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10. Theorems on Existence of Solutions of Boundary-Value Problems

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In Sections 6–9, using the principle of contracting mappings, we have proved

the existence and uniqueness of solutions of certain boundary-value problems for

multifrequency systems. This has been done on the basis of the fact that, for the

oscillation system (6.1) with ω = ω(τ ) or system (8.1) with ω = ω(x, τ, ε), we

have efficient estimates for the difference of solutions of the original and averaged

equations and their partial derivatives with respect to the initial data [inequalities

(2.5), (2.7), and (8.8)]. For multifrequency systems of the general form (4.1) in

which a(x, ϕ, τ ) depends on angular variables and the frequencies depend on the

variables x, the justification of the averaging method can be reduced to the proof

of the estimate _x−x_ ≤ c(ε), where c(ε) → 0 as ε → 0. In this case, for time

τ ∈ [0, L], the difference of the angular variables ϕ − ϕ can reach an arbitrarily

large value as ε → 0 [Arn4, Bak1, GrR3, Kha2]; the same is true for the behavior

of the partial derivatives of the functions x − x and ϕ − ϕ with respect to the

initial data. Therefore, the combination of the principle of contracting mappings

and the averaging method in the solution of boundary-value problems for systems

of the form (4.1) loses its sense.

110 Averaging Method in Multipoint Problems Chapter 2

In the present section, we prove only the existence of solutions of boundaryvalue

problems by using the Schauder fixed-point theorem [Har, Sch]. According

to this theorem, for the existence of a (not necessarily unique) solution of the

equation Ty = y it is sufficient that the mapping T of the ball K ⊂ Rn into

itself be continuous.

Consider a nonlinear system

dτ

= a(x, ϕ, τ, ε),

dϕ

dτ

= ω(x, τ )

+ b(x, ϕ, τ, ε) (10.1)

whose right-hand side is defined for (x, ϕ, τ, ε) ∈ D×Rm × [0, L] × (0, ε0] ≡

D is a bounded domain of the real Euclidean space Rn

and continuously

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

class of almost periodic (with respect to ϕν, ν = 1,m) functions

a(x, ϕ, τ, ε) =

∞_

ν=0

aν(x, τ, ε)ei(λν,ϕ),

b(x, ϕ, τ, ε) =

∞_

ν=0

bν(x, τ, ε)ei(λν,ϕ),

λ0 = 0, λν = (λ(1)

ν , . . . , λ(m)

ν ) _= 0 ∀ν ≥ 1,

i2 = −1, (λν, ϕ) =

j=1

λ(j)

ν ϕj ,

for which

∞_

ν=1

_λν_

sup

_aν_+

_λν_

sup

___∂

aν

∂τ

___

+sup

___

∂aν

∂x

___

≤ c1. (10.2)

Here, c1 is a constant independent of ε and G = D ×[0, L] × (0, ε0]. We also

assume that the first-order partial derivatives of the functions a, b, and ω with

respect to x, ϕ, and τ are uniformly bounded in G by the constant c1.

For Eqs. (10.1), we introduce boundary conditions of the form

F(x|τ=τ1, . . . , x|τ=τr, ε) = 0, ϕ|τ=τν0

= ϕ0, (10.3)

where 0 ≤ τ1 < τ2 < ... < τr ≤ L, r ≥ 2, ν0 is fixed (1 ≤ ν0 ≤ r),

ϕ0 ∈ Rm is a constant vector, and F(p1, . . . , pr, ε) is an n-dimensional vector

Section 10 Theorems on Existence of Solutions 111

function of the variables pj ∈ D, j = 1, r, and ε ∈ (0, ε0] that has continuous

and bounded (by the constant c1) first-order partial derivatives with respect to all

variables pj, j = 1, r, for every ε.

To investigate the solvability of the multipoint problem (10.1), (10.3), we use

the method of averaging with respect to all fast variables ϕ. Parallel with (10.1),

(10.3), we consider the averaged problem

dτ

= a0(x, τ, ε), (10.4I)

F(x|τ=τ1, . . . , x|τ=τr, ε) = 0, (10.4II)

dϕ

dτ

= ω(x, τ )

+ b0(x, τ, ε), (10.4III)

ϕ|τ=τν0

= ϕ0, (10.4IV)

where

[a0; b0] = lim

T→∞

−m

. . .

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

In order that the averaging method correctly describe the evolution of the slow

variables x on the time interval [0, L], it is necessary to impose certain restrictions

on the frequency vector ω(x, τ) = (ω1(x, τ ), . . . , ωm(x, τ )). Assume that,

for any (x, ϕ, τ, ε) ∈ G and ν ≥ 1 and certain α ∈

, the following

inequality holds:

|(λν, ω(x, τ ))| + |(λν, Ω(x, ϕ, τ, ε))| ≥ c2_λν_, c2 = const > 0, (10.5)

where

Ω = ∂ω(x, τ )

∂τ

+ ∂ω(x, τ )

∂x

a0(x, τ, ε) +

∞_

j=1

aj(x, τ, ε)hεα((λj, ω(x, τ )))ei(λj,ϕ)

(λν, ω), (λν, Ω), and (λj, ϕ) are the scalar products of vectors, and hd(t) for

d = εα is the function defined in Section 4. Note that, by virtue of the finiteness

of the function hd(t), conditions (10.5) are imposed not on all harmonics

112 Averaging Method in Multipoint Problems Chapter 2

of the function a(x, ϕ, τ, ε), but only on its resonance harmonics. Under these

assumptions, according to the results of Section 4, we have

_xτ (t, y, ψ, ε) − xτ (t, y, ε)_ ≤ σ

√

ε, σ = const, (10.6)

for all τ ∈ [0, L], y ∈ D1, ψ ∈ Rm, and ε ∈ (0, ε0]. In this estimate,

(xτ (t, y, ψ, ε); ϕτ (t, y, ψ, ε)) and (xτ (t, y, ε); ϕτ (t, y, ψ, ε)) are the solutions of

Eqs. (10.1) and the averaged equations (10.4I) and (10.4III) that take the value

(y; ψ) for τ = t, and the curve x = xτ (t, y, ε) lies in D together with its

ρ-neighborhood ∀(τ, y, ε) ∈ [0, L]×D1 × (0, ε0].

Theorem 10.1. Suppose that the following conditions are satisfied:

(i) conditions (10.2) and (10.5) and the restrictions imposed on a, b, ω, and

F are satisfied;

(ii) the matrices

∂a0(x, τ, ε)

∂x

and

∂F(p, ε)

∂p

are uniformly

with respect to

ε ∈ (0, ε0]

uniformly continuous in x ∈ D, τ ∈ [0, L], and p =

(p1, . . . , pr) ∈ D×. . .×D ≡ Dr ;

(iii) for every ε ∈ (0, ε0], there exists a solution x = xτ (τν0, x0, ε), x0 =

x0(ε), of problem (10.4I), (10.4II) that lies in D together with its ρ-

neighborhood;

(iv) _S−1(ε)_ ≤ c3 = const ∀ε ∈ (0, ε0], where

S(ε) =

j=1

∂F0

∂pj

∂xτj (τν0, x0, ε)

∂x0 ,

and

∂F0

∂pj

denotes the matrix of the first-order partial derivatives of the

function F(p1, . . . , pr, ε) with respect to pj for pμ = xτμ(τν0, x0, ε),

μ = 1, r.

Then one can find constants c1 > 0 and ε1 ∈ (0, ε0] such that, for every

ε ∈ (0, ε1], problem (10.1), (10.3) has at least one solution (x(τ, ε); ϕ(τ, ε)) for

which

_x(τ, ε) − xτ (τν0, x0, ε)_ ≤ c1

√

ε ∀(τ, ε) ∈ [0, L] × (0, ε1]. (10.7)

Section 10 Theorems on Existence of Solutions 113

Proof. We seek a solution of problem (10.1), (10.3) in the form (xτ (τν0, x0+

y,ϕ0, ε); ϕτ (τν0, x0 + y,ϕ0, ε)) and determine the unknown parameter y ∈ Rn

from the boundary conditions (10.3):

y = −S

−1(ε)

F(xτ1(τν0, x0 + y,ϕ0, ε), . . . , xτr (τν0, x0 + y,ϕ0, ε), ε)

− F(xτ1(τν0, x0 + y, ε), . . . , xτr (τν0, x0 + y, ε), ε)

F(xτ1(τν0, x0 + y, ε), . . . , xτr (τν0, x0 + y, ε), ε) − S(ε)y

≡ Mε(y). (10.8)

Taking into account the restrictions for F and estimate (10.6), we get

_F(xτ1(τν0, x0 + y,ϕ0, ε), . . . , ε) − F(xτ1(τν0, x0 + y, ε), . . . , ε)_

≤ c1rσ

√

ε. (10.9)

We now fix arbitrary a positive μ ≤ [2(1 + Lc1)nre2nc1Lc3]−1 ≡ Δ. Then it

follows from condition (ii) of Theorem 10.1 that there exists δ = δ(μ) such that,

for _z_ + __p_ < δ, we have

___

∂

∂x

a0(x + z, τ, ε) − ∂

∂x

a0(x, τ, ε)

___

∂

∂p

F(p + _p, ε) − ∂

∂p

F(p, ε)

___

< μ (10.10)

We choose δ <

−nc1Lρ and rewrite the averaged equations (10.4I) in the form

xτ (τν0, x0 + y, ε) = x0 + y +

_τ

τν0

a0(xt(τν0, x0 + y, ε), t, ε)dt.

Differentiating this equality with respect to x0 and using relation (10.10) and the

Gronwall–Bellman inequality, we get

___

∂

∂x0 (xτ (τν0, x0 + y, ε) − xτ (τν0, x0, ε))

___

≤ nLe2c1Lnμ

for any (τ, ε) ∈ [0, L] × (0, ε0] and _y_ < δ. Therefore,

xτ (τν0, x0 + y, ε) = xτ (τν0, x0, ε) + ∂

∂x0 xτ (τν0, x0, ε)y + h1(τ, y, ε), (10.11)

114 Averaging Method in Multipoint Problems Chapter 2

where

h1(τ, y, ε) =

_ ∂

∂x0 xτ (τν0, x0 + ty, ε) − ∂

∂x0 xτ (τν0, x0, ε)

dty, (10.12)

_h1(τ, y, ε)_ ≤ nLe2c1Lnμ_y_

for all τ ∈ [0, L], ε ∈ (0, ε0], and _y_ < δ.

The boundary condition (10.4II) and relations (10.10)–(10.12) yield the representation

F(xτ1(τν0, x0 + y, ε), . . . , ε) = S(ε)y + h2(y, ε), (10.13)

where

_h2(y, ε)_ ≤ nr(1 + Lc1)e2c1Lnμ_y_ ∀ε ∈ (0, ε0], _y_ < δ. (10.14)

Thus, it follows from (10.8), (10.9), (10.13), and (10.14) that

_Mε(y)_ < c3

c1rσ

√

ε +

c32Δμ_y_

∀ε ∈ (0, ε0], _y_ < δ.

This implies that Mε(y) maps the set _y_ ≤ 2c1c3rσ

√

ε ≡ c

√

ε into itself,

provided that c

√

ε < δ. Also note that, for every ε, the vector function Mε(y)

is continuous with respect to y ; therefore, according to the Schauder theorem,

there exists a solution y = y(ε), _y(ε)_ ≤ c

√

ε, of Eq. (10.8), and, hence, there

exists a solution

(x(τ, ε); ϕ(τ, ε)) = (xτ (τν0, x0 + y(ε), ϕ0, ε); ϕτ (τν0, x0 + y(ε), ϕ0, ε))

of the multipoint problem (10.1), (10.3). Estimate (10.7) with the constant c1 =

σ + cnec1nL follows from estimate (10.6) and the inequality

_xτ (τν0, x0 + y(ε), ε) − xτ (τν0, x0, ε)_ ≤ nenc1L_y(ε)_ ≤ ncenc1L

√

ε.

To complete the proof of the theorem, we impose the condition c1

√

ε <

2ρ,

which guarantees that the curve x = x(τ, ε) lies in D together with its

2ρ-

neighborhood ∀τ ∈ [0, L].

Condition (iv) is an essential assumption in Theorem 10.1. In what follows,

we consider the case where this condition is not satisfied, namely, we assume that

−1(ε)_ ≤ Kε

−l1, l1 = const > 0, K = const > 0. (10.15)

Section 10 Theorems on Existence of Solutions 115

Theorem 10.2. Suppose that the following conditions are satisfied:

(a) conditions (i)–(iii) of Theorem 10.1 and inequality (10.15) are satisfied;

(b) the matrices

∂

∂x

a0(x, τ, ε) and

∂

∂p

F(p, ε) satisfy the HЁolder conditions

___

∂

∂x

a0(x, τ, ε) − ∂

∂x

a0(x, τ, ε)

___

≤ M_x − x_l2 , 0 < l2 ≤ 1,

___

∂

∂p

F(p, ε) − ∂

∂p

F(p, ε)

___

≤ M_p − p_l2

for all x, x ∈ D, p,p ∈ Dr, τ ∈ [0, L], and ε ∈ (0, ε0], and the constant

M is independent of ε;

2(1 + l2) .

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

least one solution (x(τ, ε); ϕ(τ, ε)) of problem (10.1), (10.3) that satisfies the

inequality

_x(τ, ε) − xτ (τν0, x0, ε)_ ≤ c1ε

−l1

∀(τ, ε) ∈ [0, L] × (0, ε0], c1 = const.

Proof. We follow the scheme of the proof of Theorem 10.1. To determine a

solution

(xτ (τν0, x0 + y,ϕ0, ε); ϕτ (τν0, x0 + y,ϕ0, ε))

of problem (10.1), (10.3), i.e., to find y, we write equality (10.8) and inequality

(10.9). It is easy to verify that the fact that

∂

∂x

a0(x, τ, ε) belongs to the HЁolder

class guarantees that

___

∂

∂x0 xτ (τν0, x0 + y, ε) − ∂

∂x0 xτ (τν0, x0, ε)

___

≤ M1_y_l2 ,

M1 = MLn1+l2enc1(2+l2)L

for _y_ <

_ ρ

−nc1L, τ ∈ [0, L], and ε ∈ (0, ε0]. Therefore, for the function

h1(τ, y, ε) defined by equality (10.12), the following estimate is true:

_h1(τ, y, ε)_ ≤ 1

1 + l2

M1_y_1+l2 (10.16)

116 Averaging Method in Multipoint Problems Chapter 2

Taking into account equality (10.11), estimate (10.16), and the fact that the

matrix

∂

∂p

F(p, ε) belongs to the HЁolder class, we obtain relation (10.13) in

which

_h2(y, ε)_ ≤ M2_y_1+l2, M2 = 3rc1M1 + rn1+l2Menc1(2+l2)L.

Thus,

_Mε(y)_ ≤ Kε

−l1

rc1σ

√

ε +M2_y_1+l2

∀ε ∈ (0, ε0), _y_ ≤

_ ρ

−nc1L.

The analysis of the last inequality shows that if condition (c) of Theorem 10.2

is satisfied, then Mε(y) maps the set

{y : _y_ ≤ 2Krc1σε

−l1}

into itself for every ε ∈ (0, ε0], provided that

ε0 ≤ min

__4

nc1rσKenc1L

_ 2

2l1

−1 ;

_ 1

2KM2

_ 1

2rc1σK

_l2_ 2

−2(1+l2)l1

Since the mapping Mε(y) is continuous in y, there exists a solution y = y(ε)

of Eq. (10.8) that satisfies the inequality _y(ε)_ ≤ 2Krc1σε

−l1 . Therefore,

(x(τ, ε); ϕ(τ, ε)) = (xτ (τν0, x0 + y(ε), ϕ0, ε); ϕτ (τν0, x0 + y(ε), ϕ0ε))

is a solution of problem (10.1), (10.3), and

_x(τ, ε) − xτ (τν0, x0, ε)_ ≤ _x(τ, ε) − xτ (τν0, x0 + y(ε), ε)_

+ _xτ (τν0, x0 + y(ε), ε) − xτ (τν0, x0, ε)_

≤ c1ε

−l1 ,

c1 = σ + 2Krc1σnenc1L.

Theorem 10.2 is proved.

Section 10 Theorems on Existence of Solutions 117

For Eqs. (10.1), we now introduce boundary conditions of the form

x|τ=τν0

= y0 ∈ D,

j=1

Bj(ε)ϕ|τ=τj = f(x|τ=τ1, . . . , x|τ=τr, ε). (10.17)

Here, Bj(ε) are quadratic m-dimensional matrices and f(p1, . . . , pr, ε) is an

m-dimensional vector function.

Theorem 10.3. Suppose that the following conditions are satisfied:

(a) condition (i) of Theorem 10.1 is satisfied;

(b) f(p1, . . . , pr, ε) is continuous in pj ∈ D, j = 1, r, and

det

j=1

Bj(ε) _= 0 ∀ε ∈ (0, ε0];

for τ ∈ [0, L] and ε ∈ (0, ε0].

Then a solution (x(τ, ε); ϕ(τ, ε)) of problem (10.1), (10.17) exists, and the

slow variables x(τ, ε) of every solution lie in a σ

√

ε-neighborhood of the curve

x = xτ (τν0, y0, ε) ∀(τ, ε) ∈ [0, L] × (0, ε0].

Proof. We represent the fast variables ϕ(τ, ε) of the required solution

(x(τ, ε); ϕ(τ, ε)) = (xτ (τν0, y0, ψ, ε); ϕτ (τν0, y0, ψ, ε)) (10.18)

of the multipoint (10.1), (10.17) in the form

ϕ(τ, ε) = ψ +

θ(τ,ψ, ε),

θ(τ,ψ, ε) =

_τ

τν0

[ω(xt(τν0, y0, ψ, ε), t)

+ εb(xτ (τν0, y0, ψ, ε), ϕt(τν0, y0, ψ, ε), t, ε)]dt,

_θ(τ,ψ, ε)_ ≤ c1L(1 + ε) ∀(τ,ψ, ε) ∈ [0, L] × Rm × (0, ε0].

118 Averaging Method in Multipoint Problems Chapter 2

Here, ψ is unknown. To determine ψ, we use the boundary conditions (10.17).

As a result, we get

ψ =

j=1

Bj(ε)

'−1#

_ f(ψ, ε) − 1

j=1

Bj(ε)θ(τj, ψ, ε)

≡ Tε(ψ), (10.19)

where

_ f(ψ, ε) = f(xτ1(τν0, y0, ψ, ε), . . . , xτr (τν0, y0, ψ, ε), ε).

Taking into account the continuity of f(p1, . . . , pr, ε) in pj ∈ D, j = 1, r,

condition (c) of Theorem 10.3, and an estimate of the form (10.6), namely

_x(τ, ε) − xτ (τν0, y0, ε)_ ≤ σ

√

ε,

and choosing ε0 ≤

_ ρ

2σ

, we establish the existence of a constant c(ε) such

that _ _ f(ψ, ε)_ ≤ c(ε) ∀ψ ∈ Rm, ε ∈ (0, ε0]. Then relation (10.19) yields

_Tε(ψ)_ ≤

______

⎛

⎝

j=1

Bj(ε)

⎞

⎠

−1______

⎡

⎣c(ε) +

Lc1(1 + ε)

j=1

_Bj(ε)_

⎤

⎦ ≡ c(ε).

This inequality, together with the condition of the continuity of the function Tε(ψ)

with respect to ψ, guarantees the existence of a solution ψ = ψ(ε), _ψ(ε)_ ≤

c(ε), of Eq. (10.19) and, hence, the existence of a solution (10.18) of problem

(10.1), (10.17). Theorem 10.3 is proved.

The linear dependence of the boundary conditions (10.17) on ϕ|τ=τj, j =

1, r, is an essential assumption in Theorem 10.3. Below, we establish sufficient

conditions for the solvability of a multipoint problem for a one-frequency system

in the case where the boundary conditions contain nonlinearities indicated above.

Consider the case of the one-frequency (m = 1) system (10.1) with the

boundary conditions

x|τ=τν0

= y0, g(ϕ|τ=τ1, . . . , ϕ|τ=τr, ε) = f(x|τ=τ1, . . . , x|τ=τr, ε). (10.20)

Here, g(q1, . . . , qr, ε) and f(p1, . . . , pr, ε) are scalar functions of the variables

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

Section 10 Theorems on Existence of Solutions 119

Theorem 10.4. Suppose that the following conditions are satisfied:

(i) conditions (a) and (c) of Theorem 10.3 are satisfied;

(ii) f(p1, . . . , pr, ε) and g(q1, . . . , qr, ε) are continuous in pj ∈ D and qj ∈

Rm, j = 1, r;

(iii) for every N > 0, the following limits exist uniformly in c(j) = const,

|c(j)| ≤ N, j = 1, r:

lim

t→∞

g(t + c(1), . . . , t + c(r), ε) = ∞,

lim

t→−∞

g(t + c(1), . . . , t + c(r), ε) = −∞, (10.21)

lim

t→∞

g(t + c(1), . . . , t + c(r), ε) = −∞,

lim

t→−∞

g(t + c(1), . . . , t + c(r), ε) = ∞. (10.22)

Then a solution (x(τ, ε); ϕ(τ, ε)) of the multipoint problem (10.1), (10.20)

exists, and the slow variables x(τ, ε) of every solution satisfy the estimate

_x(τ, ε) − xτ (τν0, y0, ε)_ < σ

√

ε ∀(τ, ε) ∈ [0, L] × (0, ε0]. (10.23)

Proof. To find solution (10.18) of problem (10.1), (10.20), we rewrite the

boundary conditions in the form

_g(ψ, ε) ≡ g

ψ +

θ(τ1, ψ, ε), . . . , ψ +

θ(τr, ψ, ε), ε

− _ f(ψ, ε) = 0,

where

_ f(ψ, ε) (| _ f(ψ, ε)| ≤ c(ε) ∀ψ ∈ Rm)

and

θ(τ,ψ, ε) (|θ(τ,ψ, ε)| ≤ c1L(1 + ε) ∀τ ∈ [0, L], ψ ∈ R)

are defined in the proof of Theorem 10.3. We use condition (iii) of Theorem 10.4

for N =

+ 1

Lc1 and consider, e.g., case (10.21). According to condition

(iii), there exists N1(ε) > 0 such that, for ψ(1) < −N1(ε) and ψ(2) >

N1(ε), the following inequalities are true:

120 Averaging Method in Multipoint Problems Chapter 2

ψ(1) +

θ(τ1, ψ(1), ε), . . . , ψ(1) +

θ(τr, ψ(1), ε), ε

< −2c(ε),

ψ(2) +

θ(τ1, ψ(2), ε), . . . , ψ(2) +

θ(τr, ψ(2), ε), ε

> 2c(ε).

Taking into account that | _ f(ψ, ε)| < c(ε), we get

_g(ψ(1), ε) ≤ −c(ε) < 0, _g(ψ(2), ε) ≥ c(ε) > 0.

Since (according to the assumptions made) the function _g(ψ, ε) is continuous in

ψ ∈ R, there exists ψ = ψ(ε) such that _g(ψ(ε), ε) = 0. This yields the existence

of solution (10.18) of problem (10.1), (10.20), and estimate (10.20) follows

from estimate (10.6). Theorem 10.4 is proved.

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