we can deduce that x ∈ DA implies lim λ → ∞ λλ − A−1Cx Cx, and from
λ − A−1Cx λ n
∞
0
e −λt U tx dt, 2.36
assertion4 is immediate if we note that limλ → ∞ λλ − A−1Cx 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 Province2009ZC054M
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