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

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Non-autonomous Dissipative Dynamical Systems 67

δ(ε) > 0 there exists L(ε) = L(δ(ε)) > 0 such that

β(π

B(J

, γ), J

) < δ(ε) (2.34)

for all t ≥ L(ε). The inequalities (2.31) and (2.34) imply

β(σ

B(J

, α), J

) < ε

for all t ≥ L(ε). The theorem is completely proved. 

Corollary 2.1 Let h(X, T

, π), (Y, T

, σ), hi be a compact dissipative non-

autonomous dynamical system and Y be a compact invariant set (i.e., σ

Y = Y

for all t ∈ T

), then:

(1) h(J

) = Y and, consequently, J

= J

6= ∅ for all y ∈ Y , where X

−1

(y);

(2) J

= {x ∈ X

| there exists whole relatively compact trajectory ϕ

of the

dynamical system (X, T

, π) such that ϕ(0) = x}.

Remark 2.3 We don’t know whether or not the condition of the openness of h is

so essential in Theorem 2.5, but apparently we can’t completely reject it.

Deﬁnition 2.7 Recall that ϕ : Y → X is called a continuous section of the

homomorphism h : X → Y , if ϕ is continuous and h ◦ ϕ = id

Deﬁnition 2.8 A continuous section is called invariant if ϕ ◦ σ

= π

◦ ϕ for all

t ∈ T

Theorem 2.6 Let the homomorphism h : X → Y admits a continuous invariant

section, then:

(1) if (X, T

, π) is point dissipative, then (Y, T

, σ) also is point dissipative,

h(Ω

) = Ω

and h(D

(Ω

)) = D

(Ω

);

(2) if (X, T

, π) is compact dissipative, then (Y, T

, σ) also is compact dissipative

and h(J

) = J

;

(3) if (X, T

, π) is locally dissipative, then (Y, T

, σ) also is locally dissipative.

Proof. Let ϕ be a continuous invariant section of h. Since h ◦ ϕ = id

, then

h(X) = Y and the ﬁrst and the second statements follow from Theorem 2.5, except

the equality h(D

(Ω

)) = D

(Ω

). To prove it note that by virtue of compactness

Ω

from Lemma 2.9 the inclusion h(D

(Ω

)) ⊆ D

(ω

) holds.

Let us show that within the conditions of Theorem 2.6 the reverse inclusion also

holds. In fact, according to Theorem 2.5 the dynamical system (Y, T

, σ) is point

dissipative and Ω

= h(Ω

) is compact. Since ϕ : Y → X is a homomorphism

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

68 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(Y, T

, σ) onto (X, T

, π), then according to Lemma 2.9 ϕ(D

(Ω

)) ⊆ D

(Ω

)

and, consequently, D

(Ω

) = h ◦ϕ(D

(Ω

)) ⊆ h(D

(ω

)).

Let (X, T

, π) be locally dissipative, then from the foregoing proof the dynamical

system (Y, T

, σ) is compact dissipative. According to Theorem 1.18 for the locally

dissipativity of (Y, T

, σ) it is suﬃcient to show that its Levinson’s center J

is a

uniformly attracting set. As well as in Theorem 2.5, for ε > 0 (η > 0) we will select

δ(ε) > 0 (ξ(η) > 0) such that

ρ(h(x), J

) < ε (ρ(ϕ(y), J

) < η) (2.35)

for all x ∈ B(J

, δ) (y ∈ B(J

, ξ)). Since X, T

, π) is locally dissipative then

there exists γ > 0 such that

lim

t→+∞

β(π

B(J

, γ), J

) = 0. (2.36)

Assume that ν = ξ(γ) > 0 and let us show that

lim

t→+∞

β(σ

B(J

, ν), J

) = 0.

Let ε > 0, then according to (2.36) there exists L(ε) > 0 such that

B(J

, γ) ⊆ B(J

, δ) (2.37)

for all t ≥ L(ε). By virtue of the choice of ν = ξ(γ) > 0 we have

ϕ(B(J

, γ)) ⊆ B(J

, γ). (2.38)

From the inclusions (2.37) and (2.38) it follows that

ϕ(B(J

, γ)) ⊆ B(J

, δ) (2.39)

for all t ≥ L(ε). From (2.35) and (2.39) we obtain

h(π

ϕ(B(J

, ν))) ⊂ B(J

, ε) (2.40)

for all t ≥ L(ε). Since h ◦π

◦ϕ = σ

◦h ◦ϕ = σ

(h ◦ϕ = id

) for all t ∈ T

, then

from (2.40) we have

B(J

, ν) ⊂ B(J

, ε)

for all t ≥ L(ε). The theorem is completely proved. 

2.4 Non-autonomous dynamical systems with convergence

Let (X, ρ) and (Y, d) be two complete metric spaces, R(Z) be a group of real (integer)

numbers, S = R or Z, S

= {t ∈ S | t ≥ 0} and T(S

⊆ T) be a subgroup of group

S .

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Non-autonomous Dissipative Dynamical Systems 69

By (X, T, π) we denote a dynamical system on X and xt = π(t, x) = π

Recall, that a dynamical system (X, T, π) is called compactly dissipative if there

exists a nonempty compact K ⊆ X such that

lim

t→+∞

ρ(xt, K) = 0 (2.41)

for all x ∈ X; moreover equality (2.41) holds uniformly with respect to x ∈ X on

each compact subset from X. In this case the set K is called an attractor of the

family of all compact subsets C(X) from the space X.

We denote by

J = Ω(K) =

t≥0

[

τ≥t

then the set J does not depend of the choice of the attractor K and is characterized

by the properties of the dynamical system (X, T, π) . The set J is called a Levinson’s

center of the dynamical system (X, T, π).

Let us mention some facts that we will use below.

Let Y be a compact metric space, (X, T

, π) ((Y, T

, σ)) be a dynamical system

on X (Y ), (T

⊆ T

) and h : X → Y be a homomorphism of (X, T

, π) onto

(Y, T

, σ). Then the triple h(X, T

, π), (Y, T

, σ), hi is called a non-autonomous

dynamical system.

Let W and Y be complete metric spaces, (Y, S, σ) be a group dynamical system

on Y and hW, ϕ, (Y, S, σ)i be a cocycle over (Y, S, σ) with the ﬁber W (or, by

short, ϕ), i.e. ϕ is a continuous mapping of W × Y × T into W satisfying the

following conditions: ϕ(0, w, y) = w and ϕ(t + τ, w, y) = ϕ(t, ϕ(τ, w, y), σ(τ, y)) for

all t, τ ∈ T, w ∈ W and y ∈ Y .

We denote X = W × Y and deﬁne on X a skew product dynamical system

(X, T, π) by the equality π = (ϕ, σ), i.e. π(t, (w, y)) = (ϕ(t, w, y), σ(t, y)) for all

t ∈ T and (w, y) ∈ W × Y . Then the triple h(X, T, π), ((Y, S, σ), hi, where h = pr

is a non-autonomous dynamical system.

For any two bounded subsets A and B from X we denote by β(A, B) a semi-

deviation of A to B, i.e. β(A, B) = sup{ρ(a, B)|a ∈ A} and ρ(a, B) = inf{ρ(a, b)|b ∈

B}.

Deﬁnition 2.9 A cocycle ϕ over (Y, S, σ) with the ﬁber W is called compactly

dissipative if there exists a nonempty compact K ⊆ W such that

lim

t→+∞

sup{β(U(t, y)M, K) : y ∈ Y } = 0

for all M ∈ C(W ) where U(t, y) = ϕ(t, ·, y).

Deﬁnition 2.10 By a whole trajectory of the semigroup dynamical system

(X, T, π) (of the cocycle hW, ϕ, (Y, S, σ)i over (Y, T, σ) with the ﬁber W ) pass-

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70 Global Attractors of Non-autonomous Dissipative Dynamical Systems

ing through the point x ∈ X ( (u, y) ∈ W × Y ) we mean a continuous mapping

γ : S → X(ν : S → W ) satisfying the conditions : γ(0) = x(ν(0) = w) and

γ(t + τ) = π

γ(τ) (ν(t + τ) = ϕ(t, ν(τ ), yτ)) for all t ∈ T and τ ∈ S.

Deﬁnition 2.11 h(X,T

,π),(Y,T

,σ),hi is said to be convergent if the following

conditions are valid:

(1) the dynamical systems (X, T

, π) and (Y, T

, σ) are compactly dissipative;

(2) the set J

contains no more than one point for all y ∈ J

where X

−1

(y) := {x|x ∈ X, h(x) = y} and J

) is the Levinson’s center of the

dynamical system (X, T

, π)((Y, T

, σ)).

Let M ⊆ X and M

×M := {(x

, x

) | x

, x

∈ M, h(x

) = h(x

)}.

Lemma 2.10 Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical

system, K ⊆ X be a compact invariant set and M := h(K). If the equality

lim

t→+∞

sup

)∈K

×K

ρ(x

t, x

t) = 0 (2.42)

holds, then the set K

:= K

contains a single point for all y ∈ M .

Proof. Suppose that there exists y

∈ M such that K

contains at least two points

¯x

and ¯x

(¯x

6= ¯x

). Since the set K is invariant, then there exists a trajectory ϕ

passing trough the point ¯x

(i = 1, 2) such that ϕ

(S) ⊆ K. Let 0 < ε <

ρ(¯x

,¯x

)

and

L(ε) > 0, so that

ρ(x

t, x

t) < ε

for all t ≥ L(ε) and (x

, x

) ∈ K

×K. Thus, we have

ρ(¯x

, ¯x

) = ρ(π

(−t), π

(−t)) < ε

for all t ≥ L(ε). The obtained contradiction shows that K

contains a single point

for all y ∈ M. The lemma is proved. 

Deﬁnition 2.12 A dynamical system (X, T, π) is said to be satisfying the con-

dition (A) if the set

{π

K | t ≥ 0} is relatively compact for every K ∈ C(X) =

{K | K ⊆ X and K is compact }.

We denote by L

:= {x | x ∈ X is so that at least one entire trajectory of the

dynamical system (X, T, π) passes through x}.

Remark 2.4 For a compactly dissipative system (X, T, π) we have L

= J

where J

is the Levinson’s center of (X, T, π) .

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Non-autonomous Dissipative Dynamical Systems 71

Theorem 2.7 Let (X, T

, π) be a dynamical system satisfying the condition (A)

and let (Y, T

, σ) be compactly dissipative, then the following conditions are equiva-

lent:

1. the set L

contains no more than one point for all y ∈ J

;

2. every semi-trajectory Σ

= {xt | t ≥ 0} is asymptotically stable, i.e.

2.a. for all ε > 0 and p ∈ X there exists δ(ε, p) > 0 such that ρ(x, p) <

δ (h(x) = h(p)) implies ρ(xt, pt) < ε for any t ≥ 0;

2.b. there exists γ(p) > 0 such that ρ(x, p) < γ(p) (h(x) = h(p)) implies

lim

t→+∞

ρ(xt, pt) = 0.

3. 3.a for all ε > 0 and K ∈ C(X) there exists δ(ε, K) > 0 such that ρ(x

, x

) <

δ (h(x

) = h(x

); x

, x

∈ K) implies ρ(x

t, x

t) < ε for all t ≥ 0;

3.b lim

t→+∞

ρ(x

t, x

t) = 0 for all (x

, x

) ∈ X

×X.

4. the equality (2.42) holds for all K ∈ C(X).

Proof. We will prove that condition 1. implies condition 2. If we suppose that it is

not so, then there are p

∈ X, ε

> 0, p

→ p

(h(p

) = h(p

)) and t

→ +∞ such

that

ρ(p

, p

) ≥ ε

. (2.43)

Since (X, T

, π) satisﬁes the condition (A), then we may suppose that the sequences

} and {p

} are convergent. Letting ¯p = lim

n→+∞

, ¯p

= lim

n→+∞

and taking into consideration (2.43), we will have ¯p 6= ¯p

. On the other hand,

h(¯p) = lim

n→+∞

h(p

= lim

n→+∞

h(p

= h(¯p

) = ¯y ∈ J

and according to Lemma

1.3 ¯p, ¯p

∈ L

¯y

, but by virtue of condition 1. we have ¯p = ¯p

. The obtained

contradiction proves the necessary assertion.

Now we will note that condition 1. implies condition 2.b. To prove this impli-

cation is suﬃcient to show that

lim

t→+∞

ρ(x

t, x

t) = 0

for all (x

, x

) ∈ X

×X. Assuming the contrary we obtain

ρ(x

, x

) ≥ ε

. (2.44)

The dynamical system (X, T

, π) satisﬁes the condition (A) and, consequently, we

may assume that the sequences {x

}(i = 1, 2) and {y

} (y

= h(x

) = h(x

))

are convergent. We denote by ¯x

:= lim

n→+∞

and ¯y

:= lim

n→+∞

, then

¯x

, ¯x

∈ L

¯y

and according to the condition 1. ¯x

= ¯x

. The last equality

and the inequality (2.44) are contradictory. This contradiction proves the necessary

assertion.

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72 Global Attractors of Non-autonomous Dissipative Dynamical Systems

We will show that condition 2. implies condition 3. Note that

lim

t→+∞

ρ(xt, pt) = 0 (2.45)

for all p ∈ X and x ∈ X

(q = h(p)). In fact, we denote by G

:= {x | x ∈ X

such that the equality (2.45) holds } and suppose that G

6= X

. By virtue of the

Condition 2, G

is open in X

. Let Γ

:= ∂G

(∂G

is the boundary of G

) and

¯p ∈ Γ

, then B(¯p, γ(

b))

\ G

) 6= ∅ (B(¯p, γ(

b)) := {x | h(x) = h(¯p), ρ(x, ¯p) <

γ(¯p)}). It is easy to see that the last relations are not satisﬁed simultaneously and,

consequently, Γ

= ∅ for all q ∈ Y , i.e. X

= G

. Let K ∈ C(X) and ε > 0, then

there exists δ(ε, K) > 0 such that ρ(x

, x

) < δ(h(x

) = h(x

); x

, x

∈ K) implies

ρ(x

t, x

t) < ε for any t ≥ 0. Assuming the contrary, we obtain K

∈ C(X), ε

0, δ

→ 0 (δ

> 0), {x

} ⊆ K

(i = 1, 2) and t

→ +∞ such that ρ(x

, x

) < δ

and

ρ(x

, x

) ≥ ε

. (2.46)

Since K

is a compact subset of X we may suppose that the sequences {x

}(i = 1, 2)

are convergent and we denote by ¯x := lim

n→+∞

= lim

n→+∞

(¯x ∈ K

). According to

the condition 2., for ε

> 0 and ¯x ∈ K

there exists δ(

, ¯x) > 0 such that ρ(x, ¯x) <

δ(

, ¯x) (h(x) = h(¯x)) implies ρ(xt, ¯xt) <

for all t ≥ 0. Since x

→ ¯x (i = 1, 2),

then there exists ¯n such that ρ(x

, ¯x) < δ(

, ¯x) (n ≥ ¯n) and, consequently,

ρ(x

t, x

t) ≤

2ε

(2.47)

for all t ≥ 0 and n ≥ ¯n. But the inequalities (2.47) and (2.46) are contradictory.

Thus, we showed that condition 2. implies condition 3.

We will prove that condition 3. implies condition 4. If we suppose the contrary,

then there exist ε

> 0, K

∈ C(X), t

→ +∞ and {x

} ⊆ K

(i = 1, 2; h(x

) =

h(x

)) such that the inequality (2.46) holds. We may assume without loss of gen-

erality that the sequences {x

} (i = 1, 2) are convergent, because K

is compact.

Let x

:= lim

n→+∞

, 0 < ε < ε

and δ(

, K

) > 0 be chosen according to condition

3.a. Since h(x

) = h(x

) and x

, x

∈ K

, then for

there exists L(

, x

) > 0

such that ρ(x

t, x

t) <

for all t ≥ L(

, x

) and, consequently,

ρ(x

, x

) ≤ ρ(x

, x

) + ρ(x

, x

) + ρ(x

, x

) < ε (2.48)

for suﬃciently large n. The inequalities (2.47) and (2.48) are contradictory. Hence,

the necessary assertion is proved.

Finally, we note that 4. implies 1. In fact, if we suppose that there exists

∈ J

such that L

contains at least two points x

and x

6= x

)

and denoting by K a compact invariant set such that x

, x

∈ K, we will have

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Non-autonomous Dissipative Dynamical Systems 73

, x

∈ K

= K

. On the other hand, according to Lemma 2.10 K

contains

no more than one point. The obtained contradiction proves Theorem 2.7. 

Corollary 2.2 Let (X, T

, π) and (Y, T

, σ) be two compactly dissipative

dynamical systems, then the following conditions are equivalent:

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

, π), (Y, T

, σ), hi is convergent;

(2) every semi-trajectory

(x ∈ X) is asymptotically stable;

(3) 3.a and 3.b from Theorem 2.7 are fulﬁlled;

(4) the equality (2.42) holds for all K ∈ C(X).

Deﬁnition 2.13 Recall that the dynamical system (X, T, π) is called locally com-

pact if for every x ∈ X there exist δ = δ

> 0 and l = l

> 0 such that π

B(x, δ) is

relatively compact.

Theorem 2.8 Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical

system, (Y, T

, σ) be compactly dissipative and (X, T

, π) be locally compact. For

h(X, T

, π), (Y, T

, σ), hi to be convergent, it is necessary and suﬃcient that every

semi-trajectory Σ

of the system (X, T

, π) would be relatively compact and that the

dynamical system h(h

−1

), T

, π), (J

, T

, σ), hi, where J

is a Levinson center

of (Y, T

, σ), would be convergent.

Proof. The necessity of the theorem is evident.

Suﬃciency. We will prove that the dynamical system (X, T

, π) is point dissi-

pative. For this aim under the conditions of the theorem it is suﬃcient to show

that Ω

{ω

| x ∈ X} is compact. Note that h(ω

) ⊆ ω

h(x)

⊆ J

and,

consequently, ω

⊆ h

−1

). Since ω

is compact and invariant, then ω

⊆

where

J is the center of Levinson of the system (h

−1

), T

, π). Thus, Ω

⊆

and, consequently, Ω

is compact. According to Theorem 1.10 for a locally com-

pact dynamical system point dissipativity and compact dissipativity are equivalent.

Hence, the dynamical system (X, T

, π) is compactly dissipative. Let J

be the

center Levinson of (X, T

, π), then h(J

) ⊆ J

and, consequently, J

⊆ h

−1

Since

J is a maximal compact invariant set in (h

−1

), T

, π), then J

⊆

J and

⊆

for all y ∈ J

. From this inclusion follows that the set J

contains at most one point for any y ∈ J

. The theorem is proved. 

Theorem 2.9 Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical

system, (Y, T

, σ) be a compactly dissipative dynamical system and let there ex-

ists y

∈ Y such that Y = H

). For the non-autonomous dynamical system

h(X, T

, π), (Y, T

, σ), hi to be convergent, it is necessary and suﬃcient that the

following conditions would be fulﬁlled:

(1) the dynamical system (X, T

, π) satisﬁes the condition (A);

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74 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(2) L

contains at most one point for every y ∈ J

= ω

Proof. The necessity of the conditions 1 and 2 is evident.

Suﬃciency. Let x

∈ X

, then h(H

)) = H

) and h(ω

) = ω

. Note

that h(Ω

) ⊆ Ω

⊆ J

= ω

and since ω

⊆ Ω

, then h(Ω

) = ω

. On the

other hand, Ω

⊆ L

, L

contains at most one point for all y ∈ J

= ω

Thus, Ω

= ω

is compact and, consequently, (X, T

, π) is pointwise dissipative.

Since (X, T

, π) is point dissipative and satisﬁes the condition (A), then by Theorem

1.15 (X, T

, π) is compactly dissipative. Let J

be the center of Levinson of the

dynamical system (X, T

, π), then J

⊆ L

and, consequently, J

contains

at most one point for every y ∈ J

. The theorem is proved. 

Deﬁnition 2.14 A point y

∈ Y is called

[

103

]

[

301

]

asymptotically stationary

(asymptotically τ-periodic, asymptotically almost periodic, asymptotically recur-

rent) if there exists a stationary (τ− periodic, almost periodic, recurrent) point

q ∈ Y such that

lim

t→+∞

d(y

t, qt) = 0.

Remark 2.5 a. Let Y = H

) =

t | t ≥ 0} be compact, then the dynamical

system (Y, T, σ) is compactly dissipative and J

= ω

b. Let y

be asymptotically stationary (asymptotically τ-periodic, asymptotically

almost periodic, asymptotically recurrent) and Y = H

), then (Y, T, σ) is com-

pactly dissipative and the set J

= ω

is minimal.

Deﬁnition 2.15 A point x ∈ X is called

[

]

[

103

]

[

301

]

comparable in limit with

regard to the recurrence property with a point y ∈ Y if the inclusion L

⊆ L

holds,

where L

:= {{t

}|t

→ +∞ and {yt

} is convergent}.

It is known

[

49, 103, 301

]

that if L

⊆ L

, then the point x possesses the same

character of the recurrence property in limit as the point y ∈ Y. In particular, if

the point y ∈ Y is asymptotically stationary (asymptotically τ -periodic, asymptot-

ically almost periodic, asymptotically recurrent) and L

⊆ L

, then the point x

will be asymptotically stationary (asymptotically τ-periodic, asymptotically almost

periodic, asymptotically recurrent) .

Theorem 2.10 Let y

∈ Y be asymptotically stationary (asymptotically τ-

periodic, asymptotically almost periodic, asymptotically recurrent) and Y = H

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

, π), (Y, T

, σ), hi will be con-

vergent if and only if the following conditions are fulﬁlled:

a. the dynamical system (X, T

, π) satisﬁes the condition (A);

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Non-autonomous Dissipative Dynamical Systems 75

b. every point x ∈ X is comparable in limit with regard to the recurrence prop-

erty with the point y = h(x) and, in particular, x is asymptotically stationary

(asymptotically ω- periodic, asymptotically almost periodic, asymptotically re-

current);

c. for any ε > 0 and K ∈ C(X) there exists δ = δ(ε, K) > 0 such that ρ(x

, x

) <

δ (h(x

) = h(x

); x

, x

∈ K) implies ρ(x

t, x

t) < ε for all t ≥ 0;

d. the equality lim

t→+∞

ρ(x

t, x

t) = 0 holds for all (x

, x

) ∈ X

×X.

Proof. The necessity of the conditions a,c and d is assured by Remark 2.5. Now, let

us show that under the conditions of the theorem the condition b. holds. Let x ∈ X

and y = h(x), then, according to the convergence of the non-autonomous dynamical

system h(X, T

, π), (Y, T

, σ), hi, the set H

(x) :=

{xt | t ≥ 0} is compact. We note

that ω

⊆ J

for all q ∈ ω

and since h(X, T

, π), (Y, T

, σ), hi is conver-

gent, then ω

contains a single point. According to Theorem 1

[

]

the point

x is comparable in limit with regard to the recurrence property with the point y.

If y ∈ H

), then it is evident that the point y will be asymptotically stationary

(asymptotically ω- periodic, asymptotically almost periodic, asymptotically recur-

rent) and, consequently, the point x possesses the same character of recurrence

property in limit as the point y does.

We will show that the Conditions a, b, c and d imply the convergence of the

non-autonomous dynamical system h(X, T

, π), (Y, T

, σ), hi. First of all, according

to the Condition b we have that ω

6= ∅ is compact, minimal and h(ω

) = ω

for

all x ∈ X. We note that ω

contains a single point for every q ∈ ω

. In the

opposite case there exist q

∈ ω

, p

∈ ω

6= p

) and t

→ +∞ (i =

1, 2) such that xt

→ p

(i = 1, 2) as n → +∞. We note that yt

→ q

(i = 1, 2)

as n → +∞, where y = h(x). Let

2n−1

= t

and

= t

for every n ∈ N, then

{

} ∈ L

and, consequently, {

} ∈ L

, i.e. {xt

} is convergent; therefore p

= p

The last equality contradicts the choice of the points p

and p

. The obtained

contradiction proves the necessary assertion. Now we will prove that ω

for all x

, x

∈ X and q ∈ ω

. Let q ∈ ω

, {p

} = ω

(i = 1, 2) and

} ∈ L

be such that qt

→ q. By virtue of the Condition c and the minimality

of ω

(i = 1, 2) we have ρ(p

, p

) → 0 as n → +∞ and, consequently, p

= p

Thus, ω

= ω

for all x

, x

∈ X and, consequently, (X, T

, π) is point dissipative

and since (X, T

, π) satisﬁes the condition (A), then according to Theorem 2.9

(X, T

, π) is compactly dissipative. To ﬁnish the proof of the theorem is suﬃcient

to apply Theorem 2.7 and Remark 2.4. 

Corollary 2.3 Under the conditions of Theorem 2.11 if the space X is locally

compact, then the Condition a results from the Conditions b, c and d.

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

76 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Theorem 2.11 Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical

system, (Y, T

, σ) be compactly dissipative and let its Levinson’s center J

be min-

imal (i.e., every semi-trajectory

(y ∈ J

) is dense in J

). Then the following

conditions are equivalent:

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

, π), (Y, T

, σ), hi is convergent;

(2) the dynamical system (X, T

, π) satisﬁes the condition (A) and for every K ∈

C(X) the equality (2.42) holds.

Proof. By virtue of Corollary 2.2 Condition 1 implies Condition 2. Let us show

that the converse assertion holds. Let K ∈ C(X), then

{

|x ∈ K} is

relatively compact and according to Lemma 1.3 the set

Ω(K) =

t≥0

[

τ≥t

is non-empty, compact, invariant and, consequently, h(Ω(K)) ⊆ Ω(h(K)) ⊆ J

because J

is the maximal compact invariant set in Y . Thus, J

is minimal, then

the equality

h(Ω(K)) = J

holds. We note that Ω(K

) = Ω(K

) for all K

and K

from C(X). In fact,

since M := Ω(K

)

Ω(K

) is compact and invariant and J

is minimal, we have

h(M) = J

. On the other hand, according to Lemma 2.10 the set M

= M

contains a single point for every y ∈ J

. We have Ω(K

)

⊆ M

(i = 1, 2)

and Ω(K

)

= Ω(K

)

= M

for any y ∈ J

and, consequently,

Ω(K

) = Ω(K

) for all K

and K

from C(X). From this follows that (X, T

, π)

is compactly dissipative and according to Theorem 2.7 h(X, T

, π), (Y, T

, σ), hi is

convergent. 

Corollary 2.4 Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical

system, (Y, T

, σ) be compactly dissipative, J

be minimal and (X, T

, π) satisfy

the condition (A). Then the Conditions 1-4 from Corollary 2.2 are equivalent.

Theorem 2.12 Let (Y, T

, σ) be compactly dissipative and h(L

) = J

. For the

non-autonomous dynamical system h(X, T

, π), (Y, T

, σ), hi to be convergent, it is

necessary, and if J

= Y it is suﬃcient, that there would be fulﬁlled the following

conditions:

1. Σ

is relatively compact for all x ∈ X and L

is relatively compact;

2. L

contains only one point for any y ∈ Y ;

3. for every ε > 0 there exists δ(ε) > 0 such that ρ(x, x

) < δ ({x

} = L

and h(x) = y ∈ J

) implies ρ(xt, x

) < ε for all t ≥ 0 and x ∈ X.