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 7

Upper semi-continuity of attractors 267

holds for every λ

∈ Λ and ω

∈ Ω and

lim

λ→λ

β(I

, I

) = 0 (7.6)

for every λ

∈ Λ.

Proof. Let Y := Λ × Ω and µ : T × Y 7→ Y be the mapping deﬁned by the

equality µ(t, (λ, ω)) := (λ, σ(t, ω)) for every t ∈ T, λ ∈ Λ and ω ∈ Ω. It is clear that

the triplet (Y, T, µ) is the group dynamical system on Y and ϕ : T

× W × Y 7→

W (ϕ(t, x, (λ, ω)) := ϕ(t, x, λ, ω)) is the continuous cocycle on (Y, T, µ) with ﬁber

W . Under the conditions of Lemma 7.3 the cocycle ϕ admits a maximal compact

invariant set {I

| y ∈ Y } (where I

= I

(λ,ω)

= I

) because

[

| y ∈ Y } =

[

| λ ∈ Λ, ω ∈ Ω} =

[

| λ ∈ Λ}.

According to Lemma 7.2, the function F : Y 7→ C(W ), deﬁned by the equality

F (λ, ω) := I

, is upper semi-continuous and in particular the equality (7.6) holds.

We suppose that the equality (7.6) is not true, then there exist ε

> 0, λ

∈

Λ, λ

→ λ

, ω

∈ Ω and x

∈ I

such that

ρ(x

, I

) ≥ ε

. (7.7)

Without loss of generality we can suppose that ω

→ ω

, x

→ x

because the sets

Ω and

|λ ∈ Λ} are compact. According to the inequality (7.7) we have

ρ(x

, I

) ≥ ε

On the other hand x

∈ I

and from the equality (7.6) we have

∈ I

⊂ I

and, consequently,

≤ ρ(x

, I

) ≤ β(I

, I

) = 0.

This contradiction shows that the equality (7.6) holds. 

Corollary 7.1 Under the conditions of Lemma 7.3 the equality

lim

λ→λ

β(I

, I

) = 0

holds for each ω ∈ Ω.

Remark 7.2 The article

[

]

contains a statement close to Corollary 7.1 in the

case when the non-perturbed cocycle ϕ

is autonomous, i.e. the mapping ϕ

×W ×Ω → W does not depend of ω ∈ Ω .

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 7

268 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Lemma 7.4 Let the conditions of Lemma 7.3 and the following condition be ful-

ﬁlled:

5. for certain λ

∈ Λ the application F : Ω 7→ C(W ), deﬁned by equality F (ω) :=

is continuous, i.e. α(F (ω), F (ω

)) → 0 if ω → ω

for every ω

∈ Ω, where

α is the full metric of Hausdorﬀ, i.e. α(A, B) := max{β(A, B), β(B, A)}.

Then

lim

λ→λ

sup

ω∈Ω

β(I

, I

) = 0 . (7.8)

Proof. Suppose that the equality (7.8) is not correct, then there exist ε

> 0, λ

→

, ω

∈ Ω such that

β(I

, I

) ≥ ε

. (7.9)

On the other hand we have

≤ β(I

, I

) ≤ β(I

, I

) + β(I

, I

)

≤ β(I

, I

) + α(I

, I

). (7.10)

According to Lemma 7.3 (see the equality (7.5))

lim

k→+∞

β(I

, I

) = 0 . (7.11)

Under the condition 5. of Lemma 7.4 we have

lim

k→+∞

α(I

, I

) = 0. (7.12)

From (7.10)-(7.12) passing to the limit as k → +∞, we obtain ε

≤ 0. This

contradiction shows that the equality (7.8) holds. 

Deﬁnition 7.3 The family of cocycle {ϕ

}

λ∈Λ

is called collectively compact dis-

sipative (uniformly collectively compact dissipative), if there exists a nonempty

compact set K ⊆ W such that

lim

t→+∞

sup{β(U

(t, ω)M, K)|ω ∈ Ω} = 0 ∀ λ ∈ Λ (7.13)

(respectively lim

t→+∞

sup{β(U

(t, ω)M, K) | ω ∈ Ω, λ ∈ Λ} = 0)

for all M ∈ C(W ), where U

(t, ω) := ϕ

(t, ·, ω).

Lemma 7.5 The following conditions are equivalent:

1. the family of cocycles {ϕ

}

λ∈Λ

is collectively compact dissipative;

2.2.a. every cocycle ϕ

(λ ∈ Λ) is compact dissipative;

2.b. the set

| λ ∈ Λ} is compact.

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 7

Upper semi-continuity of attractors 269

Proof. According to the equality (7.13) every cocycle ϕ

(λ ∈ Λ) is compact

dissipative and

| λ ∈ Λ} ⊆ K.

Suppose that the conditions 2.a. and 2.b. hold. Let K :=

| λ ∈ Λ}, then

the equality (7.13) holds. 

Let {ϕ

}

λ∈Λ

be a family of cocycles on (Ω, T, σ) with ﬁber W and

Ω := Ω×Λ. On

Ω, we deﬁne a dynamical system (

Ω, T, ˜σ) by equality ˜σ(t, (ω, λ)) := (σ(t, ω), λ) for

all t ∈ T, ω ∈ Ω and λ ∈ Λ. By family of cocycles {ϕ

}

λ∈Λ

is generated a cocycle ˜ϕ

on (

Ω, T, ˜σ) with ﬁber W , deﬁned in the following way: ˜ϕ(t, w, (ω, λ)) := ϕ

(t, w, ω)

for all t ∈ T

, w ∈ W, ω ∈ Ω and λ ∈ Λ.

Lemma 7.6 The following conditions are equivalent:

(1) the family of cocycles {ϕ

}

λ∈Λ

is uniformly collectively compact dissipative;

(2) the cocycle ˜ϕ is compact dissipative.

Proof. This assertion follows from the fact that

sup{β(

U(t, ˜ω)M, K) | ˜ω ∈

Ω}

= sup{β(U

(t, ω)M, K) | ω ∈ Ω, λ ∈ Λ},

where

U(t, ˜ω) := ˜ϕ(t, ·, ˜ω), and from the corresponding deﬁnitions. 

Theorem 7.3 Let Λ be a compact metric space and {ϕ

}

λ∈Λ

be a family of uni-

formly collectively compact dissipative cocycles on (Ω, T, σ) with ﬁber W , then the

following statements are true:

(1) every cocycle ϕ

(λ ∈ Λ) is compact dissipative;

(2) the family of compacts {I

| ω ∈ Ω} = I

is a compact global attractor of

cocycle ϕ

, where I

:= I

(ω,λ)

and I := {I

(ω,λ)

| (ω, λ) ∈

Ω} is a Levinson

center of cocycle ˜ϕ;

(3) the set

| λ ∈ Λ} is compact.

Proof. Consider the cocycle ˜ϕ generated by the family of cocycles {ϕ

}

λ∈Λ

. Ac-

cording to Lemma 7.6 ˜ϕ is compact dissipative and in virtue of the Theorem 2.24

the following assertions take place:

1. I

˜ω

= Ω

˜ω

(K) 6= ∅, is compact, I

˜ω

⊆ K and

lim

t→+∞

β(

U(t, ˜ω

−t

)M, I

˜ω

) = 0 (7.14)

for every ˜ω ∈

Ω, where

Ω

˜ω

(K) =

t≥0

[

τ≥t

U(τ, ˜ω

−τ

)K, (7.15)

˜ω

−τ

:= ˜σ(−τ, ˜ω) and K is a nonempty compact appearing in the equality (7.13);

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 7

270 Global Attractors of Non-autonomous Dissipative Dynamical Systems

U(t, ˜ω)I

˜ω

= I

˜ωt

for all ˜ω ∈

Ω and t ∈ T

;

3. the set I =

˜ω

| ˜ω ∈

Ω} is compact.

To ﬁnish the proof we note that from the collective compact dissipativity of

the family of cocycles {ϕ

}

λ∈Λ

it follows that every cocycle ϕ

will be compact

dissipative. Let {I

| ω ∈ Ω} = I

be a Levinson center of the cocycle ϕ

, then

according to Theorem 2.24

t≥0

[

τ≥t

(τ, ω

−τ

)K . (7.16)

From (7.15) and (7.16) it follows that I

= Ω

˜ω

(K) = I

˜ω

and, consequently, I

| ω ∈ Ω} ⊆

| ω ∈ Ω, λ ∈ Λ} = I for all λ ∈ Λ. Thus

| λ ∈ Λ} ⊆ I

and, consequently, it is compact. 

Deﬁnition 7.4 The family {(X, T

, π

)}

λ∈Λ

of autonomous dynamical systems is

called collectively ( uniformly collectively) asymptotic compact if for every bounded

positive invariant set M ⊆ X (i.e. π

(t, M) ⊆ M for all t ∈ T

and λ ∈ Λ) there

exists a nonempty compact K such that

lim

t→+∞

β(π

M, K) = 0 ∀ λ ∈ Λ (7.17)

( lim

t→+∞

sup

λ∈Λ

β(π

M, K) = 0).

Deﬁnition 7.5 The bounded set K ⊂ X is called absorbing (uniformly absorb-

ing) for the family {(X, T

, π

)}

λ∈Λ

of autonomous dynamical systems if for any

bounded subset B ⊂ X there exists a number L = L(λ, B) > 0 (L = L(B) > 0)

such that π

B ⊆ K for all t ≥ L(λ, B) (t ≥ L(B)) and λ ∈ Λ.

Theorem 7.4 Let Λ be a complete metric space. If the family {(X, T

, π

)}

λ∈Λ

of autonomous dynamical systems admits an absorbing bounded set K ⊂ X and

is collectively asymptotic compact, then {(X, T

, π

)}

λ∈Λ

admits a global compact

attractor, i.e. there exists a nonempty compact set K ⊂ X such that

lim

t→+∞

β(π

B, K) = 0 (7.18)

for all λ ∈ Λ and bounded B ⊂ X.

Proof. Let the family {(X, T

, π

)}

λ∈Λ

of autonomous dynamical systems be

collectively asymptotic compact and a bounded M be its absorbing set. According

to Theorem 1.5 the nonempty set K = Ω(M ) is compact and the equality (7.18)

holds. The theorem is proved. 

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 7

Upper semi-continuity of attractors 271

Theorem 7.5 Let Λ be a complete compact metric space. If the family

{(X, T

, π

)}

λ∈Λ

of autonomous dynamical systems admits a uniformly absorb-

ing bounded set K ⊂ X and it is uniformly collectively asymptotic compact, then

{(X, T

, π

)}

λ∈Λ

admits a uniform compact global attractor, i.e. there exists a

nonempty compact set K ⊂ X such that

lim

t→+∞

sup

λ∈Λ

β(π

B, K) = 0 (7.19)

for all bounded B ⊂ X.

Proof. Consider the autonomous dynamical system (

X, T

, ˜π) on

X := X × Λ

deﬁned by equality ˜π(t, (x, λ)) := (π

(t, x), λ) for all t ∈ T

, x ∈ X and λ ∈ Λ. We

note that under the conditions of Theorem 7.5 the bounded set K ×Λ is absorbing

for dynamical system (

X, T

, ˜π) if the set K is uniformly absorbing for the family

{(X, T

, π

)}

λ∈Λ

and (

X, T

, ˜π) is asymptotically compact. According to Theorem

1.24 the dynamical system (

X, T

, ˜π) admits a compact global attractor

K ⊂

X :=

X × Λ. To ﬁnish the proof it is suﬃcient to note that the set K := pr

K ⊂ X is

compact and

sup

λ∈Λ

β(π

B, K) ≤ β(˜π

B, K

) → 0

as t → +∞, where K

:= K ×Λ ⊃

K, for all bounded subset B ⊂ X. 

Let ϕ be a cocycle on (Ω, T, σ) with ﬁber W and (X, T

, π) be a skew-product

dynamical system, where X := W × Ω and π(t, (w, ω)) := (ϕ(t, w, ω), ωt) for all

t ∈ T

, w ∈ W and ω ∈ Ω.

Deﬁnition 7.6 The cocycle ϕ is called asymptotically compact (a family

of cocycles {ϕ

}

λ∈Λ

is called collectively asymptotically compact) if a skew-

product dynamical system (X, T

, π) (a family of skew-product dynamical systems

(X, T

, π

)

λ∈Λ

) is asymptotically compact.

Theorem 7.6 Let Ω and Λ be compact metric spaces, W be a Banach space and

{ϕ

}

λ∈Λ

be a family of cocycles on (Ω, T, σ) with ﬁber W . If there exist r > 0 and

the function V

: W ×Ω → R

for all λ ∈ Λ, with the following properties:

(1) the family of cocycles {ϕ

}

λ∈Λ

is collectively asymptotically compact;

(2) the family of functions {V

}

λ∈Λ

is collectively bounded on bounded sets and for

every c ∈ R

the sets {x ∈ X

| V

(x) ≤ c} uniformly bounded;

(3) V

(w, ω) ≤ −c(|w|) for all w ∈ W

= {w ∈ W | |w| ≥ r}, ω ∈ Ω

and λ ∈ Λ, where c : R

→ R

is positive on [r, +∞), V

(w, ω) :=

lim sup

t→0+

−1

(ϕ

(t, w, ω), ωt) − V

(w, ω)] if T = R

and V

(w, ω) :=

(ϕ

(1, w, ω), σ(1, ω)) − V

(w, ω) if T = Z

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 7

272 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Then every cocycle ϕ

(λ ∈ Λ) admits a uniform compact global attractor I

(λ ∈ Λ)

and the set

| λ ∈ Λ} is compact.

Proof. Let X := W × Ω and (X, T, π

) be a skew-product dynamical system,

generated by the cocycle ϕ

, then (X, h, Ω) , where h := pr

: X → Ω, is a trivial

ﬁber bundle with ﬁber W . Under the conditions of Theorem 7.6 and according

to Theorem 5.2 the non-autonomous dynamical system h(X, T

, π

), (Ω, T, σ), hi

admits a compact global attractor J

and according to Theorem 2.24 the cocycle

admits a compact global attractor I

= {I

| ω ∈ Ω}, where I

:= pr

and

= pr

−1

(ω)

Let

Ω := Ω × Λ, (

Ω, T, ˜σ) be a dynamical system on

Ω deﬁned by the

equality ˜σ(t, (ω, λ)) := (σ(t, ω), λ) ( for all t ∈ T, ω ∈ Ω and λ ∈ Λ),

X :=

W ×

Ω and (

X, T

, ˜π) be an autonomous dynamical system deﬁned by equality

˜π(t, (w, ˜ω)) := (π

(t, w), (ωt, λ)) for all ˜ω := (ω, λ) ∈

Ω := Ω × Λ. Note that the

triplet (

X, h, Ω), where h := pr

X →

Ω, is a trivial ﬁber bundle with ﬁber

W, h(

X, T

, ˜π), (

Ω, T, ˜σ), hi is a non-autonomous dynamical system. The function

V :

:= W

Ω → R

, deﬁned by the equality

V (˜x) := V

(w, ω) for all

˜x := (w, (ω, λ)) ∈

under the conditions of Theorem 7.6, all the conditions

of Theorem 5.3 hold and, consequently, the dynamical system (

X, T

, ˜π) admits

a compact global attractor. To ﬁnish the proof of the theorem it is suﬃciently to

note that if the dynamical system (

X, T

, ˜π) admits a compact global attractor

J, then the family of cocycles {ϕ

}

λ∈Λ

is uniformly collectively compact dissipa-

tive and according to Theorem 7.3 the set I =

| λ ∈ Λ} is compact, where

= {I

| ω ∈ Ω} is the compact global attractor of cocycle ϕ

. The theorem is

proved. 

7.4 Connectedness

Recall that the space W possesses the property (S) if for every compact K ∈ C(W )

there exists a compact connected set V ∈ C(W ) such that K ⊆ V .

If M ⊆ W , for each ω ∈ Ω, we write

Ω

(M) =

t≥0

[

τ≥t

ϕ(τ, M, ω

−τ

) .

Lemma 7.7 Suppose that the cocycle ϕ admits a compact pullback attractor

|ω ∈ Ω} , then the following assertions hold:

a. ∅ 6= Ω

(M) ⊆ I

for every M ∈ C(W ) and ω ∈ Ω;

b. the family {Ω

(M) | ω ∈ Ω} is compact and invariant w.r.t. cocycle ϕ for every

M ∈ C(W );

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 7

Upper semi-continuity of attractors 273

c. if I =

| ω ∈ Ω} ⊆ M, then the following inclusion I

⊆ Ω

(M) holds for

every ω ∈ Ω.

Proof. The ﬁrst and second assertions follow from the deﬁnition of pullback at-

tractor and from the equalities (2.83)-(2.84).

Let I be a subset of M, then

= ϕ(t, I

−t

, ω

−t

) ⊆ ϕ(t, I, ω

−t

) ⊆ ϕ(t, M, ω

−t

) (7.20)

and according to the equality (2.83) we have I

⊆ Ω

(M). 

Theorem 7.7 Let W possess the property (S) and let the cocycle ϕ admit a

compact pullback attractor {I

| ω ∈ Ω}, then:

(1) the set I

is connected for every ω ∈ Ω;

(2) if the space Ω is connected, then the set I =

|ω ∈ Ω} also is connected.

Proof. 1. Since the equality (7.2) holds and the space W possesses the property

(S), then there exists a connected compact V ∈ C(W ) such that I ⊆ V and

lim

t→+∞

β(ϕ(t, V, ω

−t

), I

) = 0, (7.21)

for every ω ∈ Ω. We shall show that the set I

is connected. If we suppose

that it is not true, then there are A

, A

6= ∅, closes and A

= I

. Let

0 < ε

< d(A

, A

) and L = L(ε

) > 0 be such that

β(ϕ(t, V, ω

−t

), I

) <

(7.22)

for all t ≥ L(ε

). We note that the set ϕ(t, V, ω

−t

) is connected and according to

the inclusion (7.20) and the inequality (7.22) the following condition

ϕ(t, V, ω

−t

)

(W \[B(A

)

B(A

)]) 6= ∅

is fulﬁlled for every t ≥ L(ε

) and ω ∈ Ω, where B(A, ε) := {u ∈ W | ρ(u, A) < ε}.

Then there exists t

→ +∞ and u

∈ W such that

∈ ϕ(t

, V, ω

−t

)

(W \ [B(A

)

B(A

)]). (7.23)

According to the equality (7.21) it is possible to suppose that the sequence {u

} is

convergent. We denote by u := lim

n→+∞

, then from Lemma 2.15 follows that u ∈

Ω

(V ). Since I ⊆ V , then according to Lemma 2.15 we have u ∈ Ω

(V ) ⊆ I

⊆ I.

On the other hand according to (7.23) we have u /∈ B(A

)

B(A

). This

contradiction shows that the set I

is connected.

2. Let the space Ω be connected. According to Lemma 7.3 the function F :

Ω 7→ C(W ), deﬁned by equality F (ω) = I

is upper semi-continuous and from the

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 7

274 Global Attractors of Non-autonomous Dissipative Dynamical Systems

corollary 1.8.13

[

]

(see also Lemma 3.1

[

148

]

) follows that the set I =

| ω ∈

Ω} = F (Ω) is connected. 

Corollary 7.2 Let W be a metric space with the property (S) and let the cocycle

ϕ admit a compact global attractor {I

| ω ∈ Ω}, then:

(1) the set I

is connected for every ω ∈ Ω;

(2) if the space Ω is connected, then the set I =

| ω ∈ Ω} also is connected.

Proof. This assertion follows from Theorems 7.2, 7.7 and Lemma 7.2. 

7.5 Applications

7.5.1 Quasi-homogeneous systems

Let E and G be two ﬁnite dimensional spaces.

Deﬁnition 7.7 The function f ∈ C(E ×G, E) is called

[

94, 102

]

homogeneous of

order m with respect to variable u ∈ E if the equality f(λu, ω) = λ

f(u, ω) holds

for all λ ≥ 0, u ∈ E and ω ∈ G.

Theorem 7.8 Let f ∈ C

(E), Φ ∈ C

(G), Ω ⊆ G be a compact invariant set of

dynamical system

= Φ(ω), (7.24)

the function f be homogeneous (of order m > 1) and a zero solution of equation

= f(u) (7.25)

be uniformly asymptotically stable. If F ∈ C

(E × G, E) and

|F (u, ω)| ≤ c|u|

for all |u| ≥ r and ω ∈ Ω, where r and c are certain positive numbers, then there

exists a positive number λ

such that for all λ ∈ Λ = [−λ

, λ

] the following holds:

(1) a set I

:= {u ∈ E | sup{|ϕ

(t, u, ω)| : t ∈ R} < +∞} is not empty, compact

and connected for each ω ∈ Ω, where ϕ

(t, u, ω) is a unique solution of equation

= f (u) + λF (u, ωt) satisfying the initial condition ϕ

(0, u, ω) = u;

(2) ϕ

(t, I

, ω) = I

σ(t,ω)

for all t ∈ R

and ω ∈ Ω;

(3) the set I

| ω ∈ Ω} is compact and connected;

(4) the equalities

lim

t→+∞

β(ϕ

(t, M, ω

−t

), I

) = 0 (7.26)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 7

Upper semi-continuity of attractors 275

and

lim

t→+∞

β(ϕ

(t, M, ω), I

) = 0

take place for all λ ∈ Λ, ω ∈ Ω and bounded subset M ⊆ E.

(5) the set

| λ ∈ Λ} is compact;

(6) the equality

lim

λ→0

sup

ω∈Ω

β(I

, 0) = 0

holds.

Proof. Under the condition of Theorem 7.8 according to Theorem 5.34 by the

equality

V (u) =

+∞

|π(t, u)|

dt,

where π(t, u) is a solution of equation (7.25) with condition π(0, u) = u, is deﬁned a

continuously diﬀerentiable function V : E → R

, verifying the following conditions:

a. V (µu) = µ

k−m+1

V (u) for all µ ≥ 0 and u ∈ E;

b. there exist positive numbers α and β such that α|u|

k−m+1

≤ V (u) ≤ β|u|

k−m+1

for all u ∈ E;

c. V

(u) = DV (u)f (u) = −|u|

for all u ∈ E, where DV (u) is a derivative of

Frechet of function V in the point u.

Let us deﬁne a function V : X → R

(X := E × Ω) in the following way:

V(u, ω) := V (u) for all (u, ω) ∈ X. Note that

(u, ω) :=

V (ϕ

(t, u, ω))|

t=0

= −|u|

+ DV (u)λF (u, ω)

and there exists λ

> 0 such that the inequality

(u, ω) ≤ −ν|u|

holds for all ω ∈ Ω and |u| ≥ r, where ν = 1 −λ

cL > 0 ( see the proof of Theorem

4.3

[

102

]

For ﬁnishing the proof of the theorem it is suﬃcient to refer to Theorem 7.6 and

Lemma 7.4. 

Theorem 7.9 Let f ∈ C

(E × G, E), Φ ∈ C

(F ), Ω ⊆ G be a compact invariant

set of dynamical system (7.24), the function f be homogeneous (of order m = 1)

w.r.t. variable u ∈ E and a zero solution of equation

= f (u, ωt) (ω ∈ Ω) (7.27)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 7

276 Global Attractors of Non-autonomous Dissipative Dynamical Systems

be uniformly asymptotically stable. If |F (u, ω)| ≤ c|u| for all |u| ≥ r and ω ∈ Ω,

where r and c are certain positive numbers, then there exists a positive number λ

such that for all λ ∈ Λ = [−λ

, λ

] the following assertions take place:

(1) a set I

:= {u ∈ E | sup{|ϕ

(t, u, ω)| : t ∈ R} < +∞} is not empty, compact

and connected for each ω ∈ Ω, where ϕ

(t, u, ω) is a unique solution of equation

= f(u, ωt) + λF (u, ωt)

verifying the initial condition ϕ

(0, u, ω) = u;

(2) ϕ

(t, I

, ω) = I

σ(t,ω)

for all t ∈ R

and ω ∈ Ω;

(3) a set I

| ω ∈ Ω} is compact and connected;

(4) the equalities

lim

t→+∞

β(ϕ

(t, M, ω

−t

), I

) = 0

and

lim

t→+∞

β(ϕ

(t, M, ω), I

) = 0 (7.28)

take place for all λ ∈ Λ, ω ∈ Ω and bounded subset M ⊆ E.

(5) the set

| λ ∈ Λ} is compact;

(6) the following equality holds:

lim

λ→0

sup

ω∈Ω

β(I

, 0) = 0 .

Proof. The proof of this assertion is similar to the proof of the theorem 7.8. 

7.5.2 Monotone systems

Let f ∈ C(E × Ω, E) be a function satisfying

Rehf(u

, ω) −f(u

, ω), u

−u

i ≤ −k|u

−u

(7.29)

for all ω ∈ Ω and u

, u

∈ E (k > 0 and α ≥ 2).

Theorem 7.10

[

316, 92

]

If the function f veriﬁes the condition (7.29), then the

following statements are true:

(1) the set I

= {u ∈ E | sup

t∈R

|ϕ(t, u, ω)| < +∞} contains a single point {ϕ(ω)}

for all ω ∈ Ω, where ϕ(t, u, ω) is a solution of equation (7.26) with condition

ϕ(0, u, ω) = u;

(2) the inequalities

|ϕ(t, u, ω) − ϕ(ωt)| ≤ e

−kt

|u − ϕ(u)| (if α = 2),