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

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

Non-autonomous Dissipative Dynamical Systems 77

Proof. Necessity. Let h(X, T

, π), (Y, T

, σ), hi be convergent, then (X, T

, π) is

compactly dissipative and J

= L

. It is easy to see that the Conditions 1 and 2

are fulﬁlled. We will show that Condition 3 is fulﬁlled too. Suppose that it is not

so, then there exist ε

> 0, δ

↓ 0, {x

} from X, {y

} from J

and t

→ +∞ such

that ρ(x

, x

) < δ

= h(x

)) and

ρ(x

, x

) ≥ ε

. (2.49)

Since the sets J

and J

are compact, then we can suppose that the sequences {y

}

and {x

} are convergent. Let y

= lim

n→+∞

, then x

= lim

n→+∞

= lim

n→+∞

By the compact dissipativity of the dynamical system (X, T

, π) we can suppose

that the sequence {x

} is convergent and let ¯x = lim

n→+∞

. Since y

∈ J

then the sequence {y

} can be considered convergent too. Let ¯y = lim

n→+∞

and we note that h(¯x) = lim

n→+∞

h(x

= lim

n→+∞

= ¯y, ¯x ∈ J

and hence

¯x ∈ J

¯y

= {x

¯y

}, i.e., ¯x = x

¯y

. On the other hand, passing to limit in (2.49) as

n → +∞ we obtain ρ(¯x, x

¯y

) ≥ ε

. The obtained contradiction proves the statement

required.

Suﬃciency. Suppose that Conditions 1, 2 and 3 of the theorem are fulﬁlled. For

the convergence of the system h(X, T

, π), (Y, T

, σ), hi it is suﬃcient to show that

under the conditions of the theorem the dynamical system (X, T

, π) is compactly

dissipative. By the condition of the theorem (X, T

, π) is point dissipative and

Ω

{ω

| x ∈ X} ⊆ L

. Note, that the set L

is closed. We will show that

the set L

is orbitally stable. If we suppose that it is not true, then there exist

> 0, x

→ x

∈ L

and t

→ +∞ such that

ρ(x

, L

) ≥ ε

Since y

→ y

= h(x

), where y

= h(x

), then under the conditions of the theorem

→ x

= x

and, consequently, ρ(x

, x

) ≤ ρ(x

, x

) + ρ(x

, x

) → 0. From

this relation and Condition 3 of the theorem follows that ρ(x

, x

) → 0. This

relation and the inequality (2.49) are contradictory. Thus, L

is compact, invariant

and orbitally stable. Since Ω

⊆ L

, then J

(Ω

) ⊆ L

. Let us show that

(Ω

) = J

. In fact, let ¯x ∈ L

and ϕ : S → L

be a whole trajectory

of (X, T

, π) passing through point ¯x for t = 0. Denote by α

¯x

t≤0

τ≤t

ϕ(τ),

then Ω

¯x

6= ∅. Let p ∈ α

¯x

Ω

, then there exists t

→ −∞ such that

ϕ(t

) → p and, consequently, π

−t

ϕ(t

) = ϕ(0) = ¯x, i.e., ¯x ∈ J

⊆ J

(Ω

Thus, L

⊆ J

(Ω

), i.e. L

= J

(Ω

) is compact and orbitally stable and

by Theorem 1.13 the dynamical system (X, T

, π) is compactly dissipative. The

theorem is proved. 

Theorem 2.13 Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical

system and Y be a compact minimal set, then the following conditions are equivalent:

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

78 Global Attractors of Non-autonomous Dissipative Dynamical Systems

1. h(X, T

, π), (Y, T

, σ), hi is convergent;

2. every semi-trajectory

(x ∈ X) is relatively compact and asymptotically sta-

ble;

3. (a) every semi-trajectory

(x ∈ X) is relatively compact;

(b) lim

t→+∞

ρ(x

t, x

t) = 0 for all (x

, x

) ∈ X

×X;

, x

) <

δ(h(x

) = h(x

); x

, x

∈ K) implies ρ(x

t, x

t) < ε for all t ≥ 0.

4. every semi-trajectory

(x ∈ X) is relatively compact and the equality (2.42)

holds for all K ∈ C(X).

Proof. The implications 1. → 2. → 3. → 4. are proved by a slight modiﬁcation of

the proof of Theorem 2.7. To ﬁnish the proof of our theorem is suﬃcient to establish

that Condition 4 implies Condition 1. Let x

∈ X. Since Σ

is relatively compact,

then ω

6= ∅, is compact and all the motions in ω

are extendable to the left. We

set M := ω

. Since h(Ω

) ⊆ Ω

⊆ Y and Y is minimal, then h(M) = Y and,

consequently, M

:= M

6= ∅ for all y ∈ Y . By Lemma 2.1 the set M

consists

of the one point for any y ∈ Y . We will show now that for every x ∈ X the equality

= M is true. In fact, let K := ω

M, then by Lemma 2.10 K

:= K

contains only one point for every y ∈ Y . Since h(ω

) = h(ω

) = h(K) = Y , then

= ω

= K

for all y ∈ Y and, consequently, ω

= ω

= M for

any x ∈ X. Thus, (X, T

, π) is point dissipative. Let now K ∈ C(X), then Σ

(K)

is relatively compact. In fact, if {x

} ⊆ K and t

→ +∞, then by the condition 4.

we have

lim

n→+∞

ρ(x

, M

h(x

) = 0,

and, consequently, the sequence {x

} is relatively compact. According to

Theorem 1.15 the dynamical system (X, T

, π) is compact dissipative. Let J

the center of Levinson of the dynamical system (X, T

, π). By Lemma 2.10 the set

contains only one point for each y ∈ Y . The theorem is completely proved.

Theorem 2.14 Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical

system, M 6= ∅ be a compact positively invariant set. Suppose that the following

conditions are fulﬁlled:

(1) h(M ) = Y ;

(2) M

contains a single point for all y ∈ Y ;

(3) M is globally asymptotically stable, i.e. for any ε > 0 there exists δ(ε) > 0 such

that ρ(x, p) < δ (x ∈ X

, p ∈ M

:= M

) implies ρ(xt, pt) < ε for all t ≥ 0

and lim

t→+∞

ρ(xt, M

h(x)t

) = 0 for all x ∈ X.

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

, π), (Y, T

, σ), hi is convergent.

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

Non-autonomous Dissipative Dynamical Systems 79

Proof. We note that under the conditions of the theorem the dynamical system

(X, T

, π) is point dissipative and Ω

⊆ M. We will show that set M is orbitally

stable in (X, T

, π) . Suppose that it is not true, then there exist ε

> 0, δ

→

0, x

∈ B(M, δ

) and t

→ +∞ such that

ρ(x

, M) ≥ ε

. (2.50)

Since M is compact, then we may suppose that the sequence {x

} is convergent.

Let x

:= lim

n→+∞

, x

∈ M

, ρ(x

, M) = ρ(x

, x

) and y

= h(x

), then x

lim

n→+∞

and x

∈ M

. Let q

= h(x

) and note that

ρ(x

, x

) ≤ ρ(x

, x

) + ρ(x

, x

) → 0 (2.51)

as n → +∞, because q

→ y

and x

→ x

. Taking into account (2.51) and the

asymptotic stability of the set M, we have

ρ(x

, x

) = ρ(x

, x

) → 0. (2.52)

But the equalities (2.50) and (2.52) are contradictory. Hence, the set M is orbitally

stable in (X, T

, π) and by virtue of Theorem 1.13 the dynamical system (X, T

, π)

is compactly dissipative and J

⊆ M. To ﬁnish the proof of Theorem it is suﬃcient

to note that h(J

) = J

and for all y ∈ J

we have J

⊆ M

and,

consequently, J

contains a single point for any y ∈ J

. The theorem is

proved. 

Remark 2.6 If there exists y

∈ Y such that Y = H

), then it is evident that

Theorem 2.14 is invertible. For this aim we may take set H

), where x

∈ X

in the quality of set M appearing in the theorem.

2.5 Tests for convergence

Note that in Theorems 2.5 and 2.6 there are contained some conditions under which

the property of compact dissipativity is kept under the homomorphisms.

Below we make some assertions which ensure dissipativity of the system

(X, T

, π) if the dynamical system (Y, T

, σ) has this property and there exists

a homomorphism h of (X, T

, π) onto (Y, T

, σ).

Deﬁnition 2.16 Let h be a homomorphism of (X, T

, π) onto (Y, T

, σ). A set

M ⊆ X is called uniformly stable in the positive direction with respect to the homo-

morphism h if for all ε > 0 there exists δ = δ(ε, M) > 0 such that for any x

, x

⊆ M

for which h(x

) = h(x

) the inequality ρ(x

, x

) < δ implies ρ(x

t, x

t) < ε for every

t ≥ 0.

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

80 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Deﬁnition 2.17 A dynamical system (X, T

, π) is called uniformly stable (in the

positive direction) with respect to the homomorphism h on compact subsets from

X, if every compact M ∈ C(X) is uniformly stable in the positive direction with

respect to the homomorphism h.

Lemma 2.11 Suppose that a homomorphism h satisﬁes the following conditions:

(1) there is a continuous invariant section ϕ : Y → X of the homomorphism h;

(2) lim

t→+∞

ρ(x

t, x

t) = 0 for all x

, x

∈ X (h(x

) = h(x

));

(3) the dynamical system (X, T

, π) is uniformly stable in the positive direction on

compact subsets from X with respect to the homomorphism h.

Then:

(1) if (Y, T

, σ) is point dissipative then (X, T

, π) is point dissipative too. More-

over, Ω

and Ω

are homeomorphic;

(2) if (Y, T

, σ) is compactly dissipative then (X, T

, π) is also compactly dissipative

and, moreover, Ω

and Ω

are homeomorphic.

Proof. Let (Y, T

, σ) be point dissipative. Then Ω

is a nonempty, compact and

invariant set and, consequently, M := ϕ(Ω

) ⊆ Ω

is also non-empty, compact

and invariant. For x ∈ X and y := h(x) we have lim

t→+∞

ρ(xt, ϕ(y)t) = 0. Hence,

is relatively compact compact and ω

⊆ ω

ϕ(y)

⊆ ϕ(Ω

) = M. Thus, Ω

⊆ M

and (X, T

, π) is point dissipative and ϕ(Ω

) = Ω

. Since ϕ : Ω

→ Ω

separates

points and the set Ω

is compact, then Ω

and Ω

are homeomorphic.

Now we will prove the second statement of the lemma. Let (Y, T

, σ) be com-

pactly dissipative. By the statement above, the dynamical system (X, T

, π) is

point dissipative. Let M := ϕ(J

) = ϕ(D

(Ω

)) ⊆ D

(Ω

). We will show that

the set M is orbitally stable. If we suppose that it is not true, then there are

, x

→ x

∈ M and t

→ +∞ such that

ρ(x

, M) ≥ ε

. (2.53)

Note, that h(x

) = y

→ y

= h(x

) ∈ h(M ) = h(ϕ(J

)) = J

and by compact

dissipativity of (Y, T

, σ) we can suppose that the sequence {y

} is convergent.

Let y := lim

n→+∞

, then y ∈ J

and ϕ(y) = lim

n→+∞

ϕ(h(x

))t

. Since ϕ : J

→

ϕ(J

) = M separates points and h ◦ ϕ = id

, then ϕ : J

→ M = ϕ(J

) is a

homomorphism and ϕ ◦ h(x) = x for all x ∈ M and, consequently, ϕ(h(x

)) →

ϕ(h(x

)) = x

∈ M . From this relation follows that

lim

n→+∞

ρ(x

, ϕ ◦ h(x

)) = 0. (2.54)

Let K := M

}

{ϕ ◦ h(x

)}, ε > 0 and δ(ε, K) > 0 be the numbers from

the condition of uniform stability in the positive direction on compact subsets from

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

Non-autonomous Dissipative Dynamical Systems 81

X of the dynamical system (X, T

, π) with respect to the homomorphism h. The

equality (2.54) implies that for suﬃciently large n the inequality ρ(x

, ϕ◦h(x

)) < δ

holds and, consequently, ρ(x

t, ϕ ◦ h(x

)t) < ε for all t ≥ 0. In particular,

ρ(x

, ϕ ◦ h(x

) < ε (2.55)

for suﬃciently large n. Taking into account that ε is arbitrary, we get that from

(2.55) follows lim

n→+∞

= lim

n→+∞

ϕ ◦ h(x

= ϕ(y) ∈ M . This equality and

(2.53) are contradictory. The obtained contradiction shows that M is orbitally

stable. Thus, (X, T

, π) is point dissipative, Ω

⊆ M , the set M is non-empty,

compact, invariant and orbitally stable. By Theorem 1.13, the dynamical system

(X, T

, π) is compactly dissipative and J

⊆ M = ϕ(J

) ⊆ D

(Ω

). According

to Theorem 1.11 and Corollary 1.5, we have J

= ϕ(J

) = D

(Ω

). Thus, ϕ is a

homeomorphism of J

onto J

. The lemma is proved. 

Corollary 2.5 Under the conditions of Lemma 2.11, if (Y, T

, σ) is compactly

dissipative, then h(X, T

, π), (Y, T

, σ), hi is convergent.

Proof. This assertion follows from Lemma 2.11 and Corollary 2.2. 

Let (Y, S, σ) be a two-sided dynamical system, (X, S

, π) be a semi-group

dynamical system and h : X → Y be a homomorphism of (X, S

, π) onto (Y, S, σ).

Consider a non-autonomous dynamical system h(X, S

, π), (Y, S, σ), hi and denote

by Γ

(Y, X) the family of all continuous and bounded sections of the homomorphism

h. By equality

d(ϕ

, ϕ

) := sup

y∈Y

ρ(ϕ

(y), ϕ

(y)) (2.56)

there is deﬁned a metric on Γ

(Y, X).

Let X

×X := {(x

, x

) : x

, x

∈ X, h(x

) = h(x

)} and let V : X

×X → R

a mapping satisfying the following conditions:

1. a(ρ(x

, x

)) ≤ V (x

, x

) ≤ b(ρ(x

, x

)) for all (x

, x

) ∈ X

×X, where a, b

are two functions from A (A := {a | a : R

→ R

, a is continuous, strongly

increasing and a(0) = 0 }) and Im(a) = Im(b) ;

2. V (x

, x

) = V (x

, x

) for all (x

, x

) ∈ X

×X;

3. V (x

, x

) ≤ V (x

, x

) + V (x

, x

) for all x

, x

∈ X such that h(x

) =

h(x

) = h(x

From the conditions 1–3 follows that the function V on each ﬁber X

= h

−1

(y)

deﬁnes some metric which is topologically equivalent to ρ.

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

82 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Lemma 2.12 Suppose that the function V : X

×X → R

satisﬁes the conditions

1–3. Then by equality

p(ϕ

, ϕ

) := sup{V (ϕ

(y), ϕ

(y))| y ∈ Y } (2.57)

on Γ

(Y, X) there is deﬁned a complete metric that is topologically equivalent to

(2.56).

Proof. From Condition 1 it follows that the function V vanishes only on the

diagonal of X

×X, i.e. V (x

, x

) = 0 if and only if x

= x

. Taking into account

(2.57), we have d(ϕ, ψ) = 0 if and only if ϕ = ψ.

From Property 2 of the function V it follows that the function d is symmetric

(i.e., d(ϕ, ψ) = d(ψ, ϕ)). Finally, from Condition 3. it follows that the function d

satisﬁes the triangle inequality.

From the condition 1. follows that the metrics (2.56) and (2.57) are topologically

equivalent.

Thus, to complete the proof of the lemma it is suﬃcient to establish that the

space (Γ

(Y, X), d) is complete. Let {ϕ

} be a Cauchy’s sequence in (Γ

(Y, X), d),

i.e., d(ϕ

, ϕ

) → 0 as n, m → +∞. For every ε > 0 there exists N

(ε) > 0 such

that

ρ(ϕ

(y), ϕ

(y)) < ε (2.58)

for any y ∈ Y and n, m > N

(ε). From this relation and from the completeness

of (X, ρ) it follows that for each y ∈ Y there exists lim

n→+∞

(y) in (X, ρ). We set

ϕ(y) := lim

n→+∞

(y). We will show that this equality holds uniformly with respect

to y ∈ Y and, consequently, ϕ ∈ Γ

(Y, X). In fact, passing to limit in the equality

(2.58) as m → +∞ we have ρ(ϕ

(y), ϕ(y)) ≤ ε for all n > N

(ε) and y ∈ Y. Thus

ϕ ∈ (Γ

(Y, X), d) and d(ϕ

, ϕ) → 0 as n → +∞. The lemma is proved. 

Denote by S

: Γ

(Y, X) → Γ(Y, X) the mapping, deﬁned by the equality

ϕ)(y) = π

ϕ(σ

−1

y) for all t ∈ S

, ϕ ∈ Γ

(Y, X) and y ∈ Y . It is easy to

check that the family of mappings {S

}

t≥0

is a commutative semigroup.

Lemma 2.13 If there is a function V : X

×X → R

satisfying Conditions 1–3

and

4. V (x

t, x

t) ≤ Ne

−νt

V (x

, x

) (∀(x

, x

) ∈ X

×X, t ≥ 0), where N, ν > 0,

then the semigroup {S

}

t≥0

has a unique ﬁxed point ϕ ∈ Γ

(Y, X) which is an in-

variant section of h.

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

Non-autonomous Dissipative Dynamical Systems 83

Proof. Note that

p(S

, S

) = sup{V (π

(σ

−t

y), π

(σ

−t

y))| y ∈ Y }

≤ Ne

−νt

sup{V (ϕ

(σ

−t

y), ϕ

(σ

−t

y)) | y ∈ Y } ≤ Ne

−νt

p(ϕ

, ϕ

)

and, consequently, S

is a contraction for suﬃciently large t. From this fact and

taking into consideration that {S

} is commutative we obtain the existence of a

unique ﬁxed point ϕ of the semigroup {S

}

t≥0

, which is an invariant section of h.

The lemma is proved. 

Theorem 2.15 Let h(X, S

, π), (Y, S, σ), hi be a non-autonomous dynamical

system and let the following conditions be fulﬁlled:

1. (Y, S, σ) is compactly dissipative;

2. Γ

(Y, X) 6= ∅;

3. there exists V : X

×X → R

satisfying Conditions 1–4 of Lemma 2.13.

Then (X, S

, π) is compactly dissipative and the sets J

and J

are homeomorphic.

Proof. Under the conditions of the theorem, by Lemma 2.13 the semi-group {S

}

t≥0

has a unique ﬁxed point ϕ that is an invariant section of h.

In addition,

a(ρ(x

t, x

t)) ≤ V (x

t, x

t) ≤ Ne

−νt

V (x

, x

) ≤ Ne

−νt

b(ρ(x

, x

)).

Therefore, lim

t→+∞

a(ρ(x

t, x

t)) = 0 for all (x

, x

) ∈ X

×X and, consequently,

lim

t→+∞

ρ(x

t, x

t) = 0. Note, that the system (X, S

, π) is uniformly stable with

respect to h. In fact, let ε > 0 and δ(ε) = b

−1

(Na(ε)). Then ρ(x

, x

) < δ(ε)

(h(x

) = h(x

)) implies ρ(x

t, x

t) < ε for all t ≥ 0. To ﬁnish the proof of the

theorem it is suﬃcient to refer to Lemma 2.11. 

Corollary 2.6 Under the conditions of Theorem 2.15 the non-autonomous

dynamical system h(X, S

, π), (Y, S, σ), hi is convergent.

Proof. This assertion follows from Theorem 2.15 and Corollary 2.5. 

The function V : X

×X → R

is said to be continuous, if x

→ x

(i = 1, 2 and

h(x

) = h(x

)) implies V (x

, x

) → V (x

, x

Theorem 2.16 Let (X, T

, π) and (Y, T

, σ) be compactly dissipative. The

dynamical system h(X, T

, π), (Y, T

, σ), hi is convergent if and only if there is a

continuous function V : X

×X → R

satisfying the following conditions:

(1) V is positively deﬁned, i.e. V (x

, x

) = 0 if and only if x

= x

;

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

84 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(2) V (x

t, x

t) ≤ V (x

, x

) for all t ≥ 0 and (x

, x

) ∈ X

×X;

(3) V (x

t, x

t) = V (x

, x

) for all t ≥ 0 if and only if x

= x

Proof. Suppose that the conditions of the theorem are fulﬁlled. We will show that

for all ε > 0 and compact K ⊆ X there is δ(ε, K) > 0 such that ρ(x

, x

) < δ

, x

∈ K and h(x

) = h(x

)) implies ρ(x

t, x

t) < ε for all t ≥ 0. Indeed, if we

suppose that it is not true, then there exist ε

> 0, compact K

⊆ X, sequences

↓ 0, {x

} ⊆ K

(i = 1, 2 and h(x

) = h(x

)) and t

→ +∞ such that

ρ(x

, x

) < δ

and ρ(x

, x

) ≥ ε

. (2.59)

By the compact dissipativity of (X, T

, π) we can assume that the sequences {x

}

(i = 1, 2) are convergent. Let us put ¯x

:= lim

n→+∞

(i = 1, 2). It is clear that

h(¯x

) = h(¯x

). Since {x

} ⊆ K, then we may suppose that they are convergent too.

By virtue of (2.59), lim

n→+∞

= lim

n→+∞

= ¯x and

0 ≤ V (¯x

, ¯x

) = lim

n→+∞

V (x

, x

) ≤ lim

n→+∞

V (x

, x

) = V (¯x, ¯x) = 0,

from which it follows that ¯x

= ¯x

. The last equality contradicts (2.59).

We will show now that for all (x

, x

) ∈ X

×X (h(x

) = h(x

)) the equality

lim

t→+∞

ρ(x

t, x

t) = 0

holds. If we suppose that it is not true, then there exist y

∈ Y, ¯x

, ¯x

∈ X

, ε

> 0

and t

→ +∞ such that

ρ(¯x

, ¯x

) ≥ ε

. (2.60)

Let (X ×X, T

, π×π) := (X, T

, π)×(X, T

, π) be a direct product of (X, T

, π) and

(X, T

, π). In view of the compact dissipativity of (X, T

, π), the point (¯x

, ¯x

) ∈

X × X is L

stable in (X × X, T

, π × π), and from Condition 2 of the theorem

follows the existence of a ﬁnite limit

= lim

t→+∞

V (¯x

t, ¯x

t). (2.61)

Let (p, q) ∈ ω

(¯x

,¯x

)

, then from (2.61) it follows that V (p, q) = V

. By the invariance

of ω

(¯x

,¯x

)

we have V (pt, qt) = V (p, q) for all t ∈ T, and, according to Condition 3

of the theorem, p = q, i.e.,

(¯x

,¯x

)

⊆ ∆

:= {(x, x) | x ∈ X}. (2.62)

We may suppose that the sequences {¯x

} (i = 1, 2) are convergent. Let us put

¯p = lim

n→+∞

¯x

and ¯q = lim

n→+∞

¯x

, then (¯p, ¯q) ∈ ω

(¯x

,¯x

)

and from (2.60) it

follows that ¯p 6= ¯q. The last equality contradicts the inclusion (2.62). The obtained

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

Non-autonomous Dissipative Dynamical Systems 85

contradiction proves the required statement. By Theorem 2.7, the non-autonomous

dynamical system (2.1) is convergent.

Conversely, let h(X, T

, π), (Y, T

, σ), hi be convergent. We will deﬁne the func-

tion V : X

×X → R

by the equality

V (x

, x

) := sup{ρ(x

t, x

t) | t ≥ 0}. (2.63)

It is clear that this function satisﬁes Conditions 1 and 2 of the theorem. Let us

show that V satisﬁes Condition 3 of the theorem. Let V (x

t, x

t) = V (x

, x

) for

all t ≥ 0. If we suppose that x

6= x

, then δ := V (x

, x

) > 0. In view of Theorem

2.7, ρ(x

t, x

t) → 0 as t → +∞. Therefore, there exists t

= t

(δ) > 0 such that

ρ(x

t, x

t) <

for all t ≥ t

. Hence,

V (x

, x

) = sup{ρ(x

t, x

t) | t ≥ t

} ≤

< V (x

, x

The last inequality contradicts our assumption. We will prove that for every com-

pact K ⊆ X

lim

t→+∞

sup{ρ(x

t, x

t) | x

, x

∈ K, h(x

) = h(x

)} = 0. (2.64)

In fact, if suppose that it is not true, then there exit ε

> 0, compact K ⊆ X,

↓ 0 and sequences {x

} ⊆ K (i = 1, 2 and h(x

) = h(x

)) t

→ +∞ such

that Condition (2.46) holds. Without loss of generality we may suppose that the

sequences {x

} and {x

} (i = 1, 2) are convergent. Let lim

n→+∞

= x

(i = 1, 2).

We put ¯x

:= lim

n→+∞

(i = 1, 2). Note that h(¯x

) = h(¯x

) = y, ¯x

, ¯x

∈ ω(K) ⊆

and, hence, ¯x

= ¯x

. The last equality contradicts (2.46). Thus, the equality

(2.64) is proved. From the equalities (2.63) and (2.64), it follows that

V (x

, x

) = ρ(x

τ, x

τ) (2.65)

for some τ(x

, x

) ∈ [0, l(K)].

Let (x

, x

) → (x

, x

) ∈ X

×X. The sequence {V (x

, x

)} = {ρ(x

, x

)}

is bounded, where τ

= τ(x

, x

). We will show that it has a unique limit point.

Denote by

V one of the limit points of the sequence {V (x

, x

)}. There exists a

subsequence {V (x

, x

)} such that V (x

, x

) →

V as n → +∞. Since {τ

} is

bounded, then we may suppose that it is convergent. Then

V = ρ(x

, x

), where

= lim

n→+∞

. Let us show that ρ(x

τ, x

τ) = ρ(x

, x

) (τ = τ(x

, x

) ≥ 0 is

chosen out of Condition (2.65)). If we suppose that ρ(x

τ, x

τ) 6= ρ(x

, x

), then

ρ(x

τ, x

τ) > ρ(x

, x

). Let ε > 0 be such that ρ(x

τ, x

τ) + 2ε < ρ(x

τ, x

τ).

Then for a suﬃciently large k ∈ N we have |ρ(x

τ, x

τ) − ρ(x

τ, x

τ)| < ε and

|ρ(x

, x

) − ρ(x

, x

)| < ε, and, consequently,

ρ(x

τ, x

τ) > ρ(x

, x

) = V (x

, x

). (2.66)

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

86 Global Attractors of Non-autonomous Dissipative Dynamical Systems

The inequality (2.66) contradicts the choice of the number τ

. Thus, V (x

, x

) =

ρ(x

, x

) and, consequently, lim

n→+∞

V (x

, x

) = V (x

, x

), i.e., the function V

is continuous. The theorem is proved. 

Theorem 2.16 is a generalization for the abstract non-autonomous dynamical

systems of the theorems 7.2 and 7.3 from

[

270

]

Theorem 2.17 Let dynamical systems (X, T

, π) and (Y, T

, σ) be compactly dis-

sipative. A non-autonomous dynamical system h(X, T

, π),(Y, T

, σ), hi is conver-

gent if and only if there exists a continuous function V : X

×X → R

, satisfying

the following conditions:

(1) V is positively deﬁned;

(2) V (x

t, x

t) < V (x

, x

) for all t > 0 and (x

, x

) ∈ X

×X \ ∆

, where ∆

{(x, x) | x ∈ X}.

Proof. The suﬃciency of Conditions of the theorem it follows from the previous

theorem. It is suﬃcient to remark that from the second condition of Theorem 2.17

it follows the second and third conditions of Theorem 2.60.

Necessity. Let h(X, T

, π), (Y, T

, σ), hi be convergent and the function V :

×X → R

be deﬁned by the equality (2.63). Then it is continuous and sat-

isﬁes Conditions 1–3 of Theorem 2.16. Let us put

V (x

, x

) :=

+∞

V(x

t, x

t)e

−t

dt. (2.67)

From the deﬁnition of the function V follow its continuity and positive deﬁniteness.

The function V satisﬁes Condition 2 of Theorem 2.17. In fact, if we suppose the

contrary, then there exit (¯x

, ¯x

) ∈ X

×X and t

> 0 such that V (¯x

, ¯x

) =

V (¯x

, ¯x

) and ¯x

6= ¯x

. Then from (2.67) and from the last equality, it follows that

V(¯x

+ t), ¯x

+ t)) = V(¯x

t, ¯x

t) (2.68)

for all t ≥ 0. By virtue of (2.68), the function ϕ(t) = V(¯x

t, ¯x

t) is t

periodic.

It is obvious that ϕ is continuous and non-increasing. Therefore, ϕ is stationary

and, hence, V(¯x

t, ¯x

t) = V(¯x

, ¯x

) for all t ≥ 0. Since the function V satisﬁes

Conditions 1–3 of Theorem 2.16, then the equality V(¯x

t, ¯x

t) = V(¯x

, ¯x

) for all

t ≥ 0 implies ¯x

= ¯x

. The last equality contradicts the choice of (¯x

, ¯x

). The

obtained contradiction completes the proof of the theorem. 

Theorem 2.18 Let (X, T

, π) and (Y, T

, σ) be compactly dissipative and there

exist a continuous function V : X

×X → R

, satisfying the following conditions:

(1) V is positively deﬁned;