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

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Non-autonomous Dissipative Dynamical Systems 87

(2) V (x

t, x

t) ≤ ω(V (x

, x

), t) for all (x

, x

) ∈ X

×X and t ≥ 0, where ω :

× R

→ R

is a non-increasing in ﬁrst variable function and ω(x, t) → 0,

as t → +∞, for every x ∈ R

Then h(X, T

, π), (Y, T

, σ), hi is convergent.

Proof. According to Corollary 2.2 for the convergence of h(X, T

, π), (Y, T

, σ), hi

it is suﬃcient to show that the equality (2.42) holds for every compact K ∈ C(X).

First of all, we will show that the equality

lim

t→+∞

sup

)∈K

×K

V (x

t, x

t) = 0 (2.69)

holds for all compact K ∈ C(X). Indeed, since K is compact, then there exists

α > 0 such that V (x

, x

) ≤ α for all (x

, x

) ∈ K

×K and V (¯x

, ¯x

) = α for some

(¯x

, ¯x

) ∈ K

×K, and, consequently,

V (x

t, x

t) ≤ ω(V (x

, x

), t) ≤ ω(α, t). (2.70)

Since ω(α, t) → 0 as t → +∞, then from (2.70) it follows (2.69). Let us show that

(2.69) implies (2.42). If we suppose the contrary, then there exit K ∈ C(X), ε

0, {x

} ⊆ K (i = 1, 2) and t

→ +∞ such that

ρ(x

, x

) ≥ ε

. (2.71)

In view of the compact dissipativity of (X, T

, π) we may suppose that the sequences

} (i = 1, 2) are convergent. Let us set ¯x

= lim

n→+∞

, then from the

inequality (2.71), it follows that ¯x

6= ¯x

. On the other hand, by the inequality

(2.70)

0 ≤ V (x

, x

) ≤ V (α, t

) → 0

as n → +∞ and, hence,

V (¯x

, ¯x

) = lim

n→+∞

V (x

, x

) = 0.

From the last equality, it follows that ¯x

= ¯x

. The obtained contradiction com-

pletes the proof of the theorem. 

Remark 2.7 a. Condition 2 of Theorem 2.18 holds, if the function V : X

×X →

satisﬁes one of the following conditions:

(1) V (x

t, x

t) ≤ Ne

−νt

V (x

, x

) for all t ≥ 0 and (x

, x

) ∈ X

×X (ω(x, t) =

Nxe

−νt

);

(2) V (x

t, x

t) ≤

2V (x

)

2+V (x

for all t ≥ 0 and (x

, x

) ∈ X

×X (ω(x, t) =

2+xt

);

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88 Global Attractors of Non-autonomous Dissipative Dynamical Systems

b. All results of this section are true also in the case when the spaces X and Y

are not metric but pseudo-metric.

In conclusion, we note that convergent systems are in some sense the simplest

dissipative dynamical systems. If h(X, T

, π), (Y, T

, σ), hi is a convergent non-

autonomous dynamical system and J

) is a Levinson center of the dynamical

system (X, T

, π) ((Y, T

, σ)), then J

and J

are homeomorphic. Although the

center of Levinson of a convergent system can by completely described, it may

be suﬃciently complicated. We will give an example which illustrates the above

cooment.

Example 2.2 Let Y := R and (Y, Z

, σ) be a cascade generated by positive

powers of the odd function g, deﬁned on R

in the following way:

g(y) =











−2y , 0 ≤ y ≤

2(y − 1) ,

< y ≤ 1

(y − 1) , 1 < y < +∞.

It is easy to check that (Y, Z

, σ) is dissipative and J

⊆ [−1, 1]. Let us put

X := R

and denote by (X, Z

, π) a cascade generated by the positive powers of

the mapping P : R

→ R

f(u, y)

g(y)

, (2.72)

where f(u, y) :=

u +

y. Finally, let h = pr

: X → Y . From (2.72), it

follows that h is a homomorphism of (X, Z

, π) onto (Y, Z

, σ) and, consequently,

h(X, Z

, π), (Y, Z

, σ), hi is a non-autonomous dynamical system. Note that

|(u

, y) −(u

, y)| = |u

−u

| = 10|P (u

, y) − P (u

, y)|. (2.73)

From (2.73), it follows that

, y) −P

, y)| ≤ Ne

−νn

|hu

, yi−hu

, yi| (2.74)

for all n ∈ Z

, where N = 1 and ν = ln 10. We will show that the dynamical

system h(X, Z

, π), (Y, Z

, σ), hi is convergent. It is possible to check directly

that the dynamical system (X, Z

, π) is dissipative; therefore, for convergence of

h(X, Z

, π), (Y, Z

, σ), hi it is suﬃcient to show that J

contains only one

point for all y ∈ J

. If we suppose that it is not true, then there exit y

∈ J

and x

, x

∈ J

with x

6= x

. Let ϕ

be a motion of the dynamical system

(X, Z

, π) deﬁned on Z and passing through the point x

(i = 1, 2). From (2.74),

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Non-autonomous Dissipative Dynamical Systems 89

it follows that

|ϕ

(−n) − ϕ

(−n)| ≥ N

−1

νn

−u

| (2.75)

for all n ∈ Z

, where (u

, y

) = x

(i = 1, 2). On the other hand,

sup

n∈Z

|ϕ

(n) − ϕ

(n)| < +∞. (2.76)

The inequalities (2.75) and (2.76) are contradictory. The obtained contradic-

tion proves the required statement. Thus, the dynamical system h(X, Z

, π),

(Y, Z

, σ), hi is convergent and, consequently, J

and J

are homeomorphic.

Note that this example is an just a slight modiﬁcation of an example from

[

297, p.39–42

]

. In addition, they have the same attractors and the corresponding

systems on the attractors act equally. Hence, the Levinson centers J

and J

are

intermixing (strange) attractors.

Thus, the Levinson center J

of the convergent dynamical system h(X, T

, π),

(Y, T

, σ), hi is completely deﬁned by the structure of J

, but the last one can be

organized in a very complicated way.

2.6 Global attractors of non-autonomous dynamical systems

Let Y be a compact metric space, (X, h, Y ) be a locally trivial Banach ﬁbering

[

190

]

and | · | be a norm on (X, h, Y ) coordinated with the metric ρ on X (that is

ρ(x

, x

) = |x

−x

| for any x

, x

∈ X such that h(x

) = h(x

) ).

Theorem 2.19 Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical

system and for any bounded set M ∈ B(X) there exists l = l(M) > 0 such that

(M) would be relatively compact (that is the dynamical system (X, T

, π) is com-

pletely continuous). Then the following conditions are equivalent:

1. there exits a positive number r such that for any x ∈ X there will be τ = τ(x) ≥

0 for which |xτ| < r;

2. the dynamical system h(X, T

, π), (Y, T

, σ), hi is compactly dissipative and

lim

t→+∞

sup

|x|≤R

ρ(xt, J) = 0 (2.77)

for any R > 0, where J is the Levinson center of (X, T

, π), that is the non-

autonomous system h(X, T

, π), (Y, T

, σ), hi admits a compact global attractor.

Proof. That Condition 2 implies Condition 1 is evident. Let us show that under

conditions of Theorem 2.19, the converse also holds. Suppose that A(r) := {x ∈

E | |x| ≤ r} where r > 0 is the number appearing in Condition 1. As Y is

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90 Global Attractors of Non-autonomous Dissipative Dynamical Systems

compact and the Banach ﬁbering (X, h, Y ) is locally trivial, then its null section

Θ = {θ

| y ∈ Y , where θ

is the null element of the ﬁber E

:= h

−1

(y)} is compact

and, hence, the set A(r) is bounded, as A(r) ⊆ S(Θ, r) = {x ∈ E| |ρ(x, θ) ≤ r}.

According to the condition of the theorem for a bounded set M there exits a positive

number l such that π

M is relatively compact. Let x ∈ M and τ = τ(x) ≥ 0 be

such that xτ ∈ M, then x(τ + l) ∈ K :=

M. Thus, the non-empty compact

K is a weak attractor of the system (X, T

, π) and, according to Theorem 1.10,

the dynamical system (X, T

, π) is compactly dissipative. Let J be the Levinson

center of (X, T

, π) and R > 0, then the set A(R) := {x ∈ E | |x| ≤ R}, as it

was noticed above, is bounded, and for it there exists a number l > 0 such that

A(R) is relatively compact and since (X, T

, π) is compactly dissipative, then its

Levinson center J, according to Theorem 1.6, attracts the set π

A(R) and, hence,

the equality (2.77) holds. The theorem is proved. 

Corollary 2.7 Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical

system and suppose that the vector ﬁbering of (X, T

, π) is ﬁnite-dimensional, then

Conditions 1. and 2. of Theorem 2.19 are equivalent .

Proof. This assertion it follows from Theorem 2.19 as for any r > 0 the set

{x ∈ E| |x| ≤ r} is compact, if the vector ﬁbering (X, h, Y ) is ﬁnite-dimensional

and Y is compact. 

Recall that the dynamical system (X, T

, π) is called asymptotically compact, if

for any bounded closed positively invariant set M ∈ B(X) there exits a non-empty

compact K such that the equality

lim

t→+∞

β(π

M, K) = 0

holds.

Theorem 2.20 Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical

system and for every bounded set A ⊆ X there exists a nonempty compact K

⊆ X

such that

lim

t→+∞

β(π

A, K

) = 0. (2.78)

Then the conditions 1 and 2 of Theorem 2.19 are equivalent.

Proof. Since Y is compact and (X, h, Y ) is locally trivial, then for any R > 0 the set

{x ∈ E | |x| ≤ R} is bounded. According to Condition 1 of Theorem 2.19, for any

x ∈ E there exits τ = τ(x) ≥ 0 such that xτ ∈ A(r) := {x ∈ E | |x| ≤ r}. According

to Theorem 1.24 the dynamical system (X, T

, π) is compactly dissipative. Let J

be the Levinson center of (X, T

, π) and R > 0. As the set M := A(R) := {x ∈

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Non-autonomous Dissipative Dynamical Systems 91

E | |x| ≤ R} is bounded, then according to Condition of the theorem and of Lemma

1.3 the set Ω(M) 6= ∅, is compact, invariant and the equality

lim

t→+∞

β(π

A, ω(A)) = 0

holds. As J is the maximal compact invariant set in (X, T

, π), then Ω(M) ⊆ J

and, hence, the equality

lim

t→+∞

β(π

A, J) = 0

holds. The theorem is proved. 

Corollary 2.8 Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical

system and suppose that at least one of the following conditions is fulﬁlled:

(1) for every bounded set A ⊆ X there exits a positive number l such that π

A is

relatively compact;

(2) the dynamical system h(X, T

, π) is asymptotically compact.

Then Conditions 1 and 2 of Theorem 2.19 are equivalent.

Proof. This assertion it follows directly from Theorems 2.19 and 2.20 . 

Theorem 2.21 Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical

system and (X, T

, π) be asymptotically compact, then the following conditions are

equivalent:

1. there exits a positive number R

and for any R > 0 there exists l(R) > 0 such

that

|π

x| ≤ R

(2.79)

for all t ≥ l(R) and |x| ≤ R;

2. the dynamical system h(X, T

, π), (Y, T

, σ), hi admits a compact global attrac-

tor, i.e it is compactly dissipative and for its Levinson center J the equality

(2.77) holds for any R > 0.

Proof. Evidently Condition 2 implies Condition 1, that is why for proving the

theorem it is suﬃcient to show that from Condition 1 follows Condition 2. Let

∈ B(X), then there exits R > 0 such that M

⊆ A(R) := {x ∈ E| |x| ≤ R}.

According to Condition 1, for a given R there exists l = l(R) > 0 such that (2.79)

holds and, in particular, the set M :=

{π

|t ≥ l(R)} is bounded and positively

invariant. As (X, T

, π) is asymptotically compact, for the set M there exists a

nonempty compact K for which the equality (2.78) holds. To ﬁnish the proof of the

theorem it is suﬃcient to cite Theorem 2.20. The theorem is proved. 

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92 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Theorem 2.22 Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical

system and let mappings π

:= π(t, ·) : X → X(t ∈ T

) be represented like a

sum π(t, x) = ϕ(t, x) + ψ(t, x) for all t ∈ T

and x ∈ X Let also the following

conditions be fulﬁlled:

(1) |ϕ(t, x)| ≤ m(t, r) for all t ∈ T

, r > 0 and |x| ≤ r, where m : T

× R

→ R

and m(t, r) → 0 as t → +∞;

(2) the mappings ψ(t, ·) : X → X(t > 0) are conditionally completely continuous,

that is ψ(t, A) is relatively compact for any t > 0 and a bounded positively

invariant set A ⊆ X.

Then the dynamical system (X, T

, π) is asymptotically compact.

Proof. Let A ⊆ X be a bounded set such that Σ

(A) :=

{π

A | t ≥ 0} is also

bounded, r > 0 and A ⊆ {x ∈ X| |x| ≤ r}. Let us show that for any {x

} ⊆ A and

→ +∞, the sequence {x

} is relatively compact. We will convince ourselves

that the set M := {x

} may be covered by a compact ε net for any ε > 0.

Let ε > 0 and l > 0 be such that m(l, r) < ε/2 and let us represent M in the

form of the union M

∪ M

where M

:= {x

}

k=1

, M

:= {x

}

+∞

k=k

and

:= max{k | t

< l}. The set M

is a subset of the set π

(Σ

(A)), the elements

of which we can represent in the form of ϕ(l, x) + ψ(l, x)(x ∈ Σ

(A)). As the set

ψ(Σ

(A), l) is relatively compact, then it may be covered by a ﬁnite ε/2 net. Let

us notice that for any y ∈ ϕ(l, Σ

(A)) there exits x ∈ Σ

(A) such that y = ϕ(l, x)

and |y| = |ϕ(l, x)| ≤ m(l, r) < ε/2. That is why the null section Θ of the ﬁbering

of (X, h, Y ) is an ε/2 net of the set ϕ(l, Σ

(A)). Thus, M

, and, subsequently,

M is covered by a compact ε net and as the space X is complete, then the set

M = {x

} is relatively compact. Now to ﬁnish the proof of the theorem it is

suﬃcient to cite Lemma 1.3. The theorem is proved. 

At the end of this section we will study dissipative dynamical systems in ﬁnite-

dimensional space. We will give some conditions which are equivalent to dissipativ-

ity.

Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical system, Y be com-

pact, (X, h, Y ) be a ﬁnite-dimensional vectorial ﬁbering

[

190

]

and |·| be a Rieman-

nian metric on (X, h, Y ).

Lemma 2.14 Under the conditions mentioned above all the types of dissipativity

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

, π), (Y, T

, σ), hi are equivalent.

Proof. Since the space Y is compact and (X, h, Y ) is ﬁnite-dimensional and locally

trivial

[

190

]

, then the space X is locally compact and possesses the Heine-Borel

property. To ﬁnish the proof of the lemma it is suﬃcient to cite Theorem 1.10. 

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Non-autonomous Dissipative Dynamical Systems 93

Theorem 2.23 Let h(X, T

, π), (Y, T

, σ), hi be a ﬁnite-dimensional non-

autonomous dynamical system, then the following conditions are equivalent:

1. there exits a positive number R such that for all x ∈ X the following inequality

lim sup

t→+∞

|xt| < R (2.80)

holds;

2. there exits a positive number r such that for all x ∈ X there exists τ ≥ 0 such

that |xτ| < r;

3. there exists a nonempty compact K

⊂ X such that ω

6= ∅ for any x ∈ X;

4. there exists a nonempty compact K

⊂ X such that ∅ 6= ω

⊂ K

for every

x ∈ X;

5. there exits a positive number R

such that for all R

> 0 there exits l(R

) > 0

such that

|xt| < R

(2.81)

for all t ≥ l(R

) and |x| ≤ R

Proof. It is clear that 5. ⇒ 1. ⇒ 4. ⇒ 3. ⇒ 2. According to Theorem 2.19 from 2.

it follows 5. The theorem is completely proved. 

2.7 Global attractor of cocycles

Deﬁnition 2.18 The family {I

| y ∈ Y } (I

⊂ W ) of nonempty compact subsets

W is called (see, for example,

[

]

and

[

153

]

) a compact pullback attractor (uniform

pullback attractor ) of a cocycle ϕ, if the following conditions hold:

(1) the set I :=

| y ∈ Y } is relatively compact;

(2) the family {I

| y ∈ Y } is invariant with respect to the cocycle ϕ, i.e.

ϕ(t, I

, y) = I

σ(t,y)

for all t ∈ T

and y ∈ Y ;

(3) for all y ∈ Y (uniformly in y ∈ Y ) and K ∈ C(W )

lim

t→+∞

β(ϕ(t, K, y

−t

), I

) = 0,

where β(A, B) := sup{ρ(a, B) : a ∈ A} is a semi-distance of Hausdorﬀ.

Below in this section we suppose that T

= S.

Deﬁnition 2.19 The family {I

| y ∈ Y }(I

⊂ W ) of nonempty compact subsets

is called a compact global attractor of the cocycle ϕ, if the following conditions are

fulﬁlled:

(1) the set I :=

| y ∈ Y } is relatively compact;

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94 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(2) the family {I

| y ∈ Y } is invariant with respect to the cocycle ϕ;

(3) the equality

lim

t→+∞

sup

y∈Y

β(ϕ(t, K, y), I) = 0

holds for every K ∈ C(W ).

Let M ⊆ W and

(M) :=

t≥0

[

τ≥t

ϕ(τ, M, σ

−τ

y) (2.82)

for all y ∈ Y, where y

−t

:= σ(−t, y).

Lemma 2.15 The following assertions hold:

(1) The point p ∈ ω

(M) if and only if there exit t

→ +∞ and {x

} ⊆ M such

that p = lim

n→+∞

ϕ(t

, x

, y

−t

);

(2) U (t, y)ω

(M) ⊆ ω

(M) for all y ∈ Y and t ∈ T

, where U (t, y) := ϕ(t, ·, y);

(3) for any point w ∈ ω

(M) the motion ϕ(t, w, y) is deﬁned on S;

(4) if there exits a nonempty compact K ⊂ W such that

lim

t→+∞

β(ϕ(t, M, y

−t

), K) = 0,

then ω

(M) 6= ∅, is compact,

lim

t→+∞

β(ϕ(t, M, y

−t

), ω

(M)) = 0 (2.83)

and

U(t, y)ω

(M) = ω

(M) (2.84)

for all y ∈ Y and t ∈ S

Proof. The ﬁrst assertion of the lemma directly it follows from the equality (2.82).

Let w ∈ ω

(M), then there exit t

→ +∞ and x

⊆ M such that w =

lim

n→+∞

ϕ(t

, x

, y

−t

) and, hence,

ϕ(t, w, y) = lim

n→+∞

ϕ(t, ϕ(t

, x

, y

−t

), y) = lim

n→+∞

ϕ(t + t

, x

, y

−t

). (2.85)

Thus, ϕ(t, w, y) ∈ ω

(M), that is U(t, y)ω

(M) ⊆ ω

(M) for all y ∈ Y and

t ∈ T

From the equality (2.85), it follows that the motion ϕ(t, w, y) is deﬁned on S

like ϕ(t + t

, x

, y

−t

) is deﬁned on [−t

, +∞) and t

→ +∞.

The fourth assertion of the lemma is proved as in Theorem 2.4 and Lemma 1.1.3

from

[

217

]

. 

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Non-autonomous Dissipative Dynamical Systems 95

Deﬁnition 2.20 A cocycle ϕ over (Y, S, σ) with the ﬁber W is said to be com-

pactly dissipative, if there exits a nonempty compact K ⊆ W such that

lim

t→+∞

sup{β(U(t, y)M, K) | y ∈ Y } = 0 (2.86)

for any M ∈ C(W ).

Lemma 2.16 Let Y be compact and hW, ϕ, (Y, S, σ)i be a cocycle over (Y, S, σ)

with the ﬁber W . For hW, ϕ, (Y, S, σ)i to be compactly dissipative, it is necessary

and suﬃcient that the semigroup autonomous system (X, S

, π) should be compactly

dissipative.

Proof. This assertion follows from the corresponding deﬁnitions. 

Theorem 2.24 Let Y be compact, hW, ϕ, (Y, S, σ)i be compactly dissipative and

K be the nonempty compact subset of W appearing in the equality (2.86), then:

1. I

= ω

(K) 6= ∅, is compact, I

⊆ K and

lim

t→+∞

β(U(t, y

−t

)K, I

) = 0

for every y ∈ Y ;

2. U (t, y)I

= I

for all y ∈ Y and t ∈ S

;

lim

t→+∞

β(U(t, y

−t

)M, I

) = 0 (2.87)

for all M ∈ C(W ) and y ∈ Y ;

lim

t→+∞

sup{β(U(t, y

−t

)M, I) | y ∈ Y } = 0 (2.88)

for any M ∈ C(W ), where I := ∪{I

| y ∈ Y };

5. I

= pr

for all y ∈ Y , where J is the Levinson center of (X, T

, π), and

hence I = pr

6. the set I is compact;

7. the set I is connected if one of the next two conditions is fulﬁlled:

(a) S

= R

and the spaces W and Y are connected;

(b) S

= Z

and the space W ×Y possesses the (S)-property or it is connected

and locally connected.

Proof. The ﬁrst two assertions of the theorem follow from Lemma 2.15. If we

suppose that the equality (2.87) does not hold, then there exist ε

> 0, y

∈

Y, M

∈ C(W ), {x

} ⊆ M

and t

→ +∞ such that

ρ(U(t

, y

−t

, I

) ≥ ε

. (2.89)

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96 Global Attractors of Non-autonomous Dissipative Dynamical Systems

According to the equality (2.87), for ε

and y

∈ Y there exists t

= t

(ε

, y

) > 0

such that

β(U(t, y

−t

)K, I

) <

(2.90)

for all t ≥ t

. Let us notice that

U(t

, y

−t

= U(t

, y

−t

)U(t

−t

, y

−t

As hW, ϕ, (Y, S, σ)i is compactly dissipative, then the sequence {U(t

−

, y

−t

} may be considered convergent. Suppose

x = lim

n→+∞

ϕ(t

−t

, x

, y

−t

then, according to Lemma 2.15,

x ∈ ω

−t

) and U(t

, y

−t

)

x ∈ ω

). From

the equality (2.86), it follows that

x ∈ K. Passing to limit in (2.89) as n → +∞,

we get

U(t

, y

−t

)

x /∈ B(I

, ε

). (2.91)

On the other hand, as

x ∈ K, from (2.90) we have

U(t

, y

−t

)

x ∈ B(I

This contradicts (2.91). The obtained contradiction proves the assertion we need.

Let us prove now the equality (2.88). If we suppose that it does not hold, then

there exist ε

> 0, M

∈ C(W ), y

∈ Y, {x

} ⊆ M

and t

→ +∞ such that

ρ(U(t

, y

−t

, I) ≥ ε

. (2.92)

As Y is compact, then we may consider the sequences {y

} and {y

} convergent.

Suppose y

:= lim

n→+∞

and

y := lim

n→+∞

. According to (2.87), for given ε

> 0

and y

∈ Y , there exists t

= t

(ε

, y

) such that

β(U(t

, y

−t

, I

) <

(2.93)

for all t ≥ t

(ε

, y

). Let us note that

U(t

, y

−t

= U(t

, y

−t

)U(t

−t

, y

−t

. (2.94)

As hW, ϕ, (Y, T, σ)i is compactly dissipative, then the sequence {U(t

−t

, y

−t

}

may be considered a convergent one. Suppose x

= lim

n→+∞

ϕ(t

−t

, x

, y

) and let

us notice that according to (2.93) x

∈ K. From the equality (2.94), it follows that

U(t

, y

−t

→ U(t

, y

−t

and, hence, from (2.92) we have

U(t

, y

−t

/∈ B(I

, ε

On the other hand, from (2.93) and from x

∈ K, it follows that

U(t

, y

−t

∈ B(I