6. Boundary-Value Problems for Oscillation Systems with Frequencies Dependent on Time Variable

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Consider the multifrequency system

dx

dτ

= a(x, ϕ, τ, ε),

dϕ

dτ

= ω(τ )

ε

+ b(x, ϕ, τ, ε) (6.1)

whose right-hand side is defined in G = D × Rm × [0, L] × (0, ε0]. For this

system, we introduce the boundary conditions

x|τ=0 = y ∈ D1, ϕ|τ=L = f(x|τ=0, x|τ=L, ε), (6.2)

where f(y, z, ε) is a known m-dimensional vector function of the variables

(y, z, ε) ∈ D1 ×D ×(0, ε0] ≡ A and D1 is a certain domain (D1 ⊂ D).

Parallel with (6.1), (6.2), we consider the following boundary-value problem

averaged over all angular variables ϕ :

dx

dτ

= a(x, τ, ε), x|τ=0 = y, (6.3)

dϕ

dτ

= ω(τ )

ε

+ b(x, τ, ε), ϕ|τ=L = f(x|τ_____________=0, x|τ=L, ε). (6.4)

It is obvious that the solution of problem (6.3), (6.4) is much simpler than the

solution of problem (6.1), (6.2) because problem (6.3), (6.4) decomposes into two

Cauchy problems. If problem (6.3) has a solution x = x(τ, y, ε) defined and

lying in D ∀(τ, y, ε) ∈ [0, L]×D1 × (0, ε0], then a solution ϕ = ϕ(τ, y,ψ0, ε)

71

72 Averaging Method in Multipoint Problems Chapter 2

of problem (6.4) is given by the formulas

ϕ(τ, y,ψ0, ε) = ψ0 +

1

ε

_τ

0

[ω(t) + εb(x(t, y, ε), t, ε)] dt,

ψ0 = −1

ε

_L

0

[ω(t) + εb(x(t, y, ε), t, ε)]dt + f(y, x(L, y, ε), ε).

In the next section, we use the following theorem for the justification of the averaging

method on the entire axis:

Theorem 6.1. Suppose that system (6.1) satisfies all conditions of Theorem

2.2 and the function f(y, z, ε) is continuously differentiable with respect to z ∈

D for every fixed y ∈ D1 and ε ∈ (0, ε0] and such that

sup

z∈D

_f(y, z, ε)_ < ∞, sup

(y,z,ε)∈A

___

∂

∂z

f(y, z, ε)

___

< ∞. (6.5)

Then there exists a unique solution (x(τ, y, ψ, ε); ϕ(τ, y, ψ, ε)) of the boundaryvalue

problem (6.1), (6.2), and, furthermore, this solution satisfies the inequality

_x(τ, y, ψ, ε) − x(τ, y, ε)_ + _ϕ(τ, y, ψ, ε) − ϕ(τ, y,ψ0, ε)_ ≤ c1ε

1

p (6.6)

∀(τ, y, ε) ∈ [0, L]×D1 × (0, ε0] ,

where ε0 is positive and sufficiently small.

Proof. According to Theorem 2.1, the solution (x(τ, y, ψ, ε); ϕ(τ, y, ψ, ε))

of system (6.1) is defined for all τ ∈ [0, L], y ∈ D1, ψ ∈ Rm, and ε ∈ (0, ε0].

Therefore, we seek a solution of the boundary-value problem (6.1), (6.2) in the

form x = x(τ, y, ψ, ε), ϕ = ϕ(τ, y, ψ, ε), where ψ = ψ(y, ε) is unknown. To

determine ψ, we substitute this solution into the boundary conditions (6.2). As a

result, we get

ϕ (L, y, ψ, ε) = f (y, x (L, y, ψ, ε), ε),

or

ψ = f(y, x(L, y, ψ, ε), ε) − 1

ε

_L

0

[ω(t) + εb(x(t, y, ε), t, ε)]dt − Δϕ(L, y, ψ, ε)

≡ Φ(y, ψ, ε), (6.7)

Section 6 Boundary-Value Problems for Oscillation Systems 73

where

Δϕ (τ, y, ψ, ε) = ϕ (τ, y, ψ, ε) − ϕ (τ, y, ψ, ε).

It follows from the first inequality in (6.5) that there exists a constant c2 = c2(y, ε)

such that

_f (y, z, ε)_ ≤ c2 (y, ε) ∀z ∈ D

for every fixed y ∈ D1 and ε ∈ (0, ε0]. Furthermore, Theorem 2.1 yields

_Δϕ(τ, y, ψ, ε)_ ≤ c3ε

1

p ∀(τ, y, ψ, ε) ∈ [0, L]×D1 × Rm × (0, ε0],

where c3 = max{σ2; σ3}, and σ2 and σ3 are the constants defined by Theorems

2.1 and 2.2. Thus,

_Φ(y, ψ, ε)_ ≤ c2(y, ε) +

1

ε

L

_

max

[0,L]

_ω(τ )_ + ε sup _b(x, τ, ε)_

_

+ c3ε

1

p

≡ c4(y, ε).

Therefore, for every fixed y ∈ D1 and ε ∈ (0, ε0], the function Φ(y, ψ, ε) maps

the set ψ ∈ Rm into the set T = {ψ: ψ ∈ Rm, _ψ_ ≤ c4(y, ε)}. Moreover,

according to (6.5) and Theorem 2.2, we get

___

∂

∂ψ

Φ(y, ψ, ε)

___

≤ sup

A

___

∂

∂z

f (y, z, ε)

___

___

∂

∂ψ

(x (L, y, ψ, ε) − x (L, y, ε))

___

+

___

∂

∂ψ

Δϕ(L, y, ψ, ε)

___

≤ c3ε

1

p

_

1 + sup

A

___∂

∂z

f(y, z, ε)

___

_

≤ 1

2

for

ε0 ≤

_

2c3

_

1 + sup

A

___

∂

∂z

f(y, z, ε)

___

__−p

. (6.8)

Thus, the equation ψ = Φ(y, ψ, ε) has the unique solution ψ = ψ(y, ε) ∈

Rm, and the boundary-value problem (6.1), (6.2) has the unique solution

(x(τ, y,ψ(y, ε), ε); ϕ(τ, y,ψ(y, ε), ε)).

74 Averaging Method in Multipoint Problems Chapter 2

It remains to prove estimate (6.6). Using Theorem 2.1 and Eq. (6.7), we obtain

the inequality

_ψ(y, ε) − ψ0_ ≤ _f(y, x(L, y, ψ(y, ε), ε), ε) − f(y, x(L, y, ε), ε)_

+ _Δϕ(L, y, ψ(y, ε), ε)_

≤

_

1 + sup

A

___

∂

∂z

f(y, z, ε)

___

_

c3ε

1

p ≡ c5ε

1

p ,

which yields

_ϕ(τ, y,ψ(y, ε), ε) − ϕ(τ, y,ψ0, ε)_

≤ _Δϕ(τ, y,ψ(y, ε), ε)_ + _ϕ(τ, y,ψ(y, ε), ε) − ϕ(τ, y,ψ0, ε)_

≤ c3ε

1

p + _ψ(y, ε) − ψ0_ ≤ (c3 + c5)ε

1

p

∀(τ, y, ε) ∈ [0, L]×D1 × (0, ε0] .

Combining the last inequality and the inequality

_x(τ, y,ψ(y, ε), ε) − x(τ, y, ε)_ ≤ c3ε

1

p ,

we obtain estimate (6.6) for c1 = 2c3 + c5. The restrictions for ε0 are specified

by Theorems 2.1 and 2.2 and condition (6.8). Theorem 6.1 is proved.

Remark 1. Assume that, the function f(y, z, ε) in the boundary conditions

(6.2) is independent of y, i.e., f(y, z, ε) ≡ _ f(z, ε). Then, differentiating (6.7)

with respect to y, we get

∂ψ

∂y

=

_

Em − ∂

∂ψ

Δϕ(L, y, ψ, ε) − ∂

∂x

_ f (x(L, y, ψ, ε), ε) ∂

∂ψ

x(L, y, ψ, ε)

_−1

×

_ ∂

∂x

_ f (x(L, y, ψ, ε), ε) ∂

∂y

x (L, y, ψ, ε)

+ ∂

∂y

Δϕ(L, y, ψ, ε) −

_L

0

∂

∂x

b(x(t, y, ε), t, ε) ∂

∂y

x (t, y, ε) dt

_

,

Section 6 Boundary-Value Problems for Oscillation Systems 75

which yields

___

∂ψ(y, ε)

∂y

___

≤ 2m

____ ∂

∂x

_ f(x(L, y, ψ, ε), ε)

___

___

∂

∂y

x (L, y, ε)

___

+ c3ε

1

p + Lsup

___

∂

∂x

b(x, τ, ε)

___

sup

___

∂

∂y

x (τ, y, ε)

___

_

(6.9)

for

ε0 ≤

_

2c3

_

1 + sup

___

∂ _ f(x, ε)

∂x

___

__−p

, c3 = c3

_

1 + sup

___

∂ _ f(x, ε)

∂x

___

_

.

We now consider more general [as compared with (6.2)] boundary conditions

of the form [VaB]

F(x|τ=0, ϕ|τ=0, x|τ=L, ϕ|τ=L, ε) = 0, (6.10)

where F(y, ψ, z, θ, ε) is an (n+m)-dimensional vector function. Problem (6.2),

(6.10) is a two-point boundary-value problem that contains slow and fast variables

and possesses resonance properties. Note that there is a fairly complete theory of

singularly perturbed boundary-value problems (see [VaD]), which is based on the

method of boundary-layer functions developed in [VaB].

Assume that the following conditions are satisfied:

(a) for every ε ∈ (0, ε0], the averaged boundary-value problem

dx

dτ

= a(x, τ, ε),

dϕ

dτ

= ω(τ )

ε

+ b(x, τ, ε),

F(x|τ=0, ϕ|τ=0, x|τ=L, ϕ|τ=L, ε) = 0 (6.11)

has a unique solution

(x(τ, x0(ε), ε); ϕ(τ, x0(ε), ϕ0(ε), ε) ≡ (x(τ, ε); ϕ(τ, ε)),

which lies in D×Rm together with its ρ1-neighborhood;

(b) there exist constants c6 > 0 and c7 > 0 independent of ε and such that

F(y, ψ, z, θ, ε) ∈ C2

y,ψ,z,θ(B, c6), B= B × (0, ε0],

where B denotes the c7-neighborhood of the point (x0(ε), ϕ0(ε),

x(L, ε), ϕ(L, ε)) ∈ R2(n+m) ;

76 Averaging Method in Multipoint Problems Chapter 2

(c) _S−1(x0(ε), ϕ0(ε), ε)_ ≤ c8 = const ∀ε ∈ (0, ε0], where S is the quadratic

(n + m)-dimensional matrix defined by the equality

S(x0(ε), ϕ0(ε), ε)

=

&

∂F0

∂y

+ ∂F0

∂z

∂x(L, x0(ε), ε)

∂x0

+ ∂F0

∂θ

_L

0

∂

∂x

b(x(t, x0(ε), ε), t, ε)∂x(t, x0(ε), ε)

∂x0 dt,

∂F0

∂ψ

+ ∂F0

∂θ

'

.

In this case, the values of the derivatives of the function F(y, ψ, z, θ, ε)

are taken for y = x0(ε), ϕ = ϕ0(ε), z = x(L, x0(ε), ε), and θ =

ϕ(L, x0(ε), ϕ0(ε), ε).

Theorem 6.2. Suppose that the following conditions are satisfied:

(i) ω(τ ) ∈ Cp−1

[0,L], p ≥ m, and det(WT

p (τ )Wp(τ )) _= 0 ∀τ ∈ [0, L];

(ii) conditions (a)–(c) and inequality (2.6) are satisfied.

Then, for every ε ∈ (0, ε0] (ε0 is sufficiently small), the boundary-value

problem (6.1), (6.10) has a unique solution (x(τ, ε); ϕ(τ, ε)), which lies in a

c9ε

1

p -neighborhood of the solution (x(τ, ε); ϕ(τ, ε)) of problem (6.11).

Proof. According to condition (a), the curve x = x(τ, x0, ε) lies in D

together with its ρ1-neighborhood ∀(τ, ε) ∈ [0, L] × (0, ε0]. We now determine

from which set one must choose _x in order that the curve x = x(τ, x0 + _x, ε)

belong to D together with its

1

2ρ1-neighborhood. Using the averaged equations

for slow variables, we get

_x(τ, x0 + _x, ε) − x(τ, x0, ε)_

≤ __x_ +

_τ

0

_x(t, x0 + _x, ε) − x(t, x0, ε)_ sup

G

___

∂

∂x

a(x, τ, ε)

___

dt

Section 6 Boundary-Value Problems for Oscillation Systems 77

or

_x(τ, x0 + _x, ε) − x(τ, x0, ε)_ ≤ __x_eLσ1 ∀(τ, ε) ∈ [0, L] × (0, ε0].

Here, σ1 is the constant defined by inequality (2.6). The condition __x_ ≤ c10 =

1

2ρ1e

−Lσ1 guarantees that x = x(τ, x0 + _x, ε) lies in D together with its

1

2ρ1-

neighborhood. For every solution (x(τ, x0 + _x, ε); ϕ(τ, x0 + _x, ϕ0 + _ ψ, ε)),

__x_ ≤ c10, _ ψ ∈ Rm, of the averaged equations, we write the representation

x(τ, x0 + _x, ε) = x(τ, x0, ε) + ∂x(τ, x0, ε)

∂x0 _x + X(τ, _x, ε),

ϕ(τ, x0 + _x, ϕ0 + _ ψ, ε)

= ϕ(τ, x0, ϕ0, ε) +

_τ

0

∂

∂x

b(x(t, x0, ε), t, ε)∂x(τ, x0, ε)

∂x0 dt _x

+ _ ψ + Y (τ, _x, ε),

where [according to conditions (2.6)] the functions X and Y satisfy the following

inequality for all τ ∈ [0, L], __x_ ≤ c10, and ε ∈ (0, ε0] :

_X(τ, _x, ε)_ + _Y (τ, _x, ε)_ ≤ c11__x_2,

where the constant c11 is independent of τ, _x, and ε.

We seek a solution of problem (6.1), (6.10) in the form

x(τ, ε) = x(τ, x0 + _x, ϕ0 + _ ψ, ε), ϕ(τ, ε) = ϕ(τ, x0 + _x, ϕ0 + _ ψ, ε),

where the unknown parameters _x, __x_ ≤ c10, and _ ψ ∈ Rm can be determined

from the boundary conditions (6.10). After the substitution of the solution thus

chosen in (6.10), we obtain

F

_

x0 + _x, ϕ0 + _ ψ, x(L, x0, ε) + ∂x(L, x0, ε)

∂x0 _x

+ X(L, x0, ε) + Δx(L, x0 + _x, ϕ0 + _ ψ, ε), ϕ(L, x0, ϕ0, ε)

+ _ ψ + Y (L, x0, ε)+Δϕ(L, x0 + _x, ϕ0 + _ ψ, ε)

+

_L

0

∂

∂x

b(x(t, x0, ε), t, ε) ∂

∂x0 x (t, x0, ε) dt _x, ε

_

= 0, (6.12)

78 Averaging Method in Multipoint Problems Chapter 2

where

Δx (τ, y, ψ, ε) = x (τ, y, ψ, ε) − x (τ, y, ε),

Δϕ (τ, y, ψ, ε) = ϕ (τ, y, ψ, ε) − ϕ (τ, y, ψ, ε).

Taking into account condition (b), we specify admissible values of _x and _ ψ for

staying within the domain of definition of the function F. Since Δx and Δϕ

satisfy estimate (2.5), it suffices to impose the restrictions

__x_

_

1 + sup

___

∂

∂x0 x(τ, x0, ε)

___

_

1 + Lsup

___

∂

∂x

b(x, τ, ε)

___

__

+ 2_ _ ψ_ + _Δx_ + _Δϕ_ + _X_ + _Y _ < c7

or

__x_ + _ _ ψ_ ≤ c12 = min

_ 1

c11

; c10;

1

2c7[3 + (1 + Lσ1)neσ1L]−1

_

for ε0 ≤ cp

7(2c3)−p.

For such (_x; _ ψ) = ξ, we expand the function F on the left-hand side of

(6.12) into a Taylor series, taking into account the smoothness condition (b). After

obvious transformations, we get

ξ = S

−1(x0(ε), ϕ0(ε), ε) _ F (ξ, ε), (6.13)

where _ F(ξ, ε) is defined for any ε ∈ (0, ε0] and all ξ satisfying the inequality

_ξ_ ≤ c12 and

_ _ F(ξ, ε)_ ≤ c13(_ξ_2 + ε

1

p ),

___

∂

∂ξ

_ F(ξ, ε)

___

≤ c13(_ξ_ + ε

1

p ). (6.14)

Here, c13 is a constant independent of ξ and ε. The presence of the term ε

1

p

on the right-hand sides of the inequalities is a consequence of estimates (2.5) and

(2.7) for Δx and Δϕ and their derivatives with respect to _x and _ ψ.

Condition (c) imposed on the matrix S and the first inequality in (6.14) guarantee

that S−1(x0(ε), ϕ0(ε), ε) _ F(ξ, ε) maps the set Mε = {ξ : ξ ∈ Rn+m,

_ξ_ ≤ 2c8c13ε

1

p } into itself for ε ∈ (0, ε0],

ε0 ≤

_

min

_ 1

4c28

c2

13

; c12

2c8c13

__p

.

Section 6 Boundary-Value Problems for Oscillation Systems 79

Moreover, the second inequality in (6.14) yields

___

∂

∂ξ

(S

−1(x0(ε), ϕ0(ε), ε) _ F(ξ, ε))

___

≤ c8c13(_ξ_ + ε

1

p ) ≤ 1

2

for all ξ ∈ Mε and ε ∈ (0, ε0], ε0 ≤ [2c8c13(1 + 2c8c13)]−p. By virtue of

the fixed-point theorem, for every ε ∈ (0, ε0] Eq. (6.13) has a unique solution

ξ = ξ(ε) = (_x(ε); _ ψ(ε)) ∈ Mε, and the boundary-value problem (6.1), (6.10)

has the unique solution

x(τ, ε) = x(τ, x0(ε) + _x(ε), ϕ0(ε) + _ ψ(ε), ε), ϕ(τ, ε)

= ϕ(τ, x0(ε) + _x(ε), ϕ0(ε) + _ ψ(ε), ε),

whose initial data x0(ε)+ _x(ε), ϕ0(ε)+ _ ψ(ε) lie in the 2c8c13ε

1

p -neighborhood

of the initial data (x0(ε); ϕ0(ε)) of the solution of the averaged boundary-value

problem (6.11). Also note that, according to Theorem 2.1 and conditions (2.6),

the following inequalities are true:

_x(τ, ε) − x(τ, ε)_ + _ϕ(τ, ε) − ϕ(τ, ε)_

≤ _x(τ, x0 + _x, ϕ0 + _ ψ, ε) − x(τ, x0 + _x, ε)_

+ _ϕ(τ, x0 + _x, ϕ0 + _ ψ, ε) − ϕ(τ, x0 + _x, ϕ0 + _ ψ, ε)_

+ _x(τ, x0 + _x, ε) − x(τ, x0, ε)_

+ _ϕ(τ, x0 + _x, ϕ0 + _ ψ, ε) − ϕ(τ, x0, ϕ0, ε)_

≤ {c3 + [(1 + Lσ1)neσ1L + 1]2c8c13}ε

1

p ≡ c9ε

1

p

∀(τ, ε) ∈ [0, L] × (0, ε0].

Theorem 6.2 is proved.

Example. Consider the boundary-value problem

dx

dτ

= x(1 + x2 cos(ϕ1 − ϕ2) + cos ϕ2),

dϕ1

dτ

= τ

ε

+ x2 cos ϕ2,

dϕ2

dτ

= τ 2

ε

+ x3 sin ϕ1, (6.15)

x|τ=0 + x|τ=1 = 1, ϕ1|τ=0 + x|τ=1 = 0, ϕ2|τ=1 + x|τ=0 = 0

80 Averaging Method in Multipoint Problems Chapter 2

and the corresponding problem averaged with respect to ϕ1 and ϕ2 :

dx

dτ

= x,

dϕ1

dτ

= τ

ε

,

dϕ2

dτ

= τ 2

ε

,

x|τ=0 + x|τ=1 = 1, ϕ1

|τ=0 + x|τ=1 = 0, ϕ2

|τ=1 + x|τ=0 = 0.

It can easily be verified that the last problem has the unique solution

x(τ, ε) = eτ

1 + e

,

ϕ1(τ, ε) = τ 2

2ε

− e

e + 1, ϕ2(τ, ε) = τ 3 − 1

3ε

− 1

e + 1,

and, for this solution, we have

S(x0(ε), ϕ0(ε), ε) =

⎛

⎝

1 + e 0 0

e 1 0

1 0 1

⎞

⎠,

_S

−1(x0(ε), ϕ0(ε), ε)_ = 3+

1

e + 1.

Since det(WT

3 (τ )W3(τ)) = (τ 2 + 2)2 _= 0 ∀τ ∈ [0, 1], all conditions of

Theorem 6.2 are satisfied. Thus, for every sufficiently small ε > 0, there exists

a unique solution (x(τ, ε); ϕ1(τ, ε); ϕ2(τ, ε)) of problem (6.15) that satisfies the

inequality

___

x(τ, ε)− eτ

e + 1

___

+

___

ϕ1(τ, ε)−τ 2

2ε

+ e

e + 1

___

+

___

ϕ2(τ, ε)−τ 3 − 1

3ε

+

1

e + 1

___

≤ c9ε

1

3

for all τ ∈ [0, 1].

Finally, note that Theorem 6.1 guarantees the global uniqueness of a solution

of the boundary-value problem (6.1), (6.2), whereas Theorem 6.2 establishes the

uniqueness of a solution of problem (6.1), (6.10) only in a certain small neighborhood

of the solution of the averaged problem (6.11).

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