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

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Non-autonomous Dissipative Dynamical Systems 97

This last inclusion contradicts (2.93), and this ﬁnishes the proof of the fourth

assertion of the theorem.

Let us prove the ﬁfth assertion of the theorem. In order to do it let us notice

that w ∈ I

, if ϕ(t, w, y) is deﬁned on S and ϕ(S, w, y) is relatively compact. In

fact, as w = ϕ(t, ϕ(−t, w, y), y

−t

) for all t ∈ S, then from the equality (2.87) it

follows the inclusion we need. Thus, we get the following description of the set

: I

= {w ∈ W | there exists at least one whole trajectory of hW, ϕ, (Y, S, σ)i,

passing through the point (x, y)}. Now it remains to notice that the Levinson’s

center J is compact and consists of the whole trajectories of (X, S

, π), and, hence,

⊆ I

for all y ∈ Y .

The compactness of the set I it follows from the equality I = pr

J, from com-

pactness of J and from continuity of pr

: X → W .

The last assertion it follows from the next observation: under the conditions of

the theorem Levinson’s center J of the dynamical system (X, S

, π), according to

Corollary 1.11 and Theorem 1.33, is connected. Hence, I, as a continuous image of

a connected set, also is connected. The theorem is completely proved. 

Remark 2.8 Theorem 2.24 reﬁnes and generalises the main results of

[

110

]

and

[

217

]

Theorem 2.25 Under the conditions of Theorem 2.24 the following statements

hold true:

(1) w ∈ I

(y ∈ Y ) if and only if there exits a whole trajectory ν : S → W of the

cocycle ϕ, satisfying the following conditions: ν(0) = w and ν(S) is relatively

compact;

(2) I

(y ∈ Y ) is connected, if the space W possesses the property (S).

Proof. To prove the ﬁrst assertion we note that the continuous function ν : S → W

is a whole trajectory of the cocycle hW, ϕ, (Y, S, σ)i if and only if γ = (ν, Id

) is a

whole trajectory of the semi-group dynamical system (X, S, π) ( X = W × Y, π =

(ϕ, σ)). By Lemma 2.16, the dynamical system (X, S, π) is compactly dissipative

and according to Theorem 1.6 the set J is compact and invariant and, consequently,

the point (w, y) = x ∈ J if and only if through the point (w, y) = x passes the whole

trajectory γ = (ν, Id

) of the dynamical system (X, S, π), which belongs to J, i.e.

γ(0) = (ν(0), y) = (w, y) and γ(s) ∈ J for all s ∈ S. To ﬁnish the proof of the ﬁrst

statement of the theorem it is suﬃcient to cite Theorem 2.24 (item 5.).

To prove the second statement of the theorem we note that under the conditions

of Theorem 2.24 the set I

6= ∅ and is compact. Since the space W possesses the

property (S), then there exists a connected compact V ⊇ I. By (2.86), the following

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98 Global Attractors of Non-autonomous Dissipative Dynamical Systems

equality

lim

t→+∞

β(U(t, σ

−t

y)(V ), I

) = 0

holds. Note, that I

⊆ U(t, σ

−t

y)(V ) for all y ∈ Y and t ∈ S

; the mapping

U(t, σ

−t

y) : W → W is continuous and, consequently, the set U(t, σ

−t

y)(V ) is

compact and connected. To ﬁnish the proof of the theorem it is suﬃcient to cite

Lemma 3.12 from

[

126

]

. 

2.8 Global attractors of non-autonomous dynamical system with

minimal base

Everywhere in this paragraph we suppose that h(X, T

, π), (Y, T

, σ), hi is a non-

autonomous dynamical system, Y is a compact minimal set and (X, h, Y ) is a locally

trivial Banach ﬁbering.

Theorem 2.26 Let the following conditions be satisﬁed:

(1) (X, T

, π) is completely continuous, that is, for any bounded set A ⊆ X there

exits l = l(A) > 0 such that π

(A) is relatively compact;

(2) all motions (X, T

, π) are bounded on T

, that is, sup{|xt| |t ∈ T

} < +∞

for any x ∈ E ;

(3) there exit y

and R

> 0 such that for any x ∈ X

there exist τ = τ (x) ≥ 0

for which

|xτ| < R

. (2.95)

Then the non-autonomous dynamical system h(X, T

, π), (Y, T

, σ), hi admits a

compact global attractor.

Proof. Let R > R

, then for any x ∈ X there exits τ = τ (x) ≥ 0 such that

|xτ| < R. If it were not so, then there would be R

> R

any x

∈ E such that

τ| > R

(2.96)

for all τ ≥ 0. As the dynamical system (X, T

, π) is completely continuous and the

motion π(t, x

) is bounded on T

, the point x

is stable L

and, as Y is minimal,

the set ω

∩ X

is nonempty. Thus, according to Condition (2.96) we have

|xt| ≥ R

(2.97)

for all x ∈ ω

∩ X

and t ≥ 0. The inequality (2.97) contradicts (2.95). This

contradiction proves the assertion we need. Now to ﬁnish the proof of the theorem

it is suﬃcient to cite Theorem 2.19. 

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Non-autonomous Dissipative Dynamical Systems 99

Remark 2.9 1. For ﬁnite-dimensional systems (that is the vector ﬁbering

(X, h, Y ) is ﬁnite-dimensional) Theorem 2.26 reﬁnes Theorem 2.6.1 from

[

209

]

. To

be exact, the condition of uniform boundedness is changed for ordinary boundedness

of trajectories of (X, T

, π).

2. If the condition of minimality of Y in Theorem 2.26 is taken away, then

Theorem 2.26 is not true even in the class of linear non-autonomous systems. This

is proved by the example below.

Example 2.3 Let us consider a linear diﬀerential equation

= a(t)x, (2.98)

where a ∈ C(R, R) is deﬁned by the equality a(t) := −1 + sin t

1/3

. Let us remark

the next properties of the function a and of the equation (2.98):

1. a

(t) → 0 for t → +∞ ;

2. a(t) ∈ [−2, 0] for all t ∈ R ;

3. {a

|τ ≥ 0} is relatively compact in C(R, R), where a

(t) = a(t + τ)(t ∈ R);

4. ω

6= ∅ and is compact ;

5. all functions from ω

are constant and b(t) = c ∈ [−2, 0](t ∈ R) for any b ∈ ω

;

6. a(t

) = 0 if and only if t

= −1 + (

+ 2πn)

(n ∈ Z);

7. there exits {t

} ⊆ {t

} such that a(t + t

) → b(t) and b(t) = 0 for all t ∈ R ;

8. for any b ∈ H

(a) =

|τ ∈ R

} the inequality

|ϕ(t, x, b)| ≤ |x| (2.99)

holds for all x ∈ R and t ∈ R

, where ϕ(t, x, b) is the solution of the equation

(t) = b(t)y; (2.100)

passing through the point x ∈ R for t = 0.

9. if b ∈ ω

\{0}, then b(t) = c < 0(t ∈ R) and, hence,

lim

t→+∞

|ϕ(t, x, b)| = 0 (2.101)

for all x ∈ R ;

10. if b = 0(b ∈ ω

), then ϕ(t, x, b) = x for all t ∈ R.

Suppose Y := H

(a) and denote by (Y, R

, σ) the dynamical system of transla-

tions on Y . Let X := R×Y and (X, R

, π) be a semigroup dynamical system on X,

where π = (ϕ, σ) (that is π(t, (x, b)) = (ϕ(t, x, b), b

) for all (x, b) ∈ X and t ∈ R

Then h(X, R

, π), (Y, R

, σ)i is a non-autonomous dynamical system generated by

the equation (2.98), where h = pr

: X → Y . From Properties 1-10 it follows that

for the non-autonomous dynamical system h(X, R

, π), (Y, R

, σ), hi generated by

the equation (2.98) all the conditions of Theorem 2.26 are carried out excepting the

minimality of Y and that it has no compact global attractor.

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100 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Corollary 2.9 Let (X, T

, π) be completely continuous and for any y ∈ Y let

there exist R(y) ≥ 0 such that

lim sup

t→+∞

|xt| ≤ R(y) (2.102)

for any x ∈ X

. Then the non-autonomous dynamical system h(X, T

, π),

(Y, T

, σ), hi admits a compact global attractor.

Proof. This assertion it follows from Theorem 2.26, if we will notice that from

condition (2.102) it follows the boundedness of every motion from (X, T

, π) on T

.

Theorem 2.27 Let the next conditions are carrying out:

(1) (X, T

, π) is asymptotic compact, that is for any bounded positively invariant

set A ⊂ X there exits a nonempty compact K

such that

lim

t→+∞

β(π

A, K

) = 0; (2.103)

(2) (X, T

, π) is asymptotic bounded, that is for any bounded set A ⊂ E there exits

l = l(A) ≥ 0 such that ∪{π

A|t ≥ l} is bounded;

(3) there exit y

∈ Y and R

> 0 such that (2.95) is fulﬁlled.

Then the non-autonomous dynamical system h(X, T

, π), (Y, T

, σ), hi admits the

maximal compact attractor.

Proof. First, let us notice, that in conditions of Theorem the dynamical system

(X, T

, π) satisﬁes the condition of Ladyzhenskaya. Let R > R

, then for any x ∈ E

there will be τ = τ(x) ≥ 0 such that |xτ| < R. If we suppose that it is not so, then

there exist x

∈ E and R

> R

such that

τ| ≥ R

> R

(2.104)

for all τ ≥ 0 and, hence, ω

∩ X

6= ∅. That is why for any x ∈ ω

∩ X

the

inequality (2.97) holds, but this contradicts (2.95). Thus the assertion we need is

proved. Now for ending the proof of the Theorem it is suﬃcient to cite theorem

1.24. 

Remark 2.10 Let us notice, that Theorem 2.27 without demanding the minimal-

ity of Y does not take place even in class of linear systems. The last assertion is

proved by the example 2.3.

Theorem 2.28 Let (X, h, Y ) be a ﬁnite-dimensional vectorial ﬁbering, all the

motions (X, T

, π) are bounded on T

, Y be a compact minimal set and y

∈ Y ,

then the following conditions are equivalent:

1. the non-autonomous dynamical system h(X, T

, π), (Y, T

, σ), hi is dissipative;

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Non-autonomous Dissipative Dynamical Systems 101

2. there exits R > 0 such that

lim sup

t→+∞

|xt| < R (2.105)

for all x ∈ X

and all motions (X, T

, π) are bounded on T

;

3. there exits a positive number r such that for any x ∈ X

and l > 0 there will

be τ = τ(x) ≥ l for which

|xτ| < r (2.106)

and all the motions (X, T

, π) are bounded on T

;

4. there exits a nonempty compact K

⊂ X such that ω

∩K

6= ∅ for all x ∈ X

and all the motions (X, T

, π) are bounded on T

;

5. there exits a nonempty compact K

⊆ X such that ω

6= ∅ and ω

⊆ K

for all

x ∈ X

;

6. there exits a positive number R

such that for any R

> 0 there will be l(R

) > 0,

that

|xt| < R

(2.107)

for all t ≥ L(R

), |x| ≤ R

(x ∈ X

Proof. Implications 1.=⇒ 6. =⇒ 2. =⇒ 5. =⇒ 4. =⇒ 3. are evident. According to

Theorem 2.26 from 3. it follows 1. The theorem is proved. 

2.9 Homogeneous dynamical systems

The goal of the present section is to study the connection between uniform asymp-

totic stability and exponential asymptotic of the solutions of inﬁnite-dimensional

systems. This problem is studied and solved within the framework of general non-

autonomous dynamical system with inﬁnite-dimensional phase space.

Let (X, h, Y ) be a locally trivial ﬁber bundle

[

190

]

with the ﬁber E, (X, ρ) be a

complete metric space, T = S

:= {s ∈ S | = s ≥ 0}, where S := R or Z.

Deﬁnition 2.21 h(X, T

, π),(Y, T

, σ), hi (T

⊆ T

⊆ S) is said to be homoge-

neous of order k = 1, if π(t, λx) = λπ(t, x) for all λ > 0, x ∈ X and t ∈ T

Deﬁnition 2.22 An autonomous dynamical system (X, R

, π) is said to be ho-

mogeneous of order k (k ≥ 1), if π(t, λx) = λ

π(λ

k−1

t, x) for all λ ≥ 0, x ∈ X and

t ∈ T

If x ∈ X, then we put |x| := ρ(x, θ

h(x)

), where θ

(y ∈ Y ) is null (trivial) element

of the linear space X

and Θ := {θ

| y ∈ Y } is a null (trivial) section of the vectorial

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102 Global Attractors of Non-autonomous Dissipative Dynamical Systems

bundle (X, h, Y ). Denote by X

a stable manifold of h(X, T

, π), (Y, T

, σ)i, i.e.

= {x | x ∈ X, lim

t→+∞

|π(t, x)| = 0}.

Lemma 2.17 Let h(X, T

, π), (X, T

, σ), hi be a homogeneous dynamical system

of order k = 1. Then

(1) if Ω

is compact, then Ω

⊆ Θ;

(2) if J

(Ω

) is compact, then J

(Ω

) ⊆ Θ.

Proof. To prove the inclusion Ω

⊆ Θ it is suﬃcient to show that ω

⊆ Θ for all

x ∈ X. Let x ∈ X and p ∈ ω

, then by the homogeneity of h(X, T

, π), (Y, T

, σ), hi

of order k = 1 we have λp ∈ ω

λx

for all λ ≥ 0. If we suppose that for some x

the

inclusion ω

⊆ Θ it is not true, then there exits p ∈ ω

\Θ and, hence, λp ∈ Ω for

all λ ≥ 0. The last fact contradicts the compactness of Ω.

Let now J

(Ω

) be compact. We will show that J

(Ω

) ⊆ Θ. If p ∈ J

(Ω

then there exit q ∈ Ω

, q

→ q and t

→ ∞ such that p = lim

n→+∞

π(t

, q

) and

λp = lim

n→+∞

λπ(t

, q

) = lim

n→+∞

π(t

, λq

). Since λq ∈ Ω

along with the point q,

then λp ∈ J

λq

⊆ J

(Ω

) for all λ ≥ 0 and p ∈ J

(Ω

). If J

(Ω

) * Θ, then

should be the point p

∈ J

(Ω

) \ Θ and, consequently, λp

∈ J

(Ω

) ∀λ ≥ 0.

The last inclusion contradicts the compactness of J

(Ω

). The lemma is proved.

Corollary 2.10 Let h(X, T

, π), (Y, T

, σ), hi be a homogeneous non-autonomous

dynamical system of order k = 1. Then the following conditions are equivalent:

(1) h(X, T

, π), (Y, T

, σ), hi is compactly dissipative;

(2) h(X, T

, π), (Y, T

, σ), hi is convergent.

Theorem 2.29 Let h(X, T

, π), (Y, T

, σ), hi be homogeneous of order k = 1 and

(Y, T

, σ) be pointwise k-dissipative. Then the following conditions are equivalent:

(1) h(X, T

, π), (Y, T

, σ), hi is pointwise dissipative (i.e. (X, T, π) is pointwise dis-

sipative);

(2) X

= X.

Proof. Let h(X, T

, π), (Y, T

, σ), hi be pointwise dissipative. To prove the equality

= X it is suﬃcient to show that Ω

⊆ Θ. Since h(X, T

, π), (Y, T

, σ), hi is

pointwsice dissipative, then Ω

is compact and by Lemma 2.17 we have Ω

⊆ Θ.

Let now X

= X; we will show that (X, T

, π) is pointwise dissipative. First of

all, we note that {π(t, x) | t ≥ 0} is relatively compact for all x ∈ X. To convince

that it is so it is suﬃcient to show that from arbitrary sequence {π(t

, x)} (t

→

+∞) there can be chosen a convergent subsequence. Let y = h(x), then according to

the pointwise dissipativity of (Y, T

, h) we may suppose that the sequence {σ(t

, y)}

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Non-autonomous Dissipative Dynamical Systems 103

is convergent. Let q = lim

n→+∞

σ(t

, y), then q ∈ ω

. According to the local triviality

of the ﬁber bundle (X, h, Y ), the sequence {θ

σ(y,t

)

} → θ

and, hence,

ρ(xt

, θ

) ≤ ρ



, θ

σ(y,t

)



+ ρ



σ(y,t

)

, θ



−→ 0

for n → +∞. In addition, from the reasoning mentioned above it follows that

⊆ Θ for all x ∈ X and, consequently, Ω

⊆ Θ. Note, that by Lemma 2.9

h(Ω

) ⊆ Ω

and, hence, Ω

⊆ Θ ∩ h

−1

(Ω

). From the compactness of Ω

and

the local triviality of (X, h, Y ) it follows the compactness of Θ ∩ h

−1

(Ω

) and,

consequently, the compactness of Ω

too. The theorem is proved. 

Theorem 2.30 Let h(X, T

, π), (Y, T

, σ), hi be homogeneous of order k = 1 and

Y be compact. Then the following conditions are equivalent:

1. h(X, T

, π), (Y, T

, σ), hi is compactly dissipative, i.e. (X, T

, π) is compactly

dissipative;

2. X

= X and the set Θ ∩h

−1

) is uniformly stable, i.e. for an arbitrary ε > 0

there exists δ(ε) > 0 such that |x| < δ implies |xt| < ε for all t ≥ 0;

3. if the ﬁber bundle (X, h, Y ) is normed (i.e. the metric ρ on the ﬁbers (X, h, Y )

is compatible with the norm), then there exists a positive number N such that

|xt| ≤ N|x| (2.108)

for all x ∈ X, t ≥ 0 and X

= X.

Proof. Let h(X, T

, π), (Y, T

, σ), hi be compactly dissipative and J be the Levinson

center of the dynamical system (X, T

, π). By Theorem 2.29 X

= X. The inclusion

can be proved J

⊆ Θ in the same way that the inclusion Ω

⊆ Θ (see the proof of

Theorem 2.29). We will show that the trivial section Θ of the ﬁber bundle (X, h, Y )

is uniformly stable. If we suppose that it is not true, then there exit ε

> 0, δ

↓ 0,

| < δ

and t

→ +∞ such that

| ≥ ε

. (2.109)

By virtue of the compactness of Y and the local triviality of (X, h, Y ) the set Θ

is compact and, consequently, we may suppose that the sequence {x

} converges.

Let

x = lim

n→+∞

. Since (X, T

, π) is compactly dissipative, then we may suppose

that the sequence {x

} is convergent too and we put

x = lim

n→+∞

. Then

x ∈ J

⊆ Θ and, consequently, |x| = 0. The last equality contradicts the inequality

(2.109). The obtained contradiction proves the required statement.

To prove the implication 2. ⇒ 1. it is suﬃcient to refer to Theorem 1.13.

Let now the ﬁber bundle (X, h, Y ) be normed. We will show that in this case

every one of Conditions 1. and 2. is equivalent to Condition 3. We will prove, for

example, that 2. implies 3. Since the trivial section Θ is stable, then for ε = 1 there

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104 Global Attractors of Non-autonomous Dissipative Dynamical Systems

exits γ = δ(1) > 0 such that |π(t, x)| ≤ 1 for all t ≥ 0 and |x| ≤ q. We put N = γ

−1

then

|π(t, x)| = ||x|γ

−1

π(t, x|x|

−1

γ)| ≤ |x|γ

−1

= N|x|.

for all t ≥ 0 and x ∈ X \ Θ.

Inversely. Let X

= X and let exist N > 0 such that the inequality (2.108)

holds for all x ∈ X and t ≥ 0. Let ε > 0 and δ(ε) <

such that |x| < δ implies

|xt| < ε for all t ≥ 0. The theorem is proved. 

Theorem 2.31 Let h(X, T

, π), (Y, T

, σ), hi be a homogeneous non-autonomous

dynamical system of order k = 1 and Y be compact. Then the following conditions

are equivalent:

1. h(X, T

, π), (Y, T

, σ), hi is locally dissipative, i.e. (X, T

, π) is locally dissipa-

tive;

2. X

= X and the trivial section Θ is uniformly attracting, i.e. there exits γ > 0

such that

lim

t→+∞

sup

|x|≤γ

|π(t, x)| = 0; (2.110)

3. if the ﬁber bundle (X, h, Y ) is normed, then there exit positive numbers N and

ν such that

|π(t, x)| ≤ Ne

−νt

|x| (2.111)

for all t ≥ 0 and x ∈ X.

Proof. We will prove that 1. ⇒ 2. ⇒ 3. If the non-autonomous dynamical system

h(X, T

, π), (Y, T

, σ), hi is locally dissipative and J is its Levinson’s center, then by

Theorem 2.30 X

= X and J

⊆ Θ. In addition, according to Theorem 1.17 there

exists a positive number γ such that

lim

t→+∞

β(π

B(J, γ), J) = 0. (2.112)

We will show that for the obtained γ > 0 the equality (2.110) holds. If we suppose

that it is not true, then there exit ε

> 0, {x

} (|x

| ≤ γ) and t

→ +∞ such that

the inequality (2.109) holds. From the equality (2.112) it follows that the sequence

} can be considered convergent. Let

x = lim

n→+∞

. By (2.112) x ∈ J, and

since J

⊆ Θ, then |

x| = 0. On the other hand, passing to limit in (2.109) as

n → +∞, we obtain |

x| ≥ ε

> 0. The obtained contradiction proves the equality

(2.110).

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Non-autonomous Dissipative Dynamical Systems 105

From the homogeneity of the system h(X, T

, π), (Y, T

, σ), hi and the equality

(2.110) it follows that

lim

t→+∞

sup

|x|≤1

|π(t, x)| = 0. (2.113)

We put ψ(t) = sup{|π(t, x)| : |x| ≤ 1}. Note, that ψ(t + τ) ≤ ψ(t)ψ(τ) for all

t, τ ∈ T

and ψ(t) → 0 as t → +∞. By the equality (2.110), the trivial section Θ

of the ﬁber bundle (X, h, Y ) is uniformly stable. Indeed, if we suppose that it is

not true, then there exit ε

, δ

↓ 0, |x

| < δ

and t

→ +∞ such that the relation

(2.109) holds. We can suppose that δ

≤ γ, and according (2.110) for

there exits

> 0 such that

|xt| <

(2.114)

for all t ≥ L

. Since t

→ +∞, then for the suﬃciently large n the inequalities

(2.114) and (2.109) are contradictory. The obtained contradiction proves the re-

quired statement. By Theorem 2.30 the function ψ is bounded for t ≥ 0 and accord-

ing to Lemma 5.4

[

]

there exist positive numbers N and ν such that ψ(t) ≤ Ne

−νt

for all t ≥ 0 and, consequently, the inequality (2.111) holds too.

Now we will show, that the inverse implications 3. ⇒ 2. ⇒ 1. hold too. It is

evident that 3. implies 2., therefore to ﬁnish the proof of the theorem it is suﬃcient

to show that from 2. it follows 1. First of all we note that from the equality (2.110),

according to the reasoning above, it follows the uniform stability of the trivial section

Θ. By Theorem 2.30 the dynamical system h(X, T

, π), (Y, T

, σ), hi is compactly

dissipative and J

⊆ Θ. We will show that

lim

t→+∞

β(π

B(J, γ), J) = 0.

If we suppose that it is not true, then there exit ε

> 0, x

∈ B(J, γ) and t

→ +∞

such that

ρ(x

, Θ

) ≥ ε

(2.115)

for all y ∈ Y . From (2.110) it follows, that the sequence {x

} is convergent and

lim

n→+∞

x ∈ Θ. On the other hand, passing to limit in (2.115) as n → +∞, we

obtain ρ(

x, Θ

) ≥ ε

for all y ∈ Y , i.e. x 6∈ Θ. The obtained contradiction proves

the equality (2.111) and, hence, the local dissipativity of (X, T, π). The theorem is

proved. 

Corollary 2.11 Let h(X, T

, π), (Y, T

, σ), hi be a homogeneous non-autonomous

dynamical system and (X, T, π) be asymptotically compact, then the following con-

ditions are equivalent:

(1) h(X, T

, π), (Y, T

, σ), hi is compactly dissipative;

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 2

106 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(2) h(X, T

, π), (Y, T

, σ), hi is locally dissipative;

(3) X

= X and the trivial section Θ of the ﬁber bundle (X, h, Y ) is uniformly

stable;

(4) X

= X and the trivial section Θ of the ﬁber bundle (X, h, Y ) is uniformly

attracting;

(5) if the ﬁber bundle (X, h, Y ) is normed, then there exits a positive number N > 0

such that

|xt| ≤ N|x|

for all x ∈ X, t ≥ 0, and X

= X.

(6) if the ﬁber bundle (X, h, Y ) is normed, then there exit positive numbers N and

γ such that

|xt| ≤ Ne

−γt

|x|

for all t ≥ 0 and x ∈ X.

Proof. This statement it follows from Theorems 1.20, 2.30 and 2.31. 

Corollary 2.12 Let (X, T

, π) be completely continuous. Then the following con-

ditions are equivalent:

(1) h(X, T

, π), (Y, T

, σ), hi is pointwise dissipative;

(2) h(X, T

, π), (Y, T

, σ), hi is compactly dissipative;

(3) h(X, T

, π), (Y, T

, σ), hi is locally dissipative;

(4) X

= X;

(5) X

= X and Θ is uniformly stable;

(6) X

= X and Θ is uniformly attracting;

(7) if (X, h, Y ) is normed, then there exits N > 0 such that

|xt| ≤ N|x|

for all x ∈ X, t ≥ 0 and X

= X;

(8) if (X, h, Y ) is normed, then there exit N > 0 and γ > 0 such that

|xt| ≤ Ne

−γt

|x|

for all t ≥ 0 and x ∈ X.

Proof. Corollary 2.12 it follows from Theorems 1.10, 2.29 – 2.31. 

Remark 2.11 a) All the results of this section hold also for autonomous homo-

geneous (of order k ≥ 1) system with the continuous time (T = R

or R).

b) Note, that the results of this section take place, when the bundle (X, h, Y ) is

not necessary linear (vectorial). It is suﬃcient that there would exist a ﬁber bundle