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 9

Pullback attractors of C-analytic systems 317

(1) The compact invariant set J =

| ω ∈ Ω} of the skew-product dynamical

system (X, T

, π) (X := C

×Ω, π := (ϕ, σ)) is asymptotically stable.

(2) There exists a positive number δ

such that the cocycle ϕ is positively uniformly

stable on the compact set B[I, δ] :=

{B[I

, δ] | ω ∈ Ω}, where B[I

, δ] :=

{z ∈ C

| ρ(z, I

) ≤ δ}, for all 0 < δ < δ

(3) The skew-product dynamical system (X, T

, π) generates on J a group

dynamical system (J, T, π).

Proof. Denote by X := C

× Ω and by (X, T

, π) the skew-product dynamical

system. Then under the conditions of the theorem the set J =

|ω ∈ Ω} is

a nonempty compact invariant set and according to Theorem 4.1

[

100

]

is asymp-

totically stable with respect to (X, T

, π). In particular there exists a δ

> 0 such

that the set B[J, δ

] := {x ∈ X | ρ(x, J) ≤ δ

} is positively invariant. Since Ω

is compact and π

(u, ω) := (ϕ(t, u, ω), σ

ω), then there exists a positive number

C = C(δ

) such that |ϕ(t, u, ω)| ≤ C for all ω ∈ Ω and u ∈ B[I

, δ

]. Taking into

account the connectedness of set I

(see, for example

[

126

]

and also Theorem 2.25)

according to Cauchy’s Theorem for all δ < δ

there exists a positive number L(δ)

such that

|ϕ(t, ω, u

) − ϕ(t, ω, u

)| ≤ L(δ)|u

− u

| (9.13)

for all ω ∈ Ω, t ∈ R

and u

, u

∈ B[I

, δ]. It is easy to see that from inequality

(9.13) results the positively uniformly stability of set B[I, δ] for every 0 < δ < δ

Particularly the set I :=

| ω ∈ Ω} will be positively uniformly stable and to

ﬁnish the proof of Theorem it is suﬃciently to apply Theorem 9.1 to our situation

for the skew-product system (X, T

, π). The Theorem is completely proved. 

Deﬁnition 9.4 The cocycle hE

, ϕ, (Ω, T, σ)i is called linear (see, for example,

[

]

[

]

and

[

292

]

) if the mapping ϕ(t, ω) : E

→ E

is linear for every t ∈ T

and

ω ∈ Ω.

Theorem 9.3 Let hC

, ϕ, (Ω, T, σ)i be a linear cocycle, then the following condi-

tions are equivalent:

(1) 1. lim

t→+∞

|ϕ(t, ω, u)| = 0 for all u ∈ E

and ω ∈ Ω.

(2) 2. There exist positive numbers N, ν such that |ϕ(t, ω, u)| ≤ N exp (−νt)|u| for

all t ∈ T

, ω ∈ Ω and u ∈ E

Proof. This statement follows from Theorems 1.10 and 2.38. 

Theorem 9.4 Let hC

, ϕ, (Ω, T, σ)i be a C−analytic cocycle admitting a compact

pullback attractor {I

| ω ∈ Ω}, and let every point ω ∈ Ω be positively Poisson

stable. Then the following assertions hold:

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 9

318 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(1) For every ω ∈ Ω the set I

consists of a unique point ν(ω).

(2) ν(σ

ω) = ϕ(t, ν(ω), ω) for all ω ∈ Ω and t ∈ T

(3) The mapping ω → γ(ω) is continuous, where γ := (ν, Id

Ω

(4) Every point γ(ω) is positively Poisson stable.

(5) The continuous invariant section ν is uniformly asymptotically stable, i.e.

(a) for arbitrary ε > 0 there exists δ(ε) > 0 such that ρ(z, ν(ω)) < δ implies

ρ(ϕ(t, z, ω), ν(σ

ω)) < ε for all t ≥ 0 and ω ∈ Ω.

(b) There exists δ

> 0 such that

lim

t→+∞

ρ(ϕ(t, z, ω), ν(σ

ω)) = 0

for all ω ∈ Ω and z with the condition ρ(z, ν(ω)) ≤ δ

Proof. Under the conditions of Theorem 9.3 there exists a positive number δ

such

that the set M := B[J, δ

] is a compact and positively invariant set of skew-product

system (X, T

, π) (see the proof of Theorem 9.2), where B[I, δ

] :=

{B[I

, δ

] ×

{ω} | ω ∈ Ω}. Denote by E = E(M, T

, π) the Ellis semigroup of the dynamical

system (M, T

, π), E

:= {ξ ∈ E | ξM

⊆ M

}, where M

:= {(u, ω)| (u, ω) ∈

M} and E

:= {ξ ∈ E

| ∃{t

} ∈ N

such that π

→ ξ}. According to

Theorem 9.2 and Lemma 9.1 E

is a nonempty compact sub-semigroup of the Ellis

semigroup E. Note that every mapping ξ ∈ E

, which maps B[I

, δ

] into I

is holomorphic because, according to Theorem 9.2, the convergence π

→ ξ

is uniform on M

. Consider an idempotent v ∈ E

, then v(v(u)) = v(u) for all

u ∈ M

and, consequently, v(p) = p for every p ∈ v(M

) = v(I

). Since v is

holomorphic and v(I

) is a compact connected set, then

[

167

]

v(I

) contains only

one point ν(ω). On the other hand we have v(v(u)) = v(u) for all u ∈ M

, i.e.

v(ν(ω)) = v(u). Thus there exists a sequence t

→ +∞ such that

|ϕ(t

, u, ω) −ϕ(t

, ν(u), ω)| = 0 (9.14)

for all u ∈ M

. Taking into account the positively uniformly stability of cocycle ϕ

from (9.14) we obtain the equality

|ϕ(t, u, ω) −ϕ(t, ν(u), ω)| = 0 (9.15)

for all u ∈ B(I

, δ

) := {u ∈ C

| ρ(u, I

) < δ

}. Now we will prove that I

= {ν(ω)}

for every ω ∈ Ω. Let 0 < δ < δ

, u ∈ I

and h ∈ C

with condition |h| < δ, then

according to equality (9.15) we have

lim

t→+∞

sup

|h|≤δ

|ϕ(t, ω, u + h) −ϕ(t, ω, u)| = 0 (9.16)

for all ω ∈ Ω and u ∈ I

. In virtue of Cauchy’s formula (see

[

]

and also

[

]

)

U(t, (u, ω))w = (9.17)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 9

Pullback attractors of C-analytic systems 319

(2πi)

·... ·

ϕ(t, ω, u + v) − ϕ(t, ω, u)

·... ·v

k=1

· ... · dv

where U (t, (u, ω)) :=

∂ϕ(t,ω,u)

∂u

for all (u, ω) ∈ M and t ∈ T

. From (9.16) and (9.17)

it follows that lim

t→+∞

kU(t, (u, ω))k = 0 for all (u, ω) ∈ M. According to Theorem

9.3 there exist positive numbers N and α such that

kU(t, (u, ω))k ≤ N exp (−αt) (9.18)

for any (u, ω) ∈ M. Let now u

, u

∈ I

and ψ : [0, 1] → B(I

, δ) be a continuously

diﬀerentiable function with properties: ψ(0) = u

and ψ(1) = u

. Consider the

function ∆(s) := ϕ(t, ψ(s), ω), then according to Lagrange’s formula we have

∆(1) − ∆(0) = ∆

(τ), (9.19)

where 0 < τ < 1. Hence from (9.18) and (9.19) we have

|ϕ(t, u

, ω) −ϕ(t, u

, ω)| ≤ N

exp (−αt)|u

−u

| (9.20)

for all t ∈ T

, ω ∈ Ω and u

, u

∈ M

, where N

:= N · m and m := max

0≤s≤1

|ψ

(s)|.

To ﬁnish the proof of theorem it is suﬃcient to remark that according to Theorem

9.1 on set J there is deﬁned a two-sided dynamical system (J, T, π) and, in partic-

ular, through every point u ∈ I

passes a unique entire trajectory of cocycle ϕ, i.e.

the function ϕ(t, ω, u) (u ∈ I

and ω ∈ Ω) is deﬁned on T. If u

6= u

, u

∈ I

then from (9.20) follows that

|ϕ(−t, ω, u

) − ϕ(−t, ω, u

)| ≥ N

exp (αt)|u

−u

for all t ∈ T

. But the trajectories ϕ(t, ω, u

) ∈ I

(i = 1, 2; t ∈ T) are bounded

on T. The obtained contradiction proves our assertion. Thus we have I

= {ν(ω)}.

Now it is easy to see that the mapping ω → γ(ω) is continuous, where γ =

(ν, Id

Ω

), and π

γ(ω) = γ(σ

ω) for all ω ∈ Ω and t ∈ T

and, consequently, ν(σ

ω) =

ϕ(t, ω, ν(ω)) for all ω ∈ Ω and t ∈ T

Next we will note that every point γ(ω) is positively Poisson stable. Indeed,

let {t

} ∈ N

, then π

γ(ω) = γ(σ

ω) → γ(ω) and, consequently, {t

} ∈ N

γ(ω)

Finally, the uniformly asymptotically stability of continuous and invariant section

ν results from Theorem 9.2. The theorem is completely proved. 

9.5 The uniform dissipative cocycles with noncompact base

Let Ω be a complete metric space (generally speaking noncompact), hE

, ϕ,

(Ω, T, σ)i be a cocycle on the state space E

and (X, T

, π) be the correspond-

ing skew-product dynamical system, where X := E

×Ω and π := (ϕ, σ).

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 9

320 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Deﬁnition 9.5 The cocycle hE

, ϕ, (Ω, T, σ)i is said to be dissipative if for any

ω ∈ Ω there is a positive number r

such that

lim sup

t→+∞

|ϕ(t, ω, u)| < r

for all ω ∈ Ω and u ∈ E

, i.e. for all u ∈ E

and ω ∈ Ω there exists a positive

number L(ω, u) such that |ϕ(t, ω, u)| < r

for all t ≥ L(ω, u).

Deﬁnition 9.6 The cocycle hE

, ϕ, (Ω, T, σ)i is said to be uniformly dissipative

if there exists a positive number r (r is not depend upon ω ∈ Ω) such that for any

R > 0 there is a positive number L(R) such that |ϕ(t, ω, u)| < r for all ω ∈ Ω and

|u| ≤ R and t ≥ L(R).

Theorem 9.5 (

[

100, 127, 217

]

) Let hE

, ϕ, (Ω, T, σ)i be an uniformly dissipative

cocycle, then it admits a compact global pullback attractor {I

| ω ∈ Ω} with |u| ≤ r

for all u ∈ I

and ω ∈ Ω, where r is the positive number in deﬁnition 9.6.

Theorem 9.6 Let hC

, ϕ, (Ω, T, σ)i be a C−analytic uniformly dissipative cocycle,

then the following statements hold:

i) The cocycle ϕ admits a compact global pullback attractor {I

| ω ∈ Ω} with

|u| ≤ r for all u ∈ I

and ω ∈ Ω, where r is the positive number in condition

(9.23).

ii) For any R > 0 there exist positive constants C = C(R) and L(R) such that

ρ(ϕ(t, ω, u

), ϕ(t, ω, u

)) ≤ Cρ(u

, u

) (9.21)

for all t ≥ L(R), ω ∈ Ω and u

, u

∈ C

with the condition |u

| ≤ R (i = 1, 2).

iii) For arbitrary ε > 0 there exist L(ε) > 0 and δ(ε) > 0 such that

|ϕ(t, ω, u + h) − ϕ(t, ω, u))| < ε

for all t ≥ L(ε), u ∈ I

, ω ∈ Ω and |h| < δ.

iv) The set J of (X, T

, π) is negatively distal, i.e.

inf

t≤0

ρ(γ

(t), γ

(t)) > 0,

where γ

(i=1,2) is a entire trajectory passing through point (u

, ω) ∈ J, u

6= u

and γ

(S) ⊆ J.

v) On the set J there is deﬁned a two-sided dynamical system (J, T, π) generated

by skew-product system (X, T

, π).

Proof. The ﬁrst assertion of theorem results from Theorem 9.5. Let now R > 0

and R

> R, according to uniformly dissipativity of cocycle ϕ there exists L(R

) > 0

such that |ϕ(t, ω, u)| < r for all t ≥ L(R

), ω ∈ Ω and |u| ≤ R

. In virtue of Cauchy’s

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Pullback attractors of C-analytic systems 321

formula for R < R

there is a constant C(R) > 0 such that |

∂ϕ

∂u

(t, u, ω)| ≤ C(R) for

all t ≥ L(R), ω ∈ Ω and |u| ≤ R and, consequently the inequality (9.21) holds.

The third assertion we will prove by method of contradiction. If it is not true,

then there exist ε

> 0, δ

→ 0 (δ

> 0), |h

| < δ

∈ C

), t

≥ n, ω

∈ Ω

and u

∈ I

such that

|ϕ(t

, ω

, u

+ h

) − ϕ(t

, ω

, u

))| ≥ ε

Let now R > r and C(R), L(R) be positive constants ﬁguring in the inequality

(9.24), then we have the following inequality

≤ |ϕ(t

, ω

, u

+ h

) − ϕ(t

, ω

, u

))| ≤ C(R)|h

| ≤ C(R)δ

. (9.22)

Passing to limit in the inequality (9.22) as n → ∞ we obtain ε

≤ 0. The obtained

contradiction proves our assertion.

The fourth and ﬁfth statements follow from Theorem 9.1 (see also Remark 9.2)

because from condition iii) results that for arbitrary ε > 0 there exist two positive

constants δ(ε) and L(ε) satisfying the inequality (9.12). The Theorem is completely

proved. 

Theorem 9.7 Let hC

, ϕ, (Ω, T, σ)i be a C−analytic uniformly dissipative cocycle

and every point ω ∈ Ω be positively Poisson stable, then:

1. The set I

consists of only one point ν(ω) for every ω.

2. The mapping ω → γ(ω) is continuous, where γ := (ν, Id

Ω

3. ν(σ

ω) = ϕ(t, ν(ω), ω) for all ω ∈ Ω and t ∈ T

4. lim

t→+∞

ρ(ϕ(t, σ

−t

ω)z, ν(ω)) = 0 for every ω ∈ Ω uniformly with respect to z in

compact subsets of C

5. Every point γ(ω) is positively Poisson stable.

6. lim

t→+∞

ρ(ϕ(t, z, ω), ν(σ

ω)) = 0 for all ω ∈ Ω and z ∈ C

, i.e. every positive

semi trajectory ϕ(t, ω, z) is asymptotically Poisson stable in positive direction.

Proof. Let hC

, ϕ, (Ω, T, σ)i be a C−analytic uniformly dissipative cocycle, then

according to Theorem 9.6 this cocycle has the properties i)-v). Let {I

| ω ∈ Ω}

be the compact global pullback attractor of cocycle ϕ and let (X, T

, π) be the

skew-product dynamical system. Denote by

:= {ξ| ∃{t

} ∈ N

, π

→ ξ}.

Since the cocycle ϕ possesses the property ii), then the pointwise convergence

→ ξ coincides with uniform convergence on every compact subset of

:= C

×{ω} and, consequently, every mapping ξ ∈ E

is holomorphic. As well

as in Lemma 9.1 it is possible to show that E

is a nonempty compact semigroup

w.r.t. composition of mappings. Consider the idempotent element v of semigroup

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 9

322 Global Attractors of Non-autonomous Dissipative Dynamical Systems

. We will show that v(X

) ⊆ I

. Indeed, v ∈ E

and, consequently, there exists

a sequence {t

} ∈ N

such that v = lim

n→∞

. Let ¯x ∈ v(X

), i.e. ¯x = v(x)

for some x ∈ X

. This means that ¯x = lim

n→∞

x. According to Lemma 9.6 there

exists an entire trajectory γ of the skew-product system (X, T

, π) passing through

the point ¯x for t = 0 and γ(T) := {γ(t) | t ∈ T} is conditionally relatively compact.

Taking into account that J is a maximal invariant set of (X, T

, π) with relatively

compact pr

J we have ¯x ∈ J

, i.e. v(X

) ⊆ J

. Since X

= C

× {ω} and v is

holomorphic by virtue of Liouville’s Theorem the holomorphic function v is a con-

stant, i.e. there exists γ(ω) ∈ J

such that v(X

) = {γ(ω)}. We note that v

= v

and, consequently, v(v(x)) = v(x) for all x ∈ X

, i.e. v(γ(ω)) = v(x). Thus, there

exists a sequence t

→ +∞ such that

lim

t→+∞

ρ(π

γ(ω), π

x) = 0. (9.23)

Taking into consideration the property ii) of cocycle ϕ we obtain from (9.23) the

equality

lim

n→∞

ρ(ϕ(t, z, ω), ϕ(t, ν(ω), ω)) = 0 (9.24)

for all z ∈ C

, where γ := (ν, Id

Ω

Now we will show that there exists δ

> 0 such that for arbitrary ε > 0 there is

L(ε) > 0 with the property

|ϕ(t, ω, u + h) −ϕ(t, ω, u)| < ε (9.25)

for all (u, ω) ∈ J, t ≥ L(ε) and uniformly w.r.t. |h| ≤ δ

. If it is not true, then there

are δ → +0, ε

> 0, |h

| ≤ δ

, ω

∈ Ω, u

∈ I

and t

≥ n such that

|ϕ(t

, ω

, u

+ h

) − ϕ(t

, ω

, u

)| ≥ ε

On the other hand, according to property ii), there exists C(R) > 0 (R > sup

n∈N

)

such that for suﬃciently large n we have

≤ |ϕ(t

, ω

, u

+ h

) − ϕ(t

, ω

, u

)| ≤ C(R)δ

. (9.26)

Taking into account that δ

→ 0 from (9.26) it follows that ε

≤ 0. The obtained

contradiction prove our assertion.

From equality (9.17) and inequality (9.25) it follows that

lim

t→+∞

kU(t, (u, ω))k = 0 (9.27)

uniformly with respect to (u, ω) ∈ B[J, δ

]. Denote by

m(t) := sup{kU(t, (u, ω))k |(u, ω) ∈ B[J, δ

]},

then

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 9

Pullback attractors of C-analytic systems 323

a) m(t) → 0 as t → +∞.

b) ∃L > 0 such that m is bounded on [L, +∞).

c) m(t + τ) ≤ m(t)m(τ) for all t, τ ≥ L.

From a)-c) it follows (see, for example,

[

]

) that there exist N, α > 0 such that

m(t) ≤ N exp (−αt) for all t ≥ L and, consequently, from (9.27) we have

kU(t, (u, ω))k ≤ N exp (−αt)

for all (u, ω) ∈ B[J, δ

] and t ≥ L. Using the same arguments as well as as in the

proof of Theorem 9.4 we conclude that I

= {ν(ω)} for all ω ∈ Ω.

Now we will prove that the mapping ω → γ(ω) is continuous. Let ω

→ ω and

consider the sequence {γ(ω

)} ⊂ J. Since J is conditionally compact, this sequence

is relatively compact. Let ¯x be a limit point of {γ(ω

)}, then it is easy to see

that ¯x ∈ J

= {γ(ω)} and, consequently, γ(ω) is a unique limit point of relatively

compact sequence {γ(ω

)}. Hence γ(ω

) → γ(ω).

The equality ν(σ

ω) = ϕ(t, ν(ω), ω) follows from invariance of J and from equal-

ity J

= {γ(ω)} for all ω ∈ Ω, taking into account that ν(ω) = pr

γ(ω).

The equality 4. follows from equality J

= {γ(ω)} = {(ν(ω), ω)} and from the

fact that {ν(ω) | ω ∈ Ω} is a compact global pullback attractor of cocycle ϕ.

The stability in the sense of Poisson in the positive direction of point γ(ω) follows

from the continuity of γ and the equality π

γ(ω) = γ(σ

ω) for all t ∈ T

and ω ∈ Ω.

The sixth assertion follows from (9.24) and the equality ϕ(t, ν(ω), ω) = ν(σ

ω)

for all t ∈ T

and ω ∈ Ω. The Theorem is completely proved. 

9.6 The compact and local dissipative cocycles with noncompact base

Deﬁnition 9.7 The cocycle hE

, ϕ, (Ω, T, σ)i is said to be compact dissipative

if for any nonempty compact Ω

⊂ Ω there is a positive number r

Ω

such that for

arbitrary R > 0 there exists a positive number L(R, Ω

) with the following property

|ϕ(t, ω, u)| < r

Ω

(9.28)

for all ω ∈ Ω

, |u| ≤ R and t ≥ L(R, Ω

Deﬁnition 9.8 The cocycle hE

, ϕ, (Ω, T, σ)i is said to be locally dissipative if

for any ω ∈ Ω and R > 0 there are positive numbers r

, δ

and L(R, ω) such that

|ϕ(t, ˜ω, u)| < r

Ω

(9.29)

for all ˜ω ∈ B(ω, δ

) := {˜ω ∈ Ω | ρ(˜ω, ω) < δ

}, |u| ≤ R and t ≥ L(R, ω).

Lemma 9.7 Every locally dissipative cocycle ϕ is compact dissipative.

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324 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Proof. Suppose that ϕ is locally dissipative and let Ω

⊂ Ω be a nonempty compact

set. According to locally dissipativity of ϕ for every ω ∈ Ω, and R > 0 there exist

, L(R, ω) > 0 and δ

> 0 such that the inequality (9.29) holds. Considering the

open covering

{B(ω, δ

) | ω ∈ Ω

} of compact set Ω

, we may extract the ﬁnite

sub covering

{B(ω

, δ

) | i =

1, k}. Let L(R, Ω

) := max{L(R, ω

) | i = 1, k},

then it is clear that inequality (9.28) holds for all ω ∈ Ω

, |u| ≤ R and t ≥ L(R, Ω).

The Lemma is proved. 

Remark 9.3 Compact dissipativity, generally speaking, does not imply locally

dissipativity.

Lemma 9.8 Let hE

, ϕ, (Ω, T, σ)i be a compact dissipative C−analytic cocycle.

Then for any nonempty compact Ω

⊂ Ω and R > 0 there exist L(Ω

, R) and

C = C(Ω

, R) > 0 such that

ρ(ϕ(t, u

, ω), ϕ(t, u

, ω)) ≤ C(Ω

, R)ρ(u

, u

) (9.30)

for any ω ∈ Ω

, |u

| ≤ R (i = 1, 2) and t ≥ L(Ω

, R).

Proof. Let R > 0 and R

> R, then according to the compact dissipativity of

cocycle ϕ for nonempty compact Ω

⊂ Ω and R

> 0 there exist r

Ω

> 0 and

L(Ω

, R

) > 0 such that |ϕ(t, u, ω)| < r

Ω

for all ω ∈ Ω, |u| ≤ R

and t ≥ L(Ω

, R

In view of Cauchy‘s formula for R < R

there exists a constant C = C(R, Ω) >

0 such that |

∂ϕ

∂u

(t, u, ω)| ≤ C(R, Ω

) for all t ≥ L(R

, Ω

), ω ∈ Ω

and |u| ≤ R

and, consequently, the inequality (9.30) holds for |u

|, |u

| ≤ R, ω ∈ Ω

and t ≥

L(Ω

, R) := inf{L(Ω

, R

) | R

> R}. 

Lemma 9.9 Let hE

, ϕ, (Ω, T, σ)i be a compact dissipative C−analytic cocycle,

and γ

, γ

are two entire bounded trajectories passing through point (u

, ω) and

, ω) for t = 0 respectively, then the following assertions hold:

(1) If ω ∈ Ω is negatively Poisson stable and {t

} ∈ N

−

, then

inf

n∈N

ρ(γ

), γ

)) > 0 (9.31)

if u

6= u

(2) If ω ∈ Ω is positively Poisson stable and {t

} ∈ N

, then the equality

inf

n∈N

ρ(ϕ(t

, ω, u

), ϕ(t

, ω, u

)) = 0 (9.32)

implies

lim

t→+∞

ρ(ϕ(t, ω, u

), ϕ(t, ω, u

)) = 0. (9.33)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 9

Pullback attractors of C-analytic systems 325

Proof. Let ω ∈ Ω be a negative Poisson stable point, {t

} ∈ N

−

and u

6= u

. If the

equality (9.31) is not true, then ρ(γ

), γ

)) → 0 as n → ∞. Denote by R :=

sup

t∈T

max{|γ

(t)|, |γ

(t)|} > 0, Ω

{σ

ω | n ∈ N} ⊂ Ω and let C(Ω

, R), L(Ω

, R)

be the corresponding constants ﬁguring in the inequality (9.30), then we obtain

ρ(u

, u

) = ρ(ϕ(−t

, γ

), σ

ω), ϕ(−t

, γ

), σ

ω)) (9.34)

≤ C(Ω

, R)ρ(γ

), γ

))

for suﬃciently large n (−t

≥ L(Ω

, R)). Passing to limit in the inequality (9.34)

as n → ∞ we have ρ(u

, u

) ≤ 0, but u

6= u

. The obtained contradiction prove

the ﬁrst statement of Lemma 9.9 .

Let now ω ∈ Ω be a positive Poisson stable point and {t

} ∈ N

such that the

inequality (9.33) holds, then for arbitrary ε > 0 there exists n

= n

(ε) such that

ρ(ϕ(t, u

, ω), ϕ(t, u

, ω)) <

2C(Ω

, R)

(9.35)

and, consequently, according to Lemma 9.8 we obtain

ρ(ϕ(t, u

, ω), ϕ(t, u

, ω)) = ρ(ϕ(t −t

, ϕ

(

, u

, ω), σ

ω), (9.36)

ϕ(t − t

, ϕ(t

, u

, ω), σ

ω) ≤ C(Ω

, R)ρ(ϕ(t

, u

, ω), ϕ(t

, u

, ω))

for all t ≥ t

+ L(Ω

, R), where R := sup{max{|ϕ(t, u

, ω)|, |ϕ(t, u

, ω)|} | t ∈ T

Denote by L(ε) := t

(ε)

+ L(Ω, R), then from inequalities (9.35) and (9.36) we

obtain

ρ(ϕ(t, u

, ω), ϕ(t, u

, ω)) < ε

for all t ≥ L(ε) and, consequently, (9.33) holds. The lemma is proved. 

Theorem 9.8 Let hE

, ϕ, (Ω, T, σ)i be a compact dissipative C−analytic cocycle

admitting a compact pullback attractor {I

| ω ∈ Ω} and every point ω ∈ Ω be

two-sided Poisson stable, then the following assertions hold:

1. The set I

consists of only one point ν(ω), i.e. I

= {ν(ω)} for every ω ∈ Ω.

2. The mapping ω → γ(ω) is continuous, where γ = (ν, Id

Ω

3. ν(σ

ω) = ϕ(t, ν(ω), ω) for all ω ∈ Ω and t ∈ T

4. The point γ(ω) is Poisson’s stable for all ω ∈ Ω.

5. lim

t→+∞

β(ϕ(t, σ

−t

ω)K, ν(ω)) = 0 for all compact subsets K of C

, where β(A, B)

is the semi-distance of Hausdorﬀ between A and B.

6. lim

t→+∞

|ϕ(t, ω, z) −ν(σ

ω)| = 0 for all ω ∈ Ω and z ∈ C

Proof. To prove the ﬁrst assertion of Theorem 9.8 we consider the non-autonomous

dynamical system h(Φ, T, λ), (Ω, T, σ), Hi constructed in the proof of Theorem 9.1.

Using the same type of argument as in Theorem 9.1 and taking into consideration

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 9

326 Global Attractors of Non-autonomous Dissipative Dynamical Systems

the Lemma 9.9 we may state that the system h(Φ, T, λ), (Ω, T, σ), Hi possesses the

following properties:

a) Φ is conditionally compact invariant set.

b) for every ω ∈ Ω, γ

, γ

∈ ϕ

(γ

6= γ

) and {t

} ∈ N

−

holds

inf

n∈N

ρ(γ

, γ

) > 0,

where γ

is a t-translation of γ, i.e. γ

(s) := γ(t + s) for all s ∈ T.

Then according to Lemmas 9.3–9.5 we have that E

−

= E

is a group.

Particularly there are two sequences t

→ +∞ and t

→ −∞ such that

lim

n→∞

= γ (i = 1, 2). (9.37)

for all γ ∈ Φ

, i.e. every entire trajectory γ of global pullback attractor {I

| ω ∈ Ω}

is two-sided Poisson stable. On the other hand according to Lemma 9.9 we have

lim

t→+∞

ρ(γ

(t), γ

(t)) = 0. (9.38)

From (9.37) and (9.38) we obtain

ρ(γ

(t), γ

(t)) = lim

n→∞

ρ(γ

(t + t

), γ

(t + t

)) = 0

for all t ∈ T, i.e. γ

= γ

. Thus there exists a unique entire trajectory ˜γ

∈ Φ

, i.e.

= {˜γ

} and, consequently, I

= {˜γ

(0)} := {γ(ω)}, where γ(ω) := ˜γ

(0).

The proof of item 2.– 6. of Theorem 9.8 uses the same type of arguments as in

Theorem 9.7. The Theorem is proved. 

9.7 Applications

9.7.1 ODEs

Consider the diﬀerential equation

= f(t, z), (9.39)

where f ∈ CH(R × C

, C

) and the family of equations

= g(t, z), (9.40)

where g ∈ H(f ) :=

|τ ∈ R} and f

is a τ−translation of function f w.r.t.

variable t, i.e. f

(t, z) := f (t + τ, z) for all t ∈ R and z ∈ C

. Denote by ϕ(t, z, g)

the solution of equation (9.40) with the initial condition ϕ(0, z, g) = z, then ϕ is a

C−analytic cocycle on C