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1. AVERAGING METHOD IN OSCILLATION SYSTEMS WITH VARIABLE FREQUENCIES

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1. Uniform Estimates for One-Dimensional

Oscillation Integrals

In the present section, we study properties of oscillation integrals of the form

I(τ, ε) =

_τ

f(t) exp

a(z)dz

dt, τ ∈ [0, L], (1.1)

where f(τ) = (f1(τ ), . . . , fn(τ )), fj(τ ) and a(τ ) are real functions, j = 1, n,

L is a positive constant, ε0 ≥ ε is a positive small parameter, and i is the

imaginary unit. Integrals of this type appear in the study of oscillation phenomena

in various problems of classical and celestial mechanics, physics, and engineering

[Arn2, Arn3, Gre, Mit4]. In [PlL, Sam5, SPe3, SPe4, Kha1, Kha2], estimates of

integrals of the type (1.1) were considered for the justification of the averaging

method in multifrequency systems with slow and fast variables.

In what follows, we investigate the dependence of oscillation integrals on the

value of the small parameter ε and on properties of the functions f(τ ) and a(τ ).

The analysis of integral (1.1) shows that, for f(τ ) _≡ 0, an estimate of I(τ, ε)

substantially depends on the character of zeros of the function a(τ ). In particular,

if a(τ ) ≡ 0 and fj(τ ) ≡ 1 for any τ ∈ [0, L] and certain j, then _I(L, ε)_ ≥

L. In what follows, unless otherwise stated, the norm of a matrix is understood as

the sum of the absolute values of its elements.

Theorem 1.1. Let a(τ ) ∈ Cp

[0,L], p ≥ 1, let f(τ ) ∈ C1

[0,L], and let a(τ )

have zeros of multiplicity not higher than p on [0, L]. Then there exist a constant

10 Averaging Method in Systems with Variable Frequencies Chapter 1

ε1 > 0 and a constant c1 > 0 independent of ε such that

_I (τ, ε)_ ≤ c1ε

p+1 (1.2)

for all τ ∈ [0, L] and ε ∈ (0, ε1].

Proof. It is known [Sam5] that, under the assumptions made above, a(t)

has finitely many zeros t1 < t2 < ... < ts of multiplicities r1, r2, . . . , rs,

respectively, on [0, L]; here, rj ≤ p for all j = 1, s. Since

|a(rj ) (tj)| ≡ c(j) > 0 ∀j = 1, s, a(rj )(t) = drja(t)

dtrj

taking into account the continuity of the functions a(rj )(t) we establish that there

exists a number δ > 0 independent of j and such that

|a(rj )(t)| ≥ 1

2c(j) ≥ c2 =

min

c(j)

for |t − tj| ≤ δ, t ∈ [0, L]. We choose δ <

min

1≤j≤s−1

(tj+1 − tj) and denote

by B(τ ) the set

j=1

[tj − δ, tj + δ] ∩ [0, τ] and by A(τ ) the closure of the set

[0, τ]\B(τ ). Then [0, τ] = A(τ ) ∪ B(τ ) and, furthermore, the function a(t) is

nonzero at every point of the set A(τ ). Therefore,

min

t∈A(τ)

|a(t)| ≥ min

t∈A(L)

|a(t)| = c3 > 0 (1.3)

and the inequality

|a(rj )(t)| ≥ c2 (1.4)

holds on each segment Tj = [tj − δ, tj + δ] ∩ [0, τ], j = 1, s, of the set B(τ ).

It follows from (1.4) that the function a(rj−1)(t) vanishes on Tj at at most one

point t1,j ; moreover, for t ∈ Tj\[t1,j − μ, t1,j + μ] and 0 < μ < min{1; δ},

the inequality |a(rj−1)(t)| ≥ c2μ is satisfied. If a(rj−1)(t) does not change its

sign on Tj , then, as t1,j , we choose, respectively, the left or the right endpoint

of this segment, depending on whether the function |a(rj−1)(t)| is increasing or

decreasing. We assume that Tj ∩ [t1,j − μ, t1,j + μ] belongs to the set A(tj, μ)

and use analogous arguments for the functions a(l)(t), l = 0, rj − 1. As a result,

we establish that the set A(tj, μ) consists of d1(tj, μ) ≤ 2p − 1 segments of

length not greater than 2μ, and the set B(tj, μ), which is the closure of the set

Section 1 Uniform Estimates for One-Dimensional Oscillation Integrals 11

Tj\A(tj, μ), consists of d2(tj, μ) ≤ 2p segments on each of which the following

inequality is true:

|a(t)| ≥ c2 μp. (1.5)

Note that the function a(1)(t) does not change its sign on each segment of the set

B(tj, μ).

We represent I(τ, ε) in the form of a sum, namely,

I(τ, ε) =

A(τ)

F(t, ε) dt +

_ _

A(τj,μ)

F(t, ε)dt +

B(τj,μ)

F(t, ε) dt

, (1.6)

where F(t, ε) is the integrand of I(τ, ε). According to the definition of the set

A(τj, μ), we have

____

A(τj,μ)

F(t, ε)dt

____

≤ 2μ(2p − 1) max

[0,L]

_f(t)_ ≡ c(1)

4 μ. (1.7)

Let [α, β] be a segment from the set B(tj, μ). Then, integrating by parts and

taking (1.5) into account, we obtain

____

_β

F(t, ε)dt

____

= ε

____

_β

f(t)

a(t) d

exp

a(z)dz

____

≤ ε

c2μp +

_β

|a(1)(t)|

a2(t) dt

max

[0,L]

_f(t)_

+ ε

_β

c2μp

_f(1)(t)_dt.

Since a(1)(t) does not change its sign on [α, β], the relations

_β

|a(1)(t)|

a2(t) dt =

____

_β

_ 1

a(t)

____

≤

____

a(β)

− 1

a(α)

____

≤ 2

c2μp

yield the estimate

____

_β

F(t, ε)dt

____

≤ ε

c2μp

4 max

[0,L]

_f(t)_ + (β − α) max

[0,L]

_f(1)(t)_

12 Averaging Method in Systems with Variable Frequencies Chapter 1

Thus,

____

B(tj,μ)

F(t, ε)dt

____

≤ 2p

4 max

[0,L]

_f(t)_ + Lmax

[0,L]

_f(1)(t)_

εμ

−p

≡ c(2)

4 εμ

−p. (1.8)

It remains to estimate the first term on the right-hand side of equality (1.6).

Since the function a(t) satisfies inequality (1.3) on every segment of the set

A(τ ), we establish the following estimate by integrating by parts:

____

A(τ)

F(t, ε)dt

____

≤ s + 1

c23

2c3 + max

[0,L]

|a(1)(t)|

max

[0,L]

_f(t)_ + Lc3 max

[0,L]

_f(1)(t)_

≡ c(3)

4 ε . (1.9)

Combining (1.7)–(1.9) and using (1.6), we get

_I(τ, ε)_ ≤ sc(1)

4 μ + sc(2)

4 εμ

−p + c(3)

4 ε (1.10)

for all τ ∈ [0, L], ε ∈ (0, ε0], and 0 < μ < min{1; δ}. It is clear that the last

estimate is the best order estimate with respect to ε in the case where ε = μp+1.

Setting

c1 = (c(1)

4 + c(2)

4 )s + c(3)

4 and ε1 = min

ε0;

_δ

_p+1_

we deduce (1.2) from (1.10). Theorem 1.1 is proved.

We now consider an oscillation integral of the form

Iλ(t, .t, τ, ε) =

_t+τ

f(y) exp

_ i

(λ, ω(z)) dz

dy, (1.11)

where τ ∈ [0, L], t ∈ R = (−∞;∞), t ∈ R, λ = (λ1, . . . , λm) is a positive

nonzero m-dimensional vector, m ≥ 2, ω(t) = (ω1(t), . . . , ωm(t)) ∈ Cp−1

R ,

Section 1 Uniform Estimates for One-Dimensional Oscillation Integrals 13

p ≥ m, f(t) = (f1(t), . . . , fn(t)) ∈ C1R

, (λ, ω) is the scalar product of vectors,

and L is a positive constant.

In what follows, we establish sufficient conditions that guarantee the uniform

estimate _Iλ(t, t, τ, ε)_ ≤ cε

p with a constant c independent of λ. In

the case m = 2, the behavior of integral (1.11) is determined by the inequality

_Iλ(t, t, τ, ε)_ ≤ c

√

ε [Arn4, Neis1]. In the case m ≥ 3, the investigation of the

behavior of integral (1.11) as ε → 0 becomes more complicated because there

appear resonance relations between the components ων(t) of the vector ω(t)

[Bak1, GrR2, Kha2].

By Wp(t) and WT

p (t) we denote the matrix

(ω(j−1)

ν (t))m,p

ν,j=1

and its transpose, respectively.

Theorem 1.2. Let _(WT

p (t)Wp(t))−1WT

p (t)_ be uniformly bounded by a

constant σ1 and let the functions ω(j−1)

ν (t), ν = 1,m, j = 1, p, be uniformly

continuous for t ∈ R. Then one can indicate constants ε1 > 0 and σ2 > 0

independent of λ, t, t, τ, and ε and such that the following estimate holds for

all λ _= 0, t ∈ R, t ∈ R, τ ∈ [0, L], and ε ∈ (0, ε1] :

_Iλ(t, t, τ, ε)_

≤ σ2ε

1 +

_λ_

max

[t,t+L]

_f(y)_ +

_λ_ max

[t,t+L]

_f(1)(y)_

. (1.12)

Proof. For an arbitrary vector λ = (λ1, . . . , λm) _= 0, we consider the

obvious equality Wp(t)λ = Ω, where

Ω = (Ω0, . . . ,Ωp−1),Ωj =

ν=1

λνω(j)

ν (t) = (λ, ω(j)(t)), j= 0, p − 1.

Hence, we get

_Ω_ ≥ _λ_ _(WT

p (t)Wp(t))−1WT

p (t)_−1 ≥

_λ_

σ1

which implies that, for every y ∈ R and λ _= 0, there is an integer r = r(y, λ),

0 ≤ r ≤ p − 1, for which

|(λ, ω(r)(y))| = max

0≤j≤p−1

|(λ, ω(j)(y))| ≥

_λ_

pσ1

. (1.13)

14 Averaging Method in Systems with Variable Frequencies Chapter 1

Since the functions ω(j)

ν (t), ν = 1,m, j = 0, p − 1, are uniformly continuous

on the entire axis, it is obvious that we can choose a constant δ > 0 independent

of y, λ, and j and such that the following inequalities hold for any y ∈ [y −

δ, y + δ] and j = 0, p − 1 :

|(λ, ω(r)(y))| ≥

_λ_

2pσ1

, |(λ, ω(j)(y))| ≤ 4|(λ, ω(r)(y))|. (1.14)

Indeed, according to the definition of uniform continuity, for every ν = 1,m and

j = 0, p − 1 there exists δ(j)

ν > 0 such that, for any y_, y__ ∈ R satisfying the

inequality |y_ − y__| < δ(j)

ν , the following estimate is true:

|ω(j)

ν (y

_) − ω(j)

ν (y

__)| < σ1 ≡ 1

2pσ1

. (1.15)

Denote δ = min

ν,j

δ(j)

ν . Then estimate (1.15) is valid for _y_ − y___ < δ, 0 ≤ j ≤

p − 1, and 1 ≤ ν ≤ m, and the relations

|(λ, ω(j)(y) − ω(j)(y))| ≤

ν=1

|λν||ω(j)

ν (y) − ω(j)

ν (y)| < σ1_λ_,

which are true for |y − y| < δ and j = 0, p − 1, lead to inequalities (1.14).

We denote by s the integer part of the number

2δ

s ≤ L

2δ

and represent

integral (1.11) in the form of a sum, namely,

I (t, t, τ, ε) =

_s−1

k=0

t+2_δ(k+1)

t+2δk

Fdy +

_t+τ

t+2δs

Fdy, (1.16)

where

F = f (y) exp

(λ, ω (z)) dz

To estimate the integral

Pk =

t+2_δ(k+1)

t+2δk

Fdy, (1.17)

Section 1 Uniform Estimates for One-Dimensional Oscillation Integrals 15

we use inequalities (1.14) and the methods used in the proof of Theorem 1.1. As

a result, we obtain

_Pk_ ≤

(2p − 2) μ +

2p+1

σ1_λ_ εμ1−p

max

[t,t+L]

_f(y)_

2δ

σ1_λ_ εμ1−p max

[t,t+L]

_f(1)(y)_ (1.18)

for r(t + 2δk + δ, λ) ≥ 1 and 0 < μ < min{1, δ}. If r(t + 2δk + δ, λ) = 0,

then, integrating by parts, we get

_Pk_ ≤ 2

σ1_λ_(1 + 4δ) ε max

[t,t+L]

_f(y)_ +

2δ

σ1_λ_ ε max

[t,t+L]

_f(1)(y)_. (1.19)

Analyzing relations (1.17) and (1.18), we conclude that, in the case where

μp−1 ≤ 2p(1 + 4δ)−1, integral (1.17) satisfies inequality (1.18) for all r(t +

2δk + δ, λ) ≥ 0. The same inequality is also satisfied by the last integral on the

right-hand side of (1.16). Therefore, for

ε = μp, 0 < ε ≤ ε1 = min

ε0;

2δ

;

_ 2p

1 + 4δ

_ p

p−1

relation (1.16) yields inequality (1.12) with the constant

σ2 =

2p − 2 +

2p+1

σ1

2δ

σ1

1 + L

2δ

Theorem 1.2 is proved.

Corollary 1. If _λ_ = 1, i.e., λ is an arbitrary point of the unit sphere, then

inequality (1.12) yields a uniform estimate of the integral Iλ of the form

_Iλ(t, t, τ, ε)_ ≤ 2σ2ε

max

[t,t+L]

_f(y)_ + max

[t,t+L]

_f(1)(y)_

Corollary 2. If λ = k = (k1, . . . , km) is an arbitrary nonzero vector with

integer coordinates, then, for σ3 = 2σ2, estimate (1.12) takes the following form:

_Iλ(t, t, τ, ε)_ ≤ σ3ε

max

[t,t+L]

_f(y)_ +

_k_ max

[t,t+L]

_f(1)(y)_

. (1.20)

16 Averaging Method in Systems with Variable Frequencies Chapter 1

Note that estimate (1.20) is often used in what follows for the investigation of

properties of solutions of oscillation systems on finite and infinite time intervals.

Let us analyze in more detail the conditions imposed on the function ω(τ) =

(ω1(τ ), . . . , ωm(τ )) in Theorem 1.2. Assume that t = t = 0, τ ∈ [0, L], ε ∈

(0, ε0], λ _= 0, and Iλ(0, 0, τ, ε) ≡ Iλ(τ, ε) in integral (1.11). Since ω(τ ) ∈

Cp−1

[0,L], we conclude that the functions ων(τ ), ν = 1,m, and their derivatives

up to the order p − 1 inclusive are uniformly continuous and bounded on [0, L].

Thus, the condition of the boundedness of the matrix (WT

p (τ )Wp(τ ))−1WT

p (τ )

is equivalent in this case to the condition det (WT

p (τ )Wp(τ )) _= 0 ∀τ ∈ [0, L].

To calculate the determinant of the product of the matrices WT

p (τ ) and Wp(τ ),

we use the Binet–Cauchy formula [Gan, Lan]

det (WT

p (τ )Wp(τ ))

= det

⎛

⎜⎝

ω1(τ ) . . . ω(p−1)

1 (τ )

. . . . . . . . .

ωm(τ ) . . . ω(p−1)

m (τ )

⎞

⎟⎠

⎛

⎝

ω1(τ ) . . . ωm(τ )

. . . . . . . . .

ω(p−1)

1 (τ ) . . . ω(p−1)

m (τ )

⎞

⎠

0≤k1<k2<...<km≤p−1

Δ2

k1...km(τ ),

where

Δk1...km(τ ) = det (ω(kj )

ν (τ ))m

ν,j=1.

It follows from the above relations that

det (WT

p (τ )Wp(τ )) _= 0 ∀τ ∈ [0, L]

if and only if at least one mth-order minor of the matrix Wp(τ ) is nonzero at

every point τ ∈ [0, L]. Thus, the following statement is true:

Theorem 1.3. Suppose that ω(τ ) ∈ Cp−1

[0,L], p ≥ m, f(τ ) ∈ C1

[0,L], and,

at every point τ ∈ [0, L], at least one mth order minor of the matrix Wp(τ ) is

nonzero. Then, for all λ _= 0, τ ∈ [0, L], and ε ∈ (0, ε0] (ε0 is sufficiently

small ), the following estimate is true:

_Iλ(τ, ε)_ ≤ σ2ε

1 +

_λ_

max

[0,L]

_f(τ )_ +

_λ_ max

[0,L]

_f(1)(τ )_

where the constant σ2 is independent of λ, τ, and ε.

Section 1 Uniform Estimates for One-Dimensional Oscillation Integrals 17

The above analysis shows that the conditions

| ω(j)

ν (τ )| ≤ c ∀τ ∈ R, ν = 1,m, j = 0, p,

| det (WT

p (τ )Wp(τ ))| ≥ c > 0 ∀τ ∈ R,

(where c and c are certain constants) guarantee the uniform continuity of the

functions ω(j)

ν (τ ), ν = 1,m, j = 0, p − 1, on the entire axis and the uniform

boundedness

_(WT

p (τ )Wp(τ ))−1WT

p (τ )_ ≤ c = const ∀τ ∈ R.

The following question arises: Is the assumption on the boundedness of the functions

ων(τ ) and their derivatives necessary? The following example shows that,

generally speaking, the answer to this question is negative: For all τ ∈ R, consider

the functions

ω1(τ ) = sin τ, ω2(τ ) = cos τ, ω3(τ) =

τ, τ ∈ [π,∞),

π − sin τ, τ ∈ (−∞; π).

It is obvious that ω3(τ ) is not bounded for any τ ∈ R, and the functions ων(τ ),

ν = 1, 2, 3, and their derivatives up to the second order inclusive are uniformly

continuous on the entire axis. By direct calculation, one can easily verify that

det W3(τ) =

−τ, τ ∈ [π,∞),

−π, τ ∈ (−∞, π),

_(WT

3 (τ )W3(τ ))−1WT

3 (τ )| ≤ 6 +

∀τ ∈ R.

Hence, the indicated collection of functions ω1(τ ), ω2(τ ), and ω3(τ ) satisfies

all conditions of Theorem 1.2.

Remark 1. If p = m, then

det (WTm

(τ )Wm(τ )) = (det Wm(τ ))2.

Therefore, in this case, the condition that the Wronskian determinant of the functions

ω1(τ ), . . . , ωm(τ ) is nonzero on [0, L] is a sufficient condition for finding

an efficient estimate for the oscillation integral Iλ(τ, ε).

We now assume that this condition is not satisfied at finitely many points of

the segment [0, L] and investigate how this assumption affects the estimate of the

oscillation integral.

18 Averaging Method in Systems with Variable Frequencies Chapter 1

Theorem 1.4. Suppose that f(τ ) ∈ C1

[0,L], ω(τ ) ∈ Cm−1+r

[0,L] , and the function

Δ(τ ) = detWm(τ ) has zeros of multiplicity not higher than r, r ≥ 1,

on [0, L]. Then one can choose constants σ4 > 0 and ε1

∈ (0, ε0] independent

of λ, τ, and ε and such that the following inequality holds for all λ _= 0,

τ ∈ [0, L], and ε ∈ (0, ε1] :

_Iλ(τ, ε)_ ≤ σ4ε

m+r

1 +

_λ_

max

[0,L]

_f(τ )_ +

_λ_ max

[0,L]

_f(1)(τ )_

. (1.21)

Proof. Under the assumptions made above, the function Δ(τ ) has finitely

many zeros 0 ≤ τ1 < τ2 < ... < τs ≤ L of multiplicities r1, r2, . . . , rs, rj ≤ r

∀j = 1, s, on [0, L]. We fix an arbitrary positive δ <

min

1≤j≤s−1

(τj+1 −τj) and

divide the segment [0, τ] into two sets of points Aδ(τ ) and Bδ(τ ) such that

Bδ(τ) =

j=1

[τj − δ, τj + δ] ∩ [0, τ]

and Aδ(τ ) is the closure of the set [0, τ]\Bδ(τ ). It is obvious that the set Aδ(τ )

consists of d1 ≤ s + 1 segments on each of which we have Δ(y) _= 0. Then,

by virtue of the continuity of Δ(y), the following inequality holds for all y ∈

Aδ(τ ) :

|Δ(y)| ≥ min

y∈Aδ(τ)

|Δ(y)| ≥ min

y∈Aδ(L)

|Δ(y)| = cδ > 0.

We now consider an arbitrary nonzero vector λ = (λ1, . . . , λm) and write

the following identity for it:

(λ, ω(y))Δi0,1(y) + (λ, ω(1)(y))Δi0,2(y)

+ . . . + (λ, ω(m−1)(y))Δi0,m(y) = λi0Δ(y), y∈ [0, L], (1.22)

where |λi0

| = max

|λν|, Δi0,ν(y) is the cofactor of the element ω(ν−1)

(y) in

the determinant Δ(y), and |Δi0,ν(y)| ≤ M = const ∀y ∈ [0, L]. Differentiating

equality (1.22) rj ≤ r times with respect to y, we get

(λ, ω(y))Δi0,1,j(y) + (λ, ω(1)(y))Δi0,2,j(y)

+ . . . + (λ, ω(m−1+rj )(y))Δi0,m+rj ,j(y) = λi0Δ(rj )(y). (1.23)

Section 1 Uniform Estimates for One-Dimensional Oscillation Integrals 19

Here, Δi0,l,j(y), l = 1,m + rj , can be linearly expressed in terms of Δi0,ν(y),

ν = 1,m, and their derivatives with respect to y up to the order rj , and, therefore,

|Δi0,l,j(y)| ≤ M

∀i0 = 1,m, l = 1,m + rj, j= 1, s, y ∈ [0, L].

Since |Δ(rj )(τj)| ≡ c(j) > 0, it follows from (1.23) that there exists an integer

qj = qj(τj, λ), 0 ≤ qj ≤ m − 1 + rj , such that

|(λ, ω(qj )(τj))| = max

0≤l≤m−1+rj

|(λ, ω(l)(τj))| ≥

|λi0

|c(j)

(m + p)M

≥

_λ_

m(m + p)M

min

1≤j≤s

c(j) ≡ σ5_λ_.

It follows from the last inequality and the condition of the continuity of the functions

ω(l)

ν (y), ν = 1,m, l = 0,m − 1 + r, y ∈ [0, L], that the following

estimates hold for all y ∈ [τj − δ, τj + δ] ∩ [0, τ], 0 ≤ l ≤ m − 1 + rj :

|(λ, ω(qj )(y))| ≥ 1

2σ5_λ_, |(λ, ω(l)(y))| ≤ 4|(λ, ω(qj )(y))|, j = 1, s, (1.24)

where δ > 0 is a certain constant independent of λ and j.

We set δ = min{δ, δ} and represent the integral Iλ(τ, ε) in the form

Iλ(τ, ε) =

Aδ(τ)

F(y, 0, λ, ε) dy +

Bδ(τ)

F(y, 0, λ, ε) dy . (1.25)

For all y ∈ Aδ(τ ), we have |Δ(y)| ≥ cδ > 0, and the set Aδ(τ ) consists of

d1 ≤ s + 1 segments. Consequently, according to Theorem 1.2, we get

____

Aδ(τ)

F(y, 0, λ, ε) dy

____

≤ σ6ε

1 +

_λ_

max

[0,L]

_f(τ )_ +

_λ_ max

[0,L]

_f(1)(τ )_

(1.26)

where the constant σ6 is independent of λ and ε if ε ∈ (0, ε1] (ε1 is sufficiently

small).

20 Averaging Method in Systems with Variable Frequencies Chapter 1

The set Bδ(τ ) consists of d2 ≤ s segments on each of which inequalities

(1.24) are satisfied. Using the scheme of the proof of Theorem 1.2, we obtain

____

Bδ(τ)

F(y, 0, λ, ε)dy

____

≤ s

(2m+r − 2)μ +

2m+r+1

σ5_λ_ εμ1−(m+r)

× max

[0,L]

_f(τ )_ +

2δs

σ5_λ_εμ1−(m+r) max

[0,L]

_f(1)(τ )_. (1.27)

We set

ε = μm+r, σ4 = σ6 + s

2m+r − 2 +

2m+r−1

σ5

2δ

σ5

ε1

≤ min

__1

2δ

_m+r

;

_ 2m+r

1 + 4δ

_ m+r

m+r−1

Then, combining inequalities (1.26) and (1.27), we deduce estimate (1.21) from

(1.25). Theorem 1.4 is proved.

The example of the integral Iλ(τ, ε) for τ ∈ [0, 1], f(τ ) ≡ 1, and

ω(τ) =

1, τ,

τ 2

2! , . . . ,

τm−2

(m − 2)!, τm−1+r

, λ= (0, . . . , 0,m + r),

shows that estimate (1.21) cannot be improved in order with respect to ε under

the assumptions made in Theorem 1.4. Indeed, in this case, τ = 0 is a zero of

multiplicity r of the function Δ(τ) = τ r (m + r − 1)!

and

|Iλ(τ1, ε)| =

____

_τ1

ε ym+r

____

≥

_τ1

____

cos ym+r

____

dy ≥ 1

2τ1 =

_π

_ 1

m+r ε

m+r ,

τ1 =

_πε

_ 1

m+r .

Section 2 Justification of Averaging Method for Systems with ω = ω (τ ) 21

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