Notes on the Global Attractors for Semigroup

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.


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.0

C
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: 0 C 1) there is an absorbing set, and 2) the semigroup is  -limit compact.
A well-known result (see [2][3][4][5][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   -limit compact, and then we prove that the conditions in [1,7] are equivalent directly in

 
p   .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][9][10].We point out that our work in this paper also can be extended to pullback attractors.

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 0 admits a finite cover by sets of diameter Lemma 2.1 Let M be a complete metric space, and  be the measure of noncompactness of a set.
whenever ; M is a complete metric space, thus for any , there exists a finite subset 0 of such that the balls of radii centered at 0 form a finite covering of .By Definition 2.1, admits a finite cover by sets of diameter .The arbitrariness of  implies that .
, 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.
C is a finite cover of 1 , and is a finite cover of , then is a finite cover of , thus 3) If 1 , then the finite cover of must be a finite cover of , so .
The finite cover of 1 2 must be a finite cover of both of and .So we have and 2 and the diameter of is less than

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 M be a complete metric space.A one parameter family of maps is the identity map on M, 2) for all , Let  be a positive smooth function on R and 0 p    .Then define a weighted space as , the following holds 0 t be a semigroup in a complete metric space M, then we can have: 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 Thus there exists and Next we take hence . That is to say has no finite   Thus there exists and such that Otherwise is the finite Repeat the previous procedure, then we have the sequence which satisfies By the way of taking 0 , and has a convergent subsequence.This gives contradiction to (1).
Next, we prove the sufficiency.We need to prove that for every bounded subset , for any and any n , has a convergent subsequence.
 -limit compact, then for the bounded subset above, for any , there exists such that For , there exists , such that when .implies contains only a finite number of elements (where is fixed such that as n ).

N n
Using properties in Lemma 2.1, we have For any , has a finite cover by sets of diameter finite cover by sets of diameter    .From the arbitrariness of  and Definition 2.1, we have  