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

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Analytic dissipative systems 147

(1) the sequence {t

} converges to t

∈ R and, consequently, {t

} ∈ M

;

(2) |t

| → +∞.

Then the matrix-function B := lim

k→+∞

is ω-limit or α-limit for A and in this case

we have {t

} ∈ M

. In fact, the sequence {ϕ

} is compact in C(R, E

) because

the function ϕ is bounded. In addition, every limit function for the sequence {ϕ

}

is a bounded on the R solution of the equation

˙x = B(t)x + g(t) . (3.52)

By Lemma 4.2.1 from

[

]

the equation

˙y = B(t)y

satisﬁes the condition of the exponential dichotomy on R and, consequently, the

equation (3.52) cannot have more than one bounded on R solution. From this fact

follows that the sequence {ϕ

} converges in the space C(R, E

), i.e. {t

} ∈ M

Finally, we will shows that, in case the sequence {t

} is unbounded, it does not

contain bounded sub-sequences. If we suppose that it is not true, then there is a

subsequence {t

} ⊆ {t

} such that t

→ t

∈ R and in this case we have

B := lim

k→+∞

= lim

k→+∞

= A

∈ R).

From this equality follows that the matrix-function A is stable in the sense of Poisson

at least in one direction, but this contradicts to our condition. The theorem is

proved. 

Denote by E

:= {u ∈ E

: sup{kϕ(t, u, A)k : t ∈ R} < +∞} and by P

projection mapping the space E

onto E

. The following statement holds.

Lemma 3.2 Let A ∈ C(R, [E

]). If the matrix-function A is bounded on R and

the equation (3.49) is weakly regular, then for every f ∈ C

(R, E

) there exists

a unique solution ϕ ∈ C

(R, E

) of the equation (3.50) satisfying the following

conditions:

(1) P

ϕ(0) = 0;

(2) there exists a positive constant K (a constant of the weak regularity of (3.49)),

which does not depend on the function f, such that kϕk 6 Kkfk.

Proof. We prove the formulated statement using the same arguments as in Lemma

6.3

[

186, p.515

]

. 

Corollary 3.9 Let A ∈ C(R, [E

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

conditions are equivalent:

(1) the equation (3.49) is weakly regular;

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148 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(2) for every function f ∈ C

(R, E

) there is a unique bounded on R uniformly

compatible solution ϕ of the equation (3.50) satisfying the following conditions:

a. P

ϕ(0) = 0 and b. kϕk ≤ Kkfk,

where K is a constant of the weak regularity of the equation (3.49).

Theorem 3.19 Let A ∈ C(R, [E

]) be a bounded on R matrix-function, the equa-

tion (3.49) be weakly regular and F ∈ C(R×E

, E

) be a function with the following

property: |F (t, u)| ≤ c(|u|) (∀t ∈ R and u ∈ E

), where c : R

→ R

is a non-

decreasing function. If {r > 0 : Kc(r) ≤ r} 6= ∅, where K is a constant of the

weak regularity of the equation (3.49), then the equation

˙u = A(t)u + F (t, u) (3.53)

has at least one bounded on R solution.

Proof. For every function f ∈ C

(R, E

) the equation (3.50) has a unique solution

ψ ∈ C

(R, E

) such that

ψ(0) = 0 and kψk ≤ Kkfk , (3.54)

where K is a constant of the weak regularity of the equation (3.49). Let ϕ ∈

(R, E

). Consider a diﬀerential equation

˙u = A(t)u + F (t, ϕ(t)) . (3.55)

Since the function f (t) := F (t, ϕ(t)) is bounded on the R (|f(t)| = |F (t, ϕ(t))| ≤

c(|ϕ(t)|) ≤ c(kϕk)), then the equation (3.55) has a unique bounded on R solution

satisfying the condition (3.54) and, consequently,

kψ

k ≤ Kkf k = K sup

t∈R

|F (t, ϕ(t))| ≤ Kc(kϕk) . (3.56)

Let us deﬁne an operator Φ : C

(R, E

) → C

(R, E

) in the following way:

(Φϕ)(t) = ψ

(t) (ϕ ∈ C

(R, E

), t ∈ R ). We will show that if the number r

> 0

satisﬁes the inequality Kc(r

) ≤ r

, then the ball B[0, r

] := {ϕ ∈ C

(R, E

) :

kϕk ≤ r

} is invariant with respect to the mapping Φ. In fact, if ϕ ∈ B[0, r

], then

kΦϕk ≤ Kc(kϕk) ≤ Kc(r

) ≤ r

. Consider now the space C

(R, E

) in the quality

of the subset embedded in the space C(R, E

). First of all, we note that every ball

B[0, r] ⊂ C

(R, E

) is a convex, bounded and closed subset of the space C(R, E

Note, that the mapping Φ : B[0, r

] → B[0, r] is continuous in the topology of

the space C(R, E

). In fact, let {ϕ

} ⊆ B[0, r] and ϕ

→ ϕ in C(R, E

). Consider

the sequence (Φϕ

)(t) := ψ

(t ∈ R ). Note, that f

(t) := F (t, ϕ

(t)) → f(t) :=

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Analytic dissipative systems 149

F (t, ϕ(t)) in the topology of the space C(R, E

). If we suppose the contrary, then

there are ε

> 0 and L

> 0 such that

max

|t|≤L

|F (t, ϕ

(t)) −F (t, ϕ(t))| ≥ ε

Hence, there exists a sequence {t

} ⊆ [−L

, L

] such that

|F (t

, ϕ

)) − F (t

, ϕ(t

))| ≥ ε

. (3.57)

Since the sequence {t

} is bounded, then we may suppose that it is convergent.

Note, that ϕ

) → ϕ(t

), where t

:= lim

k→+∞

. In fact,

|ϕ

) − ϕ(t

)| ≤ |ϕ

) − ϕ(t

)|+ |ϕ(t

) − ϕ(t

≤ max

|t|≤L

|ϕ

(t) − ϕ(t)| + |ϕ(t

) − ϕ(t

)| .

Passing to limit in this inequality, as k → +∞, we obtain the necessary state-

ment. From the inequality (3.57) (taking into account that the sequence {ϕ

)}

converges to ϕ(t

) as k → +∞) we obtain ε

≤ 0 and this contradicts to the

choice of the number ε

. Thus, f

→ f in the space C(R, E

) and, in addition,

k = sup

t∈R

|F (t, ϕ

(t))| ≤ c(kϕ

k) ≤ c(r

), i.e. |f

(t)| ≤ c(r

) (t ∈ R ) and, conse-

quently, |f(t)| ≤ c(r

) (t ∈ R ). On the other hand, the function Φϕ

is a bounded

on R solution of the equation

˙u = A(t)u + f

(t) .

In addition, the functions {Φϕ

} and their derivatives are uniformly bounded on R

and, consequently, the sequence {Φϕ

} is relatively compact in the space C(R, E

Since f

→ f in C(R, E

), then every limit function of the sequence {Φϕ

} is

a bounded on R solution of the equation (3.50), satisfying the conditions (3.54).

But by Corollary 3.9 the equation (3.50) has at most one bounded on R solution

satisfying the conditions (3.54). From this fact follows that the sequence {Φϕ

} is

convergent in the space C(R, E

) and the continuity of the mapping Φ : B[0, r

] →

B[0, r

] is established.

Now we will show that the mapping Φ : B[0, r

] → B[0, r

] is completely con-

tinuous in the topology of the space C(R, E

). For this aim we note that

(Φϕ)

(t) = A(t)(Φϕ)(t) + F (t, ϕ(t))

and, consequently,

|(Φϕ)

(t)| ≤ a|(Φϕ)(t)| + |F (t, ϕ(t))| ≤ ar

+ c(r

) (t ∈ R) ,

where a = sup{kA(t)k : t ∈ R}. From this follows that the set Φ(B[0, r

]) is

relatively compact in the topology of the space C(R, E

). According to the theorem

of Tihonoﬀ-Shauder, the mapping Φ has at least one ﬁxed point ϕ ∈ B[0, r

]. It is

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

150 Global Attractors of Non-autonomous Dissipative Dynamical Systems

easy to see that the function ϕ ∈ B[0, r

] is a bounded on R solution of the equation

(3.53). The theorem is proved. 

Remark 3.9 The statement close to Theorem 3.19 was proved in the work

[

257

]

but under more strongly assumptions on the nonlinear perturbation F .

Theorem 3.20 Let A ∈ C(R, [E

]), F ∈ C(R ×E

, E

) and the following con-

ditions be hold:

a. a = sup{kA(t)k : t ∈ R} < +∞;

b. the equation (3.49) is weakly regular;

c. |F (t, u)| ≤ c(|u|) ( t ∈ R, u ∈ E

) and {r > 0 : Kc(r) ≤ r} 6= ∅, where

c : R

→ R

is a non-decreasing function and K is a constant of the weak

regularity of the equation (3.49);

d. the restriction F

of the function F on R × B[0, r

] satisﬁes the condition of

Lipschitz and Lip(F

) < K

−1

, where r

is some positive number satisfying the

inequality Kc(r

) ≤ r

Then the equation (3.53) has at least one bounded on R and uniformly compatible

solution.

Proof. Denote by B

(M) := {ϕ ∈ C

(R, E

) : |ϕ(t)| ≤ r

(t ∈ R )} and

M ⊆ M

, where M := M

(A,F )

. B

(M) is a closed subset of C

(R, E

)

[

300,

p.60

]

. The operator Φ : C

(R, E

) → C

(R, E

), deﬁned as well as in the proof

of Theorem 3.19, maps B

(M) into itself. In fact, if ϕ ∈ B

(M), then the func-

tion f (t) := F (t, ϕ(t)) (t ∈ R) belongs also to B

(M). According to Corollary

3.9, Φϕ is a unique bounded on R uniformly compatible solution of the equation

(3.50), satisfying the conditions (3.54). In addition, as well as in Theorem 3.14,

Φ(B[0, r

]) ⊆ B[0, r

] and, consequently, Φ(B

(M)) ⊆ B

(M). Let ϕ

and ϕ

the function from B

(M). Note, that

(Φϕ

−Φϕ

)

(t) = A(t)[(Φϕ

−Φϕ

)(t)] + F (t, ϕ

(t)) −F (t, ϕ

(t))

and, therefore,

kΦϕ

−Φϕ

k ≤ K sup

t∈R

|F (t, ϕ

(t)) −F (t, ϕ

(t))| ≤ KLip(F

)kϕ

−ϕ

k .

From the latter inequality follows that the mapping Φ : B

(M) → B

(M) is a

contraction, therefore it has a unique ﬁxed point, which is a uniformly compatible

solution of the equation (3.53). The theorem is proved. 

Corollary 3.10 Under the conditions of Theorem 3.20 if the functions A ∈

C(R, [E

]) and F

= F |

R×B[0,r

]

are mutually τ-periodic (almost periodic, recur-

rent, stable in the sense of Poisson, asymptotically τ-periodic, asymptotically almost

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Analytic dissipative systems 151

periodic, asymptotically recurrent). Then the equation (3.53) admits at least one

τ-periodic (almost periodic, recurrent, stable in the sense of Poisson, asymptotically

τ-periodic, asymptotically almost periodic, asymptotically recurrent) solution.

Remark 3.10 The problem of existence of uniformly compatible solutions of weak

nonlinear equations was studied before in the works

[

]

[

300

]

. The principal dif-

ference of Theorem 3.20 from the corresponding results

[

]

[

300

]

consists of the

following: under the conditions of Theorem 3.20 and Corollary 3.10 the equation

(3.49) does not obligatory satisfy the condition of the exponential dichotomy on R.

In addition, the nonlinear term is not obligatory small with respect to the variable

u ∈ E

2. In this item we will study uniformly compatible solutions of weak nonlinear

dissipative systems.

Theorem 3.21 Let A ∈ C(R, [E

]), F ∈ C(R ×E

, E

) and the following condi-

tions be fulﬁlled:

(1) a = sup{kA(t)k : t ∈ R} < +∞;

(2) there are positive numbers N and ν such that

kU(t, A)U

−1

(τ, A)k ≤ Ne

−ν(t−τ)

(t ≥ τ, t, τ ∈ R);

(3) |F (t, u)| ≤ M + ε|u| ( u ∈ E

, t ∈ R ) and 0 ≤ ε ≤ ε

< ν

(Na)

−1

Then the equation (3.53) is dissipative, i.e. there is a number R

> 0 such that

lim sup

t→+∞

|ϕ(t, ν, B, G)| < R

( ν ∈ E

, (B, G) ∈ H(A, F ) ) ,

where ϕ(·, ν, B, G) is a solution of the equation

˙ν = B(t)ν + G(t, ν) , (3.58)

satisfying the initial condition ϕ(0, ν, B, G) = ν.

Proof. For all B ∈ H(A) we will deﬁne on E

a norm |· |

by the equality

|u|

+∞

|U(t, B)u|dt .

As well as in the proof of Theorem 2.39, it is possible to check that

|u| ≤ |u|

≤

|u| ( u ∈ E

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152 Global Attractors of Non-autonomous Dissipative Dynamical Systems

We put

u(t) := |ϕ(t, u, B, G)|

+∞

|U(s, B

)ϕ(t, u, B, G)|ds .

Since

ϕ(t, u, B, G) = U(t, B)



u +

−1

(τ, B)G(τ, ϕ(τ, u, B, G))dτ



then

u(t) =

+∞

|U(s, B

)U(t, B)



u +

−1

(τ, B)G(τ, ϕ(τ, u, B, G))dτ



|ds

+∞

|U(s + t, B)(u +

−1

(τ, B)G(τ, ϕ(τ, u, B, G))dτ)|ds

≤

+∞

|U(s + t, B)u|ds +

+∞

U(s + t, B)U

−1

(τ, B) ×

G(τ, ϕ(τ, u, B, G))dτ )|ds ≤

−νt

|u|+

+∞

−ν(s+t−τ)

(M + ε|ϕ(τ, u, B, G)|)dτds

−νt

|u|+

−νt



τt

−1) + ε

|ϕ(τ, u, B, G)|e

ντ

dτ



≤

−νt

|u|+

(1 − e

−νt

) +

Nε

−νt

a|ϕ(τ, u, B, G)|

ντ

dτ

−νt

|u|+

(1 − e

−νt

) +

aNε

−νt

u(τ)e

ντ

dτ. (3.59)

Let ϕ(t) := u(t)e

νt

. From the inequality (3.59) follows that

ϕ(t) ≤

|u|+

νt

−1) +

aεN

ϕ(τ)dτ

and by Theorem 9.3 from

[

132

]

ϕ(t) ≤ ψ(t) (t ∈ R), where ψ is a solution of the

integral equation

y(t) =

|u|+

νt

−1) +

aεN

y(τ)dτ .

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Analytic dissipative systems 153

Solving the latter equation, we ﬁnd that

ψ(t) =



|u|+

−aεN



aεN

−aεN

νt

and, consequently,

u(t)e

νt

≤



|u|+

−aεN



aεN

−aεN

νt

. (3.60)

From the inequalities (3.59) and (3.60) we obtain

|ϕ(t, u, B, G)| ≤ a|ϕ(t, u, B, G)|

≤ a



|u|+

−aεN



−

−aεN

aNM

−aεN

Therefore,

lim

t→+∞

sup |ϕ(t, u, B, G)| ≤

aMN

−aεN

(u ∈ E

, (B, G) ∈ H(A, F ) and ν −

aεN

> 0). The theorem is proved. 

Remark 3.11 Simple examples show that under the conditions of Theorem 3.22

dissipativity does not reduce to convergence. This statement can be illustrated by

the following example ˙x = −x + 2x(1 + x

)

−1

Theorem 3.22 Let A ∈ C(R, [E

]), F ∈ C(R ×E

, E

) and the following con-

ditions be fulﬁlled:

(1) a := sup{ kA(t)k : t ∈ R} < +∞

(2) there exist positive numbers N and ν such that

kU(t, A)U

−1

(τ, A)k ≤ N e

−ν(t−ν)

(t ≥ τ, t, τ ∈ R) .

(3) |F (t, u)| ≤ M + ε|u| (u ∈ E

) and 0 ≤ ε < ε

:= min(

), where M is

some positive number.

(4) the restriction F

of the function F on R×B[0, r

], where r

is a positive number

appearing in Theorem 3.20 (for example, r

ν−ε

), satisﬁes the condition

of Lipschitz with the Lipschitz constant Lip(F

) < Nν

−1

Then the equation (3.52) is dissipative and in the ball B[0, r

] there exists a

unique bounded on R uniformly compatible solution of this equation.

Proof. The formulated assertion immediately follows from Theorems 3.20 and 3.21



Remark 3.12 The theorems 3.20 – 3.22 are true also for the diﬀerential equations

(3.53) in arbitrary Banach spaces.

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154 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Corollary 3.11 If under the conditions of Theorem 3.22 the functions A and F

are almost periodic, then the equation (3.53) is dissipative and its Levinson center

contains at least one almost periodic solution.

Remark 3.13 Under the conditions of Theorem 3.22 and Corollary 3.11, dissi-

pativity does not reduce to convergence.

We will give an example which conﬁrms this statement.

Example 3.7 Consider the scalar equation

˙x = −kx + αF (x) , (3.61)

where

F (x) :=

(

for |x| ≤ 10

50 + 10[1 − exp(10 −|x|)] |x| > 10

for 0 < k ≤ 5α. In this connection we can take r

. It is easy to check that for

the equation (3.61) all the conditions of Theorem 3.22 are fulﬁlled for the chosen

α, k, and F . In addition, its Levinson center contains at least two ﬁxed points

(in reality there are 3) and, consequently, the equation (3.61) is not convergent.

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 4

Chapter 4

The structure of the Levinson center of

system with the condition of the

hyperbolicity

4.1 The chain recurrent motions

Let Σ ⊆ X be a compact invariant set, Σ, ε > 0 and t > 0.

Deﬁnition 4.1 The collection {x = x

, x

, . . . , x

= y; t

, t

, . . . , t

} of the

points x

∈ Σ and the numbers t

∈ T such that t

≥ t and ρ(x

, x

i+1

) < ε (i =

0, 1, . . . , k − 1) is called a (ε, t, π)-chain joining the points x and y. We denote by

P (Σ) the set {(x, y) : x, y ∈ Σ, ∀ ε > 0 ∀ t > 0 ∃ (ε, t, π)-chain joining x and y}.

The relation P (Σ) is closed, invariant and transitive

[

]

. The point x ∈ Σ is

called chain recurrent, if (x, x) ∈ P (Σ). Let R(Σ) = {x ∈ Σ : (x, x) ∈ P (Σ)}. On

R(Σ) we will introduce a relation ∼ as follows: x ∼ y if and only if (x, y) ∈ P (Σ)

and (y, x) ∈ P (Σ). It is easy to check that the introduced relation ∼ on R(Σ) is a

relation of equivalence and, consequently, it is easy to decompose it on the classes

of equivalence {R

: λ ∈ L} i.e. R(Σ) = t{R

∈ L}. By Proposal 2.6 from

[

]

the deﬁned above components of the decomposition of the set R(Σ) are closed and

invariant.

Lemma 4.1 If the compact invariant set Σ from X contains only a ﬁnite numbers

of minimal set, then the relation ∼ decomposes the set R(Σ) on the ﬁnite numbers

of diﬀerent classes of equivalence.

Proof. Let R(Σ) = t{R

: λ ∈ L}. Since R

⊆ Σ are closed and invariant and Σ is

compact, then by the theorem of Birkhoﬀ the set R

contains at least one minimal

set. Consequently, the number of diﬀerent classes of equivalence is not more that

the number of minimal sets. The lemma is proved. 

We will indicate below the condition when the number of classes of equivalence

: λ ∈ L} is ﬁnite in case, if the set Σ contains inﬁnite number of diﬀerent

minimal sets.

155

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156 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Deﬁnition 4.2 By analogy with the work

[

]

in the collection {R

, R

, . . . , R

}

we will introduce a relation of partial order as follows: R

< R

, if there exist

, i

, . . . , i

such that i

= i, i

= j and W

)

p+1

) 6= ∅ for all p =

1, 2, . . . , r −1.

Deﬁnition 4.3 The ordered collection of r (r ≥ 2) diﬀerent indexes {i

, i

, . . . , i

}

satisfying the condition R

< R

< ··· < R

< R

is called an r-

cycle in the collection {R

, R

. . . , R

}. The 1-cycle is called such index i that

)

) \ R

6= ∅.

Note, that the introduced above notion of partial order is a slight modiﬁcation

of the corresponding notion from

[

33, p.61

]

Deﬁnition 4.4 Following the works

[

]

[

261

]

[

271

]

, the collection of points

, x

, . . . , x

} ⊆ X (or K := H(x

) ∪ H(x

) ∪ ··· ∪ H(x

)) is said to be a

generalized homoclinic contour, if ω

∩ α

i+1

6= ∅ for all i = 1, 2, . . . , n, where

n+1

= x

Lemma 4.2 Let Σ ⊆ X be a compact invariant set and {x

, x

, . . . , x

} be a

generalized homoclinic contour. Then (x

, x

) ∈ P (Σ) for all i, j = 1, 2, . . . , n.

Proof. Let i, j ∈ {1, 2, . . . , n}. Suppose, for example, that i ≤ j and 0 ≤ p =

j − i ≤ n. We will show that (x

, x

) ∈ P (Σ). Let ε > 0 and t > 0. Since

∩ α

k+1

6= ∅ for all k = i + 1, . . . , j − 1, then for the numbers ε > 0, t > 0

and the point x

(k = i, i + 1, . . . , j − 2) there exist t

> t, t

< −t and p

∈

∩ α

such that ρ(x

, p

) <

and ρ(x

, p

k−1

) <

. Let

:= x

, x

i+1

, . . . ,

p−1

:= x

, x

:= x

; t

:= t

, t

:= t

i+1

− t

i+1

, . . . , t

p−1

:= −t

. It

is clear that {

, x

, . . . , x

; t

, t

, . . . , t

p−1

} is a (ε, t, π)-chain joining the points x

and x

. The lemma is proved. 

In Lemmas 4.3-4.5 we will suppose that R(Σ) consists of ﬁnite number of dif-

ferent classes of equivalence R

, R

. . . , R

, i.e. R(Σ) = R

t R

t ··· t R

. Let

us establish some properties of the sets R

(i = 1, 2, . . . , k).

Lemma 4.3 In the collection {R

, R

. . . , R

} there is no r-cycles (r ≥ 1).

Proof. Assuming the contrary, we obtain that there exist r ≥ 1 and i

, i

, . . . , i

such that R

< R

< ··· < R

< R

. As well as in Lemma 4.2, it is possible

to prove that the set M := R

∪··· ∪ R

belongs to one class of equivalence and,

consequently, r = 1. We will show that in the collection {R

, R

. . . , R

} 1-cycles

are absent too. In fact, if we suppose that there exist 1 ≤ i ≤ r and a point x from

) ∩W

)) \R

, then H(x

) ∪R

⊆ (Σ). Moreover, H(x

) ∪R

belongs to

one class of equivalence and, consequently, H(x

)∪R

⊆ R

. The latter contradicts

to the choice of the point x

. The lemma is proved. 