8. Multipoint Problem for Resonance Multifrequency Systems

Back

Consider a nonlinear system of the form

dx

dτ

= a(x, τ, ε) + εA(x, ϕ, τ, ε),

dϕ

dτ

= ω(x, τ, ε)

ε

+ B(x, ϕ, τ, ε),

(8.1)

where a, A, ω, and B are defined for (x, ϕ, τ, ε) ∈ D×Rm×[0, L]×(0, ε0] =

G (m ≥ 2), 2π-periodic in each component ϕν, ν = 1,m, of the vector ϕ,

and l ≥ m times continuously differentiable with respect to x, ϕ, and τ for

every fixed ε ∈ (0, ε0], and all their partial derivatives are uniformly bounded in

G by a constant c1 independent of ε. In addition, we assume that

_

k

_

_k_ sup

G

_ck_ + sup

G

___

∂ck

∂τ

___

+ sup

G

___

∂ck

∂x

___

_

≤ c1. (8.2)

Here, G = D×[0, L] × (0, ε0] and ck = ck(x, τ, ε) are the Fourier coefficients

of the function [A(x, ϕ, τ, ε);B(x, ϕ, τ, ε)].

For Eqs. (8.1), we introduce the multipoint conditions

Φ(x|τ=τ1, . . . , x|τ=τr, εϕ|τ=τ1, . . . , εϕ|τ=τr, ε) = 0, (8.3)

where 0 ≤ τ1 < τ2 < ... < τr ≤ L, r ≥ 2, Φ = (Φ1, . . . ,Φn+m), and

Φj(x|τ=τ1, . . . , εϕ|τ=τr, ε), j = 1,m + n, are certain functionals.

Problem (8.1), (8.3) is a multipoint problem that possesses resonance properties.

In the case of a one-frequency system with nonzero frequency, there are no

resonance modes, but if the number of frequencies is m ≥ 2, then the resonance

phenomenon is typical of the problems under consideration. Note that, for r = 2,

τ1 = 0, and τ2 = L, problem (8.1), (8.3) is a boundary-value problem

Assume that Φ = Φ(p1, . . . , pr, q1, . . . , qr, ε) is an (n + m)-dimensional

vector function of pj ∈ D, qj ∈ Rm, j = 1, r, and ε ∈ (0, ε0], that is twice

continuously differentiable for every fixed ε and such that

_2

s=1

_DsΦ_ ≤ c2 = const (8.4)

90 Averaging Method in Multipoint Problems Chapter 2

for all pj = (p(1)

j , . . . , p(n)

j ) ∈ D, qj = (q(1)

j , . . . , q(m)

j ) ∈ Rm, and ε ∈ (0, ε].

Here, Ds is an arbitrary partial derivative with respect to p(ν)

j and q(μ)

j (j =

1, r, ν = 1, n, μ = 1,m) of order s.

Parallel with (8.1), (8.3), we consider the following problem averaged with

respect to all angular variables ϕ:

dx

dτ

= a(x, τ, ε) + εA(x, τ, ε),

dθ

dτ

= ω(x, τ, ε) + εB(x, τ, ε), (8.5)

Φ(x|τ=τ1, . . . , x|τ=τr , θ|τ=τ1, . . . , θ|τ=τr, ε) = 0, (8.6)

where θ = εϕ and

[A;B] = (2π)−m

_2π

0

. . .

_2π

0

[A(x, ϕ, τ, ε);B(x, ϕ, τ, ε)]dϕ1 . . . dϕm.

In order that the averaging operator be efficient for the investigation of oscillation

processes, it is necessary to impose certain restrictions on the components

ων(x, τ, ε), ν = 1,m, of the frequency vector ω. In what follows, we assume

that

_(WT

l (x, τ, ε)Wl(x, τ, ε))−1WT

l (x, τ, ε)_ ≤ c3 ∀(x, τ, ε) ∈ G, (8.7)

where Wl and WT

l denote the matrix

Wl(x, τ, ε) =

_ dj−1

dτj−1 ων(x, τ, ε)

_l,m

j,ν=1

and its transpose, respectively; here, the total derivatives with respect to τ of the

functions ων(x, τ, ε) are calculated with regard for the averaged system (8.5).

Conditions (8.2) and (8.7) guarantee (Theorems 5.1 and 5.2) that

_ _U_ + ε

___

∂

∂y

_U

___

+ ε

___

∂

∂ψ

_U

___

≤ c4ε1+α, α=

1

l

, (8.8)

for all τ ∈ [0, L], y ∈ D1, ψ ∈ Rm, and ε ∈ (0, ε0] for sufficiently small

ε0 > 0. Here,

_U = (x(τ, y, ψ, ε) − x(τ, y, ε); θ(τ, y, ψ, ε) − θ(τ, y, ψ, ε)), θ= εϕ,

(x; θ) and (x; θ) are solutions of systems (8.1) and (8.5), respectively, that take

the values (y; ψ) for τ = 0, and D1 is the set of points y ∈ D for which

Section 8 Multipoint Problem for Resonance Multifrequency System 91

the curve x = x(τ, y, ε) lies in D together with a certain ρ1-neighborhood

∀(τ, ε) ∈ [0, L] × (0, ε0].

Denote by P(y0, ψ0, ε) the (m + n)-dimensional square matrix

_r

j=1

_

∂Φ0

∂pj

∂x(τj, y0, ε)

∂y0 + ∂Φ0

∂qj

_τj

0

∂ω(x(τ, y0, ε), τ, ε)

∂x

∂x(τ, y0, ε)

∂y0 dτ,

∂Φ0

∂qj

 

.

Here, the values of the derivatives

∂Φ0

∂pj

and

∂Φ0

∂qj

of Φ(p1, . . . , qr, ε) are taken

for pν = x(τν, y0, ε) and qν = θ(τν, y0, ψ0, ε), ν = 1, r.

Theorem 8.1. Suppose that the following conditions are satisfied:

(i) conditions (8.2), (8.4), and (8.7) are satisfied;

(ii) for every ε ∈ (0, ε0], the averaged problem (8.5), (8.6) has a unique solution

(x(τ, y0, ε); θ(τ, y0, ψ0, ε)) that lies in D × Rm together with its

ρ-neighborhood ∀(τ, ε) ∈ [0, L] × (0, ε0];

(iii) for a given solution, the matrix P(y0, ψ0, ε) is nondegenerate and

_P

−1(y0, ψ0, ε)_ ≤ c5 = const ∀ε ∈ (0, ε0]. (8.9)

Then one can find constants c6 > 0 and ε1 > 0 such that, for all ε ∈

(0, ε0], where ε0 ≤ ε1, the multipoint problem (8.1), (8.3) has a unique solution

(x(τ, ε); θ(τ, ε)) that satisfies the inequality

_x(τ, ε) − x(τ, y0, ε)_ + _θ(τ, ε) − θ(τ, y0, ψ0, ε)_ ≤ c6ε1+α. (8.10)

Proof. It follows from the smoothness conditions for the right-hand side of

the averaged system (8.5) that, for _y_ <

1

2ρe

−2c1L, the curve x = x(τ, y +

y0, ε) lies in D together with its ρ1 =

1

2ρ-neighborhood ∀(τ, ε) ∈ [0, L] ×

(0, ε0]. Therefore, the domain D1 _= ∅ is not empty, and we can use inequality

(8.8).

We seek a solution of problem (8.1), (8.3) in the form (x(τ, y0 + y,ψ0 +

ψ, ε); θ(τ, y0 + y,ψ0 + ψ, ε)), where the unknown vector z = (y,ψ) is determined

from conditions (8.3), namely,

92 Averaging Method in Multipoint Problems Chapter 2

z = −P

−1(y0, ψ0, ε)

__

Φ(x(τ1, y0 + y, ε), . . . ,

θ(τr, y0 + y,ψ0 + ψ, ε), ε) − P(y0, ψ0, ε)z

_

+

_

Φ(x(τ1, y0 + y,ψ0 + ψ, ε), . . . , θ(τr, y0 + y,ψ0 + ψ, ε), ε)

− Φ(x(τ1, y0 + y, ε), . . . , θ(τr, y0 + y,ψ0 + ψ, ε), ε)

__

≡ M(z, ε). (8.11)

Taking into account conditions (8.4) and estimate (8.8), we get

_Φ(x(τ1, y0 + y,ψ0 + ψ, ε), . . . , ε) − Φ(x(τ1, y0 + y, ε), . . . , ε)_

≤ c2c4ε1+α. (8.12)

Using the smoothness conditions for the right-hand side of system (8.5), we get

x(τ, y0 + y, ε) = x(τ, y0, ε) + ∂x(τ, y0, ε)

∂y0 y + X(τ, y, ε),

θ(τ, y0 + y,ψ0 + ψ, ε)

=

_τ

0

∂ω(x(t, y0, ε), t, ε)

∂x

∂x(t, y0, ε)

∂y0 dty

+ θ(τ, y0, ψ0, ε) + Y (τ, y, ε), (8.13)

where

_X(τ, y, ε)_ ≤ c7_y_2,

_Y (τ, y, ε)_ ≤ c7(_y_2 + ε_y_),

and c7 is a constant independent of ε.

We expand the function

Φ(x(τ1, y0 + y, ε), . . . , θ(τ, y0 + y,ψ0 + ψ, ε), ε)

according to the Taylor formula by using equalities (8.13) and inequality (8.4).

After obvious transformations, we get

Section 8 Multipoint Problem for Resonance Multifrequency System 93

Φ(x(τ1, y0 + y, ε), . . . , θ(τr, y0 + y,ψ0 + ψ, ε), ε)

= P(y0, ψ0, ε)z + R(z, ε), (8.14)

where _R(z, ε)_ ≤ c8(_z_2 + ε_z_) and c8 is a constant. Combining (8.12)–

(8.14), we obtain

_M(z, ε)_ ≤ c5[c2c4ε1+α + c8(_z_2 + ε_z_)],

which implies that M maps the set

V = {z : z ∈ Rn+m, _z_ ≤ 2c2c4c5ε1+α}

into itself for

ε ≤ ε0 = min{(2c2c4c5)− 1

α ; (4c5c8)−1}.

Let us prove that the mapping M: V → V is contracting. For this purpose, we

represent

∂M

∂z

in the form

∂M(z, ε)

∂z

= −P

−1(y0, ψ0, ε)

__r

j=1

_ ∂Φ

∂pj

∂

∂y

(x(τj, y0 + y,ψ0 + ψ, ε)

− x(τj, y0 + y, ε)) + ∂Φ

∂qj

∂

∂y

(θ(τj, y0 + y,ψ0 + ψ, ε)

− θ(τj, y0 + y,ψ0 + ψ, ε),

∂Φ

∂pj

∂

∂ψ

(x(τj, y0 + y,ψ0 + ψ, ε))

+ ∂Φ

∂qj

∂

∂ψ

(θ(τj, y0 + y,ψ0 + ψ, ε) − θ(τj, y0 + y,ψ0 + ψ, ε))

_         

− P

−1(y0, ψ0, ε)

__r

j=1

_ ∂Φ

∂pj

∂

∂y

x(τj, y0 + y, ε)

+ ∂Φ

∂qj

∂

∂y

θ(τj, y0 + y,ψ0 + ψ, ε),

∂Φ

∂qj

_

− P(y0, ψ0, ε)

           

. (8.15)

Using inequalities (8.4) and (8.8), one can estimate the norm of the matrix in the

first braces on the right-hand side of the last equality from above by the value

(n + m)2c2c4εα. The smoothness conditions for the right-hand side of system

(8.5) yield the following representation:

94 Averaging Method in Multipoint Problems Chapter 2

∂

∂y

x(τ, y0 + y, ε) = ∂

∂y0 x(τ, y0, ε) + _X (τ, y, ε),

∂

∂ψ

θ(τ, y0 + y,ψ0 + ψ, ε) = Em, (8.16)

∂

∂y

θ(τ, y0 + y,ψ0 + ψ, ε)

=

_τ

0

∂

∂x

ω(x(t, y0, ε), t, ε) ∂

∂y0 x(t, y0, ε)dt + _Y (τ, y, ε),

where Em is the m-dimensional identity matrix,

__X (τ, y, ε)_ ≤ c9_y_, __Y (τ, y, ε)_ ≤ c9(_y_ + ε), c9 = const,

The smoothness conditions for the function Φ and inequalities (8.4) and (8.8)

yield

∂Φ

∂pj

= ∂Φ0

∂pj

+_Φj(z, ε),

∂Φ

∂qj

= ∂Φ0

∂qj

+Φ∼

j

(z, ε),

where

__Φj(z, ε)_ + _Φ∼

j

(z, ε)_ ≤ c10(_z_ + ε1+α), c10 = const.

Therefore, the norm of the matrix in the second braces on the right-hand side of

(8.15) can be estimated from above by the value

(n + m)2

_

2c2c9_y_ + c2c9ε + c10(_y_ + _ψ_ + ε1+α)

×

_

2c9_y_ + sup

G

___

∂x(τ, y, ε)

∂y

___

_

1 + Lsup

G

___

∂ω(x, τ, ε)

∂x

___

___

≤ c11ε,

c11 = const,

for z ∈ V. Thus,

___

∂M(z, ε)

∂z

___

≤ c5[c2c4(n + m)2εα + c11ε] ≤ 1

2

for

ε ≤ ε0 ≤ [2c5(c2c4(n + m)2 + c11)]− 1

α ,

Section 8 Multipoint Problem for Resonance Multifrequency System 95

i.e., the mapping M: V → V is contracting. Thus, there exists a unique solution

z = z(ε) = (y(ε), ψ(ε)) of Eq. (8.11) that satisfies the condition _z(ε)_ ≤

2c2c4c5ε1+α and, therefore, there exists a unique solution

(x(τ, ε); θ(τ, ε)) = (x(τ, y0 +y(ε), ψ0 +ψ(ε), ε); θ(τ, y0 +y(ε), ψ0 +ψ(ε), ε))

of the multipoint problem (8.1), (8.3) whose initial data lie in a small neighborhood

of the point (y0, ψ0). Estimate (8.10) follows from the inequalities

_x(τ, ε) − x(τ, y0, ε)_ + _θ(τ, ε) − θ(τ, y0, ψ0, ε)_

≤ _x(τ, ε) − x(τ, y0 + y(ε), ε)_

+ _θ(τ, ε) − θ(τ, y0 + y(ε), ψ0 + ψ(ε), ε)_

+ _x(τ, y0 + y(ε), ε) − x(τ, y0, ε)_

+ _θ(τ, y0 + y(ε), ψ0 + ψ(ε), ε) − θ(τ, y0, ψ0, ε)_

≤ c6ε1+α,

where c6 = 2c4 + 2c2c4c5(2mc1L + 1)ne2c1L.

It remains to impose the condition c6ε1+α

0

≤ 1

2ρ in order that the solution

(x(τ, ε), θ(τ, ε)) of problem (8.1), (8.3) do not leave the domain D × Rm. Theorem

8.1 is proved.

As an example, we consider the three-point problem

dx

dτ

= −x + εx2(cos ϕ2 + cos(5ϕ2 − ϕ1)),

dϕ1

dτ

=

2x2 + 2

ε

+ sin ϕ2,

dϕ2

dτ

= τ

ε

+ x sin(5ϕ2 − ϕ1),

x|τ=0 + εϕ2|τ=0 = −1.5; εϕ1|τ=1

2

= 0,

x|τ=1 + εϕ1|τ=1 + εϕ2|τ=1 = 0, (8.17)

where x ∈

_1

2, 4

_

, ϕ1 ∈ R, ϕ2 ∈ R, τ ∈ [0, 1], and ε is a small positive

parameter. The corresponding problem averaged with respect to all angular

variables

96 Averaging Method in Multipoint Problems Chapter 2

dx

dτ

= −x,

dθ1

dτ

= 2x2 + 2,

dθ2

dτ

= τ, θ1 = εϕ1, θ2 = εϕ2,

x|τ=0 + θ2|τ=0 = −1.5, θ1|

τ=1

2

= 0, x|τ=1 + θ1|τ=1 + θ2|τ=1 = 0

has the unique solution

x(τ) = e

−τ+1, θ1(τ) = 2τ − e2(1−τ) + e − 1, θ2(τ) = τ 2

2

− 1.5 − e,

which belongs to the set

_1

2, 4

_

× R2 together with its

1

2

-neighborhood. For

this solution, we have

P(y0, ψ0, ε) =

⎛

⎜⎜⎜⎝

1 0 1

2(e −1) 1 0

2e − 1

e

1 1

⎞

⎟⎟⎟⎠

, det P = −1 +

1

e

_= 0.

Moreover,

detW2(x, τ, ε) =

_____

2x2 + 2 τ

−4x2 1

_____

≥ 2.5 ∀(τ, x) ∈ [0, 1] ×

_1

2, 4

_

.

Therefore, according to Theorem 8.1, for every ε ∈ (0, ε0] (ε0 is sufficiently

small) there is a unique solution (x(τ, ε), ϕ1(τ, ε), ϕ2(τ, ε)) of problem (8.17)

that satisfies the inequality

|x(τ, ε) − e1−τ | +

___

ϕ1(τ, ε) − 2τ − e2(1−τ) + e − 1

ε

___

ε

+

___

ϕ2(τ, ε) − τ 2 − 3 − 2e

2ε

___

ε ≤ c6ε1+1

2

for all (τ, ε) ∈ [0, 1] × (0, ε0].

Remark 3. The verification of the restrictions imposed by conditions (ii) and

(iii) of Theorem 8.1 and related to the value of the small parameter ε can be a

fairly difficult problem. Assume, in addition, that the functions a, ω, and Φ are

smooth with respect to ε ∈ [0, ε0], and consider the problem

dξ

dτ

= a(ξ, τ, 0),

dη

dτ

= ω(ξ, τ, 0), Φ(ξ|τ=τ1, . . . , η|τ=τr ,0) = 0.

Section 8 Multipoint Problem for Resonance Multifrequency System 97

Assume that this problem has the unique solution

(ξ(τ, ξ0); η(τ, ξ0, η0)), ξ(0, ξ0) = ξ0, η(0, ξ0, η0) = η0,

that lies in D×Rm ∀τ ∈ [0, L] and satisfies the condition

det

_r

j=1

_∂Φ00

∂pj

∂ξ(τj, ξ0)

∂ξ0 + ∂Φ00

∂qj

_τj

0

∂ω(ξ(t, ξ0), t, 0)

∂ξ

∂ξ(t, ξ0)

∂ξ0 dt,

∂Φ00

∂qj

_

_= 0.

Here, the values of the derivatives

∂Φ00

∂pj

and

∂Φ00

∂qj

of Φ(p1, . . . , qr, 0) are taken

for pν = ξ(τν, ξ0) and qν = η(τν, ξ0, η0), ν = 1, r. It is easy to verify

that these assumptions are sufficient for the existence of a solution (x(τ, y0, ε);

θ(τ, y0, ψ0, ε)) of problem (8.5), (8.6) that satisfies condition (8.9) and the inequality

_x(τ, y0, ε) − ξ(τ, ξ0)_ + _θ(τ, y0, ψ0, ε) − η(τ, ξ0, η0)_ ≤ _c6ε.

Remark 4. It follows from estimates (8.10) that it suffices to impose restrictions

(8.2), (8.4), and (8.7) on the functions c, Φ, and ω not in the entire

domain of their definition, but only in a certain μ(ε)-neighborhood (μ(ε) → 0

as ε → 0) of the solution of the averaged problem.

An analog of Theorem 8.1 is also true for the multipoint problem

dx

dτ

= a(x, ϕ, τ, ε),

dϕ

dτ

= ω(τ )

ε

+ b(x, ϕ, τ, ε),

Φ(x|τ=τ1, . . . , x|τ=τr, ϕ|τ=τ1, . . . , ϕ|τ=τr, ε) = 0,

(8.18)

where the frequencies ω depend only on the time variable, and Φ is an (n+m)-

dimensional functional. Consider the averaged problem

dx

dτ

= a(x, ϕ, τ, ε),

dϕ

dτ

= ω(τ )

ε

+ b(x, τ, ε),

Φ(x|τ=τ1, . . . , x|τ=τr , ϕ|τ=τ1, . . . , ϕ|τ=τr, ε) = 0

(8.19)

98 Averaging Method in Multipoint Problems Chapter 2

and denote by S(y0, ψ0, ε) the (n + m)-dimensional square matrix

S(y0, ψ0, ε) =

_r

j=1

_∂Φ0

∂pj

∂x(τj, y0, ε)

∂y0

+ ∂Φ0

∂qj

_τj

0

∂b(x(t, y0, ε), t, ε)

∂x

∂x(t, y0, ε)

∂y0 dt,

∂Φ0

∂qj

_

. (8.20)

Here, the values of the derivatives of the function Φ(p1, . . . , qr, ε) with respect to

pj and qj are taken for pν = x(τν, y0, ε) and qν = ϕ(τν, y0, ψ0, ε), ν = 1, r,

and (x(τ, y0, ε); ϕ(τ, y0, ψ0, ε)) is a solution of the averaged system for which

x(0, y0, ε) = y0 and ϕ(0, y0, ψ0, ε) = ψ0.

Theorem 8.2. Suppose that the following conditions are satisfied:

(i) there exists a unique solution (x(τ, y0, ε); ϕ(τ, y0, ψ0, ε)) of the averaged

problem (8.19) whose slow variables belong to D together with their ρ-

neighborhoods;

(ii) for this solution, the matrix S(y0, ψ0, ε) is nondegenerate and, furthermore,

_S−1(y0, ψ0, ε)_ ≤ c = const ∀ε ∈ (0, ε0];

(iii) det(WT

p (τ )Wp(τ )) _= 0 for any τ ∈ [0, L] and certain p ≥ m;

(iv) conditions (2.6) and (8.4) are satisfied.

Then there exist constants c > 0 and ε > 0 such that, for any ε ∈ (0, ε],

problem (8.18) has a unique solution (x(τ, ε); ϕ(τ, ε)) that satisfies the inequality

_x(τ, ε) − x(τ, y0, ε)_ + _ϕ(τ, ε) − ϕ(τ, y0, ψ0, ε)_ ≤ cε1/p. (8.21)

Section 9 Estimates of Error of Averaging Method for Multipoint Problems 99

Навигация