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

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Global attractors of V -monotone dynamical systems 387

for all n ∈ N. Passing to limit in (12.3) as n → +∞, we obtain the equality ¯x

= ¯x

which contradicts to the inequality (12.2). This contradiction proves Theorem 12.1.



Corollary 12.1 Let h(X, S

, π), (Ω, S, σ), hi be a V -monotone compactly dissipa-

tive non-autonomous dynamical system and Ω be minimal. Then:

1. J is uniformly orbitally stable in positive direction, i.e. for ε > 0 there is

δ(ε) > 0 such that the inequality ρ(x, J

h(x)

) < δ implies that ρ(π

x, J

h(π

) < ε

for t ≥ 0;

2. J is an attractor of compact sets from X, i.e. for ε > 0 and a compact K ⊆ X

there is L(ε, K) > 0 such that π

⊆

B(J

, ε) for ω ∈ Ω and t ≥ L(ε, K),

where

B(M, ε) := {x ∈ X | ρ(x, M

h(x)

) < ε};

3. all motions on J can be continued to the left and J is bilaterally distal;

4. J

:= X

J for ω ∈ Ω and is a connected set if X

is connected, and for

distinct ω

and ω

the sets J

and J

are homeomorphic;

5. J is formed of recurrent trajectories, and every two arbitrary points x

, x

∈

(ω ∈ Ω) are mutually recurrent.

Theorem 12.2 Let h(X, S

, π), (Ω, S, σ), hi be a V -monotone compactly dissipa-

tive non-autonomous dynamical system, Ω be a minimal set and J be its Levinson’s

center. Then

V (x

t, x

t) = V (x

, x

) (12.4)

for all x

, x

∈ J such that h(x

) = h(x

Proof. Let x

, x

∈ J be such that h(x

) = h(x

). Then according to Corollary 12.1

, x

∈ J

(h(x

) = h(x

) = ω ∈ Ω) are mutually recurrent and, consequently,

the function φ ∈ C(S, R) deﬁned by the equality φ(t) := V (x

t, x

t) (t ∈ S) is

recurrent. On the other hand, since h(X, S

, π), (Ω, S, σ), hi is V -monotone, we

obtain φ(t

) ≤ φ(t

) for all t

≥ t

, t

∈ S). As the function φ is recurrent and

monotone, then it is a constant. In fact, if we suppose that there exists t

> t

such that φ(t

) < φ(t

), then in virtue of the recurrence of the function φ there

exists a sequence {s

} such that s

→ +∞ as n → +∞ and φ(s

) → φ(t

On the other hand, φ(s

) < φ(t

) for a suﬃciently large n and, consequently,

φ(t

) ≤ φ(t

) < ϕ(t

). The obtained contradiction proves Theorem 12.2. 

Corollary 12.2 Under the conditions of Theorem 12.2, if a non-autonomous

dynamical system h(X, S

, π), (Ω, S, σ), hi is strictly monotone, i.e. V (x

t, x

t) <

V (x

, x

) for all t > 0 and (x

, x

) ∈ X

×X (x

6= x

), then J

:= J

consists

of a single point for all ω ∈ Ω.

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 12

388 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Theorem 12.3 Let h(X, S

, π), (Ω, S, σ), hi be a V -monotone compactly dissipa-

tive non-autonomous dynamical system with the compact minimal base Ω and J be

its Levinson’s center. Then for every point x ∈ X

there exists a unique recurrent

point p ∈ J

such that

lim

t→+∞

ρ(xt, pt) = 0, (12.5)

i.e. every trajectory of this system is asymptotically recurrent.

Proof. Let ω ∈ Ω and x ∈ X

be an arbitrary point. Since h(X,S

,π),(Ω,S, σ),hi is

compactly dissipative, then the semi-trajectory {xt | t ∈ S

} is relatively compact.

According to Lemma 22.29 from

[

]

, there exists a recurrent point p ∈ ω

and a sequence t

→ +∞ such that

lim

n→+∞

ρ(xt

, pt

) = 0. (12.6)

According to Theorem 12.1, the non-autonomous dynamical system h(X, S

, π),

(Ω, T,σ), hi is uniformly stable in positive direction on compact K :=

{xt | t ∈ S

Let ε > 0 and δ = δ(ε, K) > 0 be chosen out of the uniform stability of K. Then

from equality (12.5) we have ρ(xt

, pt

) < δ(ε, K) for all suﬃciently large n and,

consequently,

ρ(x(t

+ t), p(t

+ t)) < ε

for all t ≥ 0 or ρ(xt, pt) < ε for every t ≥ t

, i.e. the equality (12.5) holds.

Thus, we found a recurrent point p ∈ ω

satisfying the equality (12.5). It

is clear that p ∈ J

. Now we will prove that point p with the property (12.5) is

unique. Let us suppose that p

, p

∈ J

and

lim

t→+∞

ρ(xt, p

t) = 0 (12.7)

for i = 1, 2. Then according to Corollary 12.1 the points p

and p

are mutually

recurrent. In particular, there exists a sequence t

→ +∞ such that

lim

n→+∞

= p

(i = 1, 2). (12.8)

On the other hand, from the equality (12.7) we have

lim

n→+∞

ρ(p

, p

) = 0. (12.9)

From (12.8) and (12.9) follows that p

= p

. The theorem is completely proved. 

Corollary 12.3 Under the conditions of Theorem 12.3 the following assertions

hold:

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 12

Global attractors of V -monotone dynamical systems 389

(1) ω-limit set (which is denoted as ω

) of every point x ∈ X is a compact minimal

set.

(2) if x

, x

∈ X

( ω ∈ Ω), then ω

= ω

or ω

= ∅.

12.2 On the structure of Levinson center of V -monotone NDS

Deﬁnition 12.2 (X, ρ) is called

[

189

]

a metric space with segments, if for any

, x

∈ X and α ∈ [0, 1] the intersection of B[x

, αr] (a closed ball with its center

at x and radius αr, where r = ρ(x

, x

)) and B[x

, (1 − α)r] consists of a unique

element S(α, x

, x

Deﬁnition 12.3 A metric space (X, ρ) is called

[

189

]

strictly-convex, if (X, ρ) is

a metric space with segments and for any x

, x

∈ X, x

6= x

and α ∈ (0, 1)

there holds the inequality ρ(x

, S(a, x

, x

)) < max{ρ(x

, x

), ρ(x

, x

)}.

Deﬁnition 12.4 Let X be a strictly metric-convex space. A subset M of X is

said to be metric-convex, if S(α, x

, x

) ∈ M for any α ∈ (0, 1) and x

, x

∈ M.

We note that every convex closed subset X of the Hilbert space H equipped

with the metric ρ(x

, x

) = |x

−x

| is strictly metric-convex.

Let x ∈ X. Denote by Φ

the family of all entire trajectories of the dynamical

system (X, S

, π) passing through the point x for t = 0, i.e. γ ∈ Φ

if and only if

γ : S → X is a continuous mapping with the properties: γ(0) = x and π

γ(τ) =

γ(t + τ) for all t ∈ S

and τ ∈ S.

Theorem 12.4 Let h(X, S

, π), (Ω, S, Θ), hi be a V -monotone compactly dissipa-

tive non-autonomous dynamical system, J be its Levinson center and the following

conditions be held:

1. V (x

, x

) = V (x

, x

) for all (x

, x

) ∈ X

×X.

2. V (x

, x

) ≤ V (x

, x

) + V (x

, x

) for all x

, x

∈ X with the condition

h(x

) = h(x

3. the space (X

, V

) is strictly metric-convex for all ω ∈ Ω, where X

−1

(ω) = {x ∈ X|h(x) = ω} (ω ∈ Ω) and V

:= V |

×X

If γ

∈ Φ

( i = 1, 2) and x

, x

∈ J

(ω ∈ Ω), then the function γ : S → X (γ(t) =

S(α, γ

(t), γ

(t)) for all t ∈ S) deﬁnes an entire trajectory of the dynamical system

(X, S

, π).

Proof. Let ω ∈ Ω, α ∈ [0, 1] and x

, x

∈ J

. We denote by x := S(α, x

, x

). Let

∈ Φ

(i = 1, 2). Consider the function γ : S → J deﬁned by the equality

γ(t) := S(α, γ

(t), γ

(t)) (12.10)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 12

390 Global Attractors of Non-autonomous Dissipative Dynamical Systems

for all t ∈ S. We will show that γ is an entire trajectory of (X, S

, π) with the

condition γ(0) = x. In fact, according to Theorem 12.2

V (γ

(t), γ

(t)) = V (x

, x

) = d

for all t ∈ S. Since V (γ

(t), γ(t)) = αd for all t ∈ S and h(X, S

, π), (Y, S, Θ), hi is

V -monotone, we have

V (γ

(t + τ), π

γ(τ)) = V (π

(τ), π

γ(τ)) ≤ V (γ

(τ), γ(τ)) ≤ αd

and

V (γ

(t + τ), π

γ(τ)) = V (π

(τ), π

γ(τ)) ≤ V (γ

(τ), γ(τ)) ≤ (1 − α)d

and, consequently,

γ(τ) ∈ S(α, γ

(t + τ ), γ

(t + τ )).

So, π

γ(τ) = γ(t + τ) for all τ ∈ S and t ∈ S

. Now we will prove that the function

γ is continuous. It is clear that γ is continuous on S

. Let t

∈ S, t

≤ 0 and

t = t

+ h (|h| < δ, δ > 0). Then we have

ρ(γ(t

+ h), γ(t

)) = ρ(π

+|t

|+δ+h

γ(−|t

| − δ),

+|t

|+δ

γ(−|t

|− δ)). (12.11)

Passing to limit in (12.11) as h → 0, we obtain lim

h→0

γ(t

+ h) = γ(t



We denote by K = {a ∈ C(S

, R

) | a(0) = 0, a is strictly increasing}.

Theorem 12.5 Under the conditions of Theorem 12.4, if, in addition, a non-

autonomous dynamical system h(X, S

, π), (Ω, S, Θ), hi is boundedly k-dissipative

and there exists a function a ∈ K with the property lim

t→+∞

a(t) = +∞ such that

a(ρ(x

, x

)) ≤ V (x

, x

) for all (x

, x

) ∈ X

×X, then J

will be metric-convex for

all ω ∈ Ω, where J

:= J

and J is the Levinson’s center of (X, S

, π).

Proof. Let ω ∈ Ω, x

, x

∈ J

, α ∈ (0, 1) and x := S(α, x

, x

). We will show

that x ∈ J

. Since J is invariant, there exist entire trajectories γ

and γ

such

that γ

(0) = x

(i = 1, 2) and γ

(t) ∈ J

for all t ∈ S. Consider the function

γ : S → J deﬁned by the equality (12.10). According to Theorem 12.4, the mapping

γ deﬁnes an antire trajectory of the dynamical system (X, S

, π) . We note that the

trajectory γ is bounded. In fact, by the construction of γ we have V (γ

(t), γ(t)) ≤

αd, and according to the conditions of Theorem 12.5 we obtain a(ρ(γ(t), γ

(t))) ≤

V (γ(t), γ

(t)) ≤ αd. and, consequently, ρ(γ(t), γ

(t)) ≤ a

−1

(αd) for all t ∈ S. Thus,

the trajectory γ is bounded and, taking into account that the set γ(S) is invariant

and the Levinson’s center J attracts every bounded set from X, we have x ∈ J

The theorem is proved. 

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 12

Global attractors of V -monotone dynamical systems 391

12.3 Almost periodic solutions of V -monotone systems

Let (X, ρ) be a metric space.

Deﬁnition 12.5 Recall that a function ϕ : S → X is called almost periodic (in

the sense of Bohr), if for every ε > 0 there exists a relatively dense subset A

of R

such that

ρ(ϕ(t + τ), ϕ(t)) < ε

for all t ∈ R and τ ∈ A

Lemma 12.2

[

238

]

Let Ω be a compact minimal set and M ⊆ X be a compact

invariant set of (X, S

, π). If the non-autonomous dynamical system h(X, S

, π),

(Ω, S, σ), hi is distal on M in negative direction, then the mapping ω 7−→ M

is continuous with respect to the Hausdorﬀ metric.

Lemma 12.3 Let M ⊆ X be a compact invariant set of (X, S

, π). If the non-

autonomous dynamical system h(X, S

, π), (Ω, S, σ), hi is uniformly stable in posi-

tive direction on compacts from X, then h(X, S

, π), (Ω, S, σ), hi is distal in negative

direction on the invariant set M.

Proof. We will prove that under the conditions of Lemma 12.3 a compact invariant

set M is distal in negative direction. Really, if we suppose the contrary, then there

exist ω

∈ Ω, t

≥ 0, x

6= x

, x

∈ M and h(x

) = h(x

)) and γ

∈ Φ

(i =

1, 2) such that

ρ(γ

(−t

), γ

(−t

)) → 0

as n → +∞. Let ε

= ρ(x

, x

) > 0 and δ

= δ(ε

, M) > 0 is chosen from the

condition of the uniform stability in positive direction of h(X, S

, π), (Ω, S, σ), hi on

M. Then for suﬃciently large n we have

ρ(γ

(−t

), γ

(−t

)) < δ

and, consequently,

ρ(π

(−t

), π

(−t

)) < ε

for all t ≥ 0. In particular, for t = −t

we have

ρ(x

, x

) = ρ(π

(−t

), π

(−t

)) < ε

= ρ(x

, x

The obtained contradiction proves the necessary statement. 

Corollary 12.4 Under the conditions of Lemma 12.3, if Ω is a compact minimal

set, then the mapping ω 7−→ J

is continuous with respect to the Hausdorﬀ metric.

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392 Global Attractors of Non-autonomous Dissipative Dynamical Systems

This assertion follows from Lemmas 12.2 and 12.3.

Lemma 12.4 Let (M, ρ) be a compact strictly metric-convex space and E be a

compact sub-semigroup of isometrics of the semigroup M

(i.e. E ⊆ M

and

ρ(ξx

, ξx

) = ρ(x

, x

) for all x

, x

∈ M). Then there exists a common ﬁxed point

¯x ∈ M of E, i.e. ξ(¯x) = ¯x for all ξ ∈ E.

Proof. Let x

be an arbitrary point of M . We denote by `(x) = sup

ξ∈E

ρ(ξ(x), x

)

and

= inf

x∈M

`(x). (12.12)

Let {x

} ⊆ M be a minimizing sequence for (12.12), i.e.

≤ `(x

) ≤ `

(12.13)

for all n ∈ N. Since the set M is compact, we may suppose that the sequence {x

}

is convergent. Let x

= lim

n→+∞

. Then passing to limit in the inequality (12.13),

we obtain `(x

) ≤ `

. On the other hand, `(x) ≥ `

for all x ∈ M and, consequently,

`(x

) = `

. We put M

= {x ∈ M | `(x) = `

}. Then M

6= ∅. The set M

invariant with respect to the semigroup E, i.e. ξ(M

) ⊆ M

for all ξ ∈ E. In fact,

let η ∈ E and x

∈ M

. Then

`(η(x

)) = sup

η∈E

ρ(ξ(η(x

)), x

) ≤ sup

ξ∈E

ρ(ξ(x

), x

) = `(x

) = `

and, consequently, `(η(x

)) = `

. We will show now that the set M

contains a

single point. Really, if we suppose the contrary, then there exist x

, x

∈ M

). We consider x = S(

, x

), which is an element from M, because M is

strictly metric-convex. Since the semigroup E is compact, then under the conditions

of Lemma 12.4 there exists ξ ∈ E such that `

≤ `(x) = ρ(ξ(x), x

). On the

other hand, according to the strict metric-convexity of M we have ρ(ξ(x), x

) <

max{ρ(ξ(x

), x

), ρ(ξ(x

), x

)} ≤ `

and, consequently, `(x) < `

. The obtained

contradiction proves that M

contains a unique point ¯x. Taking into account the

invariance of the set M

with respect to the semigroup E, we have ξ(¯x) = ¯x for all

ξ ∈ E. The Lemma is proved. 

Theorem 12.6 Let h(X, S

, π), (Ω, S, σ), hi be a V -monotone bounded k-

dissipative NDS, J be its Levinson’s center and the following conditions be held:

1. V (x

, x

) = V (x

, x

) for all (x

, x

) ∈ X

×X.

2. V (x

, x

) ≤ V (x

, x

) + V (x

, x

) for all x

, x

∈ X with the condition

h(x

) = h(x

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Global attractors of V -monotone dynamical systems 393

3. the space (X

, V

) is strictly metric-convex for all ω ∈ Ω, where X

−1

(ω) := {x ∈ X | h(x) = ω } (ω ∈ Ω) and V

:= V |

×X

Then the set-valued mapping ω → J

admits at least one continuous invariant

section, i.e. there exists a continuous mapping ν : Ω → J with the properties:

h(ν(ω)) = ω and ν(σ(t, y)) = π(t, ν(ω)) for all t ∈ S and ω ∈ Ω.

Proof. According to Corollary 12.1, under the conditions of Theorem 12.6 the

semi-group dynamical system (X, S

, π) deﬁnes on J a group dynamical system

(J, S, π). Let ω

∈ Ω be an arbitrary point of Ω , J be the Levinson’s center of

the dynamical system (X, S

, π) and E = E(J, S, π) be the Ellis semigroup of the

dynamical system (J, S, π), i.e. E(J, S, π) =

{π

| t ∈ S }, where by bar we denote

the close in J

is equipped with the topology of Tihonoﬀ). We denote by

:= {ξ ∈ E | ξ(J

) ⊆ J

}. Then under the conditions of Theorem 12.2,

6= ∅ is a compact sub-semigroup of Ellis of the semigroup E. According to

Theorem 12.2, we have V (ξ(x

), ξ(x

)) = V (x

, x

) for all (x

, x

) ∈ J

× J

and consequently, under the conditions of Theorem 12.6 we have a strictly metric-

convex (with respect to the complete metric V

:= V |J

× J

) compact set

and a compact semigroup of isometrics E

acting on J

. Applying Lemma

12.4, we obtain a common ﬁxed point ¯x

∈ J

. We denote by Σ := H(¯x

) =

{¯x

t | t ∈ S }. It is clear that Σ is a compactly invariant set of (J, S, π). Obviously,

:= Σ

= {x

}. We will prove that Σ

:= Σ

contains a single point

¯x

. It is evident that Σ

6= ∅ for all ω ∈ Ω, because Ω and Σ are compactly

invariant sets and Ω is minimal. Now we will prove that Σ

contains exactly one

point. If we suppose the contrary, then there exist ω ∈ Ω and x

, x

∈ Σ

such

that x

6= x

. Since the set Ω is minimal, then there exists a sequence {t

} → −∞

such that ωt

→ ω

as n → +∞. Taking into consideration the compactness of Σ,

we may suppose that the sequences {x

} (i = 1, 2) are convergent. We denote

by x

= lim

n→+∞

. It is clear that x

∈ Σ

and, consequently, x

= x

. On the

other hand, according to Corollary 12.3 the dynamical system (J, S, π) is distal in

negative direction and, consequently, x

6= x

. The obtained contradiction proves

our statement. Thus, we have a compactly invariant set Σ ⊆ J with the following

property Σ

= Σ

= {x

} for all ω ∈ Ω. Now we deﬁne a mapping ν : Ω → J

by the following equality: ν(ω) = x

for all ω ∈ Ω. It is easy to verify that ν is

a continuous and invariant section of the set-valued mapping h

−1

. The theorem is

proved. 

Corollary 12.5 Under the conditions of Theorem 12.6 the Levinson’s center of

the dynamical system (X, S

, π) contains at least one stationary (τ (τ > 0) - pe-

riodic, quasi-periodic, almost periodic) point, if the minimal set Ω consists of a

stationary (τ (τ > 0) - periodic, quasi-periodic, almost periodic) point.

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394 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Remark 12.1 a. In the proof of Lemma 12.4 we used the well known Favard

method of minimax (see, for example,

[

151

]

and

[

238

]

b. In the second part of the proof of Theorem 12.6 we used some ideas of

Zhikov

[

238

]

(Ch.VII) and Shcherbakov

[

302

]

(Ch.III).

12.4 Pullback attractors of V -monotone NDS

Let E be a Banach space and hE, ϕ, (Ω, S, σ), hi be a cocycle on the state space E.

Denote by X := E ×Ω, (X, S

, π) a skew-product dynamical system, i.e. π = (ϕ, σ)

and h(X, S

, π), (Ω, S, σ), hi is a non-autonomous dynamical system generated by

the cocycle ϕ, where h := pr

: X → Ω.

Deﬁnition 12.6 The cocycle ϕ is called V -monotone (see

[

113

]

[

114

]

[

238

]

[

333

]

), if there exists a continuous function V : E ×E ×Ω → R

with the following

properties:

a. V(u

, u

, ω) ≥ 0 for all ω ∈ Ω and u

, u

∈ Ω.

b. V(u

, u

, ω) = 0 if and only if u

= u

c. V(ϕ(t, ω, u

), ϕ(t, ω, u

), σ

ω) ≤ V(u

, u

, ω) for all u

, u

∈ E, ω ∈ Ω and t ∈

Lemma 12.5 The cocycle ϕ is V −monotone if and only if the non-autonomous

dynamical system h(X, S

, π), (Ω, S, σ), hi generated by the cocycle ϕ is V -

monotone.

Proof. Let ϕ be a V -monotone cocycle. Then we deﬁne a mapping V : X

×X → R

by the equality

V ((ω, u

), (ω, u

)) := V(u

, u

, ω) (12.14)

for all (ω, u

), (ω, u

) ∈ X

×X, where V : E ×E ×Y → R

is the function ﬁguring in

the deﬁnition of V -monotone cocycle ϕ and X

×X := {(x

, x

) ∈ X × X | h(x

) =

h(x

) }.

From the equality (12.14) and a.-c. follows that V possesses the following prop-

erties:

i) V (x

, x

) ≥ 0 for all (x

, x

) ∈ X

×X and V (x

, x

) = 0 if and only if x

= x

ii) V is continuous.

iii) V (x

t, x

t) ≤ V (x

, x

) for all (x

, x

) ∈ X

×X and t ∈ S

The inverse statement is proved by analogy. 

Theorem 12.7 Let ϕ be a V −monotone cocycle admitting a compact pullback

attractor {I

| ω ∈ Ω } and let exist a function a ∈ K with the property a(r) → +∞

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 12

Global attractors of V -monotone dynamical systems 395

as r → +∞ such that V(u

, u

, ω) ≥ a(ρ(u

, u

)) for all u

, u

∈ E and ω ∈ Ω. If

the NDS h(X, S

, π), (Ω, S, σ), hi generated by the cocycle ϕ is α-condensing, then

it admits a compact global attractor J and J

= I

× {ω} for every ω ∈ Ω, where

:= h

−1

Proof. First step: we will prove that under the conditions of Theorem 12.7 all

trajectory of the dynamical system (X, S

, π) are positively compact. Really, let

x ∈ X (x := (u, ω)) and q := (p, ω) ∈ J

:= I

× {ω} be an arbitrary point. Then

in virtue of the V −monotonicity of the cocycle ϕ we have

V(ϕ(t, u, ω), ϕ(t, p, ω), σ

ω) ≤ V(u, p, ω) (12.15)

for all t ∈ S

. From the inequality (12.15) results that {ϕ(t, ω, u) | t ∈ S

} is

bounded. If we suppose that it is not true, then there exists a sequence t

→ +∞

such that

lim

n→+∞

ρ(ϕ(t

, u, ω), ϕ(t

, p, ω)) = +∞. (12.16)

Under the conditions of Theorem 12.7 from (12.15) and (12.16) follows

V(u, p, ω) ≥ V(ϕ(t

, u, ω), ϕ(t

, p, ω), σ

ω) ≥

a(ρ((ϕ(t

, u, ω), ϕ(t

, p, ω)) → +∞

as n → +∞. The obtained contradiction proves our assertion.

Taking into consideration the asymptotic compactness of the dynamical

system (X, S

, π) and the boundedness of the positively invariant set M :=

{(ϕ(t, ω, u), σ

ω) | t ∈ S

}, we conclude that it is relatively compact. Thus, under

the conditions of Theorem 12.7 every positive semi-trajectory is relatively compact.

Second step: since ϕ admits a compact global pullback attractor, then according

to Theorems 8.4 and 7.2 the set J is asymptotically stable and it is the maximal

compact invariant set of X. We will show that W

(J) = X. Really, let x ∈ X be

an arbitrary point. Then its ω-limit set ω

is nonempty, compact and invariant,

because {xt | t ∈ S

} is relatively compact. Taking into account the maximality

of J, we have ω

⊆ J for all x ∈ X and, consequently, W

(J) = X, i.e. J is the

compact global attractor of (X, S

, π). The theorem is proved. 

12.5 Applications

12.5.1 Finite-dimensional systems

Denote by R

a real n−dimensional Euclidean space with the scalar product h, i

and the norm | · | generated by the scalar product. Let [R

] be a space of all the

linear mappings A : R

→ R

equipped with the operational norm.

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 12

396 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Theorem 12.8 Let Ω be a compact minimal set, F ∈ C(Ω × R

, R

), W ∈

C(Ω, [R

]) and the following conditions be held:

1. the matrix-function W is positively deﬁned, i.e. hW (ω)u, ui ∈ R for all ω ∈

Ω, u ∈ R

and there exists a positive constant a such that hW (ω)u, ui ≥ a|u|

for all ω ∈ Ω and u ∈ R

2. the function t → W (σ

ω) is diﬀerentiable for every ω ∈ Ω and

W (ω) ∈

C(Ω, [R

]), where

W (ω) :=

W (σ

ω)|

t=0

3. h

W (ω)(u−v)+W (ω)(F (ω, u)−F (ω, v)), u−vi ≤ 0 for all ω ∈ Ω and u, v ∈ R

4. there exist a positive constant r and a function c : [r, +∞) → (0, +∞) such that

W (ω)u + W (ω)F (ω, u), ui ≤ −c(|u|) for all |u| > r.

Then the equation

= F (u, σ

ω) (12.17)

generates a cocycle ϕ on R

which admits a compact global attractor I = {I

| ω ∈

Ω} with the following properties:

a. I

is a nonempty, compact and convex subset of R

for every ω ∈ Ω.

b. I =

| ω ∈ Ω} is connected.

c. the mapping ω → I

is continuous with respect to the metric of Hausdorﬀ.

d. I = {I

| ω ∈ Ω} is invariant, i.e. ϕ(t, ω, I

) = I

for all ω ∈ Ω and t ∈ R

e. lim

t→+∞

β(ϕ(t, σ

ω)M, I

) = 0 for all M ∈ C(R

) and ω ∈ Ω.

f. lim sup

t→+∞

{β(ϕ(t, M, σ

ω), I) | ω ∈ Ω } = 0 for any M ∈ C(R

), where I =

| ω ∈ Ω }.

g. I = {I

| ω ∈ Ω } is a uniform forward attractor ,i.e.

lim

t→+∞

sup

ω∈Ω

β(ϕ(t, ω)M, I

) = 0

for any M ∈ C(R

h. the equation (12.17) admits at least one stationary (τ−periodic, quasi-periodic,

almost periodic) solution, if the point ω ∈ Ω is stationary (τ−periodic, quasi-

periodic, almost periodic).

Proof. Since the function F ∈ C(Ω × R

, R

), then, according to the theorem of

Peano (see, for example,

[

186

]

), the equation (12.17) admits at least one solution

u(t) (t ∈ [0, t

), t

> 0) with the condition u(0) = x for every x ∈ R

. We will

show that under the conditions of Theorem 12.8 this solution is unique. In fact,

let u

(t) (i = 1, 2) be two solutions of the equation (12.17) deﬁned on [0, t

) with

the condition u

(0) = x (i = 1, 2). We consider the function hW (ωt)u

(t), u

(t)i.