Flattening Property and the Existence of Global Attractors in Banach Space

This paper analyses the existence of global attractor in infinite dimensional system using flattening property. The earlier stage we show the existence of the global attractor in complete metric space by using concept of the ω-limit compact concept with measure of non-compactness methods. Then we show that the ω-limit compact concept is equivalent with the flattening property in Banach space. If we can prove there exist an absorbing set in the system and the flattening property holds, then the global attractor exist in the system.


Introduction
Today many of the physical problems in infinite dimensional system can be represented in mathematical models studied in a strongly continuous semigroup, 0 semigroups, i.e. the operator family { } that maps a metric space M to itself and satisfies the properties [1,2] The long time dynamics of an infinite dimensional system can be described by the global attractor. Global attractors is the set of connected, compact, and invariant set which attracts all bounded set [2,3]. Normally, the existence of the global attractor can be proved by showing that: 1). the absorbing set exist in the system, and 2). the semigroup of the system is compact uniform. Unfortunately, it is very difficult to show that the semigrup is uniformly compact in many problems. Thus in [3], it is given conditions which can be easily proven by using the measure of noncompactness (MNC) called -limit compact. The existence of the global attractor for a 0 semigroup can be proved if and only if: 1). the absorbing set exist in the system, and 2). the semigroup is -limit compact.  The use of the MNC concept as well, is used by [3] to show the -limit compact of a 0 semigroup in a convex Banach space . It is shown to be equivalent with the condition known as flattening property. In [4], flattening property is defined as: Let : → be a semigroup, and B be any bounded set of X. Then for any ℰ > 0, there exist ( ) > 0 and a finite dimensional subspace 1 where : → 1 is a bounded projection, and : → . Furthermore, from the paper [2] the authors provide that the existence of a global attractors can be proved by showing that 1). the absorbing set exist in the system, and 2). the flattening property holds. From that explanation, we are interested to re-examine the concept in detail to show the existence of a global attractor by using the measure of non-compactness property. The assessment was carried out in this paper.

Measure of Non-compactness (MNC)
Some concepts of measure of non-compactness and its basic property will be given in this section, (see

Lemma 2.3 Let be a complete metric space and be the measure of non-compactness. Assume that ( ) is a sequence of bounded and closed subset of , and satisfying
is nonempty compact set.

The Concept of Global Attractor
In this section, we will provide some definitions from global attractors in dynamic systems related to our paper, ([1, 4, 5]) Definition 3.1 Let { } ≥0 be a 0 semigroup in a complete metric space . A subset 0 of is called an absorbing set in if for any bounded subset of , there exists some 1 ≥ 0 such that ⊂ 0 for all ≥ 1 .

Main Result
In this section, firstly we recall some basic lemmas in [3,5], secondly we show the existence of global attractor by concept -limit compact with measure of noncompactness methods in a metric space complete. Then we show that concept -limit compact equivalent with the flattening property in Banach space. Theorem 4.1 Let { } ≥0 be a 0 semigroup in a complete metric space . Assume that 1). is −limit compact, 2). there exists a bounded absorbing subset of . (⟹) Now we will prove the opposite. Since is a global attractor, then attractor and compact (see Definition 3.4). So that the neighborhood ( ) is absorbing set. According to the Definition 3.4 ( ) absorbs all bounded sets of . Next we will prove that the −limit compact. Since is bounded set, then ⋃ ( ) ≥ is bounded for ( ) ≥ 0, and for any ≥ ( ). Since ( ) is the absorbing set then  To show that the semigroup is compact, then it is the same as we show that Since is compact, there exists a finite number of elements 1 , 2 , … , ∈ such that ⊂ ⋃ ( , 4 ) .

=1
Let ∈ , and diameter less than then Since is uniformly convex Banach space, and let 1 = span{ 1 , … , }, there exists a projection : → 1 such that dist( , ) = ‖( − ) ‖ = dist( , 1 ), for any ∈ . Hence, Since is bounded, then ⋃ ( ) ≥ is bounded in (see theorem of close set on [6]). Hence the following equation is bound in . ( From (2) and (3), Definition 2.2 is met. Namely flattening property holds true. ■ Theorem 4.4 Let { } ≥0 be a 0 semigroup in a Banach space . If the following conditions 1). There exist a bounded absorbing set ⊂ , and 2). { } ≥0 satisfies the flattening property hold true, then there is a global attractor for { } ≥0 in . Proof: Using Theorem 4.3, it has been shown that { } ≥0 meets the flattening property. In addition, referring to Theorem 4.2 we can conclude that there is a bounded absorbing set ⊂ . Therefore the conditions in Theorem 4.4 have been met. The existence of a global attractor for { } ≥0 in is guaranteed to be true.