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.

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.

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 all3) 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

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.

The author wishes to thank professor Yongluo Cao for his invaluable suggestions and encouragement.