Cheban D.N. Global Attractors Of Non-autonomous Dissipative Dynamical Systems

Подождите немного. Документ загружается.

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 1

Autonomous dynamical systems 27

As x

∈ B(M, δ

) and δ

↓ 0, the sequence {x

} can be considered convergent.

Assume x

= lim

n→+∞

. Then x

∈ M and t

≥ L(ε

) for suﬃciently large n.

According to (1.36) holds the equality

ρ(x

, M) <

. (1.37)

The inequalities (1.35) and (1.37) are contradictory. The obtained contradiction

completes the proof of the lemma. 

Theorem 1.18 For a point dissipative dynamical system (X, T, π) to be local

dissipative, it is necessary and suﬃcient that the following two conditions hold:

1. D

(Ω) (J

(Ω)) is compact;

2. D

(Ω) (J

(Ω)) is uniformly attracting set.

Proof. Let (X, T, π) be point dissipative and the conditions 1. and 2. of the

theorem are fulﬁlled. Since according to Lemma 1.19 the set D

(Ω) (J

(Ω)) is

orbital stable, then according to Theorem 1.14 the dynamical system (X, T, π) is

compact dissipative and D

(Ω) (J

(Ω)) coincides with its center of Levinson J.

Now to ﬁnish the proof of the theorem it is suﬃcient to refer to Theorem 1.17.

Let (X, T, π) be local dissipative. As according to Lemma 1.5 the dynamical

system (X, T, π) is compact dissipative. Then according to Theorem 1.14 D

(Ω)

(Ω)) is compact and orbital stable. By Theorem 1.11 and Corollary 1.5 the

equality J = D

(Ω) (J = J

(Ω)) holds, where J is the center of Levinson of

(X, T, π). From Theorem 1.17 follows that D

(Ω) (J

(ω)) is a uniformly attracting

set. The theorem is completely proved. 

Deﬁnition 1.27 We will call the dynamical system (X, T, π) local asymptotically

condensing if for every point p ∈ X there exist δ

> 0 and a nonempty compact

⊆ X such that

lim

t→+∞

β(π

B(p, δ

), K

) = 0. (1.38)

The following holds:

Theorem 1.19 Let (X, T, π) be point dissipative. For (X, T, π) to be local dissipa-

tive it is necessary and suﬃcient that (X, T, π) be locally asymptotically condensing.

Proof. If (X, T, π) is locally dissipative, then it is clear that (X, T, π) is asymp-

totically condensing too. Suppose that (X, T, π) is point dissipative and locally

asymptotically condensing and let us show that (X, T, π) is local dissipative. First

of all, let us show that for every compact K ∈ C(X) the set Σ

(K) is relatively

compact. Let p ∈ K, δ

> 0 and K

∈ C(X) be such that the equality (1.38) is

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 1

28 Global Attractors of Non-autonomous Dissipative Dynamical Systems

fulﬁlled. By compactness of K from its open covering {B(p, δ

)| p ∈ K} we can ex-

tract ﬁnite subcovering {B(p

, δ

)| 1 ≤ i ≤ n}. Assume W = K

∪K

∪. . . ∪K

Then W is compact and

lim

t→+∞

β(π

K, W ) = 0. (1.39)

From the equality (1.39) follows relative compactness of Σ

(K). According to

Theorem 1.15 the dynamical system (X, T, π) is compact dissipative. Let now J be

the center of Levinson of (X, T, π), and that p ∈ J, δ

> 0, and K

∈ C(X) are such

that the equality (1.38) holds. By Lemma 1.3 the set ω(B(p, δ

)) 6= ∅, is compact,

invariant and the equality

lim

t→+∞

β(π

B(p, δ

), ω(B(p, δ

))) = 0 (1.40)

holds. Since J is maximal compact invariant set of (X, T, π), then ω(B(p, δ

)) ⊆ J

and from the equality (1.40) follows the equality (1.29). According to Theorem 1.16

the dynamical system (X, T, π) is local dissipative. The theorem is proved. 

At the end of this paragraph, we give an example of compact dissipative

dynamical system that is not local dissipative.

Example 1.8 Consider a linear diﬀerential equation

˙x = Ax (1.41)

in the Hilbert space H := L

[0, 1] with the continuous operator A : L

[0, 1] →

[0, 1] deﬁned by the equality (Aϕ)(τ) = −τϕ(τ) for all τ ∈ [0, 1] and ϕ ∈ L

[0, 1].

Note that the spectrum of the operator A coincides with the segment [−1, 0]. Denote

by U(t) the Cauchy operator of the equation (1.41). It is clear that (U(t)ϕ)(τ ) =

−τt

ϕ(τ) for all t ∈ R, τ ∈ [0, 1] and ϕ ∈ L

[0, 1] (see

[

250, p.404

]

). Denote by

(H, R, π) the dynamical system generated by the equation (1.41) i.e. π(ϕ, t) :=

U(t)ϕ for every t ∈ R and ϕ ∈ L

[0, 1]. According to Theorem of Lebesgue on the

limit passage under the sign of integral we have

||π(t, ϕ)||

−2τt

|ϕ(τ)|

dτ → 0

as t → +∞. Hence, the dynamical system (H, R, π) is point dissipative and ω

{0} for every ϕ ∈ H and consequently Ω =

∪{ω

|ϕ ∈ H} = {0}. Further note that

||π(t, ϕ)||

−2τt

|ϕ(τ)|

dτ ≤

|ϕ(τ)|

dτ = ||ϕ||

for all t ≥ 0. According to Theorem 1.14 the dynamical system (H, R, π) is compact

dissipative and its Levinson center J = {0}.

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 1

Autonomous dynamical systems 29

Let us show that the constructed dynamical system (H, R, π) is not local dissi-

pative. In fact, if (H, R, π) were local dissipative then according to Theorem 1.17

there would be γ > 0 such that

lim

t→+∞

sup

||ϕ||≤γ

||π(t, ϕ)|| = 0. (1.42)

By virtue of the linearity of the system (H, R, π) the equality (1.42) is equivalent

to the condition

lim

t→+∞

sup

||ϕ||≤1

||π(t, ϕ)|| = 0. (1.43)

Deﬁne the function ϕ

∈ H (n = 1, 2, . . .) by the following rule: ϕ

(τ) :=

√

nχ

(τ)

for all τ ∈ [0, 1] where χ

is the characteristic function of the set [0, n

−1

] ⊆ [0, 1].

Note that ||ϕ

|| = 1 and

||π(t

, ϕ

)||

−2τt

dτ

1 − e

−

= 1 −e

−1

6→ 0, (1.44)

where t

. However, (1.43) and (1.44) cannot take place simultaneously. The

obtained contradiction shows that our assumption about the local dissipativity of

(H, R, π) is not true. The necessary example is constructed.

1.7 Global attractors

Deﬁnition 1.28 A nonempty compact set I ⊂ X we will call the global attractor

of the dynamical system (X, T, π) if the following conditions are fulﬁlled:

a. I is invariant w.r.t. (X, T, π);

b. I attracts all the bounded subsets from X.

Theorem 1.20 The following conditions are equivalent:

1. the dynamical system (X, T, π) admits a compact global attractor;

2. the dynamical system (X, T, π) is bounded k-dissipative;

3. the dynamical system (X, T, π) is compact k-dissipative and its center of Levin-

son attracts all bounded subsets of X.

Proof. The implication 1. ⇒ 2. is obvious. Let us show that from 2. follows 3. Let

(X, T, π) be bounded k-dissipative, then (X, T, π) is compact k-dissipative. Denote

by J the center of Levinson of (X, T, π) and show that J attracts all bounded subsets

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 1

30 Global Attractors of Non-autonomous Dissipative Dynamical Systems

from X. Let B ∈ B(X) and let K be a nonempty compact from X attracting all

bounded subsets from X, then

lim

t→+∞

β(π

B, K) = 0. (1.45)

From the equality (1.45) and Lemma 1.3 follows that ω(B) 6= ∅, is compact, invari-

ant and

lim

t→+∞

β(π

B, ω(B)) = 0. (1.46)

According to Theorem 1.6 J is maximal compact invariant set of (X, T, π) and

consequently ω(B) ⊆ J. From the last inclusion and the equality (1.46) follows

that

lim

t→+∞

β(π

B, J) = 0. (1.47)

At last, it is clear that from 3. follows 1. The theorem is proved. 

Deﬁnition 1.29 Following to

[

206

]

we will say that the dynamical system

(X, T, π) satisﬁes the condition of Ladyzhenskaya if for every bounded set B ⊆ X

there exists a nonempty compact K ⊆ X such that the equality (1.45) holds.

Theorem 1.21 The following conditions are equivalent:

1. (X, T, π) is bounded k-dissipative;

2. (X, T, π) is point b-dissipative and satisﬁes the condition of Ladyzhenskaya.

Proof. According to Theorem 1.20 from 1. follows 2. Let us now prove that from 2.

follows 1. First of all note that from the condition 2. follows compact k-dissipativity

of (X, T, π). Let K ∈ C(X), then from the condition of Ladyzhenskaya, Lemma 1.3

and Theorem 1.5 follows relative compactness of Σ

(K). According to Theorem

1.15 the dynamical system (X, T, π) is compact k-dissipative. Denote by J the

center of Levinson of (X, T, π) and let B ∈ B(X). Then by virtue of the condition

of Ladyzhenskaya and Lemma 1.3 ω(B) 6= ∅, is compact, invariant and the equality

(1.46) holds. As ω(B) ⊆ J, from the equality (1.46) follows that J attracts B. The

theorem is proved. 

Deﬁnition 1.30 We will call the dynamical system (X, T, π):

− completely continuous if for every B ∈ B(X) there exists l = l(B) > 0 such

that π

B is relatively compact;

− weakly b-dissipative if there exists a nonempty bounded set B

⊆ X such that

∩B

6= ∅ for every point x from X (i.e. for every x ∈ X there is τ = τ (x) ≥ 0

such that xτ ∈ B

). In this case the set B

we will call a weak b-attractor of

the dynamical system (X, T, π).

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 1

Autonomous dynamical systems 31

Theorem 1.22 Let (X, T, π) be weakly b-dissipative and completely continuous.

Then (X, T, π) admits a compact global attractor.

Proof. Let B

∈ B(X) be a weak attractor of the system (X, T, π), l

= l(B

) > 0

be such that π

is relatively compact and let K :=

. Under the conditions

of Theorem 1.22 for every point x ∈ X there exists τ = τ(x) ≥ 0 such that xτ ∈ B

and consequently x(τ + l

) ∈ K. So, the dynamical system (X, T, π) is weakly

k-dissipative and the compact K is its weak k-attractor. According to Theorem

1.10 the dynamical system (X, T, π) is compact k-dissipative. Let J be the center

of Levinson of (X, T, π), B ∈ B(X) and l = l(B) > 0 be such that π

B is relatively

compact. According to Theorem 1.6 the center of Levinson J attracts the compact

B and consequently holds the equality (1.47). The theorem is proved. 

Deﬁnition 1.31 The dynamical system (X, T, π) is called (see

[

179, 234

]

) asymp-

totically compact if for every bounded positive invariant set B ⊆ X (i.e. π

B ⊆ B

for all t ≥ 0) there exists a nonempty compact K ⊆ X such that the equality (1.45)

holds.

Theorem 1.23 Let (X, T, π) be compact k-dissipative and asymptotically com-

pact. Then (X, T, π) is local k-dissipative.

Proof. Let J be the center of Levinson of (X, T, π). According to Theorem 1.17

for (X, T, π) to be local k-dissipative it is necessary and suﬃcient to show that

the set J is uniformly attracting. Let ε

= 1 and δ

= δ(1) > 0 be such numbers

that from x ∈ B(J, δ

) follows xt ∈ B(J, 1) for all t ≥ 0 (according to Theorem 1.6

such δ

exists). It is clear that the set B := ∪{π

B(J, δ

)| t ≥ 0} is bounded and

positively invariant. By virtue of the asymptotic compactness of (X, T, π) there

exists a nonempty compact K ⊆ X such that the equality (1.45) holds . According

to Lemma 1.3 the set ω(B) 6= ∅, is compact, invariant and the equality (1.46) holds

. Since J is the maximal compact invariant set of (X, T, π), then ω(B) ⊆ J and,

in particular, from (1.46) follows (1.47). From the equality (1.47) and from the

inclusion B(J, δ

) ⊆ B follows the equality below

lim

t→+∞

β(π

B(J, δ

), J) = 0.

The theorem is proved. 

Lemma 1.20 Let (X, T, π) be b-dissipative w.r.t. the family M and asymptotically

compact. Then (X, T, π) is also k-dissipative w.r.t. the family M.

Proof. There exists a bounded closed set B

⊆ X such that for every M ∈ M

there is L > 0 for which π

M ⊆ B

for all t ≥ L. Assume B := {x ∈ B

|xt ∈ B

for all t ≥ 0}. It is clear that B 6= ∅, is bounded and positively invariant. By the

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 1

32 Global Attractors of Non-autonomous Dissipative Dynamical Systems

asymptotic compactness of (X, T, π) there exists a compact set K ⊆ B

such that

the equality (1.45) holds. According to Lemma 1.3 the set ω(B) 6= ∅, is compact

and the equality (1.46) holds. Note that M

:= ∪{π

M |t ≥ L} ⊆ B and M

is positively invariant. By virtue of the asymptotic compactness of (X, T, π) and

according to Lemma 1.3 the set ω(M

) 6= ∅, is compact, ω(M

) ⊆ ω(B) and ω(M

)

attracts M

. Now to ﬁnish the proof of the lemma it is suﬃcient to note that the

set M is attracted by the compact ω(M

), hence by the compact ω(B) too. In such

way we constructed the nonempty compact ω(B) attracting all set from the family

M. The lemma is proved. 

Corollary 1.7 Let (X, T, π) be point (compact, local, bounded) b-dissipative and

asymptotically compact. Then (X, T, π) is point (compact, local, bounded) k-

dissipative.

Corollary 1.8 Let (X, T, π) be asymptotically compact. Then local b-dissipativity

and compact b-dissipativity are equivalent.

Proof. The formulated statement follows from Corollary 1.7 and Theorem 1.23. 

Theorem 1.24 Let (X, T, π) satisfy the condition of Ladyzhenskaya. Then the

following conditions are equivalent:

1. the dynamical system (X, T, π) is weakly b-dissipative;

2. there exists a bounded set B

⊆ X absorbing all points from X, i.e. for every

x ∈ X there exists τ = τ (x) ≥ 0 such that xt ∈ B

for all t ≥ τ ;

3. the dynamical system (X, T, π) is point k-dissipative;

4. the dynamical system (X, T, π) is weakly k-dissipative;

5. the dynamical system (X, T, π) is bounded k-dissipative;

6. there exists a bounded set B

⊆ X absorbing all bounded subsets of X, i.e.

for every B ∈ B(X) there exists L = L(B) > 0 such that π

B ⊆ B

for all

t ≥ L(B).

Proof. It is clear that from 2. follows 1.. Let us show from 1. follows 3.. Let B

be a weak attractor of the system (X, T, π). As (X, T, π) satisﬁes the condition of

Ladyzhenskaya, the set K

:= ω(B

) 6= ∅, is compact and attracts the set B

. Since

for every x ∈ X there exists τ = τ(x) > 0 such that xτ ∈ B

and consequently

lim

t→+∞

ρ(xt, K

) = 0.

Obviously, from 3. follows 4. Let the condition 4. be fulﬁlled and K

⊆ X

be a nonempty compact such that ω

∩ K

6= ∅ for every x ∈ X, γ > 0 and

:= ω(B(K

, γ)). Under the conditions of the theorem the set K

6= ∅, is compact

and attracts B(K

, γ). According to the condition 4. for every x ∈ X there exists

τ(x) > 0 such that xτ ∈ B(K

, γ) and consequently x is attracted by the set K

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 1

Autonomous dynamical systems 33

So, (X, T, π) is point k-dissipative and according to Theorem 1.21 it is bounded

k-dissipative.

To ﬁnish the proof of the theorem it is suﬃcient to note that from 5. follows 6.

and from 6. follows 2. The theorem is proved. 

1.8 On a Problem of J. Hale

Let P : X → X be a continuous mapping and (X, P ) be a discrete dynamical

system, generated by the positive powers of mapping P.

Deﬁnition 1.32 A continuous mapping P : X → X is called a λ-contraction of

order k ∈]0, 1[, if λ(P (A)) ≤ kλ(A) for all A ∈ B(X), where λ is some measure of

non-compactness on X.

In the work

[

174

]

it was formulated the following problem

Problem (Problem of J. Hale). Let a discrete dynamical system (X, P ) be

pointwise b-dissipative and the mapping P be a λ-contraction. In the dynamical

system (X, P ), does a maximal compact invariant set exist?

In the relation to the problem of J. Hale it is interesting to note the following

problem.

Problem. Let a discrete dynamical system (X, P ) be pointwise b-dissipative.

To ﬁnd the conditions which guarantee existence in the dynamical system (X, P ) a

maximal compact invariant set.

Below we give some results in relation to this problem.

The example 1.6 shows, that there exist pointwise k-dissipative dynamical

systems which do not admit a maximal compact invariant set. On the other hand,

by Theorem 1.6 for the compact k-dissipative dynamical system, its Levinson center

is a maximal compact invariant set.

The following assertion holds.

Lemma 1.21 Let (X, T, π) be a pointwise k-dissipative dynamical system and

(Ω) be a compact set, then in (X, T, π) there is a maximal negatively invariant

compact set I ⊆ J

(Ω).

Proof. Denote by I

∗

the family of all nonempty negatively invariant compact

subsets A ⊆ J

(Ω). We note that I

∗

6= ∅, because according to the corollary 1.4

Ω ∈ I

∗

Denote by I a closure of union of all negatively invariant compact subsets of

(Ω). By Theorem 1.3 the set I is negative invariant and, consequently, it is

unknown. The lemma is proved. 

Corollary 1.9 Under the conditions of Lemma 1.21 the set I is invariant.

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 1

34 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Proof. In fact, since the set I is negatively invariant, then for every t ∈ T the

set π

I also is negatively invariant and compact and by Lemma 1.15 π

I ⊆ J

(Ω).

According to Lemma 1.21 I is a maximal compact negatively invariant set and,

consequently, π

I ⊆ I i.e. π

I = I for all t ∈ T. 

Thus, if (X, T, π) is pointwise k-dissipative and J

(Ω) is compact, then in

(X, T, π) there exists a maximal compact invariant set I ⊆ J

(Ω). On the other

hand according to the example 1.7, there are pointwise k-dissipative dynamical

systems, but not compact k-dissipative, for which the set J

(Ω) is compact.

Lemma 1.22 Let T = R

. Dynamical system (X, T, π) is asymptotic compact,

if and only if for some t

> 0 the discrete dynamical system (X, P ) is asymptotic

compact, where P := π

: X −→ X.

Proof. It easy to see that if the dynamical system (X, T, π) is asymptotic compact,

then the discrete dynamical system (X, P ) is asymptotic compact too, where P :=

: X −→ X and t

is a arbitrary positive number. Let M ⊆ X be a bounded

positively invariant set of (X, T, π). Then P (M) ⊆ M and according to asymptotic

compactness of discrete dynamical system (X, P ) there exists a compact K

∈ C(X)

such that

lim

n→+∞

β(P

(M), K

) = 0. (1.48)

Let K := π([0, t

], K

). Then by the continuity of mapping π : T × X −→ X and

by compactness of the set [0, t

] ×K

the set K is also compact. We will show that

the following equality lim

t→+∞

β(π

M, K) = 0 holds. In fact, if we suppose that it is

false, then there exist ε

> 0, t

→ +∞ and {x

} ⊆ M such that

ρ(x

, K) ≥ ε

. (1.49)

Let t

= k

+τ

, where k

is a integer part of the number t

and τ

∈ [0, t

then π(t

, x

) = π(τ

, π(k

, x

)). Taking into account the equality (1.48) we

may suppose that the sequence {π(k

, x

)} is convergent. Let τ

:= lim

n→+∞

and y

:= lim

n→+∞

π(k

, x

), then lim

n→+∞

π(t

, x

) = π(τ

, y

). Since y

∈ K

then π(τ

, y

) ∈ K. On the other hand, passing to limit in the inequality (1.49) as

n → +∞, we obtain ρ(π(τ

, y

), K) ≥ ε

, i.e. y

/∈ K. The obtained contradiction

completes he proof of Lemma. 

Deﬁnition 1.33 Recall (see

[

175, 179

]

and

[

278

]

) that the mapping P : X −→ X

is called condensing with respect to measure of the non-compactness λ, if λ(P (A)) <

λ(A) for all A ∈ B(X) with condition λ(A) > 0.

Lemma 1.23

[

175

]

Let P : X → X be a continuous λ-condensing mapping. Then

the discrete dynamical system (X, P ) is asymptotic compact.

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 1

Autonomous dynamical systems 35

Proof. At ﬁrst we will show, that for every bounded set B ⊂ X with condition

P (B) ⊆ B and every sequence {k

} with integer k

→ +∞ the set {P

} is

relatively compact. Let C = {{P

} : {x

} ⊆ B, {k

} are integers, 0 ≤ k

→

+∞}. We put η := sup{λ(h) : h ∈ C}. Since h ⊆ B, if h ∈ C, then η <

λ(B).We will prove the existence h

∗

∈ C such that λ(h

∗

) = η. Let {h

} be a

sequence of elements of the set C for which λ(h

) → η. We will deﬁne



: P

∈ h

, k

> l



. Then λ(

) = λ(h

). If h

∗

= ∪{

| l = 1, 2, . . .}

is ordered in arbitrary way, then h

∗

∈ C and λ(h

∗

) ≥ λ(h

) for every l. Thus

λ(h

∗

) = η. We put

∗



−1

: P

∈ h

∗



. Then

∗

∈ C and λ(

∗

) ≤ η.

ButP (

∗

) = h

∗

, so that λ(P (

∗

)) ≥ λ(

∗

) and, consequently, λ(

∗

) = 0 because

the mapping P is λ-condensing. Therefore

∗

and h

∗

are relatively compact and

η = 0. Thus, each sequence in C is relatively compact.

By Lemma 1.3 the set ω(B) 6= ∅, compact, invariant and the equality

lim

n→+∞

β(P

(B), ω(B)) = 0 (1.50)

holds. Lemma is proved. 

Deﬁnition 1.34 A dynamical system (X, T, π) is called locally bounded, if for

every x ∈ X there are δ

> 0 and l

> 0 such that the set ∪{π

B(x, δ

) : t ≥ l

} is

bounded.

The example 1.6 shows, that there exist pointwise dissipative and locally

bounded dynamical systems that are not compact dissipative.

Deﬁnition 1.35 A dynamical system (X, T, π) is called λ-condensing, if

λ(π

B) < λ(B) for all B ∈ B(X), t ∈ T, λ(B) > 0 and π

B ∈ B(X).

Theorem 1.25 Let (X, T, π) be a pointwise b-dissipative, locally bounded and

λ-condensing dynamical system, then (X, T, π) is locally k-dissipative.

Proof. Let (X, T, π) be a λ-condensing dynamical system. Then by Lemmas 1.22

and 1.23 the dynamical system (X, T, π) is asymptotic compact. From the corollary

1.7 it follows that the dynamical system (X, T, π) is pointwise k-dissipative. Let

p ∈ X, δ

> 0 and l

> 0 be numbers from the deﬁnition of local boundedness

of dynamical system (X, T, π). Consider a set M

:= ∪{π

B(p, δ

) : t ≥ l

}. By

asymptotic compactness of dynamical system (X, T, π) there exists a compact set

∈ C(X) such that

lim

t→+∞

β(π

B(p, δ

), K

) = 0. (1.51)

According to Theorem 1.19 the dynamical system (X, T, π) is local dissipative. The

theorem is proved. 

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 1

36 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Theorem 1.26 Let (X, T, π) be a pointwise b-dissipative and λ-condensing

dynamical system. If for any p ∈ Ω there exist δ

> 0 and l

> 0 such that

∪{π

B(p, δ

)| t ≥ l

} ∈ B(X), then (X, T, π) is local k-dissipative.

Proof. Let (X, T, π) be pointwise dissipative and λ-condensing. Then by Lem-

mas 1.22 and 1.23 the dynamical system (X, T, π) is asymptotic compact. Ac-

cording to Corollary 1.7 the dynamical system (X, T, π) is pointwise k-dissipative

and, consequently, the set Ω is nonempty and compact. From the open cov-

ering {B(p, δ

)| p ∈ Ω} of the compact set Ω we choice a ﬁnite sub-covering

{B(p

, δ

) | i ∈

1, m}. We put l

:= max{l

| i ∈ 1, m}. By Lemma 1.9, there

exists γ > 0 such that B(Ω, γ) ⊂ ∪{B(p

, δ

)| i =

1, m}. If x ∈ X, then there

exists l(x) > 0 such that xl ∈ B(Ω, γ). Since the set B(Ω, γ) is open, then there

exists a number ε > 0 such that B(xl, ε) ⊂ B(Ω, γ). According to the continuity of

mapping π

: X −→ X, there exists a number δ

> 0 such that yl ∈ B(xl, ε) for all

y ∈ B(x, δ) and, consequently, the set ∪{π

B(x, δ)| t ≥ l

+ l(x)} is bounded. To

ﬁnish the proof it is suﬃcient to use Theorem 1.25. 

Theorems 1.25 and 1.26 give a solution of the problem of J. Hale for the local

bounded dynamical systems.

Deﬁnition 1.36 A dynamical system (X, T, π) is called compact bounded, if for

every compact K ∈ C(X) the set Σ

(K) is bounded, i.e. Σ

(K) ∈ B(X).

The following theorem holds:

Theorem 1.27 Let (X, T, π) be a pointwise b-dissipative, compact bounded and

λ-condensing dynamical system. Then (X, T, π) is compact b-dissipative.

Proof. By Lemmas 1.22 and 1.23 the dynamical system (X, T, π) is asymptotic

compact and, consequently, (X, T, π) is pointwise k-dissipative. Since the dynamical

system (X, T, π) is compact bounded, asymptotic compact and the set Σ

(K) is pos-

itively invariant, then the set Σ

(K) is relatively compact for every K ∈ C(X) and

according to Theorem 1.15 the dynamical system (X, T, π) is compact k-dissipative.



Deﬁnition 1.37 A dynamical system (X, T, π) is called bounded, if for every

B ∈ B(X) the set Σ

(B) is bounded, i.e. Σ

(B) ∈ B(X).

Theorem 1.28 Let (X, T, π) be a bounded and λ-condensing dynamical system.

Then (X, T, π) is bounded k-dissipative.

Proof. This assertion can be proved using the same arguments as in the proof of

Theorem 1.25. 