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

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Dissipativity of some classes of equations 247

where kfk := max{|f(ω)| : ω ∈ Ω} and w(t) := |ϕ(t, x, ω)|

. Then

≤ −2αw + 2kfkw

Thus

w(t) ≤ v(t)

for all t ∈ [0, t

(x,ω)

), where v(t) is a solution of the equation

= −2αv + 2||f||v

satisfying condition v(0) = w(0) = |x|

. Hence

v(t) = [(|x| −

kfk

−αt

kfk

]

and consequently

|ϕ(t, x, ω)| ≤ (|x| −

kfk

−αt

kfk

(6.65)

for all t ∈ [0, t

(x,ω)

). It follows from the inequality (6.65) that the solution ϕ(t, x, ω)

is bounded and therefore it may be extended to a global solution on R

= [0, +∞).

Thus we have proved the following theorem.

Theorem 6.14 Let the conditions (6.59) and (6.60) are fulﬁlled. Then the follow-

ing statements hold:

(i)

|ϕ(t, x, ω)| ≤ C(|x|),

for all t ≥ 0, ω ∈ Ω and x ∈ H, where C(r) = r if r ≥ r

kfk

and C(r) = r

if r ≤ r

;

(ii)

lim sup

t→+∞

sup{|ϕ(t, x, ω)| : |x| ≤ r, ω ∈ Ω} ≤

||f||

for every r > 0.

The item (i) in this Theorem means that the non-autonomous Lorenz ﬂow is

bounded on bounded sets, while the item (ii) implies that the non-autonomous

Lorenz system is dissipative, i.e., it admits a bounded absorbing set.

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248 Global Attractors of Non-autonomous Dissipative Dynamical Systems

6.6.2 Non-autonomous dissipative dynamical systems and their

attractors

Let Ω and W be two metric spaces and (Ω, R, σ) be an autonomous dynamical

system on Ω. Let us consider a continuous mapping ϕ : R

×W ×Ω → W satisfying

the following conditions:

ϕ(0, ·, ω) = id

ϕ(t + τ, x, ω) = ϕ(t, ϕ(τ, x, ω), ωτ)

for all t, τ ∈ R

, ω ∈ Ω and x ∈ W . Here ωτ is the short notation for σ

(ω) :=

σ(τ, ω). Such a mapping ϕ ( or more explicitly hW, ϕ, (Ω, R, σ)i) is called a cocycle

over (Ω, R, σ) with ﬁber W (see

[

6, 292

]

Example 6.2 Let E be a Banach space and C(R × E, E) be a space of all con-

tinuous functions F : R × E → E equipped by the compact–open topology. Let us

consider a parameterized diﬀerential equation

= F (σ

ω, x), ω ∈ Ω

on a Banach space E with Ω := C(R ×E, E), where σ

ω := σ(t, ω). We will deﬁne

: Ω → Ω by σ

ω(·, ·) = ω(t+·, ·) for each t ∈ R and interpret ϕ(t, x, ω) as solution

of the initial value problem

x(t) = F (σ

ω, x(t)), x(0) = x. (6.66)

Under appropriate assumptions on F : Ω × E → E (or even F : R ×E → E with

ω(t) instead of σ

ω in (6.66)) to ensure forward existence and uniqueness, then ϕ

is a cocycle on (C(R × E, E), R, σ) with ﬁber E. Note that (C(R × E, E), R, σ) is

a Bebutov’s dynamical system (see for example

[

]

[

102

]

[

292

]

[

300

]

Let ϕ be a cocycle on (Ω, R, σ) with the ﬁber E. Then the mapping π : R

×E×Ω

→ E × Ω deﬁned by

π(t, x, ω) := (ϕ(t, x, ω), σ

ω)

for all t ∈ R

and (x, ω)∈ E ×Ω forms a semi-group {π(t, ·, ·)}

t∈R

of mappings of

X := Ω × E into itself, thus a semi-dynamical system on the state space X, which

is called a skew-product ﬂow

[

292

]

. The triplet h(X, R

, π), (Ω, R, σ), hi (where

h := pr

: X → Ω) is a non-autonomous dynamical system (see

[

33, 102

]

Deﬁnition 6.7 Recall that a cocycle ϕ over (Ω, R, σ) with the ﬁber W is called a

compact (bounded) dissipative cocycle, if there is a nonempty compact set K ⊆ W

such that

lim sup

t→+∞

{β(U(t, ω)M, K)|ω ∈ Ω} = 0

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Dissipativity of some classes of equations 249

for any M ∈ C(W ) (respectively M ∈ B(W )), where C(W )( B(W )) denotes the

family of all compact (bounded) subsets of W , β is the semi-distance of Hausdorﬀ

and U(t, ω) := ϕ(t, ·, ω). We can similarly deﬁne a compact or bounded dissipative

skew-product system.

Lemma 6.10 Let Ω be a compact metric space and hW, ϕ, (Ω, R, σ)i be a cocy-

cle over (Ω, R, σ) with the ﬁber W . In order for hW, ϕ, (Ω, R, σ)i to be compact

(bounded) dissipative, it is necessary and suﬃcient that the skew-product dynamical

system (X, R

, π) is compact (bounded) dissipative.

Proof. This assertion follows from the corresponding deﬁnitions. 

Deﬁnition 6.8 The cocycle hW, ϕ, (Y, R, σi is called compact (asymptotically

compact) if the associated skew-product dynamical system (X, R

, π) with X =

W × Y and π = (ϕ, σ) is compact (respectively asymptotically compact).

Let (X, R

, π) be compact dissipative and K be a compact set, which attracts

all compact subsets of X. Let

J = Ω(K), (6.67)

where Ω(K) =

t≥0

τ≥t

K. The set J deﬁned by the equality (6.67) does not

depend on selection of the attracting set K, and is characterized only by the prop-

erties of the dynamical system (X, R

, π) itself. The set J is called the Levinson’s

centre of the compact dissipative system (X, R

, π).

Remark 6.5 Let hW, ϕ, (Ω, R, σi be a compact dissipative cocycle over (Ω, R, σi

with the ﬁber W and {I

| ω ∈ Ω} is its compact global attarctor. Then the skew-

product dynamical system (X, R

, π), generated by cocycle ϕ, is compact dissipative

too and the set J := ∪{J

| ω ∈ Ω}, where J

:= I

× {ω}, is the Levinson center

of (X, R

, π).

Applying the general theorems about non-autonomous dissipative systems to

non-autonomous system constructed in the example 6.2, we will obtain series of

facts concerning the non-autonomous Lorenz system (6.58). In particular we have

the following results.

Theorem 6.15 Let Ω be a compact metric space, (Ω, R, σ) be a dynamical system

on Ω and the conditions (6.59) and (6.60) are fulﬁlled. If the cocycle ϕ generated

by non-autonomous Lorenz system (6.58) is asymptotically compact, then for every

ω ∈ Ω, there exists a non-empty compact connected set I

⊂ H such that the

following conditions hold:

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250 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(1) The set I = ∪{I

: ω ∈ Ω} is compact and connected in H, if the space Ω is

connected;

(2)

lim

t→+∞

sup

ω∈Ω

β(U(t, ω

−t

)M, I) = 0

for any bounded set M ⊂ H, where U (t, ω) := ϕ(t, ·, ω) and β is the semi-

distance of Hausdorﬀ;

(3) U (t, ω)I

= I

ωt

for all t ∈ R

and ω ∈ Ω;

(4) I

consists of those and only those points x ∈ H through which passes the

bounded solutions (on R) of the non-autonomous Lorenz system (6.58).

Proof. This assertion follows from Theorems 6.14, 2.24 and 2.25. 

This theorem states that ∪{I

: ω ∈ Ω} is the compact global attractor of the

non-autonomous Lorenz system (6.58) and also characterizes the structure of the

sections I

of the attractor.

Theorem 6.16 Under conditions of Theorem 6.15

|ϕ(t, x, ω)| ≤

kfk

(6.68)

for all t ∈ R, ω ∈ Ω and x ∈ I

, where ϕ is the cocycle generated by Lorenz

non-autonomous system (6.58).

Proof. According to Remark 6.5 the set J :=

× {ω} : ω ∈ Ω} is a Levin-

son’s centre of dynamical system (X, R

, π) and according to (6.67) for any point

, y

) = z ∈ J there exists t

→ +∞, u

∈ H and ω

∈ Ω such that the sequence

} is bounded, u

= lim

n→+∞

ϕ(t

, u

, ω

) and ω

= lim

n→+∞

. From the inequal-

ity (6.65), it follows that |u

| ≤

kfk

, i.e. ϕ(t, x, ω) ∈ I

ωt

for all ω ∈ Ω and t ∈ R,

hence |ϕ(t, x, ω)| ≤

kfk

for any t ∈ R, x ∈ I

and ω ∈ Ω. The theorem is proved.

Theorem 6.17 Let ϕ be the cocycle generated by the Lorenz non-autonomous

system (6.58). Under conditions of Theorem 6.15 and further assume that

−2

kfk < 1. Then this cocycle ϕ is convergent, i.e. for any ω ∈ Ω the set

contains a single point u

Proof. Let ω ∈ Ω and u

, u

∈ I

. We deﬁne ψ(t) := ϕ(t, u

, ω) −ϕ(t, u

, ω) and

w(t) := |ϕ(t, u

, ω) − ϕ(t, u

, ω)|

According to Theorem 6.16, the function w(t) is bounded on R. On the other hand,

in view of (6.64) and (6.59), we have

(t) ≤ −2αw(t) + 2RehB(ωt)(ψ(t), ϕ(t, u

, ω)), ψ(t)i. (6.69)

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Dissipativity of some classes of equations 251

From the inequalities (6.63), (6.69) and Theorem 6.16, it follows that

(t) ≤ −2αw(t) + 2C

kfk

w(t).

Hence, w(t) ≤ w(0)e

−2(α−C

kfk

, i.e.

|ϕ(t, u

, ω) −ϕ(t, u

, ω)| ≤ |u

−u

−(α−

kfk

for all t ≥ 0, ω ∈ Ω and u

, u

∈ I

. In particular,

−u

| ≤ |ϕ(t, ϕ(−t, u

, σ(−t, ω)), ω) − (6.70)

ϕ(t, ϕ(−t, u

, σ(−t, ω)), ω)|e

−(α−

kfk

for all t ≥ 0, ω ∈ Ω and u

, u

∈ J

. Note that |ϕ(t, u

, ω) −ϕ(t, u

, ω)| is bounded

on R. Thus from (6.70) it follows that u

= u

, where ϕ(−t, x, ω) := u

σ(−t,ω)

for

all x ∈ I

, t ≥ 0 and ω ∈ Ω. The theorem is proved. 

6.6.3 Almost periodic and recurrent solutions of non-autonomous

Lorenz systems

In this subsection, we discuss the problem of existence of almost periodic and recur-

rent solutions of non-autonomous Lorenz systems. Let T = R or R

and (X, T, π)

be a dynamical system.

Deﬁnition 6.9 The solution ϕ(t, x, ω) of non-autonomous Lorenz system (6.58)

is called recurrent (almost periodic, quasi-periodic, periodic), if the point (x, ω) ∈

H×Ω is a recurrent (almost periodic, quasi-periodic, periodic) point of skew-product

dynamical system (X, R

, π) (X := H ×Ω and π := (ϕ, σ)).

We note (see, for example,

[

238

]

[

300

]

and

[

302

]

) that if ω ∈ Ω is a stationary

(τ-periodic, almost periodic, quasi periodic, recurrent) point of dynamical system

(Ω, R, σ) and h : Ω → X is a homomorphism of dynamical system (Ω, R, σ) onto

(X, R

, π), then the point x = h(ω) is a stationary (τ-periodic, almost periodic,

quasi periodic, recurrent) point of the system (X, R

, π).

Let X := H × Ω and π := (ϕ, σ), then mapping h : Ω → X is a homomorphism

of dynamical system (Ω, R, σ) onto (X, R

, π) if and only if h(ω) = (u(ω), ω) for

all ω ∈ Ω, where u : Ω → H is a continuous mapping with the condition that

u(ωt) = ϕ(t, u(ω), ω) for all ω ∈ Ω and t ∈ R

Theorem 6.18 Let Ω be a compact metric space, the cocycle ϕ, generated by the

non-autonomous Lorenz system (6.58), is asymptotic compact and the conditions

(6.59), (6.61)-(6.62) are fulﬁlled with

kfkC

< 1. Then the set I

contains a unique

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252 Global Attractors of Non-autonomous Dissipative Dynamical Systems

point x

= {x

}) for every ω ∈ Ω, the mapping u : Ω → H deﬁned by u(ω) :=

is continuous and u(ωt) = ϕ(t, u(ω), ω) for all ω ∈ Ω and t ∈ R

Proof. According to Theorem 6.17, it is suﬃcient to show that the mapping u : Ω →

H deﬁned above is continuous. Let ω ∈ Ω, {ω

} ⊆ Ω and ω

→ ω. Consider the

sequence {x

} := {x

} ⊂ I :=

| ω ∈ Ω}. Since the set I is compact, then the

sequence {x

} is relatively compact. Let x

be a limit point of this sequence, then

there is a subsequence {x

} such that x

→ x

. Let J be a Levinson center of the

skew-product dynamical system (X, R

, π), generated by the cocycle ϕ. Note that

the point (x

, ω

) ∈ J

:= I

× {ω

} ⊆ J and taking in the consideration

that J is compact we obtain that (x

, ω) ∈ J. Thus (x

, ω) ∈ J

= I

× {ω} and,

consequently, x

∈ I

= {x

}, i.e. the relatively compact sequence {x

} has a

unique limit point x

. This means that the sequence {x

} converges to x

n → +∞. The theorem is proved. 

Corollary 6.5 Let Ω be a compact minimal (almost periodic minimal, quasi-

periodic minimal or periodic minimal) set of dynamical system (Ω, R, σ). Then

under the conditions of Theorem 6.18, the non-autonomous Lorenz system (6.58)

admits a compact global attractor I, and for all ω ∈ Ω, the section I

of the attractor

contains a unique point x

through which passes a recurrent (almost periodic, quasi-

periodic, or periodic) solution of the equation (6.58).

Let H be a d-dimensional complex Euclidean space, i.e. H = C

. Denote by

HC(C

×Ω, C

) the space of all continuous functions f : C

×Ω → C

holomorphic

in z ∈ C

and equipped with compact-open topology. Consider the diﬀerential

equation

= f (z, σ

ω), (ω ∈ Ω) (6.71)

where f ∈ HC(C

×Ω, C

). Let ϕ(t, ω, z) be the solution of equation (6.71) passing

through the point z at t = 0 and deﬁned on R

. The mapping ϕ : R

×Ω×C

→ C

has the following properties (see, for example,

[

]

and

[

186

]

a) ϕ(0, z, ω) = z for all z ∈ C

b) ϕ(t + τ, z, ω) = ϕ(t, ϕ(τ, z, ω), σ

ω) for all t, τ ∈ R

, ω ∈ Ω and z ∈ C

c) Mapping ϕ is continuous.

d) Mapping U(t, ω) := ϕ(t, ·, ω) : C

→ C

is holomorphic for any t ∈ R

and

ω ∈ Ω.

Deﬁnition 6.10 The cocycle hC

, ϕ, (Ω, T, θ)i is said to be C-analytic if the

mapping U(t, ω) : C

→ C

is holomorphic for all t ∈ R

and ω ∈ Ω.

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Dissipativity of some classes of equations 253

Example 6.3 Let (HC(R ×C

, C

), R, σ) be a dynamical system of translations

on HC(R×C

, C

) (Bebutov’s dynamical system (see, for example,

[

102

]

and

[

300

]

)).

Denote by F the mapping from C

× HC(R × C

, C

) to C

deﬁned by equality

F (z, f) := f(0, z) for all z ∈ C

and f ∈ HC(R × C

, C

). Let Ω be the hull

H(f) of given function f ∈ HC(R × C

, C

), that is Ω = H(f ) :=

|τ ∈ R},

where f

(t, z) := f(t + τ, z) for all t, τ ∈ R and z ∈ C

. Denote the restriction of

(HC(R × C

, C

), R, σ) on Ω by (Ω, R, σ). Then, under appropriate restriction on

the given function f ∈ HC(R × C

, C

), the diﬀerential equation

= f (z, t) =

F (z, σ

f) generates a C−analytic cocycle.

Theorem 6.19 Let H := C

, Ω be a compact minimal set and the conditions

(6.59), (6.61)-(6.62) are fulﬁlled. Then the non-autonomous Lorenz system admits

a compact global attractor {I

| ω ∈ Ω} and the set I

contains a unique point

= {x

}) for every ω ∈ Ω, the mapping u : Ω → H deﬁned by equality

u(ω) := x

is continuous and u(ωt) = ϕ(t, u(ω), ω) for all ω ∈ Ω and t ∈ R

, where

ϕ is a cocycle generated by the non-autonomous Lorenz system.

Proof. We note that under the conditions of Theorem 6.19 the right-hand side

f(ω, z) := A(ω)z + B(ω)(z, z) + f(ω) is C-analytic because D

f(ω, z)h = A(ω)h +

B(ω)(h, z) + B(ω)(z, h) for all ω ∈ Ω and z ∈ C

, where D

f(ω, z) is a derivative of

function f (ω, z) w.r.t. z ∈ C

. Now our statement directly results from Theorems

6.15 and 3.2. The proof is complete. 

Corollary 6.6 Let Ω be a compact minimal (almost periodic minimal, quasi-

periodic minimal or periodic minimal) set of dynamical system (Ω, R, σ). Then

under the conditions of Theorem 6.19, the non-autonomous Lorenz system (6.58)

admits a compact global attractor I and for all ω ∈ Ω, the set I

contains a unique

point x

through which passes a recurrent (almost periodic, quasi-periodic or peri-

odic) solution of equation (6.58).

6.6.4 Uniform averaging principle

Now we consider a uniform averaging principle for a general class of diﬀerential

equations. In the next subsection, we apply this averaging principle to the non-

autonomous Lorenz system (6.58).

Let C(R × H, H) be the space of all continuous functions f : R × H → H

equipped with compact open topology and let F ⊆ C(R ×H, H). In Hilbert space

H (with the norm |·| induced by the scalar product) we will consider the family of

equations

= εf(t, x), f ∈ F, (6.72)

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254 Global Attractors of Non-autonomous Dissipative Dynamical Systems

containing a small parameter ε ∈ [0, ε

] (ε

> 0).

We assume that on the set R

× B[0, r], where B[0, r] := {x ∈ H | |x| ≤ r} is

a ball of radius r > 0 in H, the functions f ∈ F are uniformly bounded, i.e. there

exists a positive constant M such that

|f(t, x)| ≤ M (6.73)

for every f ∈ F, t ∈ R

and x ∈ B[0, r], and satisﬁes the condition of Lipschitz

|f(t, x

) − f(t, x

)| ≤ L|x

−x

| (x

, x

∈ B[0, r]) (6.74)

with a constant L > 0 does not depending neither on t ∈ R

nor on f ∈ F.

Furthermore, we assume that the mean value of f is uniform with respect to

(w.r.t.) f ∈ F and x ∈ B[0, r]

(x) := lim

T →+∞

f(t, x)dt, (6.75)

i.e. for every ε > 0 there exists a l = l(ε) > 0 such that

f(t, x)dt − f

(x)| < ε

for all T ≥ l(ε), x ∈ B[0, r] and f ∈ F, and the function f

does not depend on

f ∈ F.

Lemma 6.11 The condition (6.75) holds if and only if there exists a decreasing

continuous function m : R

→ R

, satisfying the condition m(t) → 0 as t → +∞,

such that

f(t, x)dt − f

(x)| ≤ m(T ) (6.76)

for all T > 0, x ∈ B[0, r] and f ∈ F. The function m does not depend neither on

x ∈ B[0, r] nor on f ∈ F.

Proof. Denote by

k(T ) := sup

f∈F,x∈B[0,r]

f(t, x)dt − f

(x)|.

The mapping k possesses the following properties:

(1) 0 ≤ k(T ) ≤ 2M, where M := sup{|f(t, x)| : f ∈ F, |x| ≤ r};

(2) k(T ) → 0 as T → +∞.

Let

:= sup

T ≥n

k(T ),

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Dissipativity of some classes of equations 255

then c

≥ c

≥ ... ≥ c

≥ ... and c

→ 0 as n → +∞. Deﬁne now the function

m : R

→ R

by the equality

m(t) := c

n−1

+ (t − n)(c

−c

n−1

) (n ≤ t ≤ n + 1, n = 0, 1, ...),

where c

−1

:= c

+ 1. The lemma is proved. 

Lemma 6.12 Let F ⊆ C(R × E, E) be a family of functions satisfying the con-

dition (6.75), then for every L > 0

l(ε) := sup{|

, x)dt − f

(x)| : 0 ≤ τ ≤ L, f ∈ F, |x| ≤ r} → 0

as ε → 0.

Proof. According to Lemma 6.11 there exists a decreasing continuous function

m : R

→ R

with the condition m(t) → 0 as t → +∞ and such that the inequality

(6.76) holds. Let ν ∈ (0, 1), then

l(ε) ≤ sup{|

, x)dt| : 0 ≤ τ ≤ ε

, f ∈ F, |x| ≤ r} +

sup{|

, x)dt| : ε

≤ τ ≤ L, f ∈ F, |x| ≤ r} =

sup{τ|

f(t, x)dt| : 0 ≤ τ ≤ ε

, f ∈ F, |x| ≤ r} +

sup{τ|

f(t, x)dt| : ε

≤ τ ≤ L, f ∈ F, |x| ≤ r} ≤

m(0)ε

+ Lm(ε

ν−1

) → 0

as ε → 0. The lemma is proved. 

Under the assumptions above, it is expedient to consider along with equation

(6.72) the averaged equation

= εf

(x).

From (6.75) we see that the function f

also satisﬁes the conditions (6.73) and

(6.74). Let ϕ(t, x) (0 ≤ t ≤ T

) be a solution of equation

= f

(y). (6.77)

taking values in B[0, r] and passing through the point x at the initial moment

t = 0. Then, as can easily be seen, the function ϕ(t, x, ε) := ϕ(εt, x) is the solution

of equation (6.77) on the interval 0 ≤ t ≤

. We will establish below a connection

between ϕ(t, x, ε) and the solution ϕ(t, x, f, ε) of equation (6.72) with the initial

condition ϕ(0, x, f, ε) = x.

More precisely, we will prove the following assertion.

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256 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Theorem 6.20 Suppose that on R

× B[0, r], the functions f ∈ F satisfy the

conditions (6.73)-(6.75). Then for any η > 0 there exists an ε > 0 (0 < ε < ε

)

such that the estimate

|ϕ(t, x, f, ε) − ϕ(t, x, ε)| ≤ η (0 ≤ t ≤

)

holds uniformly w.r.t. f ∈ F and x ∈ B[0, r].

Denote by K the family of all solutions x : [0, T

] → B[0, r] of the equation

(6.77). Let us prove an auxiliary assertion.

Lemma 6.13 Let F ⊆ C(R × H, H) be a family of functions satisfying the con-

ditions (6.73)-(6.75). Then the equality

lim

ε→0

, x(s))ds =

(x(s))ds (0 < τ ≤ T

)

holds uniformly w.r.t. x ∈ K, τ ∈ [0, T

] and f ∈ F.

Proof. Observe that

lim

ε→0

, x)dτ = τ f

(x) (6.78)

or, equivalently,

lim

ε→0

f(t, x)dt = f

(x) (6.79)

uniformly w.r.t. x ∈ K, τ ∈ [0, T

] and f ∈ F. In fact, according to Lemma 6.12

f(t, x)dt − f

(x)| → 0

as ε → 0 uniformly w.r.t. x ∈ B[0, r], f ∈ F and τ ∈ [0, T

]. Let us note that

the equality (6.79) is equivalent to (6.75). From (6.78) it follows that for any

, τ

∈ [0, T

] we have

lim

ε→0

, x)dτ =

(x)dτ

uniformly w.r.t. x ∈ B[0, r] and f ∈ F. Hence for any 0 ≤ τ

< τ

< ...τ

n−1

< τ

, x

∈ B[0, r] (k = 1, 2, ..., n), we conclude that

lim

ε→0

k−1

f(τ, x

, ε)dτ =

k−1

)dτ (6.80)

uniformly w.r.t. x

, x

, ..., x

∈ B[0, r] and f ∈ F.