9. Estimates of the Error of Averaging Method for Multipoint Problems in Critical Case

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In this section, we consider a multipoint problem 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), (9.1)

F(x|τ=τ1, . . . , x|τ=τr, ε) = 0, (9.2)

Φ(x|τ=τ1, . . . , x|τ=τr, ϕ|τ=τ1, . . . , ϕ|τ=τr, ε) = 0, (9.3)

where F(p1, . . . , pr, ε) and Φ(p1, . . . , pr, q1, . . . , qr, ε) are, respectively, n-dimensional

and m-dimensional vector functions of the variables pj ∈ D, qj ∈

Rm, j = 1, r, and ε ∈ (0, ε0], and 0 ≤ τ1 < τ2 < ... < τr ≤ L, r ≥ 2.

The main difference between this problem and problem (8.1), (8.3) lies in the

fact that, first, in problem (9.1)–(9.3) the group of boundary conditions dependent

only on slow variables is selected, and, second, in conditions (9.3) the function Φ

depends only on the arguments ϕ|τ=τj , whereas in conditions (8.3) it depends on

εϕ|τ=τj .

We also consider the corresponding averaged problem

dx

dτ

= a(x, τ, ε) + εA(x, τ, ε), F(x|τ=τ1, . . . , x|τ=τr, ε) = 0, (9.4)

dϕ

dτ

= ω(x, τ, ε)

ε

+ B(x, τ, ε),

Φ(x|τ=τ1, . . . , x|τ=τr , ϕ|τ=τ1, . . . , ϕ|τ=τr, ε) = 0 (9.5)

and assume that there exists a solution x = x(τ, y0, ε) of problem (9.4) for which

the matrix

P1(y0, ε) =

_r

j=1

∂F0

∂pj

∂x(τj, y0, ε)

∂y0

satisfies the inequality

_P

−1

1 (y0, ε)_ ≤ c12ε

−α1 ∀ε(0, ε0] (9.6)

where c12 > 0 and α1 ≥ 0 are certain constants. Here,

∂F0

∂pj

denotes the

matrix of the partial derivatives of the function F(p1, . . . , pr, ε) with respect to

pj = (p(1)

j , . . . , p(n)

j ) for pν = x(τν, y0, ε), ν = 1, r.

100 Averaging Method in Multipoint Problems Chapter 2

To solve problem (9.5), it suffices to solve the equation

Φ(x(τ1, y0, ε), . . . , x(τr, y0, ε), ψ

+

1

ε

_τ1

0

[ω(x(t, y0, ε), t, ε) + εB(x(t, y0, ε), t, ε)]dt, . . . , ψ

+

1

ε

_τr

0

[ω(x(t, y0, ε), t, ε) + εB(x(t, y0, ε), t, ε)]dt, ε) = 0

with respect to ψ. Assume that there exists a unique solution ψ = ψ0(ε) of this

equation, i.e., there exists a unique solution

ϕ(τ, y0, ψ0, ε) = ψ0 +

_τ

0

_

ω(x(t, y0, ε), t, ε) + εB(x(t, y0, ε), t, ε)

_

dt

of problem (9.5). We also assume that the matrix

P2(y0, ψ0, ε) =

_r

j=1

∂Φ0

∂qj

satisfies the inequality

_P

−1

2 (y0, ψ0, ε)_ ≤ c13ε

−α2, c13 > 0, α2 ≥ 0, (9.7)

where

∂Φ0

∂qj

denotes the matrix of the partial derivatives of Φ(p1, . . . , qr, ε) with

respect to qj = (q(1)

j , . . . , q(m)

j ) for pν = x(τν, y0, ε) and qν = ϕ(τν, y0, ψ0, ε),

ν = 1, r.

If the numbers α1 and α2 in inequalities (9.6) and (9.7) are positive, then

the norms of the matrices P

−1

1 and P

−1

2 may tend to infinity as ε → 0. It is

natural to call this case critical. Below, we study the question of the solvability of

problem (9.1)–(9.3) in the critical case and establish estimates for the deviation of

solutions of the original and averaged problems.

In what follows, we assume that, for every fixed ε ∈ (0, ε0], the functions

F(p1, . . . , pr, ε) and Φ(p1, . . . , qr, ε) are twice continuously differentiable with

respect to pj ∈ D and qj ∈ Rm, j = 1, r, and

Section 9 Estimates of Error of Averaging Method for Multipoint Problems 101

_r

j=1

____ ∂F

∂pj

___

+

_r

ν=1

_n

k=1

___

∂2F

∂pj∂p(k)

ν

___

_

≤ c14,

_r

j=1

_

ε

___

∂Φ

∂pj

___

+

___

∂Φ

∂qj

___

+

_r

ν=1

_m

s=1

_

ε

___

∂2Φ

∂pj∂q(s)

ν

___

+

___

∂2Φ

∂qj∂q(s)

ν

___

__

≤ c14 (9.8)

for all pj ∈ D, qj ∈ Rm, j = 1, r, and ε ∈ (0, ε0].

Theorem 9.1. Suppose that the following conditions are satisfied:

(i) for every ε ∈ (0, ε0], the averaged problem (9.4), (9.5) has a unique solution

(x(τ, y0, ε), ϕ(τ, y0, ψ0, ε)) whose slow variables x(τ, y0, ε) belong

to D together with their ρ-neighborhoods;

(ii) conditions (8.2), (8.7), and (9.6)–(9.8) for α > α1 + 2α2 are satisfied.

Then, for sufficiently small ε0 > 0 and every ε ∈ (0, ε0], problem (9.1)–(9.3)

has a unique solution (x(τ, ε); ϕ(τ, ε)) that satisfies the following inequalities for

any τ ∈ [0, L]:

_x(τ, ε) − x(τ, y0, ε)_ ≤ c15ε1+α−α1 ,

_ϕ(τ, ε) − ϕ(τ, y0, ψ0, ε)_ ≤ c15εα−α1−α2 ,

(9.9)

where the constant c15 does not depend on ε.

Proof. We determine the unknown parameters (y; ψ) of the solution

(x(τ, y0 + y,ψ0 + ψ, ε); ϕ(τ, y0 + y,ψ0 + ψ, ε)) of system (9.1) from conditions

(9.2) and (9.3). We rewrite (9.2) in the form

y = −P

−1

1 (y0, ε)

__

F(x(τ1, y0 + y, ε), . . . , x(τr, y0 + y, ε), ε) − P1(y0, ε)y

_

+

_

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

− F(x(τ1, y0 + y, ε), . . . , x(τr, y0 + y, ε), ε)

__

≡ T(y, ψ, ε). (9.10)

Using the estimate of the error of the averaging method (8.8) and inequalities

(9.8), we obtain

102 Averaging Method in Multipoint Problems Chapter 2

_F(x(τ1, y0 + y,ψ0 + ψ, ε), . . . , ε) − F(x(τ1, y0 + y, ε), . . . , ε)_

≤ c4c14rε1+α. (9.11)

Further, using representations (8.13), inequalities (9.8), and condition (9.4), we

get

F(x(τ1, y0 + y, ε), . . . , ε) = P1(y0, ε)y + _ F(y, ε),

where _ _ F(y, ε)_ ≤ c16_y_2 and c16 = const. Using this equality and inequalities

(9.6) and (9.9), we obtain the following estimate for T(y, ψ, ε):

_T(y, ψ, ε)_ ≤ c12(c4c14r + c16_y_2ε

−1−α)ε1+α−α1 .

This estimate implies that T(y, ψ, ε) maps the set

_y_ ≤ c17ε1+α−α1, c17 = 2rc4c12c14,

into itself for

ε ≤ ε0 ≤ (4c4c2

12c14c16r)

1

2α1

−1−α, ψ∈ Rm.

Let us calculate

∂T

∂y

. We have

∂T

∂y

= −P

−1

1 (y0, ε)

___r

j=1

∂

∂pj

F(x(τ1, y0 + y, ε), . . . , ε)

× ∂

∂y

x(τj, y0 + y, ε) − P1(y0, ε)

_

+

__r

j=1

_ ∂

∂pj

F(x(τ1, y0 + y,ψ0 + ψ, ε), . . . , ε) ∂

∂y

x(τj, y0 + y,ψ0 + ψ, ε)

− ∂

∂pj

F(x(τ1, y0 + y, ε), . . . , ε) ∂

∂y

x (τj, y0 + y, ε)

__       

. (9.12)

In view of inequalities (8.8) and (9.8), the norm of the matrix in the second square

brackets on the right-hand side of (9.12) can be estimated from above by the value

c18εα, c18 = const. Using (8.16) and writing the equality

Section 9 Estimates of Error of Averaging Method for Multipoint Problems 103

_r

j=1

∂

∂pj

F(x(τ1, y0 + y, ε), . . . , ε) ∂

∂y

x(τj, y0 + y, ε)

=

_r

j=1

_∂F0

∂pj

+ Fj(y, ε)

__ ∂

∂y0 x(τj, y0, ε) + _X (τj, y, ε)

_

≡ P1(y0, ε) + K(y, ε),

where _K(y, ε)_ ≤ c19_y_, c19 = const, we can estimate the norm of the

matrix in the first square brackets on the right-hand side of equality (9.12) by the

value c20_y_, c20 = const. Thus,

___

∂

∂y

T(y, ψ, ε)

___

≤ c12ε

−α1(c18εα + c20_y_) ≤ 1

2

for α > α1, _y_ ≤ c17ε1+α−α1, ψ ∈ Rm, and ε ≤ ε0 ≤ [2c12(c18 +

c17c20)]

1

α1

−α . Consequently, Eq. (9.10) has a unique solution y = y(ψ, ε),

_y(ψ, ε)_ ≤ c17ε1+α−α1 , which can be determined by the method of successive

approximations:

yk+1(ψ, ε) = T(yk(ψ, ε), ψ, ε), k≥ 1,

y0(ψ, ε) ≡ 0, y(ψ, ε) = lim

k→∞

yk(ψ, ε).

Using equality (9.10), we get

∂yk(ψ, ε)

∂ψ

= ∂

∂ψ

T(yk−1(ψ, ε), ψ, ε)

= −P

−1

1 (y0, ε)

___r

j=1

∂

∂pj

F(x(τ1, y0 + yk−1, ε), . . . , ε)

× ∂

∂y

x(τj, y0 + yk−1, ε)∂yk−1

∂ψ

− P1(y0, ε)∂yk−1

∂ψ

_

+

__r

j=1

_ ∂

∂pj

F(x(τ1, y0 + yk−1, ψ0 + ψ, ε), . . . , ε)

104 Averaging Method in Multipoint Problems Chapter 2

×

_ ∂

∂y

x(τj, y0 + yk−1, ψ0 + ψ, ε)∂_____________yk−1

∂ψ

+ ∂

∂ψ

x(τj, y0 + yk−1, ψ, ε)

_

− ∂

∂pj

F(x(τ1, y0 + yk−1, ε), . . . , ε) ∂

∂y

x(τj, y0 + yk−1, ε)∂yk−1

∂ψ

__       

.

Further, using the methods proposed in the course of the investigation of the properties

of T(y, ψ, ε), we obtain

___

∂yk(ψ, ε)

∂ψ

___

≤ c21ε1+α−α1 + c22εα−α1

___

∂yk−1(ψ, ε)

∂ψ

__ _

,

where c21 and c22 are certain constants independent of ε. This yields

___

∂yk(ψ, ε)

∂ψ

___

≤ c23ε1+α−α1 ∀k ≥ 0, c23 = 2c21,

provided that ε0 ≤ (2c22)

1

α1

−α . Hence, the sequence

_ ∂

∂ψ

yk(ψ, ε)

_

is uniformly

bounded by the constant c23ε1+α−α1 . This is sufficient to guarantee that

the function y(ψ, ε) satisfies the Lipschitz condition

_y(ψ1, ε) − y(ψ2, ε)_ ≤ c23ε1+α−α1_ψ1 − ψ2_ ∀ψ1, ψ2 ∈ Rm. (9.13)

We rewrite equality (8.3) in the form

ψ = −P

−1

2 (y0, ψ0, ε){[Φ −Φ] + [Φ − P2(y0, ψ0, ε)ψ]} ≡ _ T(ψ, ε), (9.14)

where Φ = Φ(x(τ1, y0+y(ψ, ε), ψ0+ψ, ε), . . . , ϕ(τr, y0+y(ψ, ε), ψ0+ψ, ε), ε)

and Φ = Φ(x(τ1, y0 + y(ψ, ε), ε), . . . , ϕ(τr, y0 + y(ψ, ε), ψ0 + ψ, ε), ε), and

estimate each term on the right-hand side of equality (9.14). Using inequalities

(8.8) and (9.8), we get

_Φ − Φ_ ≤

_r

j=1

_1

ε

c4c14ε1+α + c4c14εα

_

= 2rc4c14εα, (9.15)

_Φ−P2(y0, ψ0, ε)ψ_

≤ rc14(ne2c1L + c24)

_y(ψ, ε)_

ε

+ rc14

_

c24

1

ε

_y(ψ, ε)_ + _ψ_

_2

,

where c24 = 2c1Lne2c1L. Taking into account that _y(ψ, ε)_ ≤ c17ε1+α−α1 ,

we finally obtain

_Φ − P2(y0, ψ0, ε)ψ_ ≤ c25(εα−α1 + _ψ_2), c25 = const. (9.16)

Section 9 Estimates of Error of Averaging Method for Multipoint Problems 105

Inequalities (9.15) and (9.16) yield

_ _ T(ψ, ε)_ ≤ c26(εα−α1−α2 + ε

−α2_ψ_2), c26 = c13(2rc4c14 + c25). (9.17)

This implies that _ T(ψ, ε) maps the set

U = {ψ: ψ ∈ Rm, _ψ_ ≤ 2c26εα−α1−α2}

into itself for α > α1 + 2α2 and 4c26εα−α1−2α2

0

≤ 1.

Let ψ(1) and ψ(2) be arbitrary points of the set U. Using the error estimates

of the averaging method (8.8) and inequalities (9.7), (9.8), and (9.13), we get

_ _ T(ψ(1), ε) − _ T(ψ(2), ε)_

≤ c27

_

εα−α1−α2 +

_r

j,ν=1

_m

s=1

sup

___

∂2Φ

∂qj∂q(s)

ν

__ _

εα−α1−

2α2

_

_ψ(1) − ψ(2)_,

(9.18)

c27 = const.

Since α > α1 +2α2, the last inequality implies that the mapping _ T : U → U is

contracting. Thus, there exists a unique solution ψ = ψ0(ε) ∈ U of Eq. (9.14),

and, hence, there exists the solution

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

ϕ(τ, y0 + y(ψ0(ε), ε), ψ0 + ψ0(ε), ε))

of problem (9.1)–(9.3). Estimates (9.9) follow from estimates (8.8) and the inequalities

_y(ψ0(ε), ε)_ ≤ c17ε1+α−α1 , _ψ0(ε)_ ≤ 2c26εα−α1−α2 ,

and the condition c15ε1+α−α1

0

≤ 1

2ρ guarantees that x = x(τ, ε) lies in D

∀(τ, ε) ∈ [0, L] × (0, ε0]. Theorem 9.1 is proved.

Remark 5. If the vector function Φ(p1, . . . , qr, ε) in condition (9.3) linearly

depends on qj, j = 1, r, i.e.,

Φ =

_r

j=1

Aj(x|τ=τ1, . . . , x|τ=τr, ε)ϕ|τ=τj + A0(x|τ=τ1, . . . , x|τ=τr, ε),

106 Averaging Method in Multipoint Problems Chapter 2

then the inequality α > α1 + 2α2 in Theorem 9.1 can be weakened to the inequality

α > α1 + α2. Indeed, in this case, the analysis of inequalities (9.17)

and (9.18) shows that all conditions of the principle of contracting mappings are

satisfied for α > α1 + α2.

We apply the results obtained to the solution of the multipoint problem

dx

dτ

= P(τ )x + εA(x, ϕ, τ, ε),

dϕ

dτ

= ω(x, τ, ε)

ε

+ B(x, ϕ, τ, ε), (9.19)

_r

j=1

Aj(ε)x|τ=τj = x0(ε),

_r

j=1

Bj(ε)ϕ|τ=τj = f(x|τ=τ1, . . . , x|τ=τr, ε),

(9.20)

where the right-hand sides of Eqs. (9.19) satisfy the same conditions as the righthand

sides of Eqs. (8.1), Aj(ε) and Bj(ε), j = 1, r, are uniformly bounded (by

a constant c28 ) n-dimensional and m-dimensional, respectively, square matrices,

and f(p1, . . . , pr, ε) is a function continuously differentiable with respect to

pj ∈ D, j = 1, r, for every fixed ε and such that

_f_ +

_r

j=1

___

∂f

∂pj

___

≤ 1

ε

c28 ∀pj ∈ D, j= 1, r, ε ∈ (0, ε0]. (9.21)

We write the problem averaged with respect to ϕ:

dx

dτ

= P(τ )x + εA(x, τ, ε),

_r

j=1

Aj(ε)x|τ=τj = x0(ε), (9.22)

dϕ

dτ

= ω(x, τ, ε)

ε

+ B(x, τ, ε),

_r

j=1

Bj(ε)ϕ|τ=τj = f(x|τ=τ1, . . . , x|τ=τr, ε). (9.23)

Let Q(τ, t) denote the normal fundamental matrix of the linear system

dx

dτ

=

P(τ )x.

Section 9 Estimates of Error of Averaging Method for Multipoint Problems 107

Lemma 9.1. Suppose that the following conditions are satisfied:

(i) for all ε ∈ (0, ε0],

___

__r

j=1

Aj(ε)Q(τj , 0)

_−1___ ≤ c29 = const, det

_r

j=1

Bj(ε) _= 0;

(ii) for all (τ, ε) ∈ [0, L] × (0, ε0], the curve

y(τ, ε) = Q(τ, 0)

__r

j=1

Aj(ε)Q(τj , 0)

_−1

x0(ε)

lies in D together with its ρ-neighborhood.

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

unique solution x = x(τ, ε), ϕ = ϕ(τ, ε) of the averaged problem (9.22), (9.23)

for which the curve x = x(τ, ε) lies in a certain small neighborhood of the curve

y = y(τ, ε).

Proof. Since the right-hand sides of Eqs. (9.22) are smooth, there exists a

solution x = x(τ, z + _x, ε) of the Cauchy problem

dx

dτ

= P(τ )x + εA(x, τ, ε),

x|τ=0 = z + _x,

_x =

__r

j=1

Aj(ε)Q(τj , 0)

_−1

x0(ε),

which satisfies the equality

x(τ, z + _x, ε) = y(τ, ε) + Q(τ, 0)z +

_τ

0

Q(τ, t)A(x(t, z + _x, ε), t, ε)dt.

The last relation yields

_x(τ, z + _x, ε) − y(τ, ε)_ ≤ K_z_ + εKLc1 <

1

2ρ

for _z_ ≤ ρ(4K)−1 and ε ≤ ε0 ≤ ρ(4KLc1)−1. Here, K is a constant that

bounds the norm of the matrix Q(τ, t) ∀(τ, t) ∈ [0, L] × [0, L]. This implies

108 Averaging Method in Multipoint Problems Chapter 2

that, for indicated z and ε ∈ (0, ε0], the solution x = x(τ, z + _x, ε) of the

Cauchy problem can be extended for any τ ∈ [0, L] and lies in D together with

its

1

2ρ-neighborhood.

Let us prove that z can be chosen so that the function x = x(τ, z + _x, ε) is a

solution of problem (9.22). Indeed, if the boundary conditions are satisfied, then

we get

z = −ε

__r

j=1

Aj(ε)Q(τj , 0)

_−1

_r

j=1

Aj(ε)

_τj

0

Q(τj, t)A(x(t, z + _x, ε), t, ε) dt

≡ T(z, ε). (9.24)

Taking into account the first inequality in condition (i) of the lemma and restrictions

imposed on the matrices Aj(ε), one can easily establish that T(z, ε) maps

the set of vectors z ∈ Rn such that _z_ ≤ cε, c = c1c28c29rKL, into itself,

provided that cε0 ≤ ρ(4K)−1. Moreover, using the inequality

___

∂

∂z

x(τ, z + _x, ε)

___

≤ ne2c1L,

we obtain

___

∂

∂z

T(z, ε)

___

≤ c30ε ≤ 1

2

∀ε ≤ ε0 ≤ 1

2c30

, c30 = cne2c1L.

Therefore, Eq. (9.24) has a unique solution z = z(ε) that satisfies the inequality

_z(ε)_ ≤ cε, and the boundary-value problem (9.22) has a unique solution

x(τ, ε) = x(τ, z(ε) + _x, ε) for which

_x(τ, ε) − y(τ, ε)_ < K(c + c1L)ε ∀(τ, ε) ∈ [0, L] × (0, ε0].

To obtain ϕ(τ, ε), we substitute the value of x = x(τ, ε) into (9.23). As a result,

we get

ϕ(τ, ε) =

1

ε

_τ

0

[ω(x(t, ε), t, ε) + εB(x(t, ε), t, ε)] dt

+

__r

j=1

Bj(ε)

_−1_

f(x(τ1, ε), . . . , x(τr, ε), ε)

− 1

ε

_r

j=1

Bj(ε)

_τj

0

(ω(x(t, ε), t, ε) + +εB(x(t, ε), t, ε))dt

_

.

Lemma 9.1 is proved.

Section 10 Theorems on Existence of Solutions 109

The proof of the theorem below, in fact, repeats the proof of Theorem 9.1,

and, therefore, we present only its formulation.

Theorem 9.2. Suppose that conditions (8.2), (8.7), and (9.21) and the conditions

of Lemma 9.1 are satisfied and, furthermore,

___

__r

j=1

Bj(ε)

_−1___ ≤ c31ε

−β ∀ε ∈ (0, ε0], β<α.

Then there exist constants c32 and ε0 ≤ ε0 such that, for every ε ∈ (0, ε0],

problem (9.19), (9.20) has a unique solution (x(τ, ε); ϕ(τ, ε)) that satisfies the

inequalities

_x(τ, ε) − x(τ, ε)_ ≤ c32ε1+α,

_ϕ(τ, ε) − ϕ(τ, ε)_ ≤ c32εα−β

∀(τ, ε) ∈ [0, L] × (0, ε0] .

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