7. Theorem on Justification of Averaging Method on Entire Axis

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In this section, we establish the existence of a solution (defined on the entire

axis) of an oscillation system using the combination of the averaging method on

Section 7 Theorem on Justification of Averaging Method on Entire Axis 81

a segment and the solution of certain boundary-value problems. Note that, in this

case, we do not use the method of integral manifolds, which requires additional

restrictions on the equations of the system.

Consider the system of n + m equations

dx

dτ

= a(x, ϕ, τ, ε),

dϕ

dτ

= ω(τ )

ε

+ b(x, ϕ, τ, ε), (7.1)

where the functions a, b, and ω are defined on the set (x, ϕ, τ, ε) ∈ D ×

Rm × R × [0, ε0] ≡ G (Rn ⊃ D is a bounded domain) and 2π-periodic in

ϕν, ν = 1,m. For this system, we write the corresponding system of equations

of the first approximation for slow variables averaged with respect to all angular

variables ϕ, namely

dx

dτ

= a(x, τ, 0), (7.2)

a(x, τ, ε) = (2π)−m

_2π

0

. . .

_2π

0

a(x, ϕ, τ, ε)dϕ1 . . . dϕm.

In this section, we denote by (xτ (t, y, ψ, ε); ϕτ (t, y, ψ, ε)) and (xτ (t, y, ε);

ϕτ (t, y, ψ, ε)), respectively, the solutions of system (7.1) and the averaged system

dx

dτ

= a(x, τ, ε),

dϕ

dτ

= ω(τ )

ε

+ b(x, τ, ε) (7.3)

that take the value (y; ψ) for τ = t. Assume that a(x, τ, ε) satisfies the inequality

_a(x, τ, ε) − a(x, τ, 0)_ +

___

∂a(x, τ, ε)

∂x

− ∂a(x, τ, 0)

∂x

___

≤ σ1ε (7.4)

∀(x, τ, ε) ∈ D×R × [0, ε0] = G.

Theorem 7.1. Suppose that the following conditions are satisfied:

(i) the function c(x, ϕ, τ, ε) = [a(x, ϕ, τ, ε); b(x, ϕ, τ, ε)] is twice continuously

differentiable with respect to x, ϕ, and τ for every fixed ε, and its

Fourier coefficients ck(x, τ, ε) satisfy inequalities (2.6) and (7.4);

(ii) _(WT

p (τ )Wp(τ ))−1WT

p (τ )_ is uniformly bounded, and

ω(j−1)

ν (τ ), ν= 1,m, j = 1, p, p ≥ m,

are uniformly continuous ∀τ ∈ R;

82 Averaging Method in Multipoint Problems Chapter 2

(iii) there exists a solution x = ξ(τ ) of the averaged equations of the first

approximation (7.2) that is defined ∀τ ∈ R and lies in D together with its

ρ-neighborhood;

(iv) the normal fundamental matrix Q(τ, t) of solutions of the variational equation

dz

dτ

= ∂

∂x

a(ξ(τ ), τ, 0)z satisfies the estimate

_Q(τ, t)_ ≤ Ke

−γ(τ−t) (7.5)

∀τ ≥ t ∈ R, K = const ≥ 1, γ= const > 0.

Then, for sufficiently small ε0 > 0 and every (ψ, ε) ∈ Rm × (0, ε0], there

exists a point x0(ψ, ε) ∈ D such that the solution

(xτ (0, x0(ψ, ε), ψ, ε); ϕτ (0, x0(ψ, ε), ψ, ε))

of system (7.1) is defined ∀τ ∈ R and satisfies the inequality

_xτ (0, x0(ψ, ε), ψ, ε) − ξ(τ )_ ≤ σ2ε

1

p ∀(ψ, τ, ε) ∈ Rm × R × (0, ε0], (7.6)

where the constant σ2 is independent of ψ and ε.

Remark 2. Inequality (7.6) can be interpreted as an estimate of the error of

the averaging method ∀τ ∈ R under the condition that the slow variables take

the value x0(ψ, ε) at the initial moment of time.

We now establish several facts necessary for the proof of Theorem 7.1.

Lemma 7.1. If the conditions of Theorem 7.1 are satisfied and

ε0 <

γσ3

4σ1K

, σ3 = min

_1

2ρ;

1

γ

(2Kn2σ1 + γ)−1

_

,

then

_xτ (t, y + ξ(t), ε) − ξ(τ )_ ≤ K

_

_y_e

−γ

2 (τ−t) +

2

γ

σ1ε

_

, (7.7)

___

∂

∂y

xτ (t, y + ξ(t), ε)

___

≤ Ke

−γ

2 (τ−t) (7.8)

for all τ ≥ t, ε ∈ (0, ε0], and _y_ ≤ σ3(4K)−1.

Section 7 Theorem on Justification of Averaging Method on Entire Axis 83

Proof. It follows from the averaged equations (7.3) and inequality (7.5) for

the function zτ (t, y + ξ(t), ε) = xτ (t, y + ξ(t), ε) − ξ(τ ) that

_zτ (t, y + ξ(t), ε)_

≤ K_y_e

−γ(τ−t) +

_τ

t

Ke

−γ(τ−l)

_

εσ1 + n2σ1_zl(t, y + ξ(t), ε)_

_

dl. (7.9)

Assume that the inequality _zτ (t, y+ξ(t), ε)_ < σ3 holds on the maximum halfinterval

[t, T). Then relation (7.9) yields the following estimate for the function

vτ (t, y, ε) = _zτ (t, y + ξ(t), ε)_eγ(τ−t) :

vτ (t, y, ε) ≤ K_y_ +

1

γ

Kσ1eγ(τ−t)ε + γ

2

_τ

t

vl(t, y, ε)dl ∀τ ∈ [t, T).

In the last inequality, we replace the sign ≤ by = . The function vτ (t, y, ε) that

is a solution of the equation constructed is determined by the formula

vτ (t, y, ε) = K

_

_y_ − 1

γ

σ1ε

_

e

γ

2 (τ−t) +

2

γ

Kσ1eγ(τ−t).

This yields

vτ (t, y, ε) ≤ vτ (t, y, ε) < K_y_e

γ

2 (τ−t) +

2ε

γ

Kσ1eγ(τ−t),

or

_zτ (t, y + ξ(t), ε)_ ≤ K

_

_y_e

−γ

2 (τ−t) +

2

γ

σ1ε

_

∀τ ∈ [t, T). (7.10)

Since

K

_

_y_ +

2

γ

σ1ε

_

≤ 3

4σ3

for _y_ ≤ σ3(4K)−1 and ε ≤ ε0 ≤ γσ3(4Kσ1)−1, we can set T = ∞ in

(7.10). Hence, inequality (7.7) is proved.

We differentiate the averaged equations for slow variables over y. Taking

into account that

∂

∂y

xt(t, y + ξ(t), ε) = En (En is the n-dimensional identity

matrix), we get

84 Averaging Method in Multipoint Problems Chapter 2

∂

∂y

xτ (t, y + ξ(t), ε)

= Q(τ, t) +

_τ

t

Q(τ, l)

__ ∂

∂x

a(xl(t, y + ξ(t), ε), l, ε

_

− ∂

∂x

a(xl(t, y + ξ(t), ε), l, 0)) +

_ ∂

∂x

a(xl(t, y + ξ(t), ε), l, 0

_

− ∂

∂x

a(ξ(l), l, 0)

_

∂

∂y

xl(t, y + ξ(t), ε) dl,

whence

___

∂

∂y

xτ (t, y + ξ(t), ε)

___

≤ Ke

−γ(τ−t) + K

_

ε + n2

_

_y_ +

2

γ

σ1ε

_

K

_

σ1

×

_τ

t

e

−γ(τ−l)

___

∂

∂y

xl(t, y + ξ(t), ε)

___

dl.

Solving this inequality, we obtain

___

∂

∂y

xτ (t, y + ξ(t), ε)

___

≤ K exp

_

− (τ − t)

_

γ − Kσ1

_

1 + Kn2σ1

2

γ

_

ε − K2σ1n2_y_

__

≤ Ke

−γ

2 (τ−t)

for τ ≥ t, _y_ ≤ σ3(4K)−1, and ε ≤ ε0 ≤ γσ3(4Kσ1)−1. Lemma 7.1 is

proved.

It follows from estimate (7.7) and the restrictions imposed on ε0 and y that

the slow variables xτ (t, y + ξ(t), ε) of every solution of the averaged equations

(7.3) lie in D together with their

1

2ρ-neighborhoods ∀τ ≥ t. Then, using Theorems

2.1 and 2.2, we can write the following inequality for the function U =

Section 7 Theorem on Justification of Averaging Method on Entire Axis 85

(xτ (t, y+ξ(t), ε)−xτ (t, y+ξ(t), ε); ϕτ (t, y+ξ(t), ψ, ε)−ϕτ (t, y++ξ(t), ψ, ε)):

_U_ +

___

∂

∂y

U

___

+

___

∂

∂ψ

U

___

≤ σ4ε

1

p (7.11)

∀τ ∈ [t, t + L], _y_ ≤ σ3(4K)−1, ε∈ (0, ε0], ψ∈ Rm,

where the constant σ4 depends on L and does not depend on t, y, ψ, and ε.

Proof of Theorem 7.1. Let

L =

2

γ

ln(8mK) and _y_ ≤ 2σ4ε

1

p .

For ε0 ≤

_ 1

σ3

8σ4K

_−p

, inequalities (7.7) and (7.8) yield

_xτ (t, y + ξ(t), ε) − ξ(τ )_ ≤ K

_

2σ4ε

1

p e

−γ

2 (τ−t) +

2

γ

σ1ε

_

∀τ ≥ t,

___

∂

∂y

xτ (t, y + ξ(t), ε)

___

≤ 1

8m

∀τ ≥ t + L.

(7.12)

We fix an arbitrary ψ ∈ Rm and consider the boundary conditions

x|τ=−L = y + ξ(−L), ϕ|τ=0 = ψ. (7.13)

According to Theorem 7.1, there exists a unique solution

(xτ (−L, y + ξ(−L), ψ(1), ε); ϕτ (−L, y + ξ(−L), ψ(1), ε)),

ψ(1) = ψ(1)(y + ξ(−L), ψ, ε),

of the boundary-value problem (7.1), (7.13), whose slow variables, with regard

for (7.11) and (7.12), satisfy the following conditions for ε0 ≤

_ γσ4

2σ1K

_ p

p−1 :

_xτ (−L,y + ξ(−L), ψ(1), ε) − ξ(τ )_

≤ _xτ (−L, y + ξ(−L), ψ(1), ε) − xτ (−L, y + ξ(−L), ε)_

+ _xτ (−L, y + ξ(−L), ε) − ξ(τ )_

≤ σ4ε

1

p + K

_2

γ

σ1ε + 2σ4ε

1

p

_

≤ 2(K + 1)σ4ε

1

p (7.14)

86 Averaging Method in Multipoint Problems Chapter 2

for all τ ∈ [−L, 0) and

_x0(−L, y + ξ(−L), ψ(1), ε) − ξ(0)_

≤ σ4ε

1

p + K

_2

γ

σ1ε + 2σ4ε

1

p e

−γ

2 L

_

< 2σ4ε

1

p . (7.15)

Note that, for ε ≤ ε0 ≤ min{(8mσ4)−p; (2σ4(1+σ5))−p}, the function ψ(1) =

ψ(1)(y + ξ(−L), ψ, ε) satisfies inequality (6.9), namely

___

∂

∂y

ψ(1)

___

≤ 2m[Lσ1K + σ4ε

1

p ] < 2m

_

Lσ1K + σ5

1

8m

+ σ4ε

1

p (1 + σ5)

_

≤ σ5 = 4mLKσ1 +

1

4. (7.16)

We now consider the boundary conditions

x|τ=−2L = y + ξ(−2L), ϕ|τ=−L = ψ(1)(x|τ=−L, ψ, ε). (7.17)

By analogy with the above reasoning, we find the unique solution

(xτ (−2L, y + ξ(−2L), ψ(2), ε); ϕτ (−2L, y + ξ(−2L), ψ(2), ε)),

ψ(2) = ψ(2)(y + ξ(−2L), ψ, ε),

of the boundary-value problem (7.1), (7.17), for which the following estimates are

true:

_xτ (−2L, y + ξ(−2L), ψ(2), ε) − ξ(τ )_ ≤ 2(K + 1)σ4ε

1

p ∀τ ∈ [−2L,−L),

_x−L(−2L, y + ξ(−2L), ψ(2), ε) − ξ(−L)_ < 2σ4ε

1

p . (7.18)

Further, we estimate

∂ψ(2)

∂y

. Taking into account inequalities (6.9), (7.11), (7.12),

and (7.16), we get

___

∂

∂y

ψ(2)

___

≤ 2m

_

σ5

___

∂

∂y

x−L(−2L, y + ξ(−2L), ε)

___

+ Lσ1 max

[−2L,−L]

___

∂

∂y

xτ (−2L, y + ξ(−2L), ε)

___

+ σ4ε

1

p + σ4σ5ε

1

p

_

≤ 2m

_ σ5

8m

+ Lσ1K + σ4(1 + σ5)

_

ε

1

p

_

≤ σ5

Section 7 Theorem on Justification of Averaging Method on Entire Axis 87

for ε ≤ ε0 ≤ min{(8mσ4)−p; (2σ4(1 + σ5))−p}. Note that the restriction ε0 ≤

(2σ4(1 + σ5))−p is determined by conditions for the validity of inequality (6.9).

Combining (7.14), (7.15), and (7.18), we establish that

(xτ (−2L, y + ξ(−2L), ψ(2), ε); ϕτ (−2L, y + ξ(−2L), ψ(2), ε))

is a solution of system (7.1) for τ ∈ [−2L, 0] and satisfies the boundary conditions

x−2L(−2L, y + ξ(−2L), ψ(2), ε) = y + ξ(−2L),

ϕ0(−2L, y + ξ(−2L), ψ(2), ε) = ψ

and the inequalities

_xτ (−2L, y + ξ(−2L), ψ(2), ε) − ξ(τ )_ ≤ 2(K + 1)σ4ε

1

p ∀τ ∈ [−2L, 0),

_x0(−2L, y + ξ(−2L), ψ(2), ε) − ξ(0)_ < 2σ4ε

1

p .

By induction, for an arbitrary integer r > 2 and τ ∈ [−rL,−(r − 1)L] we

obtain the solution

(xτ (−rL, y + ξ(−rL), ψ(r), ε); ϕτ (−rL, y + ξ(−rL), ψ(r), ε))

of Eqs. (7.1) that satisfies the boundary conditions

x|τ=−rL = y + ξ(−rL), ϕ|τ=−(r−1)L = ψ(r−1)(x|τ=−(r−1)L, ψ, ε)

and the inequalities

_xτ (−rL, y + ξ(−rL), ψ(r), ε) − ξ(τ )_ ≤ 2(K + 1)σ4ε

1

p

∀τ ∈ [−rL,−(r − 1)L),

_x−(r−1)(−rL, y + ξ(−rL), ψ(r), ε) − ξ(−(r − 1)L)_ < 2σ4ε

1

p ,

___∂

∂y

ψ(r)(y + ξ(−rL), ψ, ε)

___

≤ σ5.

Thus,

(xτ (−rL, y + ξ(−rL), ψ(r), ε); ϕτ (−rL, y + ξ(−rL), ψ(r), ε))

88 Averaging Method in Multipoint Problems Chapter 2

is a solution of system (7.1) for all τ ∈ [−rL, 0], and

_xτ (−rL, y + ξ(−rL), ψ(r), ε) − ξ(τ )_ ≤ 2(K + 1)σ4ε

1

p ∀τ ∈ [−rL, 0),

_x0(−rL, y + ξ(−rL), ψ(r), ε) − ξ(0)_ < 2σ4ε

1

p , (7.19)

ϕ0(−rL, y + ξ(−rL), ψ(r), ε) = ψ.

We now fix an arbitrary y ∈ Rn, _y_ ≤ 2σ4ε

1

p , and consider the sequence

{x0(−rL, y + ξ(−rL), ψ(r)(y + ξ(−rL)), ψ, ε), ε)}∞

r=1

≡ {x(r)(ψ, ε)}∞

r=1.

By virtue of the uniform boundedness of the norm of every element of this sequence

by the number _ξ(0)_ + 2σ4ε

1

p , we can select a convergent subsequence

of this sequence, namely

{x(rj )(ψ, ε)}∞

j=1, rj = rj(ψ, ε), lim

j→∞

x(rj )(ψ, ε) = x0(ψ, ε),

_x0(ψ, ε) − ξ(0)_ ≤ 2σ4ε

1

p .

Let us prove that a solution (xτ (0, x0(ψ, ε), ψ, ε); ϕτ (0, x0(ψ, ε), ψ, ε)) of system

(7.1) is defined ∀τ ∈ (−∞; 0] and

_xτ (0, x0(ψ, ε), ψ, ε) − ξ(τ )_ ≤ 2(K + 1)σ4ε

1

p .

Assume the contrary, i.e., let

_xτ0(0, x0(ψ, ε), ψ, ε) − ξ(τ0)_ > 2(K + 1)σ4ε

1

p (7.20)

for certain τ0 < 0. Taking into account that

xτ (−rL, y + ξ(−rL), ψ(r)(y + ξ(−rL), ψ, ε), ε) = xτ (0, x(r)(ψ, ε), ψ, ε)

for all τ ∈ [−rL, 0], we derive from (7.19) for rjL > −τ0 that

_xτ0(0, x(rj )(ψ, ε), ψ, ε) − ξ(τ0)_ ≤ 2(K + 1)σ4ε

1

p . (7.21)

Using the continuous dependence of a solution on the initial data and passing to

the limit as j →∞ in (7.21), we arrive at a contradiction with (7.20).

For τ ∈ [0,∞), estimate (7.5) follows from Theorem 2.4 and inequality

(7.7). The restriction σ2ε

1

p

0

≤ 1

2ρ, σ2 = 2(K + 1)σ4, which guarantees that

the curve x = xτ (0, x0(ψ, ε), ψ, ε) lies in D ∀τ ∈ R, completes the proof of

Theorem 7.1.

Section 8 Multipoint Problem for Resonance Multifrequency System 89

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