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

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Analytic dissipative systems 137

Remark 3.5 a. It is easily shown that Γ is a complete space if and only if X is

a complete space.

b. If (X, h, Y ) is a vector ﬁbering, then the section space Γ = Γ(Y, X) is naturally

endowed with a vector-space structure, and all operations are continuous with respect

to the topology deﬁned on Γ, i.e., in this case Γ is covered into a topological vector

space.

c. If Y is compact and (X, h, Y ) is a Banach ﬁbering, then the relation

||γ|| := max

y∈Y

|γ(y)|

deﬁnes a norm on Γ (here |·| : X → R

is a norm on (X, h, Y ), and consistent with

the metric) and the topology generated by this norm is consistent with the original

topology on Γ.

Let µ : S

× Γ → Γ denote the mapping deﬁned by the relation: µ(t, γ)(y) :=

γ(σ

−t

y) for all y ∈ Y . It is easily veriﬁed that µ(t, γ) ∈ Γ.

The following assertion holds.

Theorem 3.14 The triple (Γ, S

, µ) is a (semigroup) dynamical system on Γ,

i.e.,

(1) µ(0, γ) = γ for all γ ∈ Γ.

(2) µ(τ, µ(t, γ)) = µ(t + τ, γ) for all γ ∈ Γ and t, τ ∈ S

(3) µ : S

×Γ → Γ is continuous.

Proof. The ﬁrst two assertions can be established directly. The nontrivial part of

Theorem 3.14 is the continuity of µ. Let t

→ t

and γ

→ γ

({t

} (respectively

{γ

}) is a directedness in S

(respectively Γ)). We shall prove that µ(t

, γ

) →

µ(t

, γ

) in the space Γ, i.e.,

lim

max

y∈K

ρ(π

(σ

−t

y), π

(σ

−t

y)) = 0.

for each compact K ⊆ Y . Assume the contrary: There is ε

> 0 and a compact

⊆ Y such that

max

y∈K

ρ(π

(σ

−t

y), π

(σ

−t

y)) ≥ ε

Since K

is compact, there is {y

} ⊆ K

for which

ρ(π

(σ

−t

), π

(σ

−t

)) ≥ ε

Again since K

is compact, we assume that the sequence {y

} is convergent. Let

:= lim

, then σ

−t

→ σ

−t

and hence

lim

(σ

−t

) = γ

(σ

−t

). (3.31)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 3

138 Global Attractors of Non-autonomous Dissipative Dynamical Systems

From (3.31) it follows that

lim

(σ

−t

) = π

(σ

−t

). (3.32)

However

≤ ρ(π

(σ

−t

), π

(σ

−t

)) ≤ (3.33)

ρ(π

(σ

−t

), π

(σ

−t

))

+ρ(π

(σ

−t

), π

(σ

−t

)).

The second term on the right here tend to zero when y

→ y

. We shall prove that

the ﬁrst term also tends to zero y

→ y

and t

→ t

. At ﬁrst we shall establish

that lim

(σ

−t

) = γ

(σ

−t

). In fact

ρ(γ

(σ

−t

), γ

(σ

−t

)) ≤ ρ(γ

(σ

−t

), γ

(σ

−t

)) +

ρ(γ

(σ

−t

), γ

(σ

−t

)) ≤ max

y∈K

ρ(γ

(y), γ

(y)) + (3.34)

ρ(γ

(σ

−t

), γ

(σ

−t

)),

where K

= σ(K

, T

) and T

{−t

}. Letting n → +∞ in (3.34) and using (3.32)

and the fact that γ

→ γ

in topology Γ, we obtain the required result.

Hence γ

(σ

−t

) → γ

(σ

−t

) and t

→ t

, and thus

ρ(π

(σ

−t

, π

(σ

−t

)) → 0. (3.35)

Letting t

→ t

and y

→ y

in (3.33) and employing (3.35), we ﬁnd that ε

≤ 0,

in contradiction with the choice of ε

. The mapping µ : Γ × S

→ Γ is therefore

continuous, and this completes the proof of the theorem. 

Remark 3.6 a. If h(X, S, π), (Y, S, σ), hi is a group non-autonomous dynamical

system, then the group dynamical system (Γ, S, µ) is deﬁned on Γ.

b. A continuous section γ ∈ Γ is invariant if and only if γ ∈ Γ is a stationary

point of the system (Γ, S

, µ).

We consider a special case of the foregoing construction. Let hW, ϕ, (Y, S, σ)i

be a cocycle over (Y, S, σ) with ﬁber W and h(X, S

, π), (Y, S, σ), hi be a non-

autonomous dynamical system generated by this cocycle. Then h ◦ γ = Id

and

because h = pr

it follows: γ = (ψ, Id

), where γ ∈ Γ(Y, X) and ψ : Y → W . Hence

to each section γ there corresponds a mapping ψ : Y → W and conversely. There

being a one-to-one relation between Γ(Y, W × Y ) and C(Y, W ), where C(Y, W ) is

the space of continuous functions ψ : Y → W, with the uniform-convergence topol-

ogy on compacts of Y , we identify these two objects from now on. The dynamical

system (Γ, S

, µ) induces, in a natural fashion a dynamical system (C(Y, W ), S

, Q)

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Analytic dissipative systems 139

on C(Y, W ). Namely

(µ

γ)(y) = π

γ(σ

−t

y) = π

(ψ, Id

)(σ

−t

y) =

(ψ(σ

−t

y), σ

−t

y) = (U(t, σ

−t

y)ψ(σ

−t

y), y) = ((Q

ψ)(y), y).

Hence µ

(ψ, Id

) = (Q

ψ, Id

), and so the mapping Q : C(Y, W ) ×S

→ C(Y, W ),

deﬁned by the relation Q(ψ, t) = Q

ψ, with (Q

ψ)(y) = U(t, σ

−t

y)ψ(σ

−t

y) (y ∈ Y ),

has the following properties:

a. Q

= Id

C(Y,W )

;

b. Q

= Q

t+τ

( t, τ ∈ S

)

c. Q is continuous,

i.e., (C(Y, W ), S

, Q) is a semigroup dynamical system on C(Y, W ). By virtue of

the foregoing, we identify the systems (Γ, S

, µ) and (C(Y, W ), S

, Q) in the sequel.

3.6 Quasi-periodic solutions

In this section we investigate dissipative diﬀerential equation (3.1) with quasi-

periodic coeﬃcients, and prove that, if its right side is analytic, then there is always

at least one quasi-periodic solution with the same frequency basis as the right side.

Consider the system

(

˙x = F (x, ϕ),

˙ϕ = iωϕ,

(3.36)

where ω = (ω

, ω

, . . . , ω

) ∈ R

(ω

, ω

, . . . , ω

are linearly independent

with respect to the integers), i

= −1, ϕ = (ϕ

, ϕ

, . . . , ϕ

) ∈ C

, ωϕ =

col(w

, . . . , ω

), x ∈ C

and F ∈ C(C

× C

, C

). We denote by T

m-dimensional torus, i.e. T

:= {ϕ ∈ C

: |ϕ

| = 1, k =

1, m}. We are interested

in the system (3.36), when the parameter ϕ runs over the points of the torus T

and a small neighborhood

:= {ϕ ∈ C

: |ϕ

| < 1 + δ, k ∈

1, m},

where δ > 0 is suﬃciently small.

The system (3.36) is clearly equivalent to the equation

˙x = F (x, ϕ

iωϕ

), (3.37)

where ϕ

∈ C

, or to a subset (for example D

Deﬁnition 3.10 We will say that the system (3.36) (or equation (3.37)) is an-

alytic on T

if there is a positive number δ, for which F : C

× D

→ C

analytic.

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140 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Let (C

, R, σ) denote the dynamical system on C

determined by the second

equation of the system (3.36). Then

σ(ϕ, t) := col



iω

, ϕ

iω

, . . . , ϕ

iω



It is easily seen that, for arbitrary δ > 0 the sets D

and

, where the bar indicates

the closure of D

in C

, are invariant with respect to (C

, R, σ).

Deﬁnition 3.11 The system (3.36) (and the equation (3.37)) is said to be dis-

sipative on T

, if there exists δ > 0 (suﬃciently small) such that the dynamical

system (C

, R, σ), generated by (3.36) ( ϕ ∈

) is dissipative.

We sharpen this deﬁnition. Suppose that, for the system (3.36), conditions

for existence, uniqueness, and continuability to the right are satisﬁed. Then it is

known that (3.36) determines a semigroup system (C

, R

, π) on C

× D

;

then π((x, ϕ

), t) := (ψ(t, x, ϕ

), ϕ

iωt

), where ψ(·, x, ϕ

) is the solution of equation

(3.37) passing through the point x ∈ C

for t = 0. It is easily to verify that the triple

h(C

, R

, π), (D

, R, σ), hi (h := pr

: C

× D

→ D

) is a non-autonomous

dynamical system. Hence, if the system of diﬀerential equations (3.36) is dissipative

on T

, then the non-autonomous dynamical system h(C

, R

, π), (D

, R, σ), hi

generated by it ( for suﬃciently small δ) is dissipative, i.e. there is a positive number

r for which

lim sup

t→+∞

|ψ(t, x, ϕ)| < r

for x ∈ C

and ϕ ∈

, and r is independent of both x ∈ C

and ϕ ∈ D

Theorem 3.15 If the equation (3.37) is dissipative on T

and the right side of

(3.37) is analytic on T

, then the equation (3.37) has at least one quasi-periodic

solution with the same frequency basis ω

, ω

, . . . , ω

as the right side F .

Proof. If (3.37) is dissipative and F is analytic on T

, then there is δ > 0 (suf-

ﬁciently small) such that F : C

× D

→ C

is analytic and the non-autonomous

dynamical system h(C

, R

, π), (D

, R, σ), hi is dissipative. Let A(D

, C

)

denote the set of all analytic functions γ : D

→ C

endowed with the uniform-

convergence topology on compacts from D

. Clearly A(D

, C

) is a closed subset

of the space C(D

, C

) of all continuous functions γ : D

→ C

with the uniform-

convergence topology on compacts from D

. Since D

is an invariant subset of

(

, R, σ), the non-autonomous system h(C

×D

, R

, π), (D

, R, σ), hi is correctly

deﬁned and, by virtue of Remark 3.6, C(D

, C

) is the space of all continuous sec-

tions for this system. Theorem 3.14 implies that the triple (C(D

, C

), R

, Q),

where (Q

γ)(ϕ) = ψ(t, γ(σ

−t

ϕ), σ

−t

ϕ), is a dynamical system on C(D

, C

We shall prove that A(D

, C

) is an invariant subset of the dynamical system

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Analytic dissipative systems 141

(C(D

, C

), R

, Q). In fact, if γ ∈ A(D

, C

), then general properties

[

122

]

solutions of equations with analytic right sides imply that Q

γ ∈ A(D

, C

) for

all t ≥ 0. Hence a semigroup dynamical system (A(D

, C

), R

, Q) is deﬁned on

A(D

, C

). We note one property of this system following from the dissipativity

of equation (3.37) on T

. Let A

:= {γ ∈ A(D

, C

) : γ(D

) ⊆ B(0, r)}, where

B(0, r) is the open ball in C

of radius r centered at the origin. It follows from Vi-

tali’s theorem

[

167

]

that, for each r > 0, the sets A

are precompact in A(D

, C

It is clear that the sets A

are convex. Theorem 3.1 implies that there is b > 0 such

that, for each r > 0, there is l

≥ 0 for which

⊆ A

for all t ≥ l

. Let r > b (for example r = b + 1), and U = IntA

(the interior of

), then Q

U ⊆ A

⊂ U. According to Theorem 1

[

205

]

there exists a stationary

point γ ∈ A(D

, C

) of the dynamical system (A(D

, C

), R

, Q), i.e.

ψ(t, γ(σ

−t

ϕ), σ

−t

ϕ) = γ(ϕ) (ϕ ∈ D

, t ≥ 0). (3.38)

Hence ψ(t, γ(ϕ), ϕ) = γ(σ

ϕ) (ϕ ∈ D

, t ≥ 0), that is, ψ(t, γ(ϕ

), ϕ

) =

γ(σ

) = γ(ϕ

iωt

) is a quasi-periodic solution of (3.37) with the frequency basis

, ω

, . . . , ω

. This proves the theorem. 

Remark 3.7 a. We note that the proof of Theorem 3.15 is of a general nature,

and it can be directly applied to equations of more general form. For example,

instead of (3.36) we could consider the system

(

˙x = F (x, ϕ)

˙ϕ = G(ϕ)

(x ∈ C

, ϕ ∈ C

). (3.39)

We must assume that T

is a minimal quasi-periodic set for the second equation

of (3.39), and that there is a neighborhood D

of T

that is an invariant set of the

dynamical system generated by second equation of (3.39). The other conditions, in

Theorem 3.15, can clearly be modiﬁed for this case.

b. Theorem 3.15 (see also Theorem 3.4) established a conjecture of Bronstein

(see

[

32, p.278

]

). It should be added that the hypothesis was concerning equation

with almost periodic right sides, satisfying the uniform positive stability condition.

Hence the latter condition is superﬂuous for analytic systems.

c. In the author’s opinion, the importance of the condition that the right side F

be analytic is doubtful in Theorem 3.15. We are inclined to believe that Theorem

3.15 still applies when F is suﬃciently smooth. In this connection we note two

examples in

[

152, 332

]

. One is an example of a dissipative equation of the form

(3.1) with quasi-periodic coeﬃcients but with no almost-periodic solutions. Both in

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 3

142 Global Attractors of Non-autonomous Dissipative Dynamical Systems

[

152, 332

]

the construction is based on Denjoy’s results concerning ﬂows on a two-

dimensional torus, and analysis of the examples shows that, if the right hand side

of equation (3.1) is suﬃciently smooth, then this situation does not prevail.

Theorem 3.16 Suppose that the conditions of Theorem 3.15 are fulﬁlled and

F (x, ϕ) = F (x, ϕ) (where by bar we note the complex conjugation). Then there

exists γ ∈ A(D

, C

) such that

(1)

γ(x) = γ(x) for all x ∈ D

;

(2) γ(ϕ

iωt

) = ψ(t, γ(ϕ

), ϕ

) for all ϕ

∈ D

and t ∈ R.

Proof. The proof of this statement uses the same argument as Theorem 3.15. 

3.7 The analogy of Cameron-Johnson’s theorem

The well-known Cameron-Johnson’s theorem

[

]

[

204

]

asserts that the equation

= A(t)x (3.40)

with a recurrent (almost periodic by Bhor) matrix may be reduced by the Lyapunov-

Perron transformation to the equation y

= B(t)y with a skew-symmetric matrix

B(t), if all solutions of the equation (3.40) and all its limit equations are bounded

on the whole real axis. The generalization of this result on the linear C-analytic

equations in the Hilbert space is our main scope in this section.

Let R = (−∞, +∞), C

is an m-dimensional complex Euclidean space, G be

a domain from C

, H be a real or complex Hilbert space with the scalar product

h·, ·i and with the norm | · |

= h·, ·i. Denote by H

a space H equipped with

weak topology, and by [H] ( respectively [H

] ) the family of all linear continuous

operators acting into H (respectively H

) and equipped with operational norm

(with weak topology). Assume that H(G, [H])) (resp. H (G, [H

]), H(G, C

)) is the

family of all holomorphic functions h : G → [H] (resp. h : G → [H

], h : G → C

)

equipped with compact-open topology.

Consider the system

(

= A(z)x

= Φ(z)

(z ∈ G), (3.41)

where Φ ∈ H(G, C

) and A ∈ H(G, [H]). We suppose that the second equation of

the system (3.41) generates the dynamical system (G, R, σ) on G. Denote by U (t, z)

the Cauchy’s operator of the equation

= A(zt)x (z ∈ G), (3.42)

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Analytic dissipative systems 143

where zt = σ(t, z). From the general property of solutions of diﬀerential equations

(see, for example,

[

122

]

and

[

132

]

[

137

]

), it follows that the family of operators

{U(t, z)|t ∈ R, z ∈ G} satisﬁes the following conditions:

1. U(0, z) = I (∀z ∈ G), where I is a unit operator on H ;

2. U(t + τ, z) = U (t, zτ)U(τ, z) (∀t, τ ∈ R and z ∈ G );

3. the mapping U : R × G → [H] (U : (t, z) 7→ U(t, z)) is continuous and for

every t ∈ R the mapping U(t, ·) : G → [H] is holomorphic.

The following assertion holds.

Theorem 3.17 Suppose that there exists a positive constant C such that

kU(t, z)k ≤ C (3.43)

for all t ∈ R and z ∈ G. Then there exists P ∈ H(G, [H]) such that

a. P - is bi-holomorphic, i.e. the operator P (z) is invertible for all z ∈ G and the

mapping P

−1

: G → [H] (P

−1

: z 7→ P

−1

(z)) is holomorphic;

b. P

∗

(z) = P (z) for all z ∈ G (P

∗

(z) is an adjoint for P (z) operator);

c. the operator P(z) is positively deﬁned for all z ∈ G;

d. C

−1

|x| ≤ |P (z)x| ≤ C|x| for all z ∈ G and x ∈ H;

e. the change of variables x = P (zt)y alters the equation (3.42) into

= B(zt)y (3.44)

with a skew-Hermitian operator B ∈ H(G, [H]), i.e. B

∗

(z) = −B(z) for all z ∈ G.

Proof. Let S

: H(G, [H]) → H(G, [H]) ( resp. H(G, [H

]) → H(G, [H

])) be a

mapping deﬁned by the equality (S

f)(z) = U

∗

(t, z)f(zt)U (t, z) for all z ∈ G and

f ∈ H(G, [H]) (resp. f ∈ H(G, [H

])). It is easy to verify that the family of

mappings {S

|t ∈ R} satisﬁes the following conditions:

4. the operator S

maps H(G, [H]) (or f ∈ H(G, [H

])) into itself for each t ∈ R;

5. S

= I, where I is a unit mapping on H(G, [H]) ( resp. H(G, [H

]));

6. S

= S

t+τ

for all t, τ ∈ R;

7. the mapping S

( t ∈ R) is linear and continuous on H(G, [H]) ( resp.

H(G, [H

])).

From 4.-7. follows that the family of mappings {S

|t ∈ R} is a commutative group

of linear continuous operators on H(G, [H]) (or H(G, [H

])).

Denote by K := {A ∈ [H]| kAk ≤ C

} and note that the set K is weakly

closed. According to Tihonoﬀ theorem, the set K is weakly compact because every

bounded and closed set in H is weakly compact. From Cauchy’s integral formula

(see, for example,

[

288, p.339

]

) and the Arzela-Ascoli theorem

[

288

]

follows that the

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144 Global Attractors of Non-autonomous Dissipative Dynamical Systems

set Γ = {γ ∈ H(G, [H]| kγ(z)k ≤ C

} is compact in the space H(G, [H

]). We

suppose that V :=

conv{S

Q | t ∈ R}, where Q(z) = I(∀z ∈ G), I is a unit operator

in H and

conv is a weak closure of the convex hull of {S

Q | t ∈ R}. Note, that the

set V is invariant, i.e. S

V ⊆ V for all t ∈ R because the family of operators {S

}

is linear and continuous in H(G, [H

]). It is easy to see that

Q(z)x, xi = hU

∗

(t, z)U(t, z)x, xi = |U(t, z)x|

(3.45)

for all z ∈ G, t ∈ R and x ∈ H. From (3.43) and (3.45) follows that

−2

|x|

≤ hS

Q(z)x, xi ≤ C

|x|

and, consequently,

−2

|x|

≤ hR(z)x, xi ≤ C

|x|

(3.46)

for all R ∈ V, z ∈ G and x ∈ H. In addition, the equality R

∗

(z) = R(z) holds

for all z ∈ G and R ∈ V. According to the equality (3.46) we have kR(z)k ≤ C

for all z ∈ G and, consequently, V ⊆ Γ. According to the theorem of Markov-

Kakutany

[

]

, the group {S

| t ∈ R} admits in V at least one ﬁx point, i.e. there

exists R ∈ V such that U

∗

(t, z)R(zt)U(t, z) = R(z) for all z ∈ G and t ∈ R. We

note that the operator R(z) is self-adjoint and, according to the equality (3.46),

is positively deﬁned. From Theorem 12.33

[

274

]

(see also

[

182, p.65

]

) follows that

there exists a unique reversible self-adjoint and positively deﬁned operator M(z)

such that M

(z) = R(z) for all z ∈ G and, according to the inequality (3.46), we

have

−1

|x| ≤ |M(z)x| ≤ C|x|

for all z ∈ G and x ∈ H. Finally, according to the equality

(z) = −

2πi

(R(z) − λI)

−1

dλ

(α = ±

), where L is a simple contour, enclosing the spectrum of the operator

R(z), we have that the mapping M : G → [H] is biholomorhic.

Note, that M(zt)U (t, z)M

−1

(z) = M

−1

(zt)U

∗

−1

(t, z)M(z) since

∗

(t, z)M

(zt)U(t, z) = M

(z)

for all t ∈ R and z ∈ G. We set V(t, z) = M (zt)U(t, z)M

−1

(z) and note that the

family of operators {V(t, z) | t ∈ R, z ∈ G} satisﬁes the conditions 1.-3. and in

addition:

8. V

∗

(t, z) = V

−1

(t, z) for all t ∈ R and z ∈ G;

9. U(t, z)P (z) = P (zt)V(t, z) for all t ∈ R and z ∈ G, where P (z) = M

−1

(z).

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Analytic dissipative systems 145

Let

B(z) :=

V(t, z)|

t=0

= [

M(z) + M(z)A(z)]P (z),

where

M(z) :=

M(zt)|

t=0

dM(z)

Φ(z). Then B ∈ H(G, [H]) and

dV(t, z)

= B(zt)V(t, z) (3.47)

for all t ∈ R and z ∈ G. Thus, V(t, z) is a Cauchy operator of (3.44). On the other

hand, by the condition 8., we have

(

∗

−1

(t, z) = −B

∗

(zt)V

∗

−1

(t, z)

∗

−1

(0, z) = I

(3.48)

for all t ∈ R and z ∈ G. From the equalities (3.47), (3.48) and condition 8. follows

that B

∗

(z) = −B(z) for all z ∈ G. Finally, to ﬁnish the proof of the theorem it is

suﬃcient to remark that the change of variables x = P (zt)y reduces equation (3.42)

to the equation (3.44). The theorem is proved. 

Remark 3.8 a. In the case, when the space H is ﬁnite-dimensional, the statement

close to Theorem 3.17 was obtained by V. V. Glavan

[

165

]

b. If the point z ∈ G is stationary ( ω-periodic, quasi-periodic, almost periodic

etc.), then according to

[

300

]

, the operator-function P (zt) will also be stationary

(ω-periodic, quasi-periodic, almost periodic etc.).

c. Theorem 3.17 holds for the system of diﬀerence equations

(

x(k + 1) = A(zk)x(k)

z(k + 1) = Φ(zk)

(k ∈ Z),

where A ∈ H(G, [H]) and Φ ∈ H(G, C

) and also for the system of diﬀerential and

diﬀerence equations with multi-dimensional time.

3.8 Almost periodic solutions of the weak nonlinear dissipative

systems

1. In this item we will study bounded and uniformly compatible solutions of the

linear and nonlinear system of diﬀerential equations.

Denote by C

(R, E

) the set of all bounded on R functions ϕ : R → E

with

the sup-norm. Consider a linear homogeneous equation

˙x = A(t)x, (3.49)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 3

146 Global Attractors of Non-autonomous Dissipative Dynamical Systems

where A ∈ C

(R, [E

]). Along with the equation (3.49) we consider the non-

homogeneous equation

˙x = A(t)x + f(t) . (3.50)

Deﬁnition 3.12 Recall, that the equation (3.49) is weakly regular if for every

function f ∈ C

(R, E

) the equation (3.50) has at least one solution ϕ ∈ C

(R, E

Let ϕ ∈ C(R, E) (respectively f ∈ C(×U, E)), denote by M

(respectively M

)

the family of all sequences {t

}such that the functional sequence {ϕ

} (respectively

}) converges in the space C(R, E) (respectively C(×U, E)), where ϕ

(t) :=

ϕ(t + τ) for all t ∈ R (respectively f

(t, x) := f(t + τ, x) for all (t, x) ∈ R ×U).

Deﬁnition 3.13 Recall

[

300, 302

]

that the solution ϕ ∈ C(R, E

) of equation

= f(t, x) (f ∈ C(R × E

, E

)) (3.51)

is called uniformly compatible if M

∈ M

, f

is a restriction on R × Q of the

function f and Q :=

ϕ(R).

It is known

[

300, 302

]

that, if the function f is τ -periodic (quasi-periodic, almost

periodic, recurrent) w.r.t. variable t ∈ R uniformly w.r.t. x on the compacts from

and ϕ is a uniformly compatible solution of equation (3.51), then the function

ϕ is is τ-periodic (quasi-periodic, almost periodic, recurrent) too.

Theorem 3.18 Let A ∈ C

(R, [E

]). If the set H(A) is compact, then the follow-

ing conditions are equivalent:

1. the equation (3.49) is weakly regular;

2. the equation (3.50) has at least one bounded on R uniformly compatible solution

for every function f ∈ C

(R, E

Proof. It is easy to see that from 2. follows 1.. Therefore, to prove Theorem 3.18

it is suﬃcient to show, that the condition 1. implies 2. We will consider two cases:

a. Let A ∈ C(R, [E

]) be a Poisson stable at least in one direction matrix, i.e.

there a sequence |t

| → +∞ such that A

→ A in C(R, [E

]). In this case from

Theorem 3.18 and Lemma 4.2.1

[

]

follows that the equation (3.49) satisﬁes the

condition of exponential dichotomy on R and, according to Theorem 22.23

[

]

, the

equation (3.50) has a unique bounded on R and uniformly compatible solution.

b. The matrix-function A ∈ C(R, [E

]) is not stable in the sense of Poisson

neither in positive nor in negative direction. Let ϕ be a bounded on R solution

of the equation (3.50) and {t

} ∈ M

(A,f)

. There is (B, g) ∈ H(A, f ) such that

→ B and f

→ g. Note, that in this case for the sequence {t

} it is possible

only the following two cases: