Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2009, Article ID 415847, 15 pages
doi:10.1155/2009/415847
Research Article
On Two-Parameter Regularized Semigroups and
the Cauchy Problem
Mohammad Janfada
Department of Mathematics, Sabzevar Tarbiat Moallem University, P.O. Box 397, Sabzevar, Iran
Correspondence should be addressed to Mohammad Janfada, m janfada@sttu.ac.ir
Received 13 December 2008; Revised 16 March 2009; Accepted 15 June 2009
Recommended by Stephen Clark
Suppose that X is a Banach space and C is an injective operator in BX, the space of all bounded
linear operators on X. In this note, a two-parameter C-semigroup regularized semigroup of
operators is introduced, and some of its properties are discussed. As an application we show
that the existence and uniqueness of solution of the 2-abstract Cauchy problem ∂/∂ti ut1 , t2 Hi ut1 , t2 , i 1, 2, ti > 0, u0, 0 x, x ∈ CDH1 ∩DH2 is closely related to the two-parameter
C-semigroups of operators.
Copyright q 2009 Mohammad Janfada. 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.
1. Introduction and Preliminaries
Suppose that X is a Banach space and A is a linear operator in X with domain DA and
range RA. For a given x ∈ DA, the abstract Cauchy problem for A with the initial value
x consists of finding a solution ut to the initial value problem
⎧
dut
⎪
⎨
Aut,
dt
ACPA; x
⎪
⎩
u0 x,
t ∈ R ,
1.1
where by a solution we mean a function u : R → X, which is continuous for t ≥ 0,
continuously differentiable for t > 0, ut ∈ DA for t ∈ R , and ACP A; x is satisfied.
If C ∈ BX, the space of all bounded linear operators on X, is injective, then a oneparameter C-semigroup regularized semigroup of operators is a family {T t}t∈R ⊂ BX
2
Abstract and Applied Analysis
for which T 0 C, T s tC T sT t, and for each x ∈ X, the mapping t → T tx is
continuous. An operator A : DA → X with
T tx − Cx
exists in the range of C ,
DA x ∈ X : lim
t→0
t
1.2
and where, for x ∈ DA, Ax : C−1 limt → 0 T tx − Cx/t is called the infinitesimal
generator of T t.
Regularized semigroups and their connection with the ACP A; x have been studied
in 1–6 and some other papers. Also the concept of local C-semigroups and their relation
with the ACP A; x have been considered in 7–10.
In Section 2, we introduce the concept of two-parameter regularized semigroups
of operators and their generator. Some basic properties of two-parameter regularized
semigroups and their relation with the generators are studied in this section.
In Section 3, two-parameter abstract Cauchy problems are considered. It is proved
that the existence and uniqueness of its solutions is closely related with two-parameter
regularized semigroups of operators.
2. Two-Parameter Regularized Semigroups
In this section we introduce two-parameter regularized semigroup and its generator on
Banach spaces. Then some properties of two-parameter regularized semigroups are studied.
Definition 2.1. Suppose that X is a Banach space and C ∈ BX is an injective operator.
A family {Ws, t}s,t∈R ⊂ BX is called a two-parameter regularized semigroup or two
parameter C-semigroup if
i W0, 0 C,
ii Ws s , t t C Ws, tWs , t , for all s, s , t, t ∈ R ,
iii lims ,t → s,t Ws , t x Ws, tx, for all x ∈ X.
It is called exponentially bounded if Ws, t ≤ Mestω , for some M, ω > 0.
Suppose that {Ws, t}s,t∈R is a two-parameter C-semigroup. Put us : Ws, 0 and
vt : W0, t, then it is easy to see that these families are two commuting one-parameter Csemigroups such that Ws, tC usvt. Also us and vt commute with C. If H1 and H2
are their generators, respectively, then we will think of H1 , H2 as the generator of Ws, t.
From the one-parameter case see 8, one can prove that RC ⊆ DH1 ∩ DH2 , and
C−1 Hi C Hi , i 1, 2.
Also if {Us}s∈R and {V t}t∈R are two commuting one-parameter C-semigroups,
then one can see that Ws, t : UsV t is a two-parameter C2 -semigroup of operators.
The following is an example of a two-parameter C-semigroup which is not
exponentially bounded.
Example 2.2. Let X L2 C, and Ws, tfz : e−|z| stz fz, Cfz : e−|z| fz, then
Ws, t is a two-parameter C-semigroup which is not exponentially bounded.
2
2
In the following theorem we can see some elementary properties of a two-parameter
C-semigroup.
Abstract and Applied Analysis
3
Theorem 2.3. Suppose that Ws, t is a two-parameter C-semigroup with the infinitesimal generator
H1 , H2 . Then, one has the following.
t s
i For each x ∈ X and for every s, t ≥ 0, 0 0 Wμ, νx dμ dν, is in DH1 ∩ DH2 . Also
1
lim
h,k → 0,0 hk
th sk
t
W μ, ν x dμ dν Ws, tx.
ii For each x ∈ X, and for every s, t ∈ R ,
DH2 ; furthermore
s
H1
2.1
s
s
0
Wμ, tx dμ ∈ DH1 and
t
0
Ws, νx dν ∈
W μ, t x dμ Ws, tx − W0, tx,
0
t
H2
2.2
Ws, νx dν Ws, tx − Ws, 0x.
0
iii RC ⊆ DH1 ∩ DH2 and H1 and H2 are closed.
iv For any x ∈ DH1 ∩ DH2 , and each s, t > 0, usx and vtx are in DH1 ∩ DH2 .
Also for this x, and i 1, 2,
∂
Wt1 , t2 x Hi Wt1 , t2 x Wt1 , t2 Hi x.
∂ti
2.3
v For any a, b > 0, T t : Wta, tb is a one-parameter C-semigroup whose generator is an
extension of aH1 bH2 .
Proof. To prove i, suppose x ∈ X. First we note that for any ν ≥ 0,
1
h→0h
th
lim
t
1
W μ, ν Cx dμ W0, νlim
h→0h
th
W μ, 0 x dμ
t
2.4
W0, νWt, 0x
Wt, νCx.
Thus
1
h
s t
Wh, 0
0
W μ, ν x dμ dν − C
0
s th
s t
0
W μ, ν x dμ dν
0
1
W μ, ν x dμ dν −
W μ, ν x dμ dν
C
h
0 h
0 0
h
s th
1
W μ, ν Cx dμ − W μ, ν Cx dμ dν,
h t
0
0
s t
2.5
4
Abstract and Applied Analysis
s t
s
which tends to C 0 Wt, ν − W0, νx dν as h → 0. This implies that 0 0 Wμ, νx dμ dν is
in DH1 and
s t
H1
0
W μ, ν x dμ dν s
0
Wt, ν − W0, νx dν.
2.6
0
A similar argument implies that it is in DH2 and
s t
H2
0
W μ, ν x dμ dν t
0
W μ, s − W μ, 0 x dν.
2.7
0
For the second part, from the continuity of C we have
C
1
h,k → 0,0 hk
th sk
lim
t
W μ, ν x dμ dν
s
th sk
1
W μ, ν Cx dμ dν
h,k → 0,0 hk t
s
th
1 sk 1
W0, ν
W μ, 0 x dμ dν
lim
h,k → 0,0 h t
k s
1 th
1 sk lim
W0, ν lim
W μ, 0 x dμ dν
h→0h t
k→0k s
lim
2.8
W0, tWs, 0x
Ws, tCx.
Now the fact that C is injective completes the proof of this part.
The proof of ii has a process similar to the first part of i.
To prove iii, we first note that H1 and H2 are closed as a trivial consequence of the
one-parameter case see 2. For any x ∈ X we saw that
1
h
h h
0
W μ, ν x dμ dν ∈ DH1 ∩ DH2 ,
2.9
0
which tends to W0, 0x Cx ∈ RC, as h → 0. This implies that RC ⊆ DH1 ∩ DH2 .
To prove iv, we let x ∈ DH1 ∩ DH2 . If us Ws, 0 and vt Ws, t, there is
y ∈ X such that
usx − Cx
Cy.
s→0
s
lim
2.10
Abstract and Applied Analysis
5
Hence
lim
s→0
usvtx − Cvtx
vtCy Cvty,
s
2.11
which is in the RC, and this implies that vtx is in DH1 , similarly it is in DH2 .
Now from 2, Theorem 2.4b, for x ∈ DH1 ∩ DH2 , from the fact that vtx is in
DH1 ,
∂
d
Ws, tCx usvtx
∂s
ds
H1 usvtx
2.12
H1 Ws, tCx
CH1 Ws, tx.
On the other hand from the part ii and closedness of H1 ,
s
H1 W μ, t x dμ H1
0
s
W μ, t x dμ Ws, tx − W0, tx,
2.13
0
which implies that ∂/∂sWs, tx exists. Hence from the continuity of C
C
∂
∂
Ws, tx Ws, tCx CH1 Ws, tx.
∂s
∂s
2.14
But C is injective so
∂
Ws, tx H1 Ws, tx Ws, tH1 x.
∂s
2.15
The second one is similar.
To prove v, first we note that T t is a one-parameter C-semigroup. Now if x ∈
DaH1 bH2 DH1 ∩ DH2 ,
C lim
t→0
T tx − Cx
Wta, 0W0, tbx − Wta, 0Cx Wta, 0Cx − C2 x
lim
t→0
t
t
b lim Wta, 0
t→0
W0, tbx − Cx
Wat, 0Cx − C2 x
a lim
t→0
bt
t
bC2 H2 x aH1 C2 x.
2.16
6
Abstract and Applied Analysis
Now the fact that C is injective implies that
C−1 lim
t→0
T tx − Cx
aH1 x bH2 x.
t
2.17
For an exponentially bounded
∞one-parameter C-semigroup T t with the generator A,
from 1 the existence of Lλ Ax 0 e−λt T tx dt is guaranteed for sufficiently large λ ∈ R.
Now we have the following lemma for one-parameter C-semigroups of operators which is
similar to the Yosida-approximation theorem for strongly continuous semigroups. This will
be applied in our study of two-parameter regularized semigroups.
Lemma 2.4. Let {T t}t∈R be a one-parameter C-semigroup satisfying the condition T t ≤ Meωt ,
for some ω > 0 and M > 0, with the generator A. If for λ > ω, Aλ : λALλ A, then one has the
following.
i For any x ∈ X, Lλ Ax ≤ M/λ−ωx, Aλ λ2 Lλ A−λC, and so Aλ is bounded.
Also St : CetAλ is a one-parameter C-semigroup which is exponentially bounded.
ii For any x ∈ DA, limλ → ∞ λLλ Ax Cx and for all x ∈ DA, limλ → ∞ Aλ x CAx.
Also if RC is dense in X, then the first equality holds on X.
iii For any x ∈ DA, T tx limλ → ∞ CetAλ x.
Proof. The first inequality of i is trivial. From 2, Lemma 2.8, we know that for any x ∈ X,
λ − ALλ Ax Cx; thus,
−λλ − ALλ Ax −λCx.
2.18
This implies our desired equality.
For the second part, first we show that CAλ Aλ C. For this we note that
CLλ A C
∞
∞
e−λt T tx dx
0
Ce−λt T tx dx
0
∞
2.19
e−λt T tCx dx
0
Lλ ACx.
This and the first part imply that CAλ Aλ C. Now we prove the C-semigroup properties of
St. Trivially S0 C. Also from the last equality,
Ss tC CestAλ C CesAλ CetAλ SsSt.
2.20
The fact that Aλ , λ > ω, is a bounded operator trivially implies that S· is exponentially
bounded. Now the continuity of the mapping t → Stx at zero implies the strongly
continuity of St.
Abstract and Applied Analysis
7
To prove ii, for x ∈ DA, from i and the fact that A is closed, we have
λLλ Ax − Cx ALλ Ax
Lλ AAx
2.21
≤ Lλ AAx
≤
M
Ax −→ 0
λ − ω
as λ −→ ∞.
The continuity of C and Lλ A implies that for any x ∈ DA, limλ → ∞ λLλ Ax Cx.
Now for x ∈ DA,
lim Aλ x lim λLλ AAx CAx ACx.
λ→∞
λ→∞
2.22
For the last part of ii, if C has a dense range, then by 8, Lemma 1.1.3, RC ⊆ DA, and
so X RC ⊆ DA ⊆ X, which means that DA X.
To prove iii, for any x ∈ DA, we have
1 d tAλ
tAμ tsAλ t1−sAμ Ce
e
x Ce x − Ce x 0 ds
1 ≤ tCetsAλ et1−sAμ Aλ x − Aμ x ds
0
2.23
≤ tCAλ x − Aμ x
≤ tC Aλ x − ACx ACx − Aμ x .
This and the previous part prove the existence of limλ → ∞ CetAλ x.
Using this theorem we may find the following approximation theorem for twoparameter regularized semigroups.
Corollary 2.5. Suppose that H, K is the infinitesimal generator of an exponentially bounded twoparameter C-semigroup Ws, t, then for each x ∈ DH ∩ DK,
Ws, tx C lim esHλ tKλ x.
λ→∞
2.24
stω
For exponentially bounded C-semigroup Ws, t satisfying
, with
∞ −λ s Ws, t ≤ Me
1
Hx
:
e
Ws,
0x
ds
and
L
Kx
:
the
infinitesimal
generator
H,
K,
define
L
λ
λ
1
2
0
∞ −λ t
2
e
W0,
tx
dt,
where
Reλ
>
ω.
From
the
previous
Lemma
L
H
and
L
K
are
i
λ
λ
1
2
0
bounded operators.
8
Abstract and Applied Analysis
Theorem 2.6. i Let H, K be the generator of an exponentially bounded two-parameter C-semigroup, then for large enough λ1 , λ2
Lλ1 HLλ2 K Lλ2 KLλ1 H.
2.25
ii Let H, K be the generator of an exponentially bounded two-parameter C-semigroup, then
DH ∩ DHK ⊆ DKH, and for x ∈ DH ∩ DHK,
HKx KHx.
2.26
iii Suppose that H and K are the generators of two exponentially bounded one-parameter Csemigroups {us}s∈R and {vt}t∈R , respectively. If their resolvents commute and RC
is dense in X, then Ws, t : usvt is a two-parameter C2 -semigroup.
Proof. The proof of i follows trivially from the properties of two-parameter C-semigroups.
To prove ii, we let x ∈ DH ∩ DHK; from the strongly continuity of Ws, t and
the fact that K is closed, we have
Ws, 0Kx − CKx
s→0
s
1
W0, tx − Cx
W0, tx − Cx
Ws, 0 lim
− lim
lim
s→0s
t→0
t→0
t
t
C2 HKx C lim
lim lim
1
lim lim
1
s → 0 t → 0 st
s → 0 t → 0 st
Ws, 0W0, tx − Ws, 0Cx − W0, tx Cx
W0, tWs, 0x − Ws, 0Cx − W0, tx Cx
2.27
1
Ws, 0x − Cx
Ws, 0x − Cx
W0, t
−
s→0 t→0 t
s
s
Ws, 0x − Cx
C lim K
s→0
s
lim lim
C2 KHx.
However, C is injective, and this completes the proof of i.
To prove iii, from our hypothesis, for sufficiently large λ, λ , we know that
Lλ HLλ K Lλ KLλ H.
2.28
By Lemma 2.4, Hλ λ2 Lλ H − λC and Kλ λ 2 Lλ H − λ C, thus Hλ Kλ Kλ Hλ . From iii
of Lemma 2.4, for each x ∈ DH ∩ DK,
usx lim CesHλ x,
λ→∞
vt lim
CetKλ x.
λ →∞
2.29
Abstract and Applied Analysis
9
So
usvtx C lim esHλ vtx
λ→∞
C lim e
2
sHλ
λ→∞
lim e
tKλ
λ → ∞
x ,
esHλ is continuous C2 lim lim
esHλ etKλ x
λ→∞ λ →∞
2.30
C2 lim lim
etKλ esHλ x
λ→∞ λ →∞
C lim vtesHλ x
λ→∞
vtusx.
Now the continuity of us and vt and the fact that DH ∩ DK RC X imply that
for each x ∈ X, usvtx vtusx. Thus
Ws, tW s , t usvtu s v t
usu s vtv t
Cu s s Cv t t
2.31
W s s , t t C2 .
On the other hand W0, 0 C2 , which completes the proof.
If H and K are two closed operators on X, then X1 : DH ∩ DK with x1 x Hx Kx, x ∈ X1 , is a Banach space.
Proposition 2.7. Suppose that C ∈ BX is injective and {Ws, t} is a two-parameter C-semigroup
with the generator H, K. Then W1 s, t : Ws, t|X1 defines a two-parameter C1 -semigroup, with
the generator H1 , K1 , where C1 C|X1 , and H1 , K1 are the part of H and K on X1 , respectively.
Proof. The C1 -semigroup properties of W1 s, t are obvious. Let A, B be the generator of
W1 s, t; we show that A H1 and B H2 . First we note that
DH1 {x ∈ X1 : Hx ∈ X1 }
x ∈ DH ∩ DK : x ∈ D H 2 ∩ DKH
DK ∩ D H 2 ∩ DKH.
2.32
10
Abstract and Applied Analysis
Let x ∈ DH1 . So we have
W1 s, 0x − C1 x Ws, 0x − Cx
−→ CHx C1 H1 x,
t
t
H
W1 s, 0x − C1 x Ws, 0Hx − CHx
−→ CH 2 x HC1 H1 x,
t
t
2.33
W1 s, 0x − C1 x Ws, 0Kx − CKx
−→ CHKx
K
t
t
KCHx KC1 H1 x.
These show that W1 s, 0x − C1 x/t → C1 H1 x in · 1 , that is, x ∈ DA and Ax H1 x.
Hence H1 ⊆ A. Conversely, if x ∈ DA ⊆ X1 , then
· 1 − lim
t→0
Ws, 0x − Cx
W1 s, 0x − C1 x
· 1 − lim
t
→
0
t
t
C1 Ax
2.34
CAx,
so Hx Ax ∈ X1 . Hence x ∈ DK ∩ DH 2 ∩ DKH DH1 and H1 x Hx Ax.
A similar argument shows that K1 B, which completes the proof.
3. Two-Parameter Abstract Cauchy Problems
Suppose that Hi : DHi ⊆ X → X, i 1, 2, is linear operator. Consider the following twoparameter Cauchy problem:
⎧
∂
⎪
⎨ ut1 , t2 Hi ut1 , t2 , ti > 0, i 1, 2,
2-ACP H1 , H2 ; x ∂ti
⎪
⎩
u0, 0 x,
x ∈ CDH1 ∩ DH2 .
3.1
We mean by a solution a continuous Banach-valued function u·, · : 0, ∞ × 0, ∞ → X
which has continuous partial derivative and satisfies 2-ACP H1 , H2 ; x.
In this section first we prove that if H1 , H2 is the infinitesimal generator of a twoparameter C-semigroup of operators, then 2-ACP H1 , H2 ; x has a unique solution for any
x ∈ CDH1 ∩ DH2 . Next it is proved that under some condition on C, existence and
uniqueness of solutions of 2-ACP H1 , H2 ; Cx, for every x ∈ DH1 ∩ DH2 , imply that this
unique solution is induced by a two-parameter regularized semigroup.
Theorem 3.1. Suppose that an extension of H1 , H2 is the generator of a two-parameter Csemigroup Ws, t, then 2-ACP H1 , H2 ; x has the unique solution us, t; x : Ws, tC−1 x, for
all x ∈ CDH1 ∩ DH2 .
Abstract and Applied Analysis
11
Proof. The fact that us, t; x : Ws, tC−1 x is a solution of 2-ACP H1 , H2 ; x is obvious from
Theorem 2.3. It is enough to show that 2-ACP H1 , H2 ; x has the unique solution us, t 0,
for the initial value x 0. From one-parameter case see 2, we know that the systems
dut
H1 ut, t ∈ R ,
dt
u0 0,
3.2
dvt
H2 vt, t ∈ R ,
dt
v0 0
3.3
have the unique solution zero. Now if us, t; 0 is a solution of 2-ACP H1 , H2 ; 0, then
u1 s : Ws, 0C−1 u0, t; 0,
u2 s : us, t; 0
3.4
are two solutions of 3.2, for the initial value u0, t; 0, since
d
d
u1 s Ws, 0C−1 u0, t; 0
ds
ds
H1 Ws, 0C−1 u0, t; 0
H1 u1 s,
d
∂
u2 s us, t; 0
ds
∂s
3.5
H1 us, t; 0
H1 u2 s.
The uniqueness of solution in one-parameter case implies that u1 s u2 s. So
Ws, 0C−1 u0, t; 0 us, t; 0.
3.6
Also v1 t : W0, tC−1 us, 0; 0 and v2 t : us, t; 0 are two solutions of 3.3 for the initial
value us, 0; 0. From the uniqueness of solution in 3.3, W0, tC−1 us, 0; 0 us, t; 0, for
all s, t ≥ 0. Thus
us, t; 0 Ws, 0C−1 u0, t; 0 Ws, 0C−1 W0, tu0, 0; 0 0.
3.7
The uniqueness of solution 2-ACP H, K; Cx, for all x ∈ DH ∩ DK, also leads us
to a two-parameter C-semigroup. This will be shown in the following theorem.
12
Abstract and Applied Analysis
In this theorem X1 and C1 have their meaning in Proposition 2.7.
Theorem 3.2. Suppose that C ∈ BX is injective and H, K are two closed operators satisfying
Cx ∈ X1 ,
KCx CKx,
HCx CHx,
∀x ∈ X1 .
3.8
If, for each x ∈ X1 , the Cauchy problem 2-ACP H, K; Cx has a unique solution u·, ·; Cx, then
there exists a two-parameter C1 -semigroup W1 ·, · on X1 such that u·, ·; Cx W1 ·, ·x. Moreover,
the infinitesimal generator of W1 ·, · is a restriction of H1 , K1 , where H1 and K1 are the part of H
and K on X1 , respectively.
Proof. Suppose that, for any x ∈ X1 , 2-ACP H, K; Cx has a unique solution u·, ·; Cx ∈
C1 0, ∞ × 0, ∞, X. For x ∈ X1 and 0 < s, t < ∞, define W1 s, tx : us, t; Cx.
From the uniqueness of solution W1 s, t is a well-defined and linear operator on X1
and
W1 0, 0x u0, 0; x Cx.
3.9
By uniqueness of solutions one can see that
W1 s s , t t C1 W1 s, tW1 s , t .
3.10
We are going to show that W1 s, t is a bounded operator on X1 , · 1 . Let 0 < s, t <
∞. Define the mapping φs,t : X1 → C0, s × 0, t, X1 by φs,t x W1 ·, ·x u·, ·; Cx.
Obviously φs,t is linear. We claim that this mapping is closed. Suppose that xn ∈ X1 , xn → x
and u·, ·; Cxn φs,t xn → y in C0, s × 0, t, X1 with its usual supremum norm. From
the Cauchy problem we know that
u μ, ν; Cxn Cxn μ
Hu η, ν; Cxn dη,
0
u μ, ν; Cxn Cxn ν
3.11
Ku μ, η; Cxn dη.
0
Letting n → ∞, we obtain
y μ, ν Cx μ
Hy η, ν dη,
0
y μ, ν Cx ν
Ky μ, η dη
0
3.12
Abstract and Applied Analysis
13
for any μ, ν ∈ 0, s × 0, t. Now define y on 0, ∞ × 0, ∞ by
⎧ ⎪
⎪
⎪Cy μ, ν ,
⎪
⎪
⎪
⎪
⎨W1 0, ν − ty μ, t ,
y μ, ν ⎪
⎪
W1 μ − s, 0 ys, ν,
⎪
⎪
⎪
⎪
⎪
⎩W μ − s, ν − tys, t,
1
0 ≤ μ ≤ s, 0 ≤ ν ≤ t,
0 ≤ μ ≤ s, t < ν < ∞,
s < μ < ∞, 0 ≤ ν ≤ t,
3.13
s < μ < ∞, t < ν < ∞.
One can see that y is a solution of 2-ACP H, K; C2 x. Indeed from 3.12
y0,
0 Cy0, 0 C2 x.
3.14
Also 3.12 and the fact that C commutes with H and K imply that
⎧
⎪
Hy μ, ν ,
⎪
⎪
⎪
⎪
⎪
⎪
⎨HW1 0, ν − ty μ, t ,
∂ y μ, ν ⎪
∂μ
⎪
HW1 μ − s, 0 ys, ν,
⎪
⎪
⎪
⎪
⎪
⎩HW μ − s, ν − tys, t,
1
H y μ, ν .
0 ≤ μ ≤ s, 0 ≤ ν ≤ t,
0 ≤ μ ≤ s, t < ν < ∞,
0 < μ < ∞, 0 ≤ ν ≤ t,
3.15
0 < μ < ∞, 0 < ν < ∞,
Similarly
∂ y μ, ν K y μ, ν .
∂ν
3.16
Uniqueness of the solution implies that
y·,
· u ·, ·; Cx2 W1 ·, ·Cx CW1 ·, ·x.
3.17
In particular for 0 ≤ μ ≤ s and 0 ≤ ν ≤ s,
Cy μ, ν y μ, ν CW1 μ, ν x Cφs,t x μ, ν .
3.18
The fact that C is injective implies that y φs,t x, which shows that φs,t is closed operator.
By the Closed Graph Theorem φs,t is a continuous operator from Banach space X1
into the Banach space C0, s × 0, t, X1 . So if xn → x in X1 , then φs,t xn → φs,t x in
C0, s × 0, t, X1 ; thus for each μ, ν ∈ 0, s × 0, t,
W1 s, txn φs,t xn μ, ν −→ φs,t x μ, ν W1 μ, ν x.
3.19
14
Abstract and Applied Analysis
But s and t were arbitrary; hence W1 μ, ν is continuous for any μ, ν ∈ 0, ∞. Also for every
x ∈ X1 , W1 ·, ·x φs,t x is continuous on 0, s × 0, t; that is, W1 ·, · is strongly continuous
family of operators.
Now let A, B be its infinitesimal generator and x ∈ DA, then
· 1 − lim
s→0
W1 s, 0x − C1 x
C1 Ax,
s
3.20
which implies that lims → 0 W1 s, 0x − Cx/s CAx, but DA ⊆ DH
lim
s→0
W1 s, 0x − Cx
us, 0; Cx − Cx
lim
s→0
s
s
∂
u0, 0; Cx
∂s
3.21
HCx
CHx.
Hence CHx CAx. The injectivity of C implies that Hx Ax ∈ X1 DH ∩ DK. Thus
x ∈ DK ∩ DH 2 ∩ DKH DH1 and H1 x Ax. This shows that A is a restriction of
H1 . Similarly one can see that B is a restriction of K1 , which completes the proof.
We conclude this section with a simple example as an application of our discussion.
Consider the following sequence of initial value problems:
∂
un s, t nun s, t,
∂s
∂
un s, t n2 un s, t,
∂t
3.22
n ∈ N,
un 0, 0 e−n qn .
2
Suppose that X c0 , the space of all complex sequences in C which vanish at infinity. Now
define linear operators H and K in X and operator C on X as follows:
Hxn n∈N nxn n∈N ,
Kxn n∈N n2 xn
n∈N
,
2 Cxn n∈N e−n xn
n∈N
.
3.23
Using these operators the initial value problem 3.22 can be rewrite as follows:
∂
us, t Hus, t,
∂s
∂
us, t Kus, t,
∂t
u0, 0 Cq,
3.24
Abstract and Applied Analysis
15
where us, t un s, tn∈N and q qn n∈N . One can easily see that H, K is the generator
of the following two-parameter C-semigroup:
Ws, txn n∈N en t−1sn xn n∈N
2
3.25
on X. Hence for every q qn n∈N ∈ DH ∩ DK, by Theorem 3.1, the abstract Cauchy
problem 3.24 has the unique solution
us, t Ws, tq en t−1sn qn n∈N .
2
3.26
This implies that for each n ∈ N, un s, t en t−1tn qn is a solution of 3.22.
2
Acknowledgment
The author is grateful to the referees for their very useful suggestions which helped him to
improve the presentation considerably.
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