Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 785428, 13 pages
doi:10.1155/2010/785428
Research Article
On Regularized Quasi-Semigroups and
Evolution Equations
M. Janfada
Department of Mathematics, Sabzevar Tarbiat Moallem University, P.O. Box 397, Sabzevar, Iran
Correspondence should be addressed to M. Janfada, mjanfada@gmail.com
Received 26 November 2009; Accepted 16 April 2010
Academic Editor: Wolfgang Ruess
Copyright q 2010 M. 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.
We introduce the notion of regularized quasi-semigroup of bounded linear operators on Banach
spaces and its infinitesimal generator, as a generalization of regularized semigroups of operators.
After some examples of such quasi-semigroups, the properties of this family of operators will
be studied. Also some applications of regularized quasi-semigroups in the abstract evolution
equations will be considered. Next some elementary perturbation results on regularized quasisemigroups will be discussed.
1. Introduction and Preliminaries
The theory of quasi-semigroups of bounded linear operators, as a generalization of strongly
continuous semigroups of operators, was introduced in 1991 1, in a preprint of Barcenas
and Leiva. This notion, its elementary properties, exponentially stability, and some of its
applications in abstract evolution equations are studied in 2–5. The dual quasi-semigroups
and the controllability of evolution equations are also discussed in 6.
Given a Banach space X, we denote by BX the space of all bounded linear operators
on X. A biparametric commutative family {Rs, t}s,t≥0 ⊆ BX is called a quasi-semigroup of
operators if for every s, t, r ≥ 0 and x ∈ X, it satisfies
1 Rt, 0 I, the identity operator on X,
2 Rr, s t Rr t, sRr, t,
3 lims,t → s0 ,t0 Rs, tx − Rs0 , t0 x 0, x ∈ X,
4 Rs, t ≤ Ms t, for some continuous increasing mapping M : 0, ∞ → 0, ∞.
Also regularized semigroups and their connection with abstract Cauchy problems are
introduced in 7 and have been studied in 8–12 and many other papers.
2
Abstract and Applied Analysis
We mention that if C ∈ BX is an injective operator, then a one-parameter family
{T t}≥0 ⊆ BX is called a C-semigroup if for any s, t ≥ 0 it satisfies T s tC T sT t and
T 0 C.
In this paper we are going to introduce regularized quasi-semigroups of operators.
In Section 2, some useful examples are discussed and elementary properties of
regularized quasi-semigroups are studied.
In Section 3 regularized quasi-semigroups are applied to find solutions of the abstract
evolution equations. Also perturbations of the generator of regularized quasi-semigroups are
also considered in this section. Our results are mainly based on the work of Barcenas and
Leiva 1.
2. Regularized Quasi-Semigroups
Suppose X is a Banach space and {Ks, t}s,t≥0 is a two-parameter family of operators in BX.
This family is called commutative if for any r, s, t, u ≥ 0,
Kr, tKs, u Ks, uKr, t.
2.1
Definition 2.1. Suppose C is an injective bounded linear operator on Banach space X. A
commutative two-parameter family {Ks, t}s,t≥0 in BX is called a regularized quasisemigroups or C-quasi-semigroups if
1 Kt, 0 C, for any t ≥ 0;
2 CKr, t s Kr t, sKr, t, r, t, s ≥ 0;
3 {Ks, t}s,t≥0 is strongly continuous, that is,
lim
s,t → s0 ,t0 Ks, tx − Ks0 , t0 x 0,
x ∈ X;
2.2
4 there exists a continuous and increasing function M : 0, ∞ → 0, ∞, such that
for any s, t > 0, Ks, t ≤ Ms t.
For a C-quasi-semigroups {Ks, t}s,t≥0 on Banach space X, let D be the set of all x ∈ X for
which the following limits exist in the range of C:
lim
t → 0
Ks, tx − Cx
Ks − t, tx − Cx
lim
,
t→0
t
t
K0, tx − Cx
lim
.
t→0
t
s>0
2.3
Now for x ∈ D and s ≥ 0, define
Asx C−1 lim
t→0
Ks, tx − Cx
.
t
2.4
{As}s≥0 is called the infinitesimal generator of the regularized quasi-semigroup
{Ks, t}s,t≥0 . Somewhere we briefly apply generator instead of infinitesimal generator.
Abstract and Applied Analysis
3
Here are some useful examples of regularized quasi-semigroups.
Example 2.2. Let {Tt }t≥0 be an exponentially bounded strongly continuous C-semigroup on
Banach space X, with the generator A. Then
Ks, t : Tt ,
s, t ≥ 0,
2.5
defines a C-quasi-semigroup with the generator As A, s ≥ 0, and so D DA.
Example 2.3. Let X BUCR, the space of all bounded uniformly continuous functions on R
with the supremum-norm. Define C, Ks, t ∈ BX, by
Cfx e−x fx,
2
2
Ks, tfx e−x f t2 2st x ,
s, t ≥ 0.
2.6
One can see that {Ks, t}s,t≥0 is a regularized C-quasi-semigroup of operators on X, with the
infinitesimal generator Asf 2sf˙ on D, where D {f ∈ X : f˙ ∈ X}.
Example 2.4. Let {Tt }t≥0 be a strongly continuous semigroup of operators on Banach space X,
with the generator A. If C ∈ BX is injective and commutes with Tt , t ≥ 0, then
Ks, t : CeTst −Ts ,
s, t ≥ 0,
2.7
is a C-quasi-semigroup with the generator As ATs . Thus D DA. In fact, for x ∈ D,
boundedness of C implies that
CAsx lim
t→0
d
CeTst −Ts x − Cx
eTst −Ts x − x
C lim
C |t0 Tst − Ts x CATs x.
t→0
t
t
ds
2.8
Now injectivity of C implies that Asx ATs x, and so D DA.
Example 2.5. Let {Tt }t≥0 be a strongly continuous exponentially bounded C-semigroup of
operators on Banach space X, with the generator A. For s, t ≥ 0, define
Ks, t T gs t − gs ,
s, t ≥ 0,
2.9
t
where gt 0 asds, and a ∈ C0, ∞, with at > 0. We have Ks, 0 T 0 C and the
C-semigroup properties of {T t}t≥0 imply that
CKr, s t CT gr t s − gr
CT gr t s − gt r gt r − gr
T gr t s − gt r T gt r − gr
Kr t, sKr, t.
2.10
4
Abstract and Applied Analysis
So {Ks, t}s,t≥0 is a C-quasi-semigroup the other properties can be also verified easily. Also
D DA and for x ∈ D, Asx asAx.
Some elementary properties of regularized quasi-semigroups can be seen in the
following theorem.
Theorem 2.6. Suppose {Ks, t}s,t≥0 is a C-quasi-semigroup with the generator {As}s≥0 on Banach
space X. Then
i for any x ∈ D and s0 , t0 ≥ 0, Ks0 , t0 x ∈ D and
Ks0 , t0 Asx AsKs0 , t0 x;
2.11
∂
Kr, tCx0 Ar tKr, tCx0 Kr, tAr tCx0 ;
∂t
2.12
ii for each x0 ∈ D,
iii if As is locally integrable, then for each x0 ∈ D and r ≥ 0,
Kr, tx0 Cx0 t
Ar sKr, sx0 ds,
t ≥ 0;
2.13
0
iv let f : 0, ∞ → X be a continuous function; then for every t ∈ 0, ∞,
1
lim
h→0h
th
Ks, ufudu Ks, tft;
2.14
t
v Let C ∈ BX be injective and for any s, t ≥ 0, C Ks, t Ks, tC . Then Rs, t :
C Ks, t is a CC -quasi-semigroup with the generator {As}s≥0 ,
vi Suppose {Rs, t}s,t≥0 is a quasi-semigroup of operators on Banach space X with the
generator {As}s≥0 , and C ∈ BX commutes with every Rs, t, s, t ≥ 0. Then
Ks, t : CRs, t is a C-quasi-semigroup of operators on X with the generator {As}s≥0 .
Proof. First we note that from the commutativity of {Ks, t}s,t≥0 ;
CKs, t Ks, tC
s, t ≥ 0.
2.15
Also x ∈ D implies that
lim
t→0
Ks, tx − Cx
CAsx
t
s ≥ 0.
2.16
Abstract and Applied Analysis
5
Thus from continuity of Ks0 , t0 , we have
lim
t→0
Ks, tKs0 , t0 x − CKs0 , t0 x
Ks, tx − Cx
Ks0 , t0 lim
t→0
t
t
Ks0 , t0 CAsx
2.17
CKs0 , t0 Asx.
Thus Ks0 , t0 x ∈ D and AsKs0 , t0 Ks0 , t0 Asx.
To prove ii, consider the quotient
Kr, t sCx0 − Kr, tCx0 Kr t, sKr, tx0 − Kr, tCx0
s
s
Kr t, sx0 − Cx0
,
Kr, t
s
2.18
which tends to Kr, tCAr tx0 as s → 0 .
Also for s < 0,
Kr, t sCx0 − Kr, tCx0 Kr, tCx0 − Kr, t sCx0
s
−s
Kr t s, −sKr, t sx0 − Kr, t sCx0
−s
Kr, t s
Kr t s, −sx0 − Cx0
−s
Kr, t s
1
Kr t s, −sx0
−s
2.19
−Kr t, −sx0 Kr t, −sx0 − Cx0 .
Now the strongly continuity of {Ks, t}s,t≥0 implies that
lim Kr t s, −sx0 − Kr t, −sx0 0.
s → 0−
2.20
Thus
lim
s → 0−
Kr t s, −sx0 − Cx0
CAr tx0 .
−s
2.21
Hence by the strongly continuity of Ks, t,
lim
s → 0−
Kr, t sCx0 − Kr, tCx0
Kr, tCAr tx0 .
s
Thus ∂/∂tKr, tCx0 Kr, tCAr tx0 . The second equality holds by i.
2.22
6
Abstract and Applied Analysis
Now integrating of this equation, we have
Kr, tCx0 − Cx0 C
t
Kr, sAr sx0 ds.
2.23
0
Hence injectivity of C implies iii.
iv is trivial from continuity of f and strongly continuity of {Ks, t}s,t≥0 . In v,
obviously {Rs, t}s,t≥0 is a C C-quasi-semigroup. For x ∈ D, we have
Rs, tx − CC x
Ks, tx − Cx
C’
,
t
t
2.24
which tends to C CAs, as t → 0 . This proves v.
vi can be seen easily.
3. Evolution Equations and Regularized Quasi-Semigroups
Suppose C is an injective bounded linear operator on Banach space X and r > 0. In
this section, we study the solutions of the following abstract evolution equation using the
regularized quasi-semigroups:
ẋt At rxt,
x0 C2 x0 ,
t > 0,
x0 ∈ X.
3.1
One can see 13, 14 for a comprehensive studying of abstract evolution equations.
Theorem 3.1. Let {As}s≥0 be the infinitesimal generator of a C-quasi-semigroups {Ks, t}s,t≥0 on
Banach space X, with domain D. Then for each x0 ∈ D and r ≥ 0, the initial value problem 3.1
admits a unique solution.
Proof. Let xt Kr, tCx0 . By Theorem 2.6ii, xt is a solution of 3.1.
Now we show that this solution is unique. Suppose ys is another solution of 3.1.
Trivially ys ∈ D. Let t > 0. For s ∈ 0, t and x ∈ X, define
Fsx Kr s, t − sCx,
Gs FsCys.
3.2
From C-quasi-semigroup properties, for small enough h > 0, we have
Kr s, t − sC Kr s t − s − t − s − h, t − s − hKr s, t − s − t − s − h
Kr s h, t − s − hKr s, h.
3.3
Abstract and Applied Analysis
7
So
Fs hx − Fsx Kr s h, t − s − hCx − Kr s h, t − s − hKr s, hx
h
h
Kr s, hx − Cx
−Kr s h, t − s − h
h
−→ −Kr s, t − sCAr sx,
3.4
as h −→ 0.
This means that
Ḟsx −Kr s, t − sCAr sx.
3.5
Therefore, from this, the fact that ys satisfies 3.1, and CFs FsC, we obtain that
Ġs ḞsCys FsCẏs −Kr s, t − sCAr sCys Kr s, t − sC2 ẏs
−Kr s, t − sCAr sCys Kr s, t − sC2 Ar sys 0.
3.6
Hence for every s ∈ 0, t, Ġs 0. Consequently, Gs is a constant function on 0, t. In
particular, G0 Gt. So from y0 Cx0 , we have
G0 F0Cy0 Kr, tC2 x0 Gt FtCyt Kr t, 0C2 yt C3 yt.
3.7
Hence C2 Kr, tx0 C3 yt. Now injectivity of C implies that yt Kr, tCx0 , which proves
the uniqueness of the solution.
Now with the above notation, we consider the inhomogeneous evolution equation
ẋt Ar txt C2 ft,
x0 C2 x0 ,
0 < t ≤ T,
x0 ∈ D.
3.8
The following theorem guarantees the existence and uniqueness of solutions of 3.8 with
some sufficient conditions on f.
Theorem 3.2. Let Ks, t be a C-quasi-semigroup on Banach space X, with the generator {As}s≥0
whose domain is D. If f : 0, T → D is a continuous function, each operator As is closed, and
t
C
Kr s, t − sfsds ∈ D,
0 < t ≤ T,
3.9
0
then the initial value equation 3.8 admits a unique solution
xt Kr, tCx0 t
0
Kr s, t − sCfsds.
3.10
8
Abstract and Applied Analysis
Proof. For the existence of the solution, it is enough to show that xt in 3.10 is continuously
differentiable and satisfies 3.8.
Trivially x0 Cx0 . We know that yt Kr, tCx0 is a solution of 3.1 by
Theorem 3.1. Define
gt t
Kr s, t − sCfsds,
3.11
0
which is in D by our hypothesis. We have
gt h − gt 1
h
h
1
h
th
Kr s, t h − sCfsds −
t
0
Kr s, t − sCfsds
0
t
Kr s, t h − sCfsds −
0
t
Kr s, t − sCfsds
3.12
0
th
Kr s, t h − sCfsds .
t
On the other hand, the C-quasi-semigroup properties imply that
Kr s, t h − sCfs Kr s t h − s − h, hKr s, t h − s − hfs
Kr t, hKr s, t − sfs.
3.13
So
gt h − gt 1
h
h
t
−
Kr t, hKr t, t − sfsds
0
t
Kr s, t − sCfsds 0
t
th
Kr s, t h − sCfsds
t
Kr t, hfs − Cfs
ds
Kr t, t − s
h
0
1 th
Kr s, t h − sCfsds.
h t
3.14
Since the range of f is in D, passing to the limit when h → 0, and using Theorem 2.6v, we
have
ġt t
Kr s, t − sCAr tfsds Kr t, t − tCft
0
t
0
3.15
Kr s, t − sCAr tfsds C2 ft.
Abstract and Applied Analysis
9
Therefore, ġt exists. Also by our hypothesis Ar t is closed, and
D, thus
t
Kr s, t − sCAr tfsds Ar t
t
0
t
0
Kr s, t − sCfsds ∈
Kr s, t − sCfsds.
3.16
0
Consequently,
ġt Ar tgt C2 ft,
3.17
t ≥ 0.
Hence
ẋt t
∂
Kr, tCx0 Ar t Kr s, t − sCfsds C2 ft
∂t
0
t
Ar t Kr, tCx0 Kr s, t − sCfsds
C2 ft
3.18
0
Ar txt C2 ft.
This completes the proof.
We conclude this section with two simple perturbation theorems and some examples,
as applications of our discussion.
Theorem 3.3. a Suppose B is the infinitesimal generator of a strongly continuous semigroup
{T t}t≥0 and {As}s≥0 with domain D is the generator of a regularized C-quasi-semigroup
{Ks, t}s,t≥0 , which commutes with {T t}t≥0 . Then {As B}s≥0 with domain D ∩ DB is the
infinitesimal generator of a regularized C-quasi-semigroup.
b Suppose B is the infinitesimal generator of an exponentially bounded C-semigroup
{T t}t≥0 and {As}s≥0 with domain D is the generator of a quasi-semigroup (resp., regularized C quasi-semigroup) {Ks, t}s,t≥0 , which commutes with {T t}t≥0 . Then {As B}s≥0 with domain
D ∩ DB is the infinitesimal generator of a C-regularized quasi-semigroup (resp., regularized CC quasi-semigroup).
Proof. In a and b, define
Rs, t T tKs, t.
3.19
One can see that {Rs, t}s,t≥0 is a C-regularized quasi-semigroup in b, resp., regularized
CC -quasi-semigroup. We just prove that {As B}s≥0 is its generator. In a, let {Bs}s≥0
be the infinitesimal generator of {Rs, t}s,t≥0 and x ∈ D ∩ DB. Hence
lim
t→0
T tx − x
,
t
lim
t→0
Ks, tx − Cx
t
3.20
10
Abstract and Applied Analysis
exist in X and the range of C, respectively. Now the fact that C commutes with T t and
strongly continuity of T t implies that
lim T t
t → 0
Ks, tx − Cx
t
3.21
exists in the range of C. So
lim
t→0
Ks, tx − Cx
Rs, tx − Cx
T tKs, tx − Cx
T tx − x
lim
lim T t
C lim
t→0
t→0
t→0
t
t
t
t
3.22
exists in the range of C and
CBsx lim
t→0
Rs, tx − Cx
CAsx CBx.
t
3.23
By injectivity of C, Bsx Asx Bx.
The proof the other parts is similar.
Theorem 3.4. Let Ks, t be a C-quasi-semigroup of operator on Banach space X with the generator
{As} on domain D. If B ∈ BX commutes with Ks, t, s, t ≥ 0, and B2 B, then {BAs}s≥0 is
the infinitesimal generator of C-regularized quasi-semigroup
Rs, t BKs, t − C C.
3.24
Proof. The C-quasi-semigroup properties of {Rs, t}s,t≥0 can be easily verified. We just prove
that its generator is {BAs}s≥0 . Let x ∈ D; we have
Rs, tx − Cx BKs, t − Cx Cx − Cx
Ks, tx − Cx
B
t
t
t
3.25
which tends to BAsx, as t → 0. This completes the proof.
Example 3.5. Let r > 0. Consider the following initial value problem:
∂
∂
xt, ε 2r t xt, ε εxt, ε,
∂t
∂ε
x0, ε e
−4ε2
x0 ε,
3.26
ε, t ≥ 0.
Let X BUCR, with the supremum-norm. Define C ∈ BX by Cxε e−ε xε, x· ∈ X.
Also define B : DB → X by Bxε εxε, where DB {x ∈ X : Bx ∈ X}. It is
well known that B is the infinitesimal generator of C-regularized semigroup T t, defined by
2
T txε e−ε εt xε. Now with D {x ∈ X : ẋ ∈ X}, if As : D → X is defined by
Asx 2sẋ, then by Example 2.3, {As}s≥0 is the infinitesimal generator of the regularized
2
Abstract and Applied Analysis
11
C2 -quasi-semigroup Ks, txε e−ε xt2 2st ε. Using Theorem 3.3 and the fact that
T tKs, r Ks, rT t, s, t, t ≥ 0, we obtain that {As B} is the infinitesimal generator of
regularized C2 -quasi-semigroup Rs, t T tKs, t. Also using these operators, 3.26 can
be written as
2
ẋt Ar t Bxt,
3.27
x0 C4 x0 .
Thus by Theorem 3.1 for any x0 ∈ D ∩ DB, 3.26 has the unique solution
xt, ε Rr, tC2 x0 ε e−4ε
2
εt
x0 t2 2rt ε .
3.28
Example 3.6. For a given sequence pn n∈N of complex numbers with nonzero elements and
yn n∈N , consider the following equation:
d
xn t eint1 xn t pn xn t,
dt
xn 0 pn2 yn ,
3.29
n ∈ N.
Let X be the space c0 , the set of all complex sequence with zero limit at infinity. For a bounded
sequence p pn n∈N , define A : DA : → X and Mp on X by
Axn n∈N ein xn
n∈N
,
Mp xn n∈N pn xn .
3.30
One can easily see that DA {xn n∈N ∈ c0 : ein xn n∈N ∈ c0 } and Mp is a bounded linear
operator which is injective. It is well known that A is the infinitesimal generator of strongly
continuous semigroup
T txn n∈N ein xn
n∈N
3.31
.
Thus by Example 2.4, {At}t≥0 , defined by
Atxn n∈N : AT txn n∈N ein1t xn
n∈N
,
3.32
is the infinitesimal generator of the Mp -quasi-semigroup
Ks, t Mp eT st−T s .
3.33
12
Abstract and Applied Analysis
Using these operators, one can rewrite 3.29 as
ẋt At MP xt,
3.34
x0 Mp2 y0 ,
where x0 yn n∈N . Trivially T t commutes with Kr, s, for any r, s, t ≥ 0. Now using
Theorem 3.3 we obtain that {At Mp }t≥0 is the infinitesimal generator of of Mp -quasisemigroup
Rs, t T tKs, t.
3.35
Also from Theorem 3.1, with r 0, for any y ∈ DA, 3.34 has a unique solution
xt R0, tMp y T tK0, tMp2 x0 .
3.36
But from definition of Ks, t, for a given xn n∈N ∈ c0 ,
K0, txn n∈N e
T t−I
e
−1
∞
T k t
k0
k!
xn n∈N e
−1
∞
eiknt xn
k!
k0
e
−1
∞ iknt
e xn
k0
n∈N
k!
.
n∈N
3.37
So the solution of 3.34 is
xt R0, tMp y ∞ iktn1−1 4
e
pn yn
k!
k0
,
3.38
n∈N
or equivalently the solution of 3.29 is
xn t ∞ iktn1−1 4
e
pn yn
k0
k!
,
n ∈ N.
3.39
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