International Journal of Modern Nonlinear Theory and Application
Vol.2 No.4(2013), Article ID:40318,4 pages DOI:10.4236/ijmnta.2013.24030
Notes on the Global Attractors for Semigroup
1Suzhou Vocational University, Suzhou, China
2Department of Mathematics, Soochow University, Suzhou, China
Email: jssvcxulan@gmail.com, sudayhshi@gmail.com
Copyright © 2013 Lan Xu, Yuhua Shi. 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.
Received October 19, 2013; revised November 19, 2013; accepted November 25, 2013
Keywords: Natural Global Attractors; Measure of Noncompactness; Asymptotic Compactness; -Limit Compact
ABSTRACT
First we introduce two necessary and sufficient conditions which ensure the existence of the global attractors for semigroup. Then we recall the concept of measure of noncompactness of a set and recapitulate its basic properties. Finally, we prove that these two conditions are equivalent directly.
1. Introduction
It is well known that many mathematical physics problems can be put into the perspective of infinite dimensional systems, which can be equivalently described by semigroups in proper function spaces. One important object to describe the long time dynamics of an infinite dimensional system is the global attractor, which is a connected and compact invariant set in some function space, and which attracts all bounded sets.
To show the existence of the global attractor, one normally needs to verify:
1) there exists an absorbing set, and
2) the semigroup is uniformly compact.
However, it is difficult or even impossible to verify the uniform compactness of the semigroup for many problems. In [1], the authors use the measure of noncompactness of a set to introduce a new concept of compactness called -limit compact, then they show that there exists a global attractor for a
semigroup if and only if:
1) there is an absorbing set, and
2) the semigroup is -limit compact.
A well-known result (see [2-6]) is that a continuous semigroup has a global attractor if and only if:
1) it has a bounded absorbing set, and
2) it is asymptotically compact.
Furthermore, in [7], the author introduce the concept of asymptotically null and show that a lattice system has a global attractor if and only if:
1) it has a bounded absorbing set, and
2) it is asymptotically null.
Our main motivation of this paper is to prove that asymptotically compact Û -limit compact, and then we prove that the conditions in [1,7] are equivalent directly in
.
The concept of pullback random attractors for random dynamical systems, which is an extension of the attractors theory of deterministic systems, was introduced by the authors in [8-10]. We point out that our work in this paper also can be extended to pullback attractors.
2. Measure of Noncompactness and Its Properties
In this section, we recall the concept of measure of noncompactness and recapitulate its basic properties; see [11].
Definition 2.1 Let M be a metric space and A be a bounded subset of M. The measure of noncompactness of A is defined by
.
Lemma 2.1 Let M be a complete metric space, and be the measure of noncompactness of a set.
1) if and only if
is compact;
2) If M is a Banach space, then;
3) whenever
;
4);
5).
Proof. 1) (a) If is compact, then
is precompact.
is a complete metric space, thus for any
, there exists a finite subset
of
such that the balls of radii
centered at
form a finite covering of
. By Definition 2.1,
admits a finite cover by sets of diameter
. The arbitrariness of
implies that
.
(b) On the other hand, if, then by Definition 2.1, we have that for any
,
admits a finite cover by sets of diameter
. So for any
,
always has a finite
-net. Then
is totally bounded.
is complete, thus
is precompact, and
is compact.
2) If is a finite cover of
, and
is a finite cover of
, then
is a finite cover of
, thus
.
3) If, then the finite cover of
must be a finite cover of
, so
.
4) (a) The finite cover of must be a finite cover of both of
and
. So we have
and
. Thus
.
(b) For any, we can find finite covers
of
and
of
with the diameter of
and
less than
. But
is a cover of
and the diameter of
is less than
. Hence
. So
.
5) Since, then
. For any
,
has a finite cover by sets of diameter
. For any
,
has a finite cover by sets of diameter
. From the arbitrariness of
and Definition 2.1, we have
. Thus
. So
.
3. Main Results
In this section, firstly we recall some basic definitions in [1,7], then we show that the two necessary and sufficient conditions for the existence of global attractors for semigroups are equivalent directly.
Definition 3.1 Let be a complete metric space. A one parameter family
of maps
,
is called a
semigroup if 1)
is the identity map on M2)
for all
3) the function
is continuous at each point
.
Definition 3.2 Let be a
semigroup in a complete metric space
. A subset
of
is called an absorbing set in
, if for any bounded subset
of
, there exists some
such that
, for all
.
Definition 3.3 A semigroup
in a comple te metric space
is called
-limit compact, if for every bounded subset
of
and any
, there exists
such that
Definition 3.4 A semigroup
in a complete metric space
is called asymptotically compact if, for every bounded subset
, for any
and any
,
has a convergent subsequence.
Let be a positive smooth function on R and
. Then define a weighted
space as
with norm.
Definition 3.5 is said to be asymptotically null in
if for any
bounded in
and
, the following holds
Theorem 3.1 Let be a
semigroup in a complete metric space M, then we can have:
is
-limit compact Û
is asymptotically compact.
Proof. First, we prove the necessity.
It suffices to prove that for every bounded subset, for any
, there exists
, such that
Assume otherwise, then there exists a bounded subset and
, such that for every
we have
We take, then
. Let
and take
.
Let then
. By the definition of measure of noncompactness,
has no finite covering of balls of radii
.
Thus there exists and
such that
Otherwise is the finite
-net of
.
Next we take hence
. That is to say
has no finite
-net. Thus there exists
and
such that
Otherwise is the finite
-net of
.
Repeat the previous procedure, then we have the sequence which satisfies
(1)
By the way of taking, and
, we have
. Since
and
is a bounded subset of
,
is asymptotically compact. Therefore
has a convergent subsequence. This gives contradiction to (1).
Thus is
-limit compact.
Next, we prove the sufficiency.
We need to prove that for every bounded subset, for any
and any
,
has a convergent subsequence.
Since is
-limit compact, then for the bounded subset
above, for any
, there exists
such that
For, there exists
, such that
when
.
implies
Property (3) of the measure of noncompactness in Lemma 2.1 shows that
So. Notice that
contains only a finite number of elements (where is fixed such that
as
).
Using properties in Lemma 2.1, we have
Thus
From the arbitrariness of, it has
Hence is precompact. Thus
has a convergent subsequence. Therefore
is asymptotically compact. This completes the proof of Theorem.
Corollary 1 Let be a semigroup of continuous operators in
. Then
has a bounded absorbing set and it is asymptotically null in
has a bounded absorbing set and it is
-limit compact.
Proof. By Corollary 3.4 in [7], we have is asymptotically compact in
if and only if
is asymptotically null in
and
is bounded in
provided
is bounded and
.
Using the Theorem 3.1 above , we have is
-limit compact in
if and only if
is asymptotically null in
and
is bounded in
provided
is bounded and
. Thus the necessity of the corollary is obvious.
If has a bounded absorbing set,
is bounded and
, then there exists
such that
is contained in the bounded absorbing set.
is a finite set in
, so it is bounded. Thus
is bounded. Now we can have the sufficiency immediately. This completes the proof of Corollary.
4. Acknowledgements
The author wishes to thank professor Yongluo Cao for his invaluable suggestions and encouragement.
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