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

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Linear almost periodic dynamical systems 427

(in particular, almost periodic) matrix from asymptotic stability of null solution of

system (13.44) and all system

= B(t)x, (13.45)

where B ∈ H(A) :=

: τ ∈ R}, A

is the translation of matrix A on τ and by

bar is denoted the closure in the topology of uniform convergence uniform on every

compact from R, follows the uniform stability of null solution of system (13.44).

Finally we note that from results of author

[

]

follows the validity of above men-

tioned result for arbitrary system (13.44) with compact matrix (i.e. when H(A) is

compact). Below we study the relationship between the asymptotic stability and

uniform stability of null solution of system (13.44) in the arbitrary Banach space.

Our main result is that for linear system (13.44) with recurrent coeﬃcients in

the arbitrary Banach space the following statement takes place: if the null solution

of equation (13.44) and all the equation (13.45) are asymptotically stable, then the

null solution of equation (13.44) is uniformly stable.

From the theorem of Banach-Steinhauss follows that point dissipativity and

compact dissipativity are equivalent for autonomous linear system. One example of

linear autonomous dynamical system which is compact dissipative, but is not local

dissipative, is constructed in the section 1.6 (see example1.8).

Theorem 13.14 Let h(X, S

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

dynamical system and the following conditions hold:

1. Y is compact and minimal ( i.e. Y = H(y) :=

{yt : t ∈ S} for all y ∈ Y );

2. for any x ∈ X there exists C

≥ 0 such that

|xt| ≤ C

(13.46)

for all t ∈ S

;

3. the mapping y 7→ kπ

k is continuous, where kπ

k is a norm of linear operator

:= π

, for every t ∈ S

Then there exists M ≥ 0 such that the inequality

|π

x| ≤ M|x| (13.47)

takes place for all t ∈ S

and x ∈ X.

Proof. From condition 2. and theorem of Banach-Steinhauss follows the uniform

boundedness of the family of linear operators {π

: t ∈ S

} for every y ∈ Y , i.e.

for any y ∈ Y there exists M

≥ 0 such that kπ

k ≤ M

for all t ∈ S

. We put

d(y) := sup{kπ

k : t ∈ S

} (13.48)

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428 Global Attractors of Non-autonomous Dissipative Dynamical Systems

and claim that the function d : Y → R

, deﬁned by equality (13.48) be lower-

semicontinuous, i.e. lim inf

→y

d(y

) ≥ d(y) for all y ∈ Y and {y

} → y. Suppose that

it is not true, then there exist y ∈ Y, {y

} and ε > 0 such that

lim inf

→y

d(y

) = d(y) −ε. (13.49)

From the equality (13.48) follows d(y) = lim

n→+∞

kπ

k for some sequence {t

} ⊆ S

and, consequently, there exists k such that

|kπ

k− d(y)| <

(13.50)

for all n ≥ k. According to continuity of mapping y 7→ kπ

k there exists n(k) such

that

|kπ

k− kπ

k| <

(13.51)

for all n ≥ n(k). From (13.50) and (13.51) follows

|d(y) − kπ

k| <

(13.52)

for all n ≥ n(k). From inequality (13.52) results

|d(y) − d(y

)| ≤

(13.53)

for all n ≥ n(k). Inequality (13.53) contradicts (13.49). 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 exist a positive numbers

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

, ..., t

such that Y =

i=1

σ(S[p, δ

], t

) (see

[

238, p.134

]

). We put L := max{t

|i = 1, 2, ..., m}.

Assume that m ∈ Y, y ∈ B[p, δ

] and t

are such that m = yt

. Then

|xt| = |π

t+t

(π

−t

(x))| ≤ M

C|x| (13.54)

for all x ∈ X and t ≥ L, where

C := max{max{kπ

−t

k |y ∈ Y }, i = 1, 2, ..., m}.

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 |xt| ≤ ε for all

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

> 0, δ

→ 0 (δ

> 0), |x

| < δ

and t

∈ [0, L] such that

| ≥ ε

. (13.55)

Since (X, h, Y ) is locally trivial Banach ﬁber bundle and Y is compact, then the

zero section Θ = {θ

: y ∈ Y } of (X, h, Y ) is compact and, consequently, we can

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Linear almost periodic dynamical systems 429

assume that the sequences {x

} and {t

} are convergent. Put x

= lim

n→+∞

and

= lim

n→+∞

, then x

= θ

= h(x

)). Passing to the limit in (13.55) as

n → +∞, we obtain 0 = |x

| ≥ ε

. The last inequality contradicts the choice of

. This contradiction proves the above assertion. If γ > 0 is such that |π

x| ≤ 1

for all |x| ≤ γ and t ∈ [0, L], then

|xt| ≤

|x| (13.4.12)

for all t ∈ [0, L] and x ∈ X. We put M := max{γ

−1

, M

C}, then from (13.54) and

(13.4.12) follows the inequality (13.47) for all t ≥ 0 and x ∈ X. The theorem is

proved. 

Remark 13.6 a. If the ﬁber bundle (X, h, Y ) is ﬁnite-dimensional, then the

condition 3. of Theorem 13.14 holds.

b. Let X := E × Y, where E is a Banach space and π := (ϕ, σ), i.e. π

x :=

(ϕ(t, u, y), σ

y) for all t ∈ S

and x := (u, y) ∈ X = E × Y. Then the condition

3. of Theorem 13.14 holds, if for every t ∈ S

the mapping U(t, ·) : Y → [E] is

continuous, where U (t, y)u = ϕ(t, u, y) for any (t, u, y) ∈ S

× E × Y and [E] is

a Banach space of all continuous operators acting onto E and equipped with the

operational norm.

Theorem 13.15 Let h(X, S

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

dynamical system, Y be a compact minimal set and the mapping y 7→ kπ

k be

continuous for each t ∈ S

. Then from point dissipativity of h(X, S

, π), (Y, S, σ), hi

follows its compact dissipativity.

Proof. Assume that the conditions of Theorem 13.15 are fulﬁlled and the non-

autonomous dynamical system h(X, S

, π), (Y, S, σ), hi is point dissipative, then ac-

cording to Theorem 2.36 for every x ∈ X there exists a constant C

≥ 0 such that

the inequality (13.46) takes place for all x ∈ X and t ∈ S

. Next to complete the

proof of Theorem 13.15, it is suﬃcient to refer to Theorems 13.14 and 2.37. 

Linear ordinary diﬀerential equations.

Let Λ be a complete metric space of linear operators that act on Banach space

E and C(R, Λ) be a space of all continuous operator-functions A : R → Λ equipped

with open-compact topology and (C(R, Λ), R, σ) be a dynamical system of shifts on

C(R, Λ).

Let Λ = [E] and consider the linear diﬀerential equation

= A(t)u , (13.56)

where A ∈ C(R, Λ). Along with equation (13.56), we shall also consider its H−class,

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430 Global Attractors of Non-autonomous Dissipative Dynamical Systems

that is, the family of equations

= B(t)v , (13.57)

where B ∈ H(A) :=

: τ ∈ R}, A

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

closure in C(R, Λ). Let ϕ(t, u, B) be the solution of equation (13.57) that satis-

ﬁes 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), then the triple h(X, R

, π), (Y, R, σ), hi is a

linear non-autonomous dynamical system, where X := E × Y, π := (ϕ, σ) ( i.e.

π(τ, (v, B)) := (ϕ(τ, v, B), B

) and h := pr

: X → Y. Applying Theorem 13.15 to

this system, we obtain the following assertion.

Theorem 13.16 Let A ∈ C(R, Λ) be recurrent (i.e. H(A) is compact minimal

set of (C(R, Λ), R, σ) ) and the zero solution of equation (13.56) and all equation

(13.57) are asymptotically stable, i.e. lim

t→+∞

|ϕ(t, v, B)| = 0 for all v ∈ E and

B ∈ H(A). Then the zero solution of equation (13.56) is uniformly stable, i.e. there

exists M ≥ 0 such that |ϕ(t, v, B)| ≤ M|v| for all t ≥ 0, v ∈ E and B ∈ H(A).

Proof. According to Lemma 2

[

]

the mapping B 7→ ϕ(t, ·, B) from H(A) into [E]

is continuous for all t ∈ R. To ﬁnish the proof of the theorem it suﬃces to refer to

Theorem 13.14. 

Linear Partial diﬀerential equations. Let Λ be some complete metric

space of linear closed operators acting into Banach space E ( for example Λ =

+ B|B ∈ [E]}, where A

is a closed operator that acts on E). We assume that

the following conditions are fulﬁlled for equation (13.56) and its H− class (13.57):

a. for any v ∈ E and B ∈ H(A) equation (13.57) has exactly one solution that is

deﬁned on R

and satisﬁes the condition ϕ(0, v, B) = v;

b. the mapping ϕ : (t, v, B) → ϕ(t, v, B) is continuous in the topology of R

E × C(R; Λ);

c. for every t ∈ R

the mapping U(t, ·) : H(A) → [E] is continuous, where U(t, ·)

is the Cauchy’s operator of equation (13.57), i.e. U(t, B)v := ϕ(t, v, B) (t ∈

, v ∈ E and ∈H(A) ).

Under the above assumptions the equation (13.56) generates a linear non-

autonomous dynamical system h(X, R

, π), (Y, R, σ), hi, where X = E × Y, π =

(ϕ, σ) and h := pr

: X → Y. Applying Theorem 13.15 to this system, we will

obtain the analogue of Theorem 13.16 for diﬀerent classes of partial diﬀerential

equations.

We will consider an example of partial diﬀerential equation which satisﬁes the

above conditions a.-c. Let H be a Hilbert space with a scalar product h·, ·i =

|·|

, D(R

, H) be a set of all inﬁnite diﬀerentiable and ﬁnite into R

functions with

values into H.

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Linear almost periodic dynamical systems 431

Denote by (C(R, [H]), R, σ) a dynamical system of shifts on C(R, [H]). Consider

the equation

[hu(t), ϕ

(t)i + hA(t)u(t), ϕ(t)i]dt = 0, (13.58)

along with the family of equations

[hu(t), ϕ

(t)i + hB(t)u(t), ϕ(t)i]dt = 0, (13.59)

where B ∈ H(A) :=

|τ ∈ R}, A

(t) := (t + τ ) and the bar denotes closure in

C(R, [H]).

The function u ∈ C(R

, H) is called a solution of equation (13.58), if (13.58)

takes place for all ϕ ∈ D(R

, H).

Assume that the operator A(t) is self-adjoint. Let (H(A), R, σ) be a dynamical

system of shifts on H(A), ϕ(t, v, B) be a solution of equation (13.59) with condition

ϕ(0, v, B) = v,

X = H × H(A), X be a set of all the points hu, Bi ∈

X such that

through point u ∈ H passes a solution ϕ(t, u, A) of equation (13.58) deﬁned on

. According to Lemma 2.21

[

]

the set X is closed in

X. In virtue of Lemma

2.22

[

]

the triple (X, R

, π) is a dynamical system on X ( where π := (ϕ, σ))

and h(X, R

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

h := pr

: X → Y = H(A). Applying the results from

[

]

it is possible to show that

for every t the mapping B 7→ U (t, B) ( where U(t, B)v := ϕ(t, v, B)) from H(A)

into [H] is continuous and, consequently, for this system is applicable Theorem

13.14. Thus the following assertion takes place.

Theorem 13.17 Let A ∈ C(R, [H]) be recurrent and the zero solution of equation

(13.56) and all equation (13.57) are asymptotically stable, i.e. lim

t→+∞

|ϕ(t, v, B)| = 0

for all v ∈ E and B ∈ H(A). Then the zero solution of equation (13.56) is uniformly

stable, i.e. there exists M ≥ 0 such that |ϕ(t, v, B)| ≤ M |v| for all t ≥ 0, v ∈ H and

B ∈ H(A).

We will give the example of limit problem reducing to equation of type (13.58).

Let Ω be a bounded domain in R

, Γ be frontier of Ω, Q = R

×Ω and S = R

×Γ.

Consider in Q the ﬁrst initial limit problem for equation

∂u

∂t

= L(t)u (u|

t=0

= ϕ, u|

= 0), (13.60)

where

L(t)u :=

i,j=1

∂

∂x

(t, x)

∂u

∂x

) − a(t, x)u

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432 Global Attractors of Non-autonomous Dissipative Dynamical Systems

The operator A(t) according to theorem of Riesz is deﬁned by equality

hA(t)u, ϕi = −

Ω

[

i,j=1

(t, x)

∂u

∂x

∂ϕ

∂x

+ a(t, x)uϕ]dx.

If a

(t, x) = a

(t, x) and the functions a

(t, x) and a(t, x) are recurrent (almost

periodic) with respect to t ∈ R uniformly with respect to x ∈ Ω, then for equation

(13.60) it is applicable Theorem 13.17, if H =

(Ω)

Linear functional-diﬀerential equations. Denote by A = A(C, R

) a

Banach space of all linear continuous operators acting from C := C([−r, 0], R

) into

, equipped by operational norm. Consider the equation

= A(t)u

, (13.61)

where A ∈ C(R, A). We put H(A) :=

: τ ∈ R}, A

(t) := A(t + τ) and the bar

denotes closure in the topology of uniform convergence on every compact from R.

Along with equation (13.61) we also consider the family of equations

= B(t)u

, (13.62)

where B ∈ H(A). Let ϕ

(v, B) be a solution of equation (13.62) with condition

(v, B) = v, deﬁned on R

. We put Y := H(A) and denote by (Y, R, σ) a

dynamical system of shifts on H(A). Let X := C ×Y and π := (ϕ, σ) a dynamical

system on X, deﬁned by equality π(t, (v, B)) := (ϕ

(v, B), B

). A non-autonomous

dynamical system h(X, R

, π), (Y, R, σ), hi (h := pr

: X → Y ) is linear. The

following assertion takes place.

Lemma 13.9 Let H(A) be compact in C(R, A), then the non-autonomous

dynamical system h(X, R

, π), (Y, R, σ), hi generated by equation (13.61) is com-

pletely continuous.

Proof. Let B ⊂ C = C[−r, 0] be a bounded set and t ≥ r. According to the

continuity of mapping ϕ : R

× C × H(A) 7→ C and compactness of H(A) there

exists a positive number M such that |ϕ

(v, B)| ≤ M and |B(τ)ϕ

(v, B)| ≤ M

for all τ ∈ [0, t], B ∈ H(A) and v ∈ B, and, consequently, | ˙ϕ(τ, v, B)| ≤ M for

all τ ∈ [0, t], B ∈ H(A) and v ∈ B, i.e. the family of functions {ϕ

(v, B) : B ∈

H(A), v ∈ B} ( for t ≥ r) is uniformly continuous on [−r, 0]. Therefore this family

of functions is relatively compact. The Lemma is proved. 

Theorem 13.18 Let H(A) be compact. Then the following assertion are equiva-

lent:

(1) for any B ∈ H(A) the zero solution of equation (13.62) is asymptotically stable,

i.e. lim

t→+∞

|ϕ

(v, B)| = 0 for all v ∈ C and B ∈ H(A);

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Linear almost periodic dynamical systems 433

(2) the zero solution of equation (13.61) is uniformly asymptotically stable, i.e.

there are the positive numbers N and ν such that |ϕ

(v, B)| ≤ N e

−νt

|v| for all

t ≥ 0, v ∈ C and B ∈ H(A).

Proof. Let h(X, R

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

generated by equation (13.61). According to Lemma 13.9 this system is completely

continuous and to ﬁnish the proof it is suﬃciently to refer to Theorems 2.38 and

6.10. 

Consider the neutral functional diﬀerential equation

= A(t)x

, (13.63)

where A ∈ C(R, A) and D ∈ A is non-atomic at zero operator

[

175, p.67

]

. As well

as in the case of equation (13.61), the equation (13.63) generates a linear dynamical

system h(X, R

, π), (Y, R, σ), hi, where X := C × Y, Y := H(A) and π := (ϕ, σ).

The following statement takes place.

Lemma 13.10 Let H(A) be compact and the operator D is stable, i.e. the zero

solution of homogeneous diﬀerence equation Dy

= 0 is uniformly asymptotically

stable. Then a linear non-autonomous dynamical system h(X, R

, π), (Y, R, σ), hi,

generated by equation (13.63), is asymptotically compact.

Proof. According to

[

179

]

(see, p.119, formula (5.18)) the mapping ϕ

(·, B) : C → C

can be written as

(·, B) = S

(·) + U

(·, B)

for all B ∈ H(A), where U

(·, B) is conditionally completely continuous for t ≥ r

and there exist constants N > 0, ν > 0 such that kS

k ≤ Ne

−νt

(t ≥ 0). To ﬁnish

the proof of Lemma 13.10 it is suﬃciently to refer to Theorem 2.22. 

Theorem 13.19 Let A ∈ C(R, A) be recurrent (i.e. H(A) is compact minimal in

the dynamical system of shifts (C(R, A), R, σ) ) and D is stable, then the following

assertions are equivalent:

(1) the zero solution of equation (13.61) and all equation

= B(t)x

, (13.64)

where B ∈ H(A), are asymptotically stable, i.e. lim

t→+∞

|ϕ

(v, B)| = 0 for all

v ∈ C and B ∈ H(A) (ϕ

(v, B) is a solution of equation (13.64) with condition

(v, B) = v);

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434 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(2) the zero solution of equation (13.63) is uniformly exponentially stable, i.e. there

are a positive numbers N and ν such that |ϕ

(v, B)| ≤ Ne

−νt

|v| for all t ≥ 0, v ∈

C and B ∈ H(A).

Proof. Let h(X, R

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

generated by equation (13.63). According to Lemma 13.10 this system is asymp-

totically compact. To ﬁnish the proof of Theorem 13.19 it is suﬃciently to refer to

Theorems 2.38 and 1.25. The theorem is proved. 

13.5 Linear α-condensing systems

Let A(t) be a continuous n×n matrix-function and H(A) be the family of all matrix-

functions B = lim

n→+∞

, where {t

} ⊂ R, A

(t) = A(t

+ t) and the convergence

→ B is uniform on every compact subset of R. The following result is well

known.

Theorem 13.20

[

275, 33, 65

]

Let A be a bounded and uniformly continuous

matrix-function on R, then the following conditions are equivalent:

1. The trivial solution of equation

= A(t)x (13.65)

is uniformly exponentially stable.

2. The trivial solution of equation (13.65) is uniformly asymptotically stable.

3. The trivial solution of equation (13.65) and every equation

= B(t)y (B ∈ H(A)) (13.66)

is asymptotically stable.

For equations in inﬁnite-dimensional spaces conditions 1., 2., and 3. are not

equivalent; see example 1.8 and also

[

81, 121, 249

]

. However, in the general inﬁnite-

dimensional case condition 1. implies condition 2., and condition 2. implies condi-

tion 3.

Linear non-autonomous dynamical systems satisfying one of the conditions 1.,

or 2., or 3. are studied in

[

]

; see also Chapter 2 (section 2.11). We will show that

if the operator corresponding to the the Cauchy’s problem for (13.65) satisﬁes some

compactness condition, then condition 3 implies condition 1 (see Theorems 13.23

and 13.24).

For recurrent (almost periodic) systems this result is made precise in Theorems

13.25 and 13.26. Applications of this result to diﬀerent classes of linear evolution

equations (ordinary linear diﬀerential equations in a Banach space, retarded and

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Linear almost periodic dynamical systems 435

neutral functional diﬀerential equations, some classes of evolution partial diﬀerential

equations) are given.

Assume that X and Y are complete metric spaces, R is the set of real numbers,

Z is the set of integer numbers, S = R or Z, S

= {t ∈ S : t ≥ 0} and S

−

= {t ∈

S|t ≤ 0}.

Recall that a dynamical system (X, S

, π) is said to be conditionally β-

condensing

[

179

]

if there exists t

> 0 such that β(π

B) < β(B) for all bounded sets

B in X with β(B) > 0. The dynamical system (X, S

, π) is said to be β-condensing

if it is conditionally β-condensing and the set π

B is bounded for all bounded sets

B ⊆ X.

According to Lemma 2.3.5 in

[

179, p.15

]

and Lemma 3.3 in

[

]

the conditional

condensing dynamical system (X, S

, π) is asymptotically compact.

Let E be a metric space, X := E × Y , A ⊂ X, and A

:= {x ∈ A : pr

x = y}.

Then A = ∪{A

: y ∈ Y }. Let

:= pr

and

A = ∪{

: y ∈ Y }. Note that if

the space Y is compact, then a set A ⊂ X is bounded in X if and only if the set

is bounded in E.

Lemma 13.11 The equality α(A) = α(

A) takes place for all bounded sets A ⊂ X,

where α(A) and α(

A) are the Kuratowskii’s measure of non-compactness for the sets

A ⊂ X and

A ⊂ E.

Proof. Let ε > 0 and A be a bounded subset in X, then there are sets A

, A

, . . . , A

such that A = ∪{A

: i = 1, 2, . . . , n} and diam A

< α(A) + ε. Note that

A =

∪{

: i = 1, 2, . . . , n} and diam

≤ diam A

< α(A) + ε, and consequently,

α(

A) ≤ α(A).

Let ε be a positive constant, A be a bounded set in X,

A = ∪{

: k =

1, 2, . . . , m} and diam

< α(

A) + ε. Since Y is compact, there are sets

, Y

, . . . , Y

such that Y

∪ Y

∪ ··· ∪ Y

= Y and diam Y

< ε (j = 1, 2, . . . `).

Let A

= pr

−1

(

) ∩ A, and

= pr

−1

(pr

−1

(

) ∩ A) ∩Y

) ∩ A

Note that A

⊆

×Y

, and that

diam A

≤ diam

+ diam Y

< α(

A) + ε + ε = α(A) + 2ε .

Since A = ∪{A

: i = 1, 2, . . . , n, j = 1, 2, . . . , `}, it follows that α(A) ≤ α(

A) and

α(A) = α(

A). which concludes the present proof. 

Deﬁnition 13.5 A cocycle hE, ϕ, (Y, ,σ)i is called conditionally α-condensing

if there exists t

> 0 such that for any bounded set B ⊆ E the inequality

α(ϕ(t

, B, Y )) < α(B) holds if α(B) > 0. The cocycle ϕ is called α-condensing if

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

436 Global Attractors of Non-autonomous Dissipative Dynamical Systems

it is a conditional α-condensing cocycle and the set ϕ(t

, B, Y ) = ∪{ϕ(t

, u, Y )|u ∈

B, y ∈ Y } is bounded for all bounded set B ⊆ E.

Deﬁnition 13.6 A cocycle ϕ is called conditional α-contraction of order k ∈

[0, 1), if there exists t

> 0 such that for any bounded set B ⊆ E for

which ϕ(t

, B, Y ) = ∪{ϕ(t

, u, Y )|u ∈ B, y ∈ Y } is bounded the inequality

α(ϕ(t

, B, Y )) ≤ kα(B) holds. The cocycle ϕ is called α-contraction if it is a condi-

tional α-contraction cocycle and the set ϕ(t

, B, Y ) = ∪{ϕ(t

, u, Y )|u ∈ B, y ∈ Y }

is bounded for all bounded sets B ⊆ E.

Lemma 13.12 Let Y be compact and the cocycle ϕ be α-condensing. Then

the skew-product dynamical system (X, S

, π), generated by the cocycle ϕ, is α-

condensing too.

Proof. Let A ⊂ X be a bounded subset, t

> 0 and α(A) > 0, then

π(t

, A) = ∪{π(t

, A

|y ∈ Y }

= ∪{(ϕ(t

, A

, y), yt)|y ∈ Y } ⊆ ϕ(t

A, Y ) ×Y. (13.67)

Since A is bounded,

A is also bounded in E and according to the condition of

the lemma the set ϕ(t

A, Y ) is bounded and, consequently, π(t

, A) is bounded.

According to Lemma 13.11 and (13.67) we have

α(π(t

, A)) = α(∪{(ϕ(t

, A

, y), yt

)|y ∈ Y }) ≤ α(ϕ(t

A, Y )) < α(

A) = α(A).

The lemma is proved. 

Theorem 13.21 Let E be a Banach space, ϕ be a cocycle on (Y, S, σ) with ﬁber

E and the following conditions be fulﬁlled:

(1) ϕ(t, u, y) = ψ(t, u, y) + γ(t, u, y) for all t ∈ S

, u ∈ E and y ∈ Y.

(2) There exists a function m : R

→ R

satisfying the condition m(t, r) → 0 as

t → +∞ (for every r > 0) such that |ψ(t, u

, y) −ψ(t, u

, y)| ≤ m(t, r)|u

−u

for all t ∈ S

, u

∈ B[0, r] and y ∈ Y .

(3) γ(t, A, Y ) is compact for all bounded A ⊂ X and t > 0.

Then the cocycle ϕ is an α-contraction.

Proof. Let ε > 0 and A be a bounded set in E, then there are sets A

, A

, . . . , A

such that A = ∪{A

: i = 1, 2, . . . , n} and diam A

< α(A) + ε for i = 1, 2, . . . , n.

Since Y is compact, then there are a sets Y

, Y

, . . . , Y

such that Y

∪Y

∪···∪Y

Y with condition diam Y

< ε for all j = 1, 2, . . . , m.