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 13

Chapter 13

Linear almost periodic dynamical systems

13.1 Bounded motions of linear systems

Assume that X and Y are complete metric spaces, R (Z) is the group of real

numbers (integers), S = R or S = Z, S

= {s ∈ S | s ≥ 0}, S

−

= {s ∈ S | s ≤ 0},

and T = S

, S

−

or S. Let (X, T, π) ((Y, S, σ)) be a semigroup (group) dynamical

system on X(Y ).

Recall that a triple h(X, T, π), (Y, S, σ), hi, where h is a homomorphism of

(X, T, π) onto (Y, S, σ), is called

[

]

a non-autonomous dynamical system.

Deﬁnition 13.1 h(X, T, π), (Y, S, σ), hi is said

[

32, 333

]

to be distal on S

the ﬁber X

:= {x ∈ X | h(x) = y} if inf{ρ(x

t, x

t) | t ∈ S

} > 0 for all

, x

∈ X

, x

6= x

For group non-autonomous dynamical systems the distalness on S

−

and S on

the ﬁber X

can be deﬁned likewise.

Deﬁnition 13.2 A non-autonomous system is said to be distal on S

−

, S) if it

is distal in every ﬁber X

, y ∈ Y .

Assume that (X

, T, π

) is a dynamical system on X

, i = 1, . . . , k; let X :=

×... ×X

, and let π := (π

, ..., k) : X ×T → X be deﬁned by the formula

π(x, t) := (π

, t), ..., π

, t))

for all t ∈ T and x := (x

, ..., x

) ∈ X.

Deﬁnition 13.3 The dynamical system (X, T, π), where X := X

×... ×X

and

π := (π

, ..., π

), is called the direct product of the dynamical systems (X

, T, π

), i =

1, ..., k and denoted by (X

, T, π

)×, ..., (X

, T, π

If X

= X, i = 1, ..., k, and π

= π, i = 1, ..., k, then

(X, T, π) ×(X, T, π) ×... ×(X, T, π) := (X

, T, π).

407

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408 Global Attractors of Non-autonomous Dissipative Dynamical Systems

The direct product of group dynamical systems is deﬁned likewise.

Deﬁnition 13.4 The points x

, ..., x

∈ X are said to be jointly recurrent if the

point (x

, ..., x

) ∈ X

is recurrent in the dynamical system (X

, T, π).

Lemma 13.1

[

32, 333

]

. The following assertions hold.

(1) Assume that X is compact and (Y, S, σ) is minimal. If the group non-

autonomous dynamical system h(X, S, π), (Y, S, σ), hi is distal on S

−

), then

it is distal on S.

(2) Assume that X is compact, (Y, S, σ) is minimal, and y ∈ Y . Then the following

conditions are equivalent :

(a) the group non-autonomous system h(X, S, π), (Y, S, σ), hi is distal on S in

the ﬁber X

;

(b) for any points x

, ..., x

∈ X, where k is any positive integer ≥ 2, the point

, ..., x

) ∈ X

is recurrent in (X

, S, π).

Let C(S, X) be the space of all continuous maps ϕ : S → X equipped with the

compact-open topology and let (C(S, X), S, σ) be the dynamical system of transla-

tions (shifts) on C(S, X). Let d be a metric on C(S, X) consistent with its topology

(for example, the Bebutov metric).

Lemma 13.2

[

282

]

. Let (X, S

, π) be a semigroup dynamical system and assume

that for any t ∈ S

the map π

: X → X is a homeomorphism and ˜π is the map of

X ×S toX deﬁned by the equality

˜π(x, t) :=

(

π(x, t), (x, t) ∈ X ×S

(π

−t

)

−1

(x), (x, t) ∈ X × S

−

Then the triple (X, S, ˜π) is a group dynamical system.

Lemma 13.3

[

]

. Assume that h(X, S

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

dynamical system, Y is compact, (Y, S, σ) is minimal, and x

∈ X

has a relatively

compact semi-trajectory {x

t | t ∈ S

}. Then one can ﬁnd a recurrent point x ∈

(x ∈ X

) and a sequence t

→ +∞ such that ρ(x

, xt

) → 0 as n → +∞.

Recall that a dynamical system (X, S, π) is said to be asymptotically compact if

for any bounded positively invariant set B ⊆ X there is a non-empty compact set

K ⊆ X such that

lim

t→+∞

sup{ρ(xt, K) | x ∈ B} = 0.

Remark 13.1 (i) Assume that x ∈ X is such that {xt | t ∈ S

} is bounded and

(X, S

, π) is asymptotically compact. Then {xt | t ∈ S

} is relatively compact.

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Linear almost periodic dynamical systems 409

(ii) Let M ⊆ X be bounded and invariant. Then M is relatively compact if the

dynamical system (X, S

, π) is asymptotically compact. In particular, if x ∈ X and

γ ∈ Φ

is such that γ(S) is bounded, then γ(S) is relatively compact, where Φ

is a

set of all entire trajectories of (X, S

, π) passing through x at t = 0.

If X := E×Y, π := (ϕ, σ), that is, π((u, y), t) := (ϕ(t, x, y), σ(t, y)) for all (u, y) ∈

E×Y and t ∈ S, then the non-autonomous dynamical system h(X, T, π), (Y, S, σ), hi,

where h := pr

: X → Y, is called

[

275

]

a skew product over (Y, S, σ) with the ﬁber

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

[

]

Recall that a non-autonomous dynamical system h(X, T, π), (Y, S, σ), hi is said

[

275

]

[

]

to be linear if the map π

: X

→ X

is linear for every t ∈ T and y ∈ Y .

If h(X, T, π), (Y, S, σ), hi is a skew product over (Y, S, σ) with the ﬁber E, then

it is linear if and only if E is a Banach space and the map ϕ(t, ·, y) : E → E is

linear for every y ∈ Y and t ∈ T.

Throughout the rest of this section we assume that Y is compact, the dynamical

system (Y, S, σ) is minimal, X = E ×Y, E is a Banach space with the norm |·|, the

non-autonomous dynamical system h(X, T, π), (Y, S, σ), hi is linear, π = (ϕ, σ), and

h = pr

Let F ⊆ E × Y be a closed vectorial subset of the trivial ﬁber bundle (E ×

Y, pr

, Y ) that is positively invariant relative to (X, T, π). We put

= {(x, y) ∈ F | sup |ϕ(t, x, y)| : t ∈ S

< +∞}.

The set B

−

is deﬁned likewise. If h(X, S

, π), (Y, S, σ), hi is a semigroup non-

autonomous dynamical system, then B is the set of all points of F with the follow-

ing property: there is an entire trajectory of the dynamical system (F, S

, π)

bounded on S that passes through this point. We put B

:= B

and

:= B

, y ∈ Y.

Theorem 13.1 The following conditions are equivalent :

(i) there is an M > 0 such that

|ϕ(t, x, y)| ≤ M|x| (13.1)

for all (x, y) ∈ B

−

, B) and t ∈ S

−

, S);

(ii) B

−

, B) is closed in F.

Proof. We prove this theorem in the case when T = S

. In the case when T = S

−

or T = S it can be proved in a similar way. We claim that (i) implies (ii). Assume

that (x, y) ∈ B

. Then there is an (x

, y

) ∈ B

such that (x

, y

) → (x, y) as

n → +∞. By condition (i), the inequality

|ϕ(t, x

, y

)| ≤ M|x

| (13.2)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 13

410 Global Attractors of Non-autonomous Dissipative Dynamical Systems

is valid for all n = 1, 2, ... and t ∈ S

. Passing to the limit in (13.2) as n → +∞, we

obtain that |ϕ(t, x, y)| ≤ M|x| for all t ∈ S

, that is, (x, y) ∈ B

Now we claim that (ii) implies (i). Let B

be closed and let y ∈ Y. We put

d(y) = sup{|ϕ(t, x, y)| : t ∈ S

, (x, y) ∈ B

, |x| ≤ 1}. (13.3)

We claim that the function d : Y → R

deﬁned by formula (13.3) is lower semi-

continuous. Assume the contrary. Then there are ε > 0, y ∈ Y, and y

→ y such

that

lim

n→+∞

d(y

) = d(y) −ε. (13.4)

Formula (13.3) implies that there are |x

| ≤ 1 and {t

} ⊆ S

((x

, y) ∈ B

) such

that

d(y) = lim

n→+∞

|ϕ(t

, x

, y)|.

Therefore, for ε > 0 one can ﬁnd a k such that the inequality

||ϕ(t

, x

, y)|−d(y)| <

(13.5)

is valid for all n ≥ k. Since the map ϕ(t

, x

, ·) : Y → X is continuous, there is an

n = n(k) such that

|ϕ(t

, x

, y

) − ϕ(t

, x

, y)| <

(13.6)

for all n ≥ n(k). Inequalities (13.5) and (13.6) imply that

|d(y) − |(t

, x

, y

)|| <

(13.7)

for all n ≥ n(k). Hence,

d(y) − d(y

) ≤

(13.8)

for all n ≥ n(k). On the other hand, formula (13.4) implies that

d(y) − d(y

) ≥ 3

(13.9)

if n is suﬃciently large.

Inequality (13.9) contradicts (13.8). This contradiction proves that d : Y → R

is lower semi-continuous. Hence, this function has a set of points of continuity

D ⊆ Y of the type G

. Let p ∈ D, then there exists positive numbers δ

and M

such that d(y) ≤ M

for all

y ∈ B[p, δ

] := {y ∈ Y | ρ(y, p) ≤ δ

} ⊆ Y.

Since Y is minimal, there are negative numbers t

, ..., t

such that Y =

i=1

σ(t

, B[p, δ

]) (see

[

238, p.134

]

). We put L := max{|t

| : i = 1, ..., m}.

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Linear almost periodic dynamical systems 411

We claim that the family of operators {π

| t ∈ [0, L]} is uniformly continuous, that

is, for any ε > 0 there is a δ(ε) > 0 such that |x| ≤ δ implies that |xt| ≤ ε for all

t ∈ [0, L]. Assume the contrary. Then there are ε

> 0, δ

↓ 0, |x

| < δ

, y

∈ Y,

and t

∈ [0, L] such that

|ϕ(t

, x

, y

)| ≥ ε

. (13.10)

Since Y and [0, L] are compact, we can assume that the sequences {y

} and {t

}

are convergent. Put y

= lim

n→+∞

and t

:= lim

n→+∞

. Passing to the limit in

(13.10) as n → +∞, we obtain 0 ≥ ε

. The last inequality contradicts the choice of

. This contradiction proves the above assertion.

If α > 0 is such that |ϕ(t, x, y)| ≤ 1 for all t ∈ [0, L], |x| ≤ α, and y ∈ Y, then

|ϕ(t, x, y)| ≤ α

−1

|x| (13.11)

for all t ∈ [0, L], x ∈ E, and y ∈ Y. Assume that q ∈ Y, y ∈ B[p, δ

], and t

are such

that q = yt

. Then

|ϕ(t, x, q)| = |ϕ(t, x, yt

)| = |ϕ(t + t

, ϕ(−t

, x, yt

), y)| =

|ϕ(−t

, x, yt

)||ϕ(t + t

ϕ(−t

, x, yt

)

|ϕ(−t

, x, yt

, y)|. (13.12)

The set B

is positively invariant and contains (x, q). Therefore, B

con-

tains π

−t

(x, q), where π

−t

(x, q) = (ϕ(−t

, x, q), σ(−t

, q)). Hence, π

−t

(x, q) =

(ϕ(−t

, x, q), σ(−t

, q)) ∈ B

and

|ϕ(t + t

ϕ(−t

, x, yt

)

|ϕ(−t

, x, yt

)|, y)|

, y)| ≤ M

(13.13)

for all t ≥ L. On the other hand, inequality (13.11) implies that

|ϕ(−t

, x, yt

)| ≤ α

−1

|x| (13.14)

for all x ∈ E. Formulas (13.12)-(13.14) imply that |ϕ(t, x, q)| ≤ M|x| for all t ∈ S

and x ∈ E, where M = α

−1

max{1, M

}. This completes the proof of the theorem.

Lemma 13.4 Assume that h(X, S

, π), (Y, S, σ), hi is a linear non-autonomous

dynamical system, (X, S

, π) is asymptotic compact, and there is an M > 0 such

that

|γ(t)| ≤ M|x| (13.15)

for all γ ∈ Φ

(x,y)

, (x, y) ∈ B, and t ∈ S. Then the set Φ

{Φ

(x,y)

| (x, y) ∈

B, |x| ≤ 1} is relatively compact in C(S, X).

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 13

412 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Proof. Consider the set K := {γ(t) | t ∈ S, γ ∈ Φ

} ⊆ B. It is obvious that K

is invariant. By (13.15), it is bounded. Since (X, S

, π) is asymptotic compact,

K is relatively compact (see remark 13.1). We claim that the family of functions

⊆ C(S, X) is equicontinuous. Assume the contrary. Then there are ε

> 0, δ

↓

0, {t

}(i = 1, 2), and {γ

} ⊆ Φ

such that |t

−t

| < δ

and

|γ

) − γ

)| ≥ ε

. (13.16)

Without loss of generality we can assume that t

− n > t

. Then inequality

(13.16) implies that

|π

−x

| ≥ ε

(13.17)

for all n = 1, 2, ..., where τ

:= t

− t

and x

:= γ

(

). Since {x

} ⊆ K, we can

assume that the sequence {x

} converges. Let x

= lim

n→+∞

. Passing to the limit

in inequality (13.17) as n → +∞, we obtain the inequality 0 ≥ ε

, which contradicts

the choice of ε

. To complete the proof of the lemma, it is suﬃcient to apply the

Arzela–Ascoli- theorem. 

Theorem 13.2 Assume that h(X, S

, π), (Y, S, σ), hi is a linear non-autonomous

dynamical system, (X, S

, π) is asymptotic compact, and there is an M > 0 such

that inequality (13.15) is valid for all γ ∈ Φ

(x,y)

, (x, y) ∈ B, and t ∈ S. Then the

following assertions hold:

(i) Any two diﬀerent entire trajectories γ

and γ

(h(γ

(0)) = h(γ

(0))) are jointly

recurrent;

(ii) For any (x, y) ∈ B the set Φ

(x,y)

consists of a single entire recurrent trajectory;

(iii) B is closed in F ;

(iv) (X, S

, π) induces a group dynamical system (B, S, π) on B;

(v) For any y ∈ Y the set B

is ﬁnite-dimensional and dim B

does not depend

on y ∈ Y .

Proof. Assume that (x, y) ∈ B, and let γ ∈ Φ

(x,y)

be bounded on S. By Lemma

13.4, the set H(γ) :=

{γ

| τ ∈ S} is compact in C(S, X), since γ

∈ Φ

γ(τ)

where γ

is the shift of γ by τ and the bar denotes closure in C(S, X). Con-

sider the group non-autonomous dynamical system h(H(γ), S, λ), (Y, S, σ), hi, where

(H(γ), S, λ) is the dynamical system of shifts on H(γ) induced by the Bebutov’s

system (C(S, X), S, σ) and µ : H(γ) → Y is the map deﬁned by the equality

µ(ψ) = h(ψ(0)). Under the conditions of Theorem 13.2 (see also inequality (13.15))

the non-autonomous dynamical system h(H(γ), S, λ), (Y, S, σ), µi is negatively dis-

tal, that is, inf{dλ(t, γ

), λ(t, γ

)) : t ∈ S

−

} > 0 for any γ

, γ

∈ H(γ) such that

6= γ

and µ(ψ

) = µ(ψ

). By Lemma 1 in

[

238, p.104

]

, h(H(γ), S, λ), (Y, S, σ), hi

is distal on S. Therefore, γ

and γ

are jointly recurrent. In particular, they are

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Linear almost periodic dynamical systems 413

recurrent. Moreover, by Lemma 13.1, any two entire trajectories γ

∈ Φ

(x,y)

and

∈ Φ

(x,y)

are jointly recurrent.

We claim that for any (x, y) ∈ B the set Φ

(x,y)

consists of a single entire recurrent

trajectory. Assume the contrary. Then there are γ

, γ

∈ Φ

(x,y)

such that γ

6= γ

Putting γ(t) := γ

(t)−γ

(t), we obtain a recurrent function γ 6= 0 such that γ(t) = 0

for all t ∈ S

, which is impossible.

Now let (x, y) ∈

B. Then there is an (x

, y

) ∈ B such that (x

, y

) → (x, y).

Let γ

be the (unique) entire trajectory of (F, S

, π) bounded on S and satisfying

the condition γ

(0) = (x

, y

). By Lemma 13.4, we can assume that the sequence

{γ

} converges in C(S, X). Inequality (13.15) implies that γ = lim

n→+∞

is an entire

trajectory of (F, S

, π) bounded on S. Moreover, γ(0) = (x, y). Thus, γ ∈ Φ

(x,y)

whence (x, y) ∈ B.

Since B is closed and invariant, (X, S

, π) induces a dynamical system (B, S

, π)

on B, and π

B = B for all t ∈ S

. By assertion (ii) of the theorem, π

: B → B (t ∈

) is a one-to-one map and (π

)

−1

(b) = γ

(−t) for all t ∈ S

and b ∈ B, where

{γ

} = Φ

. Lemma 13.4 implies that π

: B → B is a homeomorphism. To prove

assertion (iv), it is suﬃcient to apply Lemma 13.2.

We now prove the last assertion of the theorem. Since (X, S

, π) is asymptotic

compact, K := {(x, y) | (x, y) ∈ B, |x| ≤ 1, y ∈ Y } is compact. Therefore, every

linear subspace B

of X

:= E × {y} is ﬁnite-dimensional. Let y ∈ Y and let

, ..., x

∈ B

be a basis in B

. Then Lemma 13.1 implies that the points x

, ..., x

are jointly recurrent. Let q be an arbitrary point of Y. Since Y is minimal, there is

a sequence {t

} ⊂ S such that yt

→ q and π(t

, x

) → ξ

(i = 1, ..., k) as n → +∞,

where ξ

, ..., ξ

∈ B

and the points ξ

, ..., ξ

are jointly recurrent.

We claim that ξ

, ..., ξ

are linearly independent. Assume the contrary. Then

there are constants c

, ..., c

such that c

+ ... + c

= 0 and

i=1

| 6= 0. Since

(ξ

, ..., ξ

) ∈ H(x

, ..., x

) and (x

, ..., x

) is recurrent, there is a {τ

} ⊂ S such that

qτ

→ y and π(ξ

, τ

) → x

(i = 1, ..., k) as n → +∞. Therefore,

+ ... + c

= lim

n→+∞

π(τ

, c

+ ... + c

) = 0.

The last relation contradicts the choice of x

, ..., x

. Thus, dimB

≥ dim B

for all

q ∈ Y. Since Y is minimal, the reverse inequality also holds. Hence, dim B

= dimB

for all q ∈ Y, which completes the proof of the theorem. 

Theorem 13.3 Let h(X, S

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

dynamical system and assume that (X, S

, π) is asymptotic compact. Then the

following conditions are equivalent :

(i) there is an M > 0 such that (13.15) is valid for all γ ∈ Φ

(x,y)

, (x, y) ∈ B, and

t ∈ S;

(ii) B is closed in F.

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414 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Proof. By Theorem 13.2, the theorem will be proved if we prove that (ii) implies

(i). By Theorem 13.1, there is an M > 0 such that (13.1) is valid for all (x, y) ∈ B

and t ∈ S

. Since B is invariant, for any (x, y) ∈ B and γ ∈ Φ

(x,y)

the inequality

|γ(t)| ≥ M

−1

|x| (13.18)

is valid for all t ∈ S

−

. Repeating the arguments used in Lemma 13.4, we can show

that H(γ) :=

{γ

| τ ∈ S} is compact in C(S, X). Consider the group dynamical

system h(H(γ), S, λ), (Y, S, σ), µi (see the proof of Theorem 13.2). Inequality (13.18)

implies that the non-autonomous system (H(γ), S, λ), (Y, S, σ), µi, is distal on S

−

By Lemma 13.1, γ is recurrent. Therefore,

sup{|γ(t)| : t ∈ S} = sup{|γ(t)| : t ∈ S

} ≤ M|x|

for all γ ∈ Φ

(x,y)

and (x, y) ∈ B. Hence, condition (i) holds and the theorem is

proved. 

Theorem 13.4 Let h(X, S

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

dynamical system. Assume that (X, S

, π) is asymptotic compact and B

is closed.

Then

(i) B is closed, and

(ii) for any (x, y) ∈ B

there is a recurrent point (x, y) ∈ B such that

lim

t→+∞

|ϕ(t, x

, y) − ϕ(t, x, y)| = 0.

Proof. Assume that B

is closed. By Theorem 13.1, there is an M > 0 such

that (13.1) is valid for all (x, y) ∈ B

and t ∈ S

. Assume that (x, y) ∈ B ⊆ B

Then inequality (13.1) implies that (13.18) is valid for all t ∈ S

−

and γ ∈ Φ

(x,y)

Repeating the arguments used in the proof of Theorem 13.3, we obtain (13.15) for

all γ ∈ Φ

(x,y)

, (x, y) ∈ B and t ∈ S. To complete the proof of the ﬁrst assertion of

the theorem, it is suﬃcient to apply Theorem 13.2.

Now we prove the second assertion of the theorem. Let (x

, y) ∈ B

. Since

(X, S

, π) is asymptotic compact, the semi-trajectory {π

, y) | t ∈ S

} of the

point (x

, y) is relatively compact. Therefore, ω

,y)

6= ∅ is compact and invariant.

By Lemma 13.3, one can ﬁnd a recurrent point (x, y) ∈ ω

,y)

and a sequence

→ +∞ such that

lim

n→+∞

|ϕ(t

, x

, y) − ϕ(t

, x, y)| = 0. (13.19)

Inequality (13.15) implies that

|ϕ(t, x

, y) − ϕ(t, x, y)| ≤ M |ϕ(t

, x

, y) − ϕ(t

, x, y)| (13.20)

for all t ≥ t

. Formulas (13.19) and (13.20) imply the desired assertion, which

completes the proof of the theorem. 

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 13

Linear almost periodic dynamical systems 415

Remark 13.2 The second assertion of Theorem 13.3 remains true even if we do

not assume that B

is closed.

We conclude this section with a condition under which a linear non-autonomous

system is asymptotic compact.

Let P : X → X be a projection of the vector bundle, that is, P

:= P |

projection in X

for every y ∈ Y. Then P is said to be completely continuous if

P (M) is relatively compact for any bounded set M ⊆ X.

Lemma 13.5 Let h(X, S

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

system. Assume that the maps π

:= π(t, ·) : X → X can be represented as sums

π(t, x) := π

(t, x) + π

(t, x) for all t ∈ S

and x ∈ X, and that the following

conditions hold.

(i) |π

(t, x)| ≤ m(t, r) for all t ∈ S

and x ∈ X, where m : R

→ R

and for

every r ≥ 0 the function m(t, r) tends to zero as t → +∞.

(ii) The maps π

(t, ·) : X → X (t > 0) are conditionally completely continuous,

that is, π

(t, A) is relatively compact for any t > 0 and any bounded positively

invariant set A ⊆ X.

Then the dynamical system (X, S

, π) is asymptotically compact.

Proof. Let A ⊆ X be a bounded positively invariant set. Then A = Σ

(A) :=

{π

A | t ∈ S

}. Since Y is compact and (X, h, Y ) is local trivial, there is an r > 0

such that A ⊂ {x ∈ X : |x| ≤ r}. We claim that for any {x

} ⊂ A and t

→ +∞

the sequence {x

} is relatively compact. We can cover M := {x

} with a ﬁnite

ε-net for any ε > 0. Assume that ε > 0 and l > 0 are such that m(l, r) <

. We

represent M as the union M

, where M

:= {x

}

k=1

, M

:= {x

}

∞

k=k1+1

and k

:= max{k | t

≤ l}. The set M

is a subset of π

(Σ

(A)) whose elements

can be represented in the form π

(t, x)+π

(t, x) (x ∈ Σ

(A)). Since π

(l, Σ

(A)) is

relatively compact, we can cover it with a ﬁnite (

)-net. For any y ∈ π

(l, Σ

(A))

there is an x ∈ Σ

(A) such that y = π

(l, x) and |y| = |π

(l, x)| ≤ m(l, r) <

. Therefore, the zero section Θ of the vector bundle (X, h, Y ) is an (

)-net of

(l, Σ

(A)). Since Y is compact and (X, h, Y ) is local trivial, the zero section Θ is

compact. Hence, M

and M are covered by the ε-net θ

. Since Θ is compact

and the space Y is complete, M = {x

}

∞

k=1

is relatively compact. We complete

the proof by applying Lemma 1.3. 

Corollary 13.1 Let h(X, S

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

dynamical system and let P : X → X be a completely continuous projection.

Assume that there are positive numbers N and ν such that |π

Q(x)| ≤ Ne

−νt

|x|

for all x ∈ X and t ∈ S

, where Q : X → X and Q

:= Q|

= I

− P

for all

y ∈ Y (I

:= id

). Then (X, S

, π) is asymptotic compact.

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 13

416 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Proof. To deduce this corollary from Lemma 13.5, it is suﬃcient to observe that

π(t, x) = π

(t, x) + π

(t, x) for all x ∈ X and t ∈ S

, where π

(t, x) := π

Q(x) and

(t, x) := π

P (x). Under the hypotheses of Corollary 13.1, for every t > 0 the map

(t, ·) := π

P is completely continuous and |π

(t, x)| ≤ Ne

−νt

|x|. Hence, Lemma

13.5 is applicable. 

Remark 13.3 In the proofs of Lemma 13.5 and Corollary 13.1 we used only the

compactness of Y . Hence, these statements are also valid in the case when the

dynamical system (Y, S, σ) is not minimal.

13.2 Bounded solutions of linear equations

Let [E] be the Banach space of all bounded linear operators that act on a Banach

space E equipped with the operator norm. Let Λ be a complete metric space of

closed linear operators that act on E (for example, Λ = [E] or Λ = {A

+ B | B ∈

[E]}, where A

is a closed operator that acts on E). Let C(R, Λ) be the space of all

continuous operator-valued functions A : R → Λ equipped with the compact-open

topology and let (C(R, Λ), R, σ) be the dynamical system of shifts on C(R, Λ).

Linear ordinary diﬀerential equations. Let Λ = [E] and consider the dif-

ferential equation

= A(t)u, (13.21)

where A ∈ C(R, [E]). Consider the H-class of equation (13.21), that is, the family

of equations

= B(t)v, (13.22)

with B ∈ H(A) :=

| τ ∈ R}, A

(t) = A(t+τ ), and t ∈ R, where the bar denotes

closure in C(R, [E]). Let ϕ(t, v, B) be the solution of equation (13.22) that satisﬁes

the condition ϕ(0, v, B) = v.

We put Y := H(A) and denote the dynamical system of shifts on H(A) by

(Y, R, σ). We put X := E × Y and deﬁne a dynamical system on X by set-

ting π(t, (v, B)) := (ϕ(t, v, B), B

) for all (v, B) ∈ E × Y and t ∈ R. Then

h(X, R, π), (Y, R, σ), hi is a linear group non-autonomous dynamical system, where

h := pr

: X → Y. Applying the results of section 13.1 to this system, we obtain

the following assertions.

Lemma 13.6

[

51, 60

]

(i) The map (t, u, A) 7→ ϕ(t, u, A) of R × E × C(R, [E]) to E is continuous, and

(ii) the map A 7→ U(·, A) of C(R, [E]) to C(R, [E]) is continuous, where U (t, A) is

the Cauchy’s operator

[

132

]

of equation (13.21).