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

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Method of Lyapunov functions 177

is asymptotically compact, we conclude that Ω is compact and, hence, the system

(X, T

, π) is pointwise k-dissipative. Taking into consideration (5.6) and Theorem

1.19 we conclude that the system (X, T

, π) is locally k-dissipative. Theorem is

proved. 

Remark 5.3 a) From condition 3. it follows that level lines of the function V do

not contain ω-limit points of the dynamical system (X, T

, π).

b) Theorem 5.4 takes place for discrete dynamical systems, too, if condition 3.

is changed by the following: level lines of the function V do not contain ω-limit

points of the dynamical system (X, T

, π).

Proof. For proving the last assertion let us notice that in conditions of Theorem

11.2 the set K is a weak attractor and that is why Ω ⊆ K. Indeed, if this were not

so, then we should have x

∈ E

such that ω

\K 6= ∅. Let p ∈ ω

\K, then there

exits c

such that ω

⊆ V

−1

). Let c

= V (p), then c

= V (p) = c

and, hence,

p ∈ V

−1

) ∩ E

∩ ω

. The last contradicts the condition of the theorem. 

Deﬁnition 5.1 h(X, T

, π), (Y, T

, σ), hi is said to be bounded, if for arbitrary

R > 0 there is C(R) > 0 such that |xt| ≤ C(R) for all |x| ≤ R and t ≥ 0.

Theorem 5.5 Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical

system, T

= R

and (X, T

, π) is asymptotically compact. If there exist r > 0

and a continuous bounded on the bounded sets from X function V : X

→ R which

satisﬁes the following conditions:

1. for any c ∈ R the set {x ∈ E

| V (x) ≤ c} is bounded;

2. if xτ ∈ E

for all τ ∈ [0, t], then V (xt) ≤ V (x);

3. level lines of V do not contain ω-limit points of the dynamical system (X, T

, π).

Then the non-autonomous dynamical system h(X, T

, π), (Y, T

, σ), hi is boundedly

k-dissipative and bounded, i.e. for any R > 0 there exists C(R) ≥ 0 such that

|x| ≤ C(R) for all |x| ≤ R and t ≥ 0.

Proof. Like in Theorem 5.3 the existence of C(R) ≥ 0 such that |xt| ≤ C(R) for

all t ≥ 0 and |x| ≤ R (R ≥ r) is proved.

Let us show that under the conditions of Theorem 5.5 the non-autonomous

system (X, T

, π), (Y, T

, σ), hi is boundedly k-dissipative. As h(X, T

, π),

(Y, T

, σ), hi is bounded and asymptotically compact, then according to Lemma 1.4

Ω(K

) 6= ∅, is compact and attracts K

:= {x ∈ E | |x| ≤ r}. Suppose K := Ω(K

)

and let us show that ω

∩ K 6= ∅ for any x ∈ X. If this were not so, we should

have x

∈ X such that ω

∩ K = ∅. Under the conditions of Theorem 5.5 the set

:= {x

t | t ≥ 0} is relatively compact and, hence, ω

6= ∅ and is compact.

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178 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Suppose d = ρ(ω

, K) > 0, then

B(ω

) ∩

B(K,

) = ∅ (5.7)

and there will be l > 0 such that

⊆ B(K,

) (5.8)

for all t ≥ l. Let us show that x

t /∈ K

for all t ≥ t

, where t

is some nonnegative

number. If it is not so then there will be t

→ +∞, such that x

∈ K

and

according to (5.8) x

+ l) ∈ B(K,

). We consider the sequence {x

+ l)} to

be convergent. Suppose p = lim

n→+∞

+ l), then p ∈ ω

∩

B(K,

). The last

contradicts (5.7), and it proves the assertion we need. Further, reasoning like in

Theorem 5.1 we will have that for some c

∈ R the set ω

∩V

−1

) 6= ∅. The last

contradicts condition 3. of the theorem. The contradiction shows that ω

∩ K 6= ∅

for all x ∈ X. As the dynamical system h(X, T

, π), (Y, T

, σ), hi is bounded and

asymptotically compact, then (X, T

, π) satisﬁes the condition of Ladyzhenskaya.

Now for ﬁnishing the proof of Theorem 5.5 it is suﬃcient to refer to Theorem 2.20.

Theorem is proved. 

Remark 5.4 a) The condition 1. of Theorem 5.5 is equivalent to positive deﬁ-

niteness V on X

and to its boundedness on bounded subsets from X;

b) The assertion which is the opposite to Theorem 5.5 takes place and can be

proved like Theorem 5.1.

c) Lemma 5.1 remains correct if the condition of complete continuity of

(X, T

, π) is changed by the condition of Ladyzhenskaya. That is why Theorem

5.5 is correct in the case when condition 3. is changed by the following: level lines

of the function V do not contain ω- limit points of the dynamical system (X, T

, π).

d) Theorem 5.5 remains valid if conditions 2. and 3. are changed by the follow-

ing: if xτ ∈ E

for all τ ∈ [0, t] (t > 0), then V (xt) < V (x), and it can be proved

like Theorem 5.2.

e) Theorem 5.5 takes place for discrete dynamical systems if the function V is

deﬁned on the whole space and condition 2. takes place for all t ≥ 0 and x ∈ E.

For autonomous systems with discrete time this assertion is contained in

[

174

]

5.2 Some criterions of dissipativity of diﬀerential equations

1. Let E

be n-dimensional Euclidean space, | · | is the norm on E

generated by

the scalar production of h·, ·i. Consider the diﬀerential equation

˙u = f (t, u), (5.9)

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Method of Lyapunov functions 179

where f ∈ C(R × E

, E

) is a regular function. Along with equation (5.9) let us

consider its H-class

˙v = f (t, v) (g ∈ H(f)). (5.10)

As it was mentioned above, the equation (5.9) naturally generates a non-

autonomous dynamical system (see example 3.1). Applying to it the results con-

cerning general non-autonomous dynamical system we obtain a series of statements

about the equation (5.9). From theorems 5.1, 5.2, Lemma 5.1 and Corollary 5.1

follow Theorems below.

Let r > 0 and E

:= {u ∈ E

| |u| ≥ r}.

Theorem 5.6 Let H(f) be compact. For the equation (5.9) to be dissipative

it is necessary and suﬃcient that would exist r > 0 and continious function V :

H(f) ×E

→ R

satisfying the conditions:

(1) V (g, v) ≥ a(|v|) for all v ∈ E

and g ∈ H(f), where a ∈ A and Im V ⊆ Im a;

(2) if |ϕ(τ, v, g)| ≥ r for all τ ∈ [0, t] then V (g

, ϕ(t, v, g)) ≤ V (g, v) ( g ∈ H(f ));

(3) the lines of the level of the function V do not contain positive semitrajectories

of equation (5.10).

Theorem 5.7 Let H(f) be compact. For the equality (5.9) to be dissipative it

is necessary and suﬃcient that there would exist r > 0 and a continious function

V : H(f) ×E

→ R

, satisfying the following conditions:

(1) V (g, v) ≥ a(|v|) for all v ∈ E

g ∈ H(f ), where a ∈ A and Im V ⊆ Im a;

(2) if |ϕ(τ, v, g)| ≥ r for all τ ∈ [0, t] then V (g

, ϕ(t, v, g)) < V (g, v) (g ∈ H(f)).

Theorem 5.8 Let H(f) be compact. Then the following conditions are equivalent:

1. the equation (5.9) is dissipative;

2. there exists a number r > 0 possessing the following property: for every v ∈ E

and g ∈ H(f ) there exists τ = τ (v, g) > 0 such that, |ϕ(τ, v, g)| < r;

3. there exists such number R

> 0 that lim

t→+∞

inf |ϕ(t, v, g)| < R

for all v ∈ E

and g ∈ H(f );

4. there exists a number R

> 0 and for every R > 0 there is l(R) > 0, such that

|ϕ(t, v, g)| ≤ R

for all t ≥ l(R), |v| ≤ R and g ∈ H(f).

Proof. The statement formulated above follows from Theorems 2.19 and 3.1. In the

case when f is periodic the equivalence of the conditions 1., 2. and 4. is established

[

270

]

. The equivalence of the conditions 1. and 2. for nonperiodic equationsis

established in

[

163

]

. 

Deﬁnition 5.2 We will say that the function F ∈ C(R × E

, E

) satisﬁes the

local condition of Lipschitz u ∈ E

uniformly on t ∈ R or just satisﬁes the condition

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 5

180 Global Attractors of Non-autonomous Dissipative Dynamical Systems

of Lipschitz if for every r > 0 there exists L(r) > 0 such that |F (t, u

) −F (t, u

)| ≤

L|u

−u

| for all t ∈ R and u

, u

∈ B[0, r].

Theorem 5.9 Let f ∈ C(R × E

, E

) satisfy the condition of Lipschitz, H(f )

be compact and let exist R > 0 and continuous diﬀerentiable function V ∈ C(R ×

, R

) satisfying the following conditions:

a. V satisﬁes the condition of Lipschitz;

b. a(|u|) ≤ V (t, u) ≤ b(|u|) (a, b ∈ A, Im a = Im b, t ∈ R and u ∈ E

);

c. M

⊆ M

∩ M ∂V

∂t

∩ M

grad

(where M

:= {{t

} : {f

} is convergent } and

by analogy for M

, M ∂V

∂t

and M

grad

);

(t, u) :=

∂V

∂t

(t, u) + hgrad

V, f(t, u)i ≤ 0 (t ∈ R and u ∈ E

);

e. whatever would be (

V , g) ∈ H(V, f) :=

{(V

, f

) : τ ∈ R} the lines of level of

the function

V do not contain positive semi-trajectories of the equation (5.10)

lying out of the ball B[0, r].

Then the equation (5.9) is dissipative.

Proof. We will execute the proof in few steps.

1. Let g ∈ H(f). There exists {t

} ⊂ R such that f

→ g. Since M

⊆ M

the sequence {V

} is also convergent. Assume

V := lim

k→+∞

. From the conditions

a.– e. follows that the function

V satisﬁes the condition of Lipschitz,

(t, v) :=

∂

∂t

(t, v) + hgrad

V , gi ≤ 0 and besides a(|v|) ≤

V (t, v) ≤ b(|v|) for all v ∈ E

and

t ∈ R.

2. Let v ∈ E

and g ∈ H(f ). Denote by ϕ(·, v, g) the solution of the equation

(5.10) passing through the point v for t = 0. Let us show that this solution is

extendable to the right onto the whole semi-axis R

. Obviously, for this it is

suﬃcient to show that it is bounded on the whole domain of its existence [0, t

Let R = R(r) > 0 be such that a(R) > b(r), T

(v) := {t ∈ [0, t

[: |ϕ(t, v, g)| ≤

r} and T

(v) := [0, t

[ \T

(v). It is clear that T

(v) is open and, consequently,

(v) =

, t

[ (β = β(α)). For every t ∈ T

(v) there exists α such that

t ∈]t

, t

[, |ϕ(t

, v, g)| = |ϕ(t

, v, g)| = r, |ϕ(t, v, g)| > r and, consequently,

a(|ϕ(t, v, g)|) ≤

V (t, ϕ(t, v, g)) ≤

V (t

, ϕ(t

, v, g))

≤ b(|ϕ(t

, v, g)|) = b(r) ≤ a(R) (5.11)

From the inequality (5.11) follows that |ϕ(t, v, g)| ≤ R and we have

sup{|ϕ(t, v, g)| : t ∈ [0, t

[} ≤ r

:= max(r, R(r)).

Since f ∈ C(R × E

, E

) satisﬁes the condition of Lipshitz, it is regular.

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Method of Lyapunov functions 181

3. Let h(X, R

, π), (Y, R

, σ), hi be a non-autonomous dynamical system gener-

ated by the equation (5.9). Deﬁne on X

:= E

×H(f) a function V : X

→ R

the equality V(v, g) =

V (0, v) where

V is the function deﬁned in the point 1. Note

that the function V(·, g) is deﬁned uniquely by g ∈ H(f) in virtue of the inclusion

⊆ M

4. Show that the introduced in the previous point function V satisﬁes all the

conditions of Corollary 5.1. Let us establish its continuity on X

= E

×H(f). Let

→ g and v

→ v ((v

, g

) ∈ E

×H(f)). Then

|V(v

, g

) − V(v, g)| ≤ |V(v

, g

) − V(v, g

)| (5.12)

+|V(v, g

) − V(v, g)| ≤ Lg

(R)|v

−v|+ |V(v, g

) − V(v, g)|,

where L

(R) is the constant of Lipschitz for V (·, g

) on B(0, R) and R := sup

k∈N

Since L

(R) ≤ L

(R) for all g ∈ H(f ) (see, for example, [95]) and M

⊆ M

V(v, g

) → V(v, g) and passing to limit in (5.12) with k → +∞ we establish the

continuity of V in the point (v, g) ∈ X

Other conditions of Corollary 5.1 under the conditions of the theorem are obvi-

ously fulﬁlled. To ﬁnish the proof it is suﬃcient to refer to Corollary 5.1. 

Note, that in the case of periodicity of f the statement of Theorem 5.9 reinforces

Theorem 2.5 from

[

270

]

2. Consider the linear homogeneous equation

˙u = A(t)u, (5.13)

where A ∈ C(R, [E

]) and [E

] is the set of all linear operators A : E

→ E

. By

U(t, A) we denote the operator of Cauchy of the equation (5.13). Along with the

equation (5.13) we will consider the perturbed equation

˙u = A(t)u + F (t, u) (F ∈ C(R ×E

, E

)). (5.14)

The following theorem takes place.

Theorem 5.10 For the equation (5.14) to be dissipative the fulﬁllment of the

following condition is suﬃcient:

(1) the function F satisﬁes the condition of Lipschitz;

(2) H(A, F ) :=

{(A

, F

) : τ ∈ R} is compact in C(R, [E

]) × C(R × E

, E

);

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

kU(t, A)U

−1

(τ, A)k ≤ Ne

−ν(t−τ)

(t ≥ τ); (5.15)

(4) |F (t, u)| ≤ A + ε|u| (∀u ∈ E

, t ∈ R) where A > 0 and 0 ≤ ε ≤ ε

< νN

−2

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182 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Proof. Let (B, G) ∈ H(A, F ). Deﬁne W

∈ [E

] by the equality

+∞

∗

(τ, B)U(τ, B)dτ. (5.16)

From (5.15) and (5.16) follows that ||W

|| ≤ N

(2ν)

−1

. Denote by a :=

sup{||A(t)|| : t ∈ R} and note that

|u| = |U(−τ, B

)U(τ, B)u| ≤ e

aτ

|U(τ, B)u|.

From this

u, ui =

+∞

|U(τ, B)u|

dτ ≥

+∞

−2aτ

|u|

dτ =

|u|

So, we established the following inequality

|u|

≤ hW

u, ui ≤

2ν

|u|

(u ∈ E

, B ∈ H(A)).

Let Y := H(A, F ), (Y, R, σ) be the dynamical system of shifts on Y, X := E

Y and π

(v; B, g) := (ϕ(t, v, B, g); B

, g

) (v ∈ E

, (B, G) ∈ H(A, F )), where

ϕ(·, v, B, G) is the solution of the equation

˙v = B(t)v + G(t, v), (5.18)

passing through the point v as t = 0. Let us consider a non-autonomous dynamical

system h(X, R, π), (Y, R, σ), hi where h := pr

: X → Y . Deﬁne on X a function

V : X → R

by the following rule:

V (v; B, G) :=



v, v



(5.19)

for all v ∈ E

and (B, G) ∈ H(A, F ). Note that this function possesses the following

properties:

|v|

≤ V (v; B, G) ≤

|v|

(v ∈ E

, (B, g) ∈ H(A, F )).

b. V is continuous.

In fact, if v

→ v and (B

, G

) → (B, G) then

|V (v

; B

, G

) − V (v; B, G)| = |



, v



−



v, v



| (5.20)

= |



−v), v





v, v

−v





−W

)v, v



≤

2ν

||v

−v| +

2ν

|v||v − v

| + ||W

−W

|||v|

From the inequality (5.20) we can easily see that for V to be continuous it is suﬃcient

to show that ||W

−W

|| → 0 if B

→ B. Let us establish the latter. From (5.16)

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Method of Lyapunov functions 183

follows that W

is a self-conjugate operator. That is why

||W

−W

|| = sup

|v|=1



−W

)v, v



| ≤ (5.21)

≤

+∞

sup

|v|≤1

||[U(τ, B

) − U(τ, B)]v||

dτ =

sup

|v|≤1

|[U(τ, B

) − U (τ, B)]v|

dτ

+∞

sup

|v|≤1

|[U(τ, B

) − U (τ, B)]v|

dτ ≤ sup

0≤τ≤l

||U(τ, B

) − U(τ, B)||

+∞

(||U(τ, B

)||

+ ||U(τ, B)||

)dτ.

From (5.15) follows the inequality ||U(τ, B)|| ≤ Ne

−ντ

for all τ ∈ R

and

B ∈ H(A) (see

[

51, p.70

]

). Hence, from (5.21) we obtain

||W

−W

|| ≤ sup

0≤τ≤l

||U(τ, B

) − U(τ, B)||

l +

−2vl

. (5.22)

Passing to limit as k → +∞ and taking in consideration that the ﬁrst term in the

right hand side of (5.22) vanishes with any ﬁxed l > 0 we establish that

lim sup

k→+∞

||W

−W

|| ≤

−2vl

. (5.23)

In this case (5.23) takes place for all l > 0. Making l → +∞ and taking in

consideration that the left hand side of the equation (5.23) does not depend on l

we will obtain the result needed. So, the continuity of V is established.

c. Let now |ϕ(τ ; v, B, G)| ≥ r (r > N

A(v − ε

)

−1

) for all τ ∈ [0, t]. Then

V (ϕ(t; v, B, G), B

, G

) < V (v; B, G). To establish this inequality we calculate the

derivative of the function

V(τ) := V (ϕ(τ; v, B, G), B

, G

) =



ϕ(τ; v, B, G), ϕ(τ; v, B, G)



in τ ∈ R. By standard reasoning, taking into account the identity

dτ

+ B

∗

(τ)W

+ W

B(τ) = −E (5.24)

(E is the unit operator in E

), we have

dτ

V (ϕ(τ ; v, B, G), B

, G

) = −|ϕ(τ; v, B, G)|

2Re



G(τ, ϕ(τ; v, B, G)), W

ϕ(τ, v, B, G)



(5.25)

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184 Global Attractors of Non-autonomous Dissipative Dynamical Systems

From (5.25) we obtain

dτ

V (ϕ(τ ; v, B, G), B

, G

) ≤ −|ϕ(τ; v, B, G)|

|ϕ(τ; v, B, G)|(A + ε

|ϕ(τ; v; B, G)|)

≤ |ϕ(τ; v, B, G)|

(−1 + ε

From this follows that V (ϕ(t; v, B, G), B

, G

) < V (v, B, G). In the same way

as in Theorem 5.9 we prove the possibility of non-local extension to the right of

all the solutions of the equation (5.18); their uniqueness follows from the condition

of Lipschitz for F . To ﬁnish the proof of the theorem it is suﬃcient to refer to

Theorem 5.9 (see also Theorem 5.2). 

Remark 5.5 a. Theorem 5.10 coincides with Theorem 3.21, which is obtained

with the help of a set of a priori estimations. However, Theorem 3.21 takes place

also for the equations deﬁned in some Banach space.

b. Theorem 5.11 remains valid also in the case when we replace the condition

4. by: |F (t, u)| ≤ A + B|u|

(∀u ∈ E

, t ∈ R) where A and B are some positive

numbers and 0 ≤ α ≤ 1. We execute the proof of this statement by the same scheme

that Theorem 5.10 choosing r such that the condition r > r

would take place, where

is the solution of the equation Ar

−1

+ Br

α−1

= ν

−2

c. Note that the statement of the point b., generally speaking, does not take place

if α > 1. This is proved by the example ˙x = −x + x

Theorem 5.11 Let f ∈ C(R ×E

, E

) satisﬁes the condition of Lipschitz, H(f)

be compact and let exist A ∈ C(R, [E

]) satisfying the following conditions:

1. M

⊆ M

∩ M

(

A(t) :=

dA(t)

);

2. A(t) = A

∗

(t) and α|u|

≤



A(t)u, u



≤ β|u|

u ∈ E

for all t ∈ R and

α, β > 0;



A(t)u, u



≤ 0 (u ∈ E

, t ∈ R);

4. there exists r > 0 such that Re



A(t)u, f(t, u)



≤ −γ(|u|) for all u ∈ E

t ∈ R

where γ(s) > 0 as s ≥ r.

Then the equation (5.9) is dissipative.

Proof. According to Theorem 5.9 the function f is regular. Let



(X, R

, π),

(Y, R, σ), h



be a non-autonomous dynamical system generated by the equation (5.9)

(see example 3.1) and g ∈ H(f). There exists {t

} ∈ M

such that g = lim

k→+∞

and under the conditions of the theorem {A

} is convergent. Denote its limit

by B

. We deﬁne now on X

:= E

× Y a function V by the following rule:

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Method of Lyapunov functions 185

V (v, g) =



(0)v, v



for every v ∈ E

and g ∈ H(f ). Note that B

is deﬁned

uniquely because M

⊆ M

. The continuity of V is proved in the same way that

in Theorems 5.7 and 5.8.

Let now |ϕ(τ, v, g)| ≥ r for all τ ∈ [0, t] (t > 0). Let us calculate the derivative

of V (ϕ(τ, v, g), g

) in τ ∈ R. For this note that from the inclusion M

⊆ M

∩M

follows the equality

(ϕ(τ, v, g), g

) =

dτ



B(τ)ϕ(τ, v, g), ϕ(τ, v, g)



(5.26)



B(τ)ϕ(τ, v, g), ϕ(τ, v, g)



+ 2Re



B(τ)ϕ(τ, v, g), g(τ, ϕ(τ, v, g))



From the conditions 1.–4. of the theorem we obtain



B(t)v, v



≤ 0 and Re



B(t)v, g(t, v)



≤ −γ(|v|) (5.27)

for all v ∈ E

, B ∈ H(A), t ∈ R and g ∈ H(f ). From (5.26) and (5.27) follows

that

dτ

V (ϕ(τ, v, g), g

) ≤ −2γ(|ϕ(τ, v, g)|), hence, V (ϕ(τ, v, g), g

) < V (v, g) for

all v ∈ E

and g ∈ H(f ). To ﬁnish the proof it is suﬃcient to refer to Corollary 5.2

and Theorem 5.2. 

Corollary 5.4 If the equation (5.9) is τ-periodic and the conditions of Theorem

5.9 are fulﬁlled then the equation (5.9) has at least one τ-periodic solution.

The particular case of Corollary 5.4 is announced in the work

[

192

]

Corollary 5.5 Let a

∈ C(R, R). If H(a

) is compact (i =

0, 2n + 1) and

2n+1

(t) ≤ −α (α > 0) for all t ∈ R

then the equation

˙v(t) = a

(t) + a

(t)v + ···+ a

2n+1

(5.28)

is dissipative.

Proof. The formulated statement follows from Theorem 5.9. For this it suﬃcient

to take as A(t) a unit operator and to notice that



A(t)v, f(t, v)



= v(a

(t) + a

(t)v + ···+ a

2n+1

(t)v

2n+1

) =

2n+2



2n+1

(t) +

(t)

+ ··· +

(t)

2n+1



≤ −

2n+2

for all |v| ≥ r and t ∈ R and when r > 0 is big enough. 

Corollary 5.6 If functions a

(i ∈

0, 2n + 1) are τ-periodic and a

2n+1

(t) > 0 for

all t ∈ [0, τ[ then the equation (5.28) has at least one τ -periodic solution.

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 5

186 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Note that in the case when a

2n+1

(t) < 0 (t ∈ [0, τ [) according to Corollary 5.6

the equation (5.28) is dissipative and, consequently, by Theorem 2.3 from

[

270

]

has at least one τ -periodic solution. The case a

2n+1

(t) > 0 (t ∈ [0, τ [) is reduced

to the previous one by the replacement of t by −t.

In the case when a

2n+1

≡ 1 Corollary 5.6 coincides with the result established

[

270, p.126

]

Corollary 5.7 Let f = (f

, f

, . . . , f

) ∈ C(R × E

, E

) be continuously dif-

ferentiable. If

∂f

∂x

≤ −m < 0 (i ∈

1, n), H(f) is compact and functions

(t, x

, x

, . . . , x

i−1

, 0, x

i+1

, . . . , x

) are bounded, then the equation (5.9) is dissi-

pative.

Proof. From the equality

(t, x

, x

, . . . , x

) =

∂f

∂x

(t, x

, x

, . . . , x

)dx

+ f

(t, x

, x

, . . . , x

i−1

, 0, x

i+1

, . . . , x

in the conditions of Corollary 5.7 follows that for all x ∈ R

the inequality

(t, x

, x

, . . . , x

) ≤ −mx

+ M|x

| holds, hence,

hx, f(t, x)i =

i=1

(t, x) ≤ −m|x|

+ Mn|x| ≤ −

|x|

for all t ∈ R and |x| ≥

2Mn

. According to Theorem 5.6 the equation (5.9) is

dissipative. 

Note that in the case of periodicity of f Corollary 5.7 is well known

[

270

]

. When

f is not periodic the statement close to Corollary 5.7 is established in

[

264

]

with

some additional restrictions on f.

Theorem 5.12 Let the condition of uniqueness be fulﬁlled for every equation

(5.10) with g ∈ Σ

:= {f

: τ ∈ R}. For the disiipaivity of the equation (5.9)

there is suﬃcient existence of a function V : R ×E

→ R

(r > 0) satisfying the

conditions:

(1) V satisﬁes the condition of Lipschitz;

(2) a(|u|) ≤ V (t, u) ≤ b(|u|) (a, b ∈ A, Im a = Im b) when u ∈ E

;

(3)

(t, u) := lim sup

τ↓0

−1

V (t + τ, u + τf(t, u)) − V (t, u)

≤ −c(|u|) for u ∈ E

where c : R

→ R

is continuous and c(s) > 0 as s > r.