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

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Linear almost periodic dynamical systems 447

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, (13.88)

along with the family of equations

[hu(t), ϕ

(t)i + hB(t)u(t), ϕ(t)i]dt = 0, (13.89)

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 (13.88), if (13.88)

takes place for all ϕ ∈ D(R

, H).

Let (H(A), R, σ) be a dynamical system of shifts on H(A), ϕ(t, v, B) be a solution

of (13.89) 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

(13.88) deﬁned on R

. According to Lemma 2.21 in

[

]

the set X is closed in

X. In virtue of Lemma 2.22 in

[

]

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 Theorem 13.22 is applicable. Thus the following

assertion takes place.

Theorem 13.32 Let A ∈ C(R, [H]) be compact, then the following assertion hold:

(1) The trivial solution of equation (13.77) is uniformly exponentially stable, i.e.

there exist positive numbers N and ν such that kU(t, B)k ≤ Ne

−νt

for all t ≥ 0

and B ∈ H(A).

(2) lim sup

t→+∞

{kU(t, B)k : B ∈ H(A)} = 0.

Proof. This statement follows directly from Theorem 13.22. 

Linear functional-diﬀerential equations. Consider the equation

= A(t)u

, (13.90)

where A ∈ C(R, A). We put H(A) :=

: τ ∈ R}, A

(t) = A(t + τ ), where the

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

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

448 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Along with equation (13.90) we also consider the family of equations

= B(t)u

, (13.91)

where B ∈ H(A). Let ϕ

(v, B) be a solution of equation (13.91) with condition

(v, B) = v deﬁned on R

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

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

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

(v, B), B

). The non-

autonomous dynamical system h(X, R

, π), (Y, R, σ), hi (h := pr

: X → Y ) is

linear. The following assertion takes place.

Theorem 13.33 Let H(A) be compact. Then the following assertions are equiv-

alent:

(1) For any B ∈ H(A) the zero solution of equation (13.91) is asymptotically stable,

i.e. lim

t→+∞

|ϕ

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

(2) The zero solution of equation (13.90) is uniformly exponentially stable, i.e.

there are 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 the linear non-autonomous dynamical

system, generated by equation (13.90). According to Lemma 13.9 this system is

completely continuous and to ﬁnish the proof it is suﬃcient to refer to Theorem

13.21. 

Consider the neutral functional diﬀerential equation

= A(t)x

, (13.92)

where A ∈ C(R, A) and D ∈ A is an operator non-atomic at zero [19, p.67]. As

well as in the case of equation (13.90), the equation (13.92) generates a linear

non-autonomous 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 13.14 Let H(A) be compact and the operator D is stable; i.e., the zero

solution of the homogeneous diﬀerence equation Dy

= 0 is uniformly asymptotically

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

, π), (Y, R, σ), hi,

generated by equation (13.92), is conditionally α-condensing.

Proof. According to [20, p.119, formula (5.18)] the mapping ϕ

(·, B) : C → C can

be written as

(·, B) = S

(·) + U

(·, B)

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

Linear almost periodic dynamical systems 449

for all B ∈ H(A), where U

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

Also there exist positive constants N, ν such that kS

k ≤ Ne

−νt

(t ≥ 0). Then the

proof is completed by referring to Theorem 13.23. 

Theorem 13.34 Let A ∈ C(R, A) be recurrent (i.e. H(A) is compact minimal

set 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 (13.90) and all equations

= B(t)x

, (13.93)

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

t→+∞

|ϕ

(v, B)| = 0 for all v ∈ C

and B ∈ H(A) (ϕ

(v, B) is the solution of equation (13.93) with condition

(v, B) = v).

(2) The zero solution of equation (13.92) is uniformly exponentially stable, i.e.

there are positive numbers N and ν such that |ϕ

(v, B)| ≤ Ne

−νt

|v| for all

t ≥ 0, v ∈ C and B ∈ H(A).

(3) All solutions of all equations (13.93) are bounded on R

and they don’t have

non-trivial solutions bounded on R .

Proof. Let h(X, R

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

system, generated by equation (13.92). According to Lemma 13.14 this system

is conditionally α condensing. To ﬁnish the proof of Theorem 13.34 it is suﬃcient

to refer to Theorems 13.23 and 13.26. 

Theorem 13.35 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) ), D is stable, and all solutions of

all equations (13.93) are bounded on R

. Let h(X, S

, π), (Y, S, σ), hi be the linear

non-autonomous dynamical system generated by equation (13.92), then there are

two positively invariant vector sub-ﬁberings (X

, h, Y ) and (X

, h, Y ) of (X, h, Y )

such that:

a. X

= X

+ X

and X

∩X

= θ

for all y ∈ Y , where θ

= (0, y) ∈ X = E ×Y

and 0 is the zero in the Banach space E.

b. The vector subﬁbering (X

, h, Y ) is ﬁnite dimensional, invariant (i.e.π

for all t ∈ S

) and every trajectory of a dynamical system (X, S

, π) be-

longing to X

is recurrent.

c. There exist two positive numbers N and ν such that |ϕ

(v, B)| ≤ Ne

−νt

|v| for

all (v, B) ∈ X

, where ϕ

(v, B) := U(t, B)v and U(t, B) is the Cauchy operator

of equation (13.93).

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450 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Proof. Let h(X, S

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

generated by equation (13.92). In virtue of Lemma 13.14 this non-autonomous

dynamical system is conditionally α-condensing. According to Theorem 13.1 there

exists a positive number M such that |ϕ

(v, B)| ≤ M|v| for all t ≥ 0, v ∈ C and

B ∈ H(A). To ﬁnish the proof of Theorem 13.35 it is suﬃcient to refer to Theorem

13.26. 

13.9 Linear periodic systems

This section is devoted to the study of linear periodic dynamical systems, possessing

the property of uniform exponential stability. It is proved that if the Cauchy’s oper-

ator of these systems possesses a certain compactness property, then the asymptotic

stability implies the uniform exponential stability. We also give the application to

diﬀerent classes of linear evolution equations, such as ordinary linear diﬀerential

equations in the space of Banach, retarded and neutral functional diﬀerential equa-

tions, some classes of evolution partial diﬀerential equations.

Let A(t) be a τ-periodic continuous n×n matrix-function. It is well-known that

the following three conditions are equivalent:

1. The trivial solution of equation

= A(t)u (13.94)

is uniformly exponentially stable.

2. The trivial solution of equation (13.94) is uniformly asymptotically stable.

3. The trivial solution of equation (13.94) is asymptotically stable.

For equations in inﬁnite-dimensional spaces the statements 1.-3. are not equiv-

alent, as shown by the examples in

[

121, 249

]

It is clear that in general for the inﬁnite-dimensional case condition 1. implies

2. and 2. implies 3. In this section we show that if the Cauchy operator of equation

(13.94) satisﬁes some compactness condition, then the condition 3. implies 1. (see

Theorem 13.37 below).

Applications to diﬀerent classes of linear evolution equations (ordinary linear

diﬀerential equations in a Banach space, retarded and neutral functional-diﬀerential

equations, some classes of evolutionary partial diﬀerential equations) are given.

The exponential dichotomy of asymptotically compact cocycles was studied by

R. Sacker and G. Sell

[

277

]

. The general case was studied by C. Chicone and Yu.

Latushkin

[

117

]

(see also their references), Yu. Latushkin and R. Schnaubelt

[

236

]

and many other authors.

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Linear almost periodic dynamical systems 451

13.9.1 Exponential stable linear periodic dynamical systems

Let X and Y be complete metric spaces, (X, h, Y ) be a locally trivial Banach ﬁber

bundle over Y

[

]

, [E] be a Banach space of the all linear continuous operators

acting onto Banach space E with the operator norm and U : S

× Y 7→ [E] be

a mapping with properties: U(0, y) = I, U (t + τ, y) = U (t, σ(τ, y))U(τ, y) for all

y ∈ Y and t, τ ∈ S

and the mapping ϕ(·, u, ·) : S

×Y → E (ϕ(t, u, y) := U (t, y)u)

is continuous for every u ∈ E.

Deﬁnition 13.7 A triplet h[E], U, (Y, S, σ)i is called a c

− cocycle on (Y, S, σ)

with ﬁber [E].

Lemma 13.15 Let h[E], U, (Y, S, σ)i be a c

− cocycle on (Y, S, σ) with ﬁber [E]

and Y be a compact, then the following assertions hold:

(1) For every ` > 0 there exists a positive number M(`) such that kU (t, y)k ≤ M (`)

for all t ∈ [0, `] and y ∈ Y ;

(2) The mapping ϕ : S

×E ×Y 7→ E (ϕ(t, u, y) = U(t, y)u) is continuous;

(3) There exist positive numbers N and ν such that kU(t, y)k ≤ Ne

νt

for all t ∈ S

and y ∈ Y .

Proof. Let ` > 0 and u ∈ E, then there exists a positive number M(`, u) such that

|U(t, y)u| ≤ M(`, u) for all (t, y) ∈ [0, `]×Y because the mapping (t, y) → U(t, y)u is

continuous. According to principle of uniformly boundedness there exists a positive

number M(`) such that kU(t, y)k ≤ M (`) for all (t, y) ∈ [0, `] × Y .

Let now (t

, u

, y

) ∈ S

× E × Y and t

→ t

, u

→ u

and y

→ y

, then we

have

|ϕ(t

, u

, y

) − ϕ(t

, u

, y

≤ |ϕ(t

, u

, y

) − ϕ(t

, u

, y

)|+ |ϕ(t

, u

, y

) − ϕ(t

, u

, y

≤ kU (t

, y

)(u

−u

)k + |(U(t

, y

) − U(t

, y

))u

|. (13.95)

In view of ﬁrst statement of Lemma 13.15 there exists the positive number M such

that

kU(t

, y

)k ≤ M (13.96)

for all n ∈ N. From inequalities (13.95) and (13.96) follows the continuity of

mapping ϕ : S

× E ×Y → E (ϕ(t, u, y) = U(t, y)u).

Denote by a := sup{kU(t, y)k : (t, y) ∈ [0, 1] × Y } and let t ∈ S

, t = n + τ(n ∈

N, τ ∈ [0, 1)), then we obtain

kU(t, y)k ≤ kU(n, yτ)kkU(τ, y)k ≤ a

n+1

≤ N e

νt

for all t ∈ S

and y ∈ Y , where N := a and ν := ln a. 

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452 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Lemma 13.16

[

121, Chapter 9

]

Let m : R

7→ R

be a positive and continuous

function. If there exists a positive constant M such that m(t + s) ≤ Mm(t) for all

s ∈ [0, 1] and t ∈ S

, then

+∞

m(t)dt < +∞ implies m(t) → 0 as t → +∞.

Theorem 13.36 Let h[E], U, (Y, S, σ)i be the C

− cocycle on (Y, S, σ) with ﬁber

[E] and (Y, S, σ) be a periodical dynamical system (i.e. there are y

∈ Y and τ ∈

S (τ > 0) such that Y = {y

t : 0 ≤ t < τ }). Then the following conditions are

equivalent:

(i)

lim

t→+∞

kU(t, y

)k = 0. (13.97)

(ii) There exist positive constants N and ν such that for all t ∈ S

and y ∈ Y ,

kU(t, y)k ≤ Ne

−νt

. (13.98)

(iii) There exists p ≥ 1 such that for all u ∈ E,

+∞

|U(t, y

)u|

dt < +∞ . (13.99)

Proof. We remark that from equality (13.97) follows the condition

lim

n→+∞

sup

0≤s≤τ

kU(s + nτ, y

)k = 0. (13.100)

In fact, by virtue of Lemma 13.15 there exists a positive constant M such that

kU(s, y)k ≤ M (13.101)

for all s ∈ [0, τ] and y ∈ Y . Therefore,

kU(s + nτ, y

)k = kU(s, y

)U(nτ, y

)k ≤ MkU (nτ, y

)k (13.102)

for all 0 ≤ s ≤ τ . Consequently, from (13.97) and (13.102) results the condition

(13.100).

We will show that under the condition (13.100) the equality

lim

t→+∞

sup

y∈Y

kU(t, y)k = 0 (13.103)

holds. In fact, let y ∈ Y then there exists a number s ∈ [0, τ ) such that y = y

and, consequently, for t ∈ S

(t = nτ +

t ∈ [0, τ)) we obtain

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

Linear almost periodic dynamical systems 453

kU(t, y)k = kU(t, y

s)k = kU(nτ +

t, y

s)k

= kU ((n −1)τ +

t + s, y

τ)U (τ − s, y

s)k

≤ M max{ sup

0≤s≤τ

kU((n −1)τ + s, y

)k, sup

0≤s≤τ

kU(nτ + s, y

)k}. (13.104)

From (13.100) and (13.104) results the equality (13.103). For ﬁnishing the proof

that (i) implies (ii) is suﬃcient to apply Theorem 13.22.

The fact that (ii) implies (iii) is obvious. Now we prove that (iii) implies (i).

Indeed, let u ∈ E and we consider the function m(t) = |U (t, y

)u|

(t ≥ 0). We

note that

m(t + s) =|U(t + s, y

)u|

= |U (s, y

t)U(t, y

)u|

≤kU(s, y

t)k

|U(t, y

)u|

≤ M

m(t)

for all t ∈ S

and s ∈ [0, 1], where M := sup

0≤s≤1,y∈Y

kU(s, y)k. By Lemma 13.16

m(t) → 0 as t → +∞ and, consequently,

lim

t→+∞

|U(t, y

)u|

= 0 (13.105)

for all u ∈ E. Let now y ∈ Y , then there exists s ∈ [0, τ) such that y = y

s and for

t ≥ τ − s we have

U(t, y)u = U(t, y

s)u = U(t − τ + s, y

)U(τ − s, y

s)u. (13.106)

From equalities (13.105) and (13.106),

lim

t→+∞

|U(t, y)u|

= 0 (13.107)

for all u ∈ E and y ∈ Y. According to Theorem 13.1 there exists a positive number

M such that kU(t, y)k ≤ M for all t ∈ S

and y ∈ Y . Let t > 0 and u ∈ E, then

we obtain

t|U(t, y

)u|

|U(t, y

)u|

ds ≤ (13.108)

|U(t −s, y

s)|

|U(s, y

)u|

≤

|U(s, y

)u|

ds ≤ M

+∞

|U(s, y

)u|

ds = C

for all t ≥ 0. By virtue of principle of uniformly boundedness there exists a positive

number C such that

tkU(t, y

≤ C

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

454 Global Attractors of Non-autonomous Dissipative Dynamical Systems

for all t > 0 and, consequently

kU(t, y

)k ≤ C

−

→ 0

as t → +∞. This completes the present proof. 

Remark 13.8 a. Theorem 13.36 (the equivalence of assertions (ii) and (iii)) is

a variant of the Datko-Pazy theorem (see

[

121

]

[

133

]

and

[

134

]

) for the cocycle over

periodic dynamical systems.

b. Periodic, almost periodic and asymptotic almost periodic mild solutions of

inhomogeneous periodic Cauchy’s problems considered recently by C. J. K. Batty,

W.Hutter and F. R¨abiger

[

]

and W. Hutter

[

191

]

Deﬁnition 13.8 The operator U(τ, y

) is called operator of monodromy for τ-

periodic cocycle U(t, y). The number 0 6= λ ∈ C is called multiplicator of operator

of monodromy U(τ, y

) if there exists u

∈ E (u

6= 0) such that U(τ, y

= λu

(or, what is the same, U(t + τ, y

= λU (t, y

for all t ∈ S

Remark 13.9 a. Condition (13.97) and the equality

lim

n→+∞

kU(nτ, y

)k = 0. (13.109)

are equivalent. We show that (13.109) implies (13.97) as follows. Let now t = nτ +

s, 0 ≤ s < τ, then U(t, y

) = U(s + nτ, y

) = U(s, y

)U(nτ, y

) and, consequently,

kU(t, y

)k ≤ max

0≤s≤τ

kU(s, y

)kkU(nτ, y

)k. (13.110)

From conditions (13.109) and (13.110) results (13.97).

b. Condition (13.98) and the inequality

kU(t, y

)k ≤ N

−ν

(∀t ∈ S

) (13.111)

are equivalent, where N

and ν

are some positive constants. Indeed, from (13.111),

taking into account (13.106), we obtain (13.98).

c. Condition (13.109) is satisﬁed if and only if σ(U (τ, y

)) ⊂ D = {z ∈ C :

|z| < 1}, where σ(U(τ, y

)) is a spectrum of operator of monodromy U(τ, y

). In

fact, from (13.98) results that r

U(τ,y

)

:= lim sup

n→+∞

(kU(nτ, y

))k)

1/n

≤ e

−ν

< 1,

because U

(τ, y

) = U(nτ, y

). If γ = r

U(τ,y

)

< 1, then for all ε > 0 there exists

a n(ε) ∈ N such that (kU (nτ, y

)k)

1/n

≤ γ + ε for all n ≥ n(ε) and, consequently,

kU(nτ, y

)k ≤ (γ + ε)

for all n ≥ n(ε). Thus kU (nτ, y

)k → 0 as n → +∞.

Theorem 13.37 Let h[E], U, (Y, S, σ)i be a c

− cocycle on (Y, S, σ) with ﬁber [E],

(Y, S, σ) be a periodic dynamical system and U(τ, y

) be asymptotically compact (i.e.

if k

→ +∞ (k

∈ N), the sequences {u

} ⊆ E and {U (k

τ, y

} are bounded;

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

Linear almost periodic dynamical systems 455

then the sequence {U(k

τ, y

} is relatively compact). Then the following condi-

tions are equivalent

1. Equality (13.97) holds.

2. For all u ∈ E,

lim

t→+∞

|U(t, y

)u| = 0 . (13.112)

Proof. It is evidently that 1. implies 2. Now, under the conditions of Theorem

13.37 the mapping P := U (τ, y

) : E 7→ E is asymptotically compact because P

U(nτ, y

). From condition (13.112) according to uniform boundedness principle it

follows that there is a positive constant M such that kP

k ≤ M for all n ∈ Z

and,

consequently, the set B = ∪{P

x : |x| ≤ 1, n ∈ Z

} is bounded and P (B) ⊂ B.

Since the mapping P is asymptotically compact in virtue of Lemma 1.3 the set

ω(B) =

n≥0

[

m≥n

(B)

is nonempty, compact, and invariant and ω(B) attracts B.

Now we will prove that lim

n→+∞

k = 0. If we suppose the contrary, then there

are ε

> 0, {x

}(|x

| ≤ 1) and n

→ +∞({n

} ⊂ Z

) such that

| ≥ ε

. (13.113)

Since P is asymptotically compact without loss of generality we can suppose that the

sequence {P

} is convergent. Let ¯x := lim

k→+∞

, then ¯x ∈ ω(B) and from

(13.113) we have |¯x| ≥ ε

> 0. According to the invariance of the set ω(B) there

exists a beside sequence {w

}

n∈Z

⊂ ω(B) such that: w

= ¯x and P (w

) = w

n+1

for all n ∈ Z. We note that

inf

n∈Z

−

| = 0. (13.114)

Suppose that it is not true, then there is a positive number ` such that

| ≥ ` (13.115)

for all n ∈ Z

−

. Let p := lim

k→+∞

and {z

} ⊆ α

, where

n≤0

[

m≤n

be a beside sequence such that z

= p and P (z

) = z

n+1

for all n ∈ Z. From the

inequality (13.115) results that |z

| ≥ ` for all n ∈ Z. On the other hand in view

of (13.112) lim

n→+∞

| = lim

n→+∞

| = 0. The obtained contradiction proves the

equality (13.114).

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456 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Let now n

→ −∞ and |w

| → 0, then w

= P

−n

for all r ∈ N and,

consequently, |w

| = 0 because |w

| ≤ kP

−n

k|w

| ≤ M |w

|. On the other hand

| = |¯x| ≥ ε

> 0. The obtained contradiction ﬁnishes the proof of our assertion.

The Theorem is proved. 

Remark 13.10 C.Bu¸se wrote several papers

[

]

[

]

and

[

]

on evolution pe-

riodic processes that are in the spirit of the current section. In particularly, in

[

]

it is proved that a trivial solution of equation u

(t) = A(t)u(t) with p−

periodic coeﬃcients on a separable Hilbert space H is uniformly exponentially

stable if the mild solution u

µx

of a well-posed inhomogeneous Cauchy’s problem

(t) = A(t)u(t) + e

iµt

x(t ≥ 0), µ ∈ R, u(0) = 0 satisﬁes the following condition

sup

µ∈R

sup

t>0

µx

(t)| < +∞, ∀x ∈ H.

13.9.2 Some classes of linear uniformly exponentially stable

periodic diﬀerential equations

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

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

with open-compact topology and (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

linear diﬀerential equation

= A(t)u , (13.116)

where A ∈ C(R, Λ). Along with equation (13.116), we shall also consider its

H−class, that is, the family of equations

= B(t)v , (13.117)

where B ∈ H(A) :=

: s ∈ R}, A

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

closure in C(R, Λ). Let ϕ(t, u, B) be the solution of equation (13.117) 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), then the triple h[E], U, (Y, R, σ)i is the linear cocycle

on (Y, R, σ), where U(t, B) := ϕ(t, ·, B) for all t ∈ R and B ∈ Y .

According to Lemma 13.13

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

(ii) the mapping U : A → U(·, A) of C(R, [E]) to C(R, [E]) is continuous, where

U(·, A) is the Cauchy’s operator

[

]

of equation (13.116).

Theorem 13.38 Let A ∈ C(R, Λ) be τ-periodic (i.e. A(t + τ ) = A(t) for all

t ∈ R), then the following conditions are equivalent: