11. Boundary-Value Problems with Parameters

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In the present section, we study boundary-value problems with parameters for

the oscillation system

dx

dτ

= a(x, ϕ, τ ),

dϕ

dτ

= ω(τ )

ε

+ b(x, ϕ, τ ). (11.1)

Most investigations of boundary-value problems with parameters relate to the case

where unknown parameters are present only in differential equations. However,

for practical purposes, it is also necessary to study problems with parameters

in boundary conditions [Luc, Sam9, SaR]. Below, we show that the averaging

method can be efficiently applied to the proof of the solvability of boundary-value

problems with parameters.

Assume that the functions a and b have continuous bounded partial derivatives

with respect to (x, ϕ, τ ) ∈ D×Rm×[0, L] up to the second order inclusive

and are almost periodic in ϕν, ν = 1,m, and such that

[a(x, ϕ, τ ); b(x, ϕ, τ )] =

∞_

s=0

[as(x, τ ); bs(x, τ )]ei(λs,ϕ),

where i is the imaginary unit, λ0 = 0, λs _= 0 for s ≥ 1, (λs, ϕ) is the

scalar product of vectors λs = (λ(1)

s , . . . , λ(m)

s ) and ϕ = (ϕ1, . . . , ϕm), and the

functions cs = [as(x, τ ); bs(x, τ )] satisfy the inequality

Section 11 Boundary-Value Problems with Parameters 121

sup _c0_ + sup

___

∂c0

∂τ

___

+ sup

___

∂c0

∂x

___

+

_n

j=1

sup

___

∂2c0

∂x∂xj

___

+

∞_

s=1

__

_λs_ +

1

_λs_

_

sup _cs_

+

_

1 +

1

_λs_

__

sup

___

∂cs

∂τ

___

+ sup

___

∂cs

∂x

___

_

+

1

_λs_

_

sup

___

∂2cs

∂τ∂x

___

+

_n

j=1

sup

___

∂2cs

∂x∂xj

___

__

≤ σ1, (11.2)

where the supremum is taken over all (x, τ ) ∈ D×[0, L].

Consider boundary conditions of the form

A1x|τ=0 + A2x|τ=μ = C1, xn|τ=0 = x0

n,

B1ϕ|τ=0 + B2x|τ=μ = C2, (11.3)

where A1 and A2 are n×n matrices, B1 and B2 are m×m matrices, C1 and

C2 are n-dimensional and m-dimensional vectors, μ ∈ (0, L) is an unknown

parameter, xn is the nth coordinate of the component x = (x1, . . . , xn) of a

solution (x; ϕ) of system (11.1), and x0

n is a given number.

Problem (11.1), (11.3) is a boundary-value problem with nonfixed right

boundary. To solve this problem, i.e., to determine the unknown parameter μ

and a solution of system (11.1) that satisfies the boundary conditions (11.3), we

use the method of averaging over all fast variables ϕ. We write the averaged

problem

dx

dτ

= a(x, τ ), (11.4I)

dϕ

dτ

= ω(τ )

ε

+ b(x, τ ), (11.4II)

A1x|τ=0 + A2x|τ=μ = C1, xn|τ=0 = x0

n, (11.4III)

B1ϕ|τ=0 + B2ϕ|τ=μ = C2, (11.4IV)

122 Averaging Method in Multipoint Problems Chapter 2

where

[a; b] = lim

T→∞

T

−m

_T

0

. . .

_T

0

[a(x, ϕ, τ ); b(x, ϕ, τ )]dϕ1 . . . dϕm

≡ [a0(x, τ ); b0(x, τ )].

As in the previous sections, we denote by (x(τ, y, ψ, ε); ϕ(τ, y, ψ, ε)) and

(x(τ, y); ϕ(τ, y, ψ, ε)), respectively, the solutions of problems (11.1) and (11.4I),

(11.4II) that take the value (y; ψ) for τ = 0. By Dρ, we denote the set of

points y ∈ D for which the curve x = x(τ, y) lies in D together with its ρ-

neighborhood ∀τ ∈ [0, L]. Assume that the set Dρ0 is nonempty for certain

ρ0 > 0.

Lemma 11.1. Suppose that the following conditions are satisfied:

(a) there exists a unique solution _x0, μ0 of the equation

A1x0 + A2x(μ0, x0) = C1

such that x0 = (_x0, x0

n) ∈ Dρ0 and μ0 ∈ (0, L);

(b) det(B1 + B2) _= 0.

Then there exists a unique solution {μ0, x(τ, x0), ϕ(τ, x0, ϕ0, ε)} of the averaged

problem (11.4I)–(11.4IV) defined ∀(τ, ε) ∈ [0, L] × (0, ε0].

Proof. It follows from condition (a) that the solution x = x(τ, x0) of

Eq. (11.4I) is defined for all τ ∈ [0, L] and satisfies the boundary condition

(11.4III) for μ = μ0. It is easy to verify that this solution is associated with the

unique solution

ϕ(τ, x0, ϕ0, ε) = ϕ0 +

1

ε

_τ

0

[ω(t) + εb(x(t, x0), t)]dt, τ ∈ [0, L], (11.5)

of problem (11.4II), (11.4IV), where

ϕ0 = (B1 + B2)−1

_

C2 − 1

ε

B2

_μ0

0

[ω(t) + εb(x(t, x0), t)]dt

_

.

Lemma 11.1 is proved.

Section 11 Boundary-Value Problems with Parameters 123

We now study the problem of the existence of a solution of the original problem

(11.1), (11.3) and establish an estimate for its deviation from a solution of the

averaged problem (11.4I)–(11.4IV). For this purpose, we denote by P the n×n

square matrix

P =

_

A11

+ A2

∂x(μ0, x0)

∂_x0 ,A2a(x(μ0, x0), μ0)

_

.

Here, A11

is the n×(n−1) rectangular matrix whose columns are the first n−1

columns of the matrix A.

Theorem 11.1. Suppose that the following conditions are satisfied:

(i) ω(τ) = (ω1(τ ), . . . , ωm(τ )) ∈ Cm−1+l

[0,L] and the Wronskian of the functions

ω1(τ ), . . . , ωm(τ ) has zeros of multiplicity not higher than l on

[0, L];

(ii) the conditions of Lemma 11.1 and inequality (11.2) are satisfied;

(iii) det P _= 0 and

σ0 = sup

ϕ∈Rm

_P

−1A2_a(x(μ0, x0), ϕ, μ0)_ < 1,

where _a(x, ϕ, τ) = a(x, ϕ, τ ) − a(x, τ ).

Then one can find positive constants ε0 ≤ ε0 and σ such that, for every

ε ∈ (0, ε0], there exists a solution {μ(ε), x(τ, ε), ϕ(τ, ε)} of the boundary-value

problem (11.1), (11.3) that satisfies the estimates

|μ(ε) − μ0| + _x(τ, ε) − x(τ, x0)_ ≤ σεα,

_ϕ(τ, ε) − ϕ(τ, x0, ϕ0, ε)_ ≤ σεα−1 ∀τ ∈ [0, L], α =

1

m + l

. (11.6)

Proof. We seek a solution of problem (11.1), (11.3) in the form

{μ0 + h, x(τ, x0 + y,ϕ0 + ψ, ε), ϕ(τ, x0 + y,ϕ0 + ψ, ε)},

where y = (_y,0) = (y1, . . . , yn−1, 0), h, and ψ = (ψ1, . . . , ψm) are unknown

parameters. For their determination, we use the boundary conditions (11.3). As a

result, we obtain

A1y + A2x(μ0 + h, x0 + y) = C1 − A1x0 − A2Δxμ0+h,

B1ψ + B2ϕ(μ0 + h, x0 + y,ψ0 + ψ, ε) = C2 − B1ϕ0 − B2Δϕμ0+h, (11.7)

124 Averaging Method in Multipoint Problems Chapter 2

where

Δxτ = x(τ, x0 + y,ϕ0 + ψ, ε) − x(τ, x0 + y),

Δϕτ = ϕ(τ, x0 + y,ϕ0 + ψ, ε) − ϕ(τ, x0 + y,ϕ0 + ψ, ε).

Note that if the conditions of Theorem 11.1 are satisfied and ε0 > 0 is sufficiently

small, then it follows from the results of Chapter 1 that the following inequality

holds for all τ ∈ [0, L], _y_ ≤ 1

2ρ0, ψ ∈ Rm, and ε ∈ (0, ε0]:

_Δxτ _ + _Δϕτ _ +

___

∂

∂y

Δxτ

___

+

___

∂

∂y

Δϕτ

___

+

___

∂

∂ψ

Δxτ

___

+

___

∂

∂ψ

Δϕτ

___

≤ σεα, (11.8)

where σ is a certain positive constant independent of ε.

Note that

x(μ0 + h, x0 + y)

= x(μ0, x0) + ∂x(μ0, x0)

∂x0 y + a(x(μ0, x0), μ0)h + X(x0, μ0, h, y),

ϕ(μ0 + h, x0 + y,ϕ0 + ψ, ε) = ϕ0 + ψ +

1

ε

μ_0+h

0

[ω(t) + εb(x(t, x0 + y), t)] dt,

_X(x0, μ0, h, y)_ ≤ σ1(_y_2 + h2), (11.9)

___

∂

∂y

X(x0, μ0, h, y)

___

+

___∂

∂h

X(x0, μ0, h, y)

___

≤ σ1(_y_ + |h|),

σ1 = const,

Therefore, denoting (_y, h) = z, we can rewrite equalities (11.7) in the form

z = −P

−1A2[Δxμ0+h + X(x0, μ0, h, y)] ≡ M(z,ψ, ε),

Section 11 Boundary-Value Problems with Parameters 125

ψ = −(B1 + B2)−1B2

_ μ_0+h

μ0

_ω(t)

ε

+ b(x(t, x0 + y), t)

_

dt

+

_μ0

0

(b(x(t, x0 + y), t) − b(x(t, x0), t))dt+Δϕμ0+h

_

≡ N(z,ψ, ε). (11.10)

It follows from (11.8) and (11.9) that, for every fixed ψ ∈ Rm and ε ∈ (0, ε0],

M(z,ψ, ε) maps the set

K = {z : z ∈ Rn, _z_ ≤ σ2εα}, σ2 = 2σ_P

−1A2_,

into itself for

ε0 ≤ (4σ1σ)− 1

α _P

−1A2_− 2

α .

In addition, we impose the restriction

εα0

≤ 1

σ2

min{μ0;L − μ0},

which guarantees that the condition μ0 + h ∈ [0, L] is satisfied.

Further, we show that M: K → K is a contracting mapping. Using estimate

(11.8) and the equality

d

dτ

Δxτ = [a(x, τ ) − a(x, τ )] + _a(x, ϕ, τ ),

from the relation

∂M

∂z

= −P

−1A2

__ ∂

∂_y

Δxμ0+h; ∂

∂h

Δxμ0+h

_

+ ∂

∂z

X(x0, μ0, h, y)

_

we derive the inequality

___

∂M

∂z

___

≤ _P

−1A2_(σ + σσ1 + σ1σ2)εα

+ _P

−1A2_a(x(μ0 + h, x0 + y,ϕ0 + ψ, ε),

ϕ(μ0 + h, x0 + y,ϕ0 + ψ, ε), μ0 + h)_.

126 Averaging Method in Multipoint Problems Chapter 2

This inequality, the restriction σ0 < 1, and the inequality

__a(x(μ0 + h, x0 + y,ϕ0 + ψ, ε), ϕ, μ0 + h) − _a(x(μ0, x0, ϕ, μ0)_

≤ σ1[σ + σ2(neσ1L +1+σ1)]εα

∀z ∈ K, ϕ ∈ Rm, ε∈ (0, ε0],

[which follows from (11.2), (11.8), and (11.9)] imply that, for all z ∈ K, ε ∈

(0, ε0], ε0 ≤ [(2σ2)−1(1 − σ0)] 1

α , and ψ ∈ Rm, the following estimate is true:

___

∂M

∂z

___

≤ σ0 + σ2εα ≤ σ0 + 1

2 < 1;

here,

σ2 = _P

−1A2_ [σ + 2σσ1 + σ1σ2 + σ1σ2(1 + neσ1L + σ1)].

Thus, M: K → K is a contracting mapping, and, therefore, there exists a unique

solution z = z0(ψ, ε) ≡ (_y0(ψ, ε), h0(ψ, ε)) ∈ K that continuously depends on

(ψ, ε) ∈ Rm × (0, ε0].

Substituting the value of z = z0(ψ, ε) into the second equation in (11.10),

we obtain

ψ = N (z0(ψ, ε), ψ, ε). (11.11)

Since the mapping N is continuous in ψ ∈ Rm and, according to (11.9),

_N(z0(ψ, ε), ψ, ε)_ ≤ σ3εα−1 ∀(ψ, ε) ∈ Rm × (0, ε0],

where

σ3 = _(B1 + B2)−1B2_(σ + σ1σ2 + 2σ1L + σ2 max

[0,L]

_ω(τ )_),

using the Schauder theorem we establish that there exists a solution ψ = ψ0(ε),

_ψ0(ε)_ ≤ σ3εα−1, of Eq. (11.11). Hence,

{μ(ε), x(τ, ε), ϕ(τ, ε)}

= {μ0 + μ0(ε), x(τ, x0 + y0(ε), ϕ0 + ψ0(ε), ε),

ϕ(τ, x0 + y0(ε), ϕ0 + ψ0(ε), ε)},

where μ0(ε) = h0(ψ0(ε), ε), y0(ε) = (_y0(ψ0(ε), ε), 0), is a solution of the

boundary-value problem (11.1), (11.3). Estimates (11.6) with the constant σ =

σ +σ3 +nσ2eσ1L max{1; σ1L} follow from estimate (11.8) and the inequalities

_y0(ε)_ + |μ0(ε)| ≤ σ2εα and _ψ0(ε)_ ≤ σ3εα−1. Theorem 11.1 is proved.

Section 11 Boundary-Value Problems with Parameters 127

Corollary 1. Suppose that B2 = 0 in Theorem 11.1, i.e., the boundary condition

for the fast variables ϕ turns into the initial condition ϕ|τ=0 = B

−1

1 C2 ≡

ϕ0. Then, in a small neighborhood of the solution {μ0, x(τ, x0), ϕ(τ, x0, ϕ0, ε)}

of the averaged problem (11.4I)–(11.4IV), there exists a unique solution {μ(ε),

x(τ, ε), ϕ(τ, ε)} of the boundary-value problem (11.1), (11.3), and this solution

satisfies the following inequality for all (τ, ε) ∈ [0, L] × (0, ε0]:

|μ(ε) − μ0| + _x(τ, ε) − x(τ, x0)_ + _ϕ(τ, ε) − ϕ(τ, x0, ϕ0, ε)_ ≤ σεα.

The method proposed above can be generalized to the case of a multipoint

problem that contains unknown parameters μ1, . . . , μr (2 ≤ r < n) in the

boundary conditions. Instead of (11.3), we consider the boundary conditions

_r

j=1

Ajx|τ=μj = C1, x∼

|τ=0 = x∼

0,

_r

j=1

Bjϕ|τ=μj = C2, (11.12)

where 0 < μ1 < μ2 < ... < μr < L, x∼

= (xn−r+1, . . . , xn) is the vector

whose coordinates are the last r coordinates of the slow component x =

(x1, . . . , xn) of the solution (x; ϕ) of system (11.1), and x∼

0 is a given r-dimensional

vector.

Lemma 11.2. If the matrix

,r

j=1

Bj is nondegenerate and there exists a unique

solution _x0 = (x01

, . . . , x0

n−r), μ0 = (μ01

, . . . , μ0r

) of the equation

_r

j=1

Ajx(μ0j

, x0) = C1

that satisfies the conditions x0 = (_x0,x∼

0) ∈ Dρ0 , 0 < μ01

< μ02

< . . . <

μ0r

< L, then there exists a unique solution {μ0, x(τ, x0), ϕ(τ, x0, ϕ0, ε)} of the

averaged problem (11.4I), (11.4II), (11.12).

Proof. It is obviously sufficient to find the ϕ-component of a solution of the

averaged problem. To do this, we use formula (11.5), in which we set

ϕ0 =

__r

j=1

Bj

_−1

_

C2 − 1

ε

_r

j=1

Bj

μ0j

_

0

(ω(t) + εb(x(t, x0), t)) dt

_

.

Lemma 11.2 is proved.

128 Averaging Method in Multipoint Problems Chapter 2

Denote by Q the n × n square matrix

__r

j=1

Aj

∂x(μ0j

, x0)

∂_x0 ,A1a(x(μ01

, x0), μ01

), . . . , Ara(x(μ0r

, x0), μ0r

)

_

.

The proof of the theorem below, in fact, repeats the proof of Theorem 11.1,

and, therefore, we present only its formulation.

Theorem 11.2. Suppose that the following conditions are satisfied:

(a) condition (i) of Theorem 11.1, inequality (11.2), and the conditions of Lemma

11.2 are satisfied;

(b) the matrix Q is nondegenerate and

sup

ϕ(j)∈Rm,1≤j≤r

__ _

Q

−1

_r

j=1

Aj_a(x(μ0j

, x0), ϕ(j), μ0j

)

___

< 1.

Then, for sufficiently small ε0 > 0 and every ε ∈ (0, ε0], there exists a solution

{μ(ε), x(τ, ε), ϕ(τ, ε)} of the multipoint problem (11.1), (11.2) that satisfies

the following inequality for any (τ, ε) ∈ [0, L] × (0, ε0]:

_μ(ε) − μ0_ + _x(τ, ε) − x(τ, x0)_ + ε_ϕ(τ, ε) − ϕ(τ, x0, ϕ0, ε)_ ≤ σ4εα,

where the constant σ4 is independent of ε.

Finally, we consider the case where an unknown scalar parameter μ ∈ R

enters into the boundary conditions in a linear manner. For the multifrequency

system (11.1), we introduce the boundary conditions

A1x|τ=0 + μA2x|τ=L = C1, xn|τ=0 = x0

n,

B1ϕ|τ=0 + B2ϕ|τ=L = C2 (A2 _= 0).

(11.13)

The solvability of the averaged problem (11.4I), (11.4II), (11.13) follows from

the lemma presented below.

Lemma 11.3. Suppose that the matrix B1 + B2 is nondegenerate and the

equation

A1x0 + μ0A2x(L, x0) = C1

Section 11 Boundary-Value Problems with Parameters 129

has a unique solution μ0, _x0 = (x01

, . . . , x0

n−1) for which x0 = (_x0, x0

n) ∈

Dρ0 . Then, for all (τ, ε) ∈ [0, L] × (0, ε0], the unique solution {μ0, x(τ, x0),

ϕ(τ, x0, ϕ0, ε)}, where

ϕ0 = (B1 + B2)−1

_

C2 − 1

ε

B2

_L

0

(ω(t) + εb(x(t, x0), t)) dt

_

,

of the boundary-value problem (11.4I), (11.4II), (11.13) is defined.

Lemma 11.3 can be proved by analogy with Lemmas 11.1 and 11.2.

Further, we choose h ∈ R, _y ∈ Rn−1, and ψ ∈ Rm so that a solution of

problem (11.1), (11.13) has the form

{μ0 + h, x(τ, x0 + y,ϕ0 + ψ, ε), ϕ(τ, x0 + y,ϕ0 + ψ, ε)},

where y = (_y, 0) ∈ Rn. Using the boundary conditions (11.13) and Lemma 11.3,

we get

z = −P

−1

A2

_

(ΔxL + X(x0, L, 0, y))(μ0 + h) + ∂x(L, x0)

∂x0 yh

_

≡ M(z,ψ, ε), (11.14)

ψ = −(B1 + B2)−1B2

_

ΔϕL +

_L

0

(b(x(t, x0 + y), t) − b(x(t, x0), t)) dt

_

≡ N(z,ψ, ε), (11.15)

where z = (_y, h), P is an n × n matrix of the form

P =

_

A11

+ μ0A2

∂x(L, x0)

∂_x0 ,A2x(L, x0)

_

,

and X and A11

are defined in Theorem 11.1.

If μ0 _= 0, then the analysis of the inequality

_M(z,ψε)_ ≤ _P

−1

A2_[(σεα + σ1_z_2)(μ0 + _z_) + neσ1L_z_2]

[which follows from (11.2), (11.8), (11.9), and (11.14)] shows that, for fixed ψ ∈

Rm, ε ∈ (0, ε0], and

ε0 ≤

_ σ5

σμ0 (σ + σ1σ5μ0 + σ1σ2

5 + neσ1Lσ5)

_− 1

α, σ5 = 2σμ0_P

−1

A2_,

130 Averaging Method in Multipoint Problems Chapter 2

M(z,ψ, ε) maps the set K = {z : z ∈ Rn, _z_ ≤ σ5εα} into itself. Moreover,

it follows from (11.14) that

___

∂

∂z

M(z,ψ, ε)

___

≤ _P

−1

A2_[σ(μ0 + 1) + σ5(σ + σ1μ0 + n2eσ1L) + 2σ1σ2

5]εα

≡ σ6εα ≤ 1

2

for εα0

≤ (2σ6)−1.

Therefore, for μ0 _= 0, there exists a unique solution

z = z(ψ, ε) = (_y(ψ, ε), h(ψ, ε)) ∈ K

of Eq. (11.14), which can be determined by the method of successive approximations:

zs+1(ψ, ε) = M(zs(ψ, ε), ψ, ε), s≥ 0,

z0(ψ, ε) ≡ 0, lim

s→∞zs(ψ, ε) = z(ψ, ε).

Differentiating the equality zs+1(ψ, ε) = M(zs(ψ, ε), ψ, ε) with respect to ψ

and taking into account estimate (11.8), we get

___

∂

∂ψ

zs+1(ψ, ε)

___

≤ σ7εα

___

∂

∂ψ

zs(ψ, ε)

___

+ σ8εα, s≥ 0, (11.16)

where

σ7 = _P

−1

A2_[(μ0 + σ5)(σ + σ1σ5) + σ + σ1σ2

5 + nσ5eσ1L],

σ8 = _P

−1

A2_σ(μ0 + σ5).

Inequality (11.16) yields

___

∂

∂ψ

zs+1(ψ, ε)

___

≤ σ8

1 − σ7εα0

εα ≡ σ9εα

∀s ≥ 0, ψ∈ Rm, ε∈ (0, ε0],

provided that εα0

≤ (2σ7)−1. This is sufficient for the function z(ψ, ε) to satisfy

the Lipschitz condition

_z(ψ, ε) − z(ψ, ε)_ ≤ σ9εα_ψ − ψ_ ∀ψ,ψ ∈ Rm. (11.17)

Section 11 Boundary-Value Problems with Parameters 131

If μ0 = 0, then it follows from Eq. (11.14) that z = z(ψ, ε) ≡ 0 is its unique

solution for small _z_.

Substituting z = z(ψ, ε) in (11.15), we obtain the equation

ψ = N(z(ψ, ε), ψ, ε). (11.18)

Inequality (11.8) and the restriction _y_ ≤ σ5εα yield

_N(z(ψ, ε), ψ, ε)_ ≤ σ10εα,

σ10 = _(B1 + B2)−1B2_(σ + Lnσ1σ5eσ1L),

and the Lipschitz condition (11.17) guarantees that the following inequality holds

for all ψ,ψ ∈ Rm and ε ∈ (0, ε0]:

_N(z(ψ, ε), ψ, ε) − N(z(ψ, ε), ψ, ε)_ ≤ σ10εα_ψ − ψ_.

If we choose ε0 > 0 so small that σ10εα0

≤ 1

2, then N(z(ψ, ε), ψ, ε) maps the

set _ψ_ ≤ σ10εα into itself and is a contracting mapping. Therefore, there exists

a unique solution ψ = ψ(ε) of Eq. (11.18), and, hence, there exists a unique

solution z = z(ε) = z(ψ(ε), ε), ψ = ψ(ε) of system (11.14), (11.15) that

satisfies the inequalities

_z(ε)_ ≤ σ5εα, _ψ(ε)_ ≤ σ10εα ∀ε ∈ (0, ε0]. (11.19)

Thus,

{μ(ε), x(τ, ε), ϕ(τ, ε)}

= {μ0 + h(ε), x(τ, x0 + y(ε), ϕ0 + ψ(ε), ε),

ϕ(τ, x0 + y(ε), ϕ0 + ψ(ε), ε)},

where z(ε) = (_y(ε), h(ε)), y(ε) = (_y(ε), 0) is the unique solution of problem

(11.1), (11.13) in a small neighborhood of the solution of the averaged problem.

Moreover, according to inequalities (11.8) and (11.9), the following estimate

holds for all (τ, ε) ∈ [0, L] × (0, ε0] :

|μ(ε)−μ0|+_x(τ, ε)−x(τ, x0)_+_ϕ(τ, ε)−ϕ(τ, x0, ϕ0, ε)_ ≤ σ11εα, (11.20)

where σ11 = σ + σ10 + neσ1Lσ5 max{1; Lσ1}.

132 Averaging Method in Multipoint Problems Chapter 2

Thus, the following statement is true:

Theorem 11.3. If condition (i) of Theorem 11.1, inequality (11.2), the conditions

of Lemma 11.3, and the condition det P _= 0 are satisfied, then, for sufficiently

small ε0 > 0 and all ε ∈ (0, ε0], in a small neighborhood of the solution

{μ0, x(τ, x0), ϕ(τ, x0, ϕ0, ε)} of the averaged problem (11.4I), (11.4II), (11.13)

the boundary-value problem (11.1), (11.13) has a unique solution {μ(ε), x(τ, ε),

ϕ(τ, ε)} that satisfies inequality (11.20).

3. INTEGRAL MANIFOLDS

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