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

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The structure of the Levinson center with the condition of the hyperbolicity 167

Then, according to Corollary 4.2, there exists an inﬁnite number of diﬀerent periodic

points p

→ p ∈ M , moreover, Σ

⊆ B(M, ε

) and ε

↓ 0. Since M

⊆ R

(i ≥ i

)

is closed and locally maximal, then starting from a number m

we have p

∈ R

The theorem is proved. 

Deﬁnition 4.7 Recall

[

]

[

]

, that an invariant set Λ of the diﬀeomorphism f :

U → M is called factor-markovian, if there is a TMC (topological markovian chain

[

]

) T : Ω

→ Ω

and a continuous mapping ϕ : Ω

→ Λ such that ϕ ◦ T = f ◦ ϕ.

If the mapping ϕ is a homeomorphism, then the set Λ is called markovian.

Corollary 4.4 Under the conditions of Theorem 4.8 in the set R there is a locally

maximal hyperbolic markovian set Λ ⊆ R and, in particular, in the set R there is a

transversal homoclynic trajectory.

Proof. The formulated statement follows from Theorem 4.8 and also from Theorems

7.7

[

]

and 2.4

[

271

]

. 

4.5 The periodic dissipative systems

1. Let

h(X, T

, π), (Y, T

, σ), hi (4.6)

be a non-autonomous dynamical system. In this paragraph we will suppose that

the dynamical system (Y, T

, σ) is periodic, i.e. there exist τ ∈ T

(τ > 0) and

q ∈ Y such that σ(q, τ) = q and Y = {σ(q, t) : 0 ≤ t < τ}. In this connection

the non-autonomous dynamical system (4.6) is said to be τ-periodic. Denote by

P : X

→ X

a mapping deﬁned by the equality P (x) = π(τ, x) and by (X

, P )

denote a cascade generated by the positive powers of the mapping P.

Lemma 4.11 A non-autonomous dynamical system (4.6) is pointwise (com-

pact,local) dissipative if and only if the cascade (X

, P ) is pointwise (compactly,

locally) dissipative. In this case the following relations hold:

J = {π

: 0 ≤ t < τ } and J

:= J ∩ X

where J is the Levinson center of the dynamical system (4.6), but J

is the Levinson

center of the dynamical system (X

, P ).

Proof. This assertion follows immediately from the corresponding deﬁnitions. 

Corollary 4.5 If (X, h, Y ) is a ﬁnite-dimensional vectorial ﬁbering, then the

center of Levinson J of the non-autonomous dynamical system (4.6) and J

are

connected.

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168 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Proof. This statement follows from Theorems 1.32, 1.33 and Corollary 1.14. 

Corollary 4.6 Under the conditions of Corollary 4.5, there exists r > 0 and for

all a > 0 there is k(a) ∈ Z

such that P

B[0, a] ⊆ B[0, r] for all k ≥ k(a), where

B[0, a] := {x ∈ X

: |x| ≤ a}.

Proof. Corollary 4.6 follows from Theorem 2.23 and Lemma 4.11. 

Corollary 4.7 Under the conditions of Corollary 4.5 there exists a ﬁxed point of

the mapping P : X

→ X

Proof. Corollary 4.7 follows from the theorem on the ﬁxed point of Brauder

[

]

and Corollary 4.6. 

Theorem 4.9 Let (X, h, Y ) be a ﬁnite-dimensional ﬁbering. There exists a

nonempty compact invariant set J (Levinson center) of the dynamical system (4.6)

possessing the following properties:

(1) for every y ∈ Y the set J

= J ∩ X

is connected;

(2) for every  > 0 there exists δ() > 0 such that the inequality ρ(x, J

) < δ

(x ∈ X

) implies ρ(xt, J

) <  for all t ≥ 0;

(3) for all y ∈ Y and x ∈ X

lim

t→+∞

ρ(xt, J

) = 0;

(4) in the center of Levinson J of the system (4.6) there is at least one τ-periodic

point.

Proof. The formulated statement follows from Lemma 2.1, Theorem 2.1 and Corol-

lary 4.7. 

Theorem 4.10 If X

is a complex n-dimensional space (i.e. X

= C

) and the

mapping P : X

→ X

is holomorphic, then J

= {p} and lim

t→+∞

ρ(xt, pt) = 0 for

all x ∈ X

, i.e. the dynamical system (4.6) is convergent.

Proof. The proof of this assertion can be obtained with the help of a slight modi-

ﬁcation of the proof of Theorem 3.3. 

2. Consider a diﬀerential equation

˙u = f (t, u), (t ∈ R, u ∈ E

) (4.7)

where f is a τ-periodic function with respect to the variable t ∈ R and f is contin-

uously diﬀerentiable w.r.t. u ∈ E

. If the function f ∈ C(R × E

, E

) is regular,

then (see Example 3.1) the equation (4.7) generates a non-autonomous dynamical

system (and this system is τ-periodic). Applying to this non-autonomous dynamical

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The structure of the Levinson center with the condition of the hyperbolicity 169

system Theorems 4.6, 4.7 and 4.8, we will obtain the corresponding statements for

the equation (4.7).

Let P : E

→ E

be a Poincare transformation for the equation (4.7), i.e.

P (u) = ϕ(τ, u, f ).

Deﬁnition 4.8 The set of chain recurrent points for the equation (4.7) is said to

be a set of chain recurrent points of the cascade (E

, P ).

It is well known (see, for example,

[

33, p.56

]

), that for the equation (4.7) with

the diﬀerentiable (w.r.t. variable u) function f the set of chain recurrent points R

coincides with the set Π that generates periodic (with the period kτ , where k ∈ N)

solutions of the equation (4.7) (for the deﬁnition of the set Π see

[

271, p.278

]

Theorem 4.11 Suppose, that for the equation (4.7) the set of chain recurrent

points R is compact and the set M

(R) is hyperbolic, then

(1) the relation ∼ decomposes the set R on ﬁnite number of classes of equivalence,

i.e. R = R

t R

t ··· t R

;

(2) the sets R

(i =

1, k) are closed, invariant, indecomposable, locally maximal

and in the collection {R

, R

, . . . , R

} there is no r (r ≥ 1)-cycles;

Corollary 4.8 Suppose, that for the equation (4.6) the set Π, which generates

periodic solutions (with the period kτ, where k ∈ N), is compact and hyperbolic,

then:

(1) the relation ∼ decomposes the set Π on ﬁnite number of classes of equivalence,

i.e. Π = Π

tΠ

t ··· t Π

;

(2) the sets Π

(i =

1, k) are closed,invariant, quasi-minimal, locally maximal and

the collection {Π

, Π

, . . . , Π

} does not contain r-cycles;

(3) in addition, if in the set Π there is an inﬁnite number of minimal sets, then in

Π there exists a nontrivial minimal set.

This assertion generalizes Theorem 3.3 from

[

271, p.280

]

Theorem 4.12 Let the equation (4.7) be dissipative and the set J be its Levinson’s

center. Suppose that one of the following two conditions is fulﬁlled:

a. in the set J there is a ﬁnite number of minimal sets;

b. in the set J there is an inﬁnite number of minimal sets and the set M

(J) is

hyperbolic.

Then:

(1) the relation ∼ decomposes the set of chain recurrent points of the equation (4.7)

R on ﬁnite number of classes of equivalence, i.e. R = R

t R

t ··· t R

;

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170 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(2) the sets R

(i =

1, k) are closed, invariant, indecomposable, locally maximal

and in the collection {R

, R

, . . . , R

} there is no r-cycles (r ≥ 1);

(3) J = ∪{W

) : i ∈

1, k}.

Theorem 4.13 Suppose that the equation (4.7) is dissipative, J is its Levinson

center. If the set J contains an inﬁnite number of minimal sets and the set M

(J)

is hyperbolic, then the statements 1.-3. of Theorem 4.12 hold. In addition, in the

set J there is an inﬁnite number of hyperbolic periodic solutions.

Corollary 4.9 Under the conditions of Theorem 4.13 in the set J there is a locally

maximal markovian set Λ ⊆ J and, in particular, in the set J there is a transversal

homoclinic periodic trajectory.

This statement follows from Corollary 4.4.

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Chapter 5

Method of Lyapunov functions

5.1 Criterions of dissipativity in term of Lyapunov functions

In this section we suppose that (X, h, Y ) is a Banach ﬁber bundle.

Theorem 5.1 Let Y be compact and (X, T

, π) be completely continuous.

The non-autonomous dynamical system h(X, T

, π), (Y, T

, σ), hi is boundedly k-

dissipative if and only if there is a number r > 0 and a continuous function

V : E

→ R (E

:= {x ∈ E | |x| ≥ r}) with the following properties:

1. the set {x | V (x) ≤ c} is bounded for any c ∈ R;

2. if xτ ∈ E

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

3. level lines of the function V do not contain ω-limit points of the dynamical

system (X, T

, π).

Proof. Let K

= {x ∈ E | |x| ≤ r}, where r is the number from the statement

of the theorem. As Y is compact and the ﬁber bundle (X, h, Y ) is locally trivial,

then the set K

is bounded. As (X, T

, π) is completely continuous, then for the

bounded set K

there will be a number l such that π

is relatively compact.

Suppose K =

and let us show that ω

∩ K 6= ∅ for any x ∈ E. Suppose it

is not so, then there will be x

∈ E for which ω

∩ K 6= ∅ and, consequently, for

some t

≥ 0 we will have x

t /∈ K

for all t ≥ t

. If this were not so, then we should

have a sequence t

→ +∞ such that |x

| ≤ r and, hence, {x

+ l)} ⊆ K. We

consider the sequence {x

+ l)} to be convergent. Let

x = lim

n→+∞

+ l), then

x ∈ ω

∩K, but this contradicts the choice of x

. Thus x

t /∈ K

for all t ≥ t

and,

hence, V (x

t) ≤ V (x

) for all t ≥ t

. According to condition 1. of the theorem,

the set Σ

= {x

t | t ≥ 0} is bounded and by virtue of complete continuity of

(X, T

, π) it is relatively compact. From here it results that ω

6= ∅ is compact,

invariant and ω

⊆ E

. Let p ∈ ω

and ϕ : S → ω

is the whole trajectory of

the system (X, T

, π) passing through the point p ∈ ω

. Consider the function

µ : S

→ R deﬁned by the equality µ(t) = V (ϕ(−t)). It is continuous, bounded

171

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172 Global Attractors of Non-autonomous Dissipative Dynamical Systems

and non-decreasing and, consequently, there is c

= lim

t→+∞

µ(t). From here it results

that on the set α

:= {q | ∃t

→ +∞, ϕ(−t

) → q} the function V is continuous,

that is α

⊆ V

−1

) ∩ ω

. This contradicts the condition 3. of the theorem.

So ω

∩ K 6= ∅ for all x ∈ E. According to Theorems 1.22 the non-autonomous

dynamical system h(X, T

, π), (Y, T

, σ), hi be boundedly k-dissipative.

Let now h(X, T

, π), (Y, T

, σ), hi be boundedly k-dissipative, J be its Levinson

center and r > r

:= sup{|x| : x ∈ J}. Let us deﬁne a function V : E

→ R by the

equality

V (x) := sup{|xt| : t ≥ 0}. (5.1)

Directly from the deﬁnition of V it results that:

(1) V (x) ≥ |x| for all x ∈ E

and, therefore, it satisﬁes the condition 1. of the

theorem;

(2) if xτ ∈ E

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

Let us show that the function V , deﬁned by the equality (5.1), is continuous. Let

→ x(x, x

∈ E

) and R := sup{|x

| | n ∈ N}, then from Theorem 2.19 for the

number R > 0 there exists l(R) > 0 such that

|xt| ≤ r (t ≥ l(R) and |x| ≤ R). (5.2)

From (5.1) and (5.2) it results that

V (x) = |xτ|

for some τ (x) ∈ [0, l(R)] (|x| ≤ R). The sequence {V (x

)} = {|x

|} is relatively

compact, where τ

= τ(x

). Let us show that it has a unique limit point. Let V

be one of the limit points of the sequence {V (x

)}, then there is a subsequence

{V (x

)} such that V (x

) → V

for k → +∞. As {τ

} is bounded, then we

can consider it to be convergent. Then V

= |xτ

|, where τ

= lim

n→+∞

. Let us

show that |xτ

| = |xτ| = V (x). If we suppose that |xτ

| 6= |xτ|, then |xτ | > |xτ

Let  > 0 be such that |xτ

| + 2 < |xτ|. Then for k suﬃciently large we have

||x

τ| − |xτ || <  and ||x

|− |xτ

|| < , hence,

τ| > |τ | − > |xτ

|+  > |x

| = V (x

). (5.3)

Inequality (5.3) contradicts the choice of the number τ

. Thus V (x) = |xτ

| and,

hence V (x) = lim

n→+∞

V (x

), so that V is continuous.

Finally, let us show that level lines of the function V do not contain ω-limit points

(X, T

, π). Really, for any x ∈ E, we have ω

⊆ J. According to the deﬁnition of

the set E

it has not common points with J and therefore, ω

∩ V

−1

any x ∈ E

and c ∈ R. This proves the theorem. 

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Method of Lyapunov functions 173

Lemma 5.1 Let (X, T

, π) be completely continuous, h(X, T

, π), (Y,T

,σ),hi be a

non-autonomous dynamical system and there exists r > 0 and a continuous function

V : E

→ R which satisﬁes conditions:

1. the set {x ∈ E : V (x) ≤ c} is bounded for any c ∈ R;

2. if xτ ∈ E

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

Then the following conditions are equivalent:

3. level lines of the function V do not contain ω-limit points of the dynamical

system (X, T

, π);

4. level lines of the function V do not contain positive semi-trajectories of the

dynamical system (X, T

, π).

Proof. Let us show that Condition 3. implies Condition 4. If we suppose that it

is not so, then there will be c

and x

∈ V

−1

) such that x

t ∈∈ V

−1

) for all

t ≥ 0. In conditions of the lemma the level lines of the function V are bounded sets,

therefore Σ

is bounded. As (X, T

, π) is completely continuous then the set Σ

is relatively compact and, hence, ω

6= ∅, is compact and ω

⊆ V

−1

). The last

is a contradiction to the condition 3. This contradiction proves the implication we

need.

Let us show that from the condition 4. follows the condition 3. Suppose the

opposite, that there is c

∈ R and x

∈ E

such that ω

∩V

−1

) 6= ∅. Reasoning

like in Theorem 5.1 we can show that ω

is compact and α

⊆ ω

∩ V

−1

) 6= ∅

for some complete motion ϕ (ϕ(0) = p). As α

is positive invariant, then V

−1

)

contains positive semi-trajectories of the system (X, T

, π), but this contradicts the

condition 4. Lemma is proved. 

Corollary 5.1 Let Y be compact and the dynamical system (X, T

, π) be

completely continuous. The non-autonomous dynamical system h(X, T

, π),

(Y, T

, σ), hi is boundedly k-dissipative if and only if there are number r > 0 and

continuous function V : X

→ R, satisfying the conditions:

(1) V (x) ≥ a(|x|) for all x ∈ X

, where a ∈ A :={a | a : R

→R

, a is a continu-

ous and strongly monotonically increasing function } and Im V ⊆ Im (a);

(2) if xτ ∈ X

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

(3) level lines of the function V do not contain positive semi-trajectories of

dynamical system (X, T

, π).

Theorem 5.2 Let (X, T

, π) be completely continuous. A non-autonomous

dynamical system h(X, T

, π), (Y, T

, σ), hi is boundedly k-dissipative if there is a

number r > 0 and a continuous function V : E

→ R with the following properties:

(1) the set {x ∈ E | V (x) ≤ c} is bounded for any c ∈ R;

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174 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(2) if xτ ∈ E

for all τ ∈ [0, t], then V (xt) < V (x).

Proof. Let K

= {x ∈ E : |x| ≤ r}, l > 0, be such that π

is relatively compact

and K =

. Let us show that ω

∩ K 6= ∅ for all x ∈ E. Like in Theorem

5.1 it can be shown that in the opposite case we should have a point x

∈ E such

that Σ

is relatively compact, ω

6= ∅ is compact and ω

∩ K = ∅. Furthermore

there will be c

∈ R and a complete motion ϕ : S → ω

(ϕ(0) = p) such that

⊆ ω

∩ V

−1

) 6= ∅. As α

is positively invariant, then together with the

point x the set α

contains xt for all t > 0, and, hence, V (xt) < V (x) = c

. The

contradiction we have got proves the assertion we need. Thus ω

∩ K 6= ∅ for

all x ∈ E and according to Theorem 2.19 the non-autonomous dynamical system

h(X, T

, π), (Y, T

, σ), hi is boundedly k-dissipative. The Theorem is proved. 

Theorem 5.3 Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical

system and T

= R

. Assume that there is a number r > 0 and function

V : E

→ R

with the following properties:

1. the function V is bounded on bounded sets and for any c ∈ R the set {x ∈

| V (x) ≤ c} is bounded;

2. V

(x) ≤ −c(|x|) for all x ∈ E

, where c : R

→ R

is strictly positive on

[r, +∞), and V

(x) = lim sup

t→+0

−1

[V (xt) − V (x)].

Then there exists R

> 0 such that for any R > 0 there will be l(R) > 0 for which

|xt| ≤ R

for all t ≥ l(R) and |x| ≤ R.

Proof. First let us show that for any x ∈ E

there will be τ = τ (x) > 0 such

that |xτ| < r. If that assertion is not true then there exists x

∈ E

such that

τ| ≥ r for all τ ≥ 0. Consequently, V (x

τ) ≤ V (x

) for all τ ≥ 0 and the

set Σ

:= ∪{x

τ | τ ≥ 0} is bounded. Let b

:= sup{|x

τ | |τ ≥ 0} and ν =

inf{c(α) | r ≤ α ≤ b

} and we note that V

t) ≤ −ν, therefore

V (x

t) ≤ V (x

) − νt (5.4)

for all t > 0. The right hand side of (5.4) is negative for t suﬃciently large, but this

contradicts positive deﬁniteness of V on E

. This contradiction proves the assertion

we need.

Let b(R) := sup{V (x) : x ∈ E

, |x| ≤ R}, M(R) := {x ∈ E

| V (x) ≤ b(R)}

and R

:= sup{|x| | x ∈ M(R)}. According to condition 1. of the theorem the

set M (R) is bounded and, hence, R

< +∞. Let us show that |xt| ≤ R

for all

|x| ≤ r and t ≥ 0. In fact, if |x| ≤ r, |xβ

| = |xγ

| = r and |xt| > r for all

t ∈]β

, γ

[, then V (xt) ≤ V (xβ

) ≤ b(R) and, hence, xt ∈ M(R). From this it

follows that |xt| ≤ R

for all t ∈]β

, γ

[ and |x| ≤ r, and therefore |xt| ≤ R

for all t ≥ 0 and |x| ≤ r. According to the proved above for any x ∈ E

there

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Method of Lyapunov functions 175

will be τ (x) > 0 such that |xτ| < r. Suppose ν

:= sup{t | xτ ∈ E

for all

τ ∈ [0, t]}, then |xν

| = r and |xt| > r for all t ∈]0, ν

[. Let us notice that for

any R > 0 the set B(R) := {xt | x ∈ E

, |x| ≤ R and t ∈]0, ν

[} is bounded.

Indeed, V (xt) ≤ V (x) ≤ b(R) for all |x| ≤ R and t ∈]0, ν

[, therefore xt ∈ M(R).

According to condition 1. of the theorem the set M(R) is bounded. Let us show

now that L(R) := sup{ν

| |x| ≤ R} is ﬁnite for any R > 0. If the assertion is

not true, then there will be

R > 0 and {x

} such that |x

| ≤ R and ν

→ +∞.

Suppose d(R) := sup{|x| | x ∈ B(R)}. As the set B(R) is bounded, then we have

d(R) < +∞. Suppose

γ(R) := inf{c(ν) | r ≤ ν ≤ d(R)} and let us notice that

(xt) ≤ −

γ for all |x| ≤ R and t ∈]0, ν

[ and, hence, V (xt) ≤ V (x) − γ(R)t. In

particular,

V (x

) ≤ b(

R) − γ(R)ν

(5.5)

for all n. The right hand side of (5.5) is negative for n suﬃciently large and this

contradicts the positive deﬁniteness of V on E

. From this contradiction it follows

that for every R > 0 the number L(R) > 0 and is ﬁnite. Now for ﬁnishing the proof

of the theorem it is suﬃcient to notice that |xt| ≤ R

for all |x| ≤ R and t ≥ L(R).

Corollary 5.2 Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical

system and T

= R

. Assume that there is a function V : E → R which satis-

ﬁes the following conditions:

(1) V is bounded on the bounded sets from E;

(2) V (x) ≥ γ

|x|

−D

(γ

, D

, m > 0) for all x ∈ E;

(3) V

(x) ≤ −γ

V (x) + D

(γ

, D

> 0) for all x ∈ E.

Then there exists R

> 0 such that for any R > 0 there will be l(R) > 0 for which

|xt| ≤ R

for all |x| ≤ R and t ≥ l(R).

Proof. This assertion follows from Theorem 5.3. To prove that it is suﬃcient to

notice that for |x| suﬃciently large conditions 1.–3. of Corollary 5.1 guarantee the

fulﬁllment of the conditions of Theorem 5.3. 

Corollary 5.3 Under the conditions of Theorem 5.3, if Y is compact and

(X, T

, π) is asymptotic compact, then the non-autonomous dynamical system

h(X, T

, π), (Y, T

, σ), hi is boundedly k-dissipativ.

Proof. This statement follows from Theorems 5.3 and 2.21. 

Remark 5.1 The statement, analogously to Corollary 5.3, holds also for the

dynamical system with discrete time T

= Z

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176 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Remark 5.2 Unlike Theorems 5.1 and 5.2 in Theorem 5.3 we do not demand the

continuity of the function V , but instead of that we impose certain condition on the

velocity of decreasing of the function V along trajectories of the system (X, T

, π).

In additional, when T

= R

the completeness of the space is not necessary (in the

applications there are examples, when Y is a fortiori incomplete).

Theorem 5.4 Let h(X, T

, π), (Y, T

, σ), hi be a non-autonomous dynamical

system, T

= R

, (X, T

, π) be asymptotically compact and there is a continuous

function V : E → R satisfying the following conditions:

1. for all c ∈ R the set {x ∈ E| V (x) ≤ c} is bounded;

2. along trajectories of the system (X, T

, π) the function V is non-increasing, i.e.

V (xt) ≤ V (x) for all x ∈ E and t ≥ 0;

3. there exists r > 0 such that V (xt) < V (x), if xτ ∈ E

for all τ ∈ [0, t] and

t > 0.

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

, π), (Y, T

, σ), hi is locally k-

dissipative.

Proof. First let us notice that as V is continuous, then in a point p ∈ X there is

∈ R and δ

> 0 such that V (x) ≤ C

for any x ∈ B(p, δ

). According to the

condition 2. V (xt) ≤ V (x) ≤ C

for all x ∈ B(p, δ

) and t ≥ 0. According to the

condition 1. the set M

:= Σ

(B(p, δ

)) = ∪{π

B(p, δ

)|t ≥ 0} is bounded and as

(X, T

, π) is asymptotically compact, then there is a nonempty compact K

such

that the equality

lim

t→+∞

β(π

, K

) = 0 (5.6)

takes place. Deﬁne K := {x ∈ X | |x| ≤ r}, where r > 0 is the number from the

condition 3. of the theorem. As Y is compact and as the ﬁber bundle (X, h, Y ) is

locally trivial, then the set K is bounded. Let us notice that ω

∩ K 6= ∅ for any

point p ∈ X. In fact, if this were not so, we should have a point p

∈ X such that

∩ K = ∅. According to the equality (5.6) Σ

:= ∪{π

| t ≥ 0} is relatively

compact and, hence, ω

6= ∅ is compact and invariant. The function µ : R

→ R

deﬁned by the equality µ(t) := V (p

t) is continuous, bounded and monotone non-

increasing and, hence, there is lim

t→+∞

V (p

t) = c

. As V is continuous, we have

⊆ V

−1

). Let x ∈ ω

, then V (xt) = V (x) = c

for all t > 0. The last

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

∩K 6= ∅

for all p ∈ X. Let us show that as a matter of fact ω

⊆ K for all p ∈ X. Indeed, if

it is not so, then there will be t

> 0 such that xτ ∈ ω

\K for all τ ∈ [0, t

] and,

hence, V (xt

) < V (x). On the other hand, reasoning as we do above, we will have

V (x) = c

for some c

∈ R and all x ∈ ω

. This contradicts the last inequality.

Thus Ω ⊆ K, where Ω :=

∪{ω

|p ∈ X}, and, as Ω is invariant and as (X, T

, π)