19. Proof of Theorem 18.1

Back

Consider iterations (18.11). Since Φ0 ≡ 0 satisfies all conditions of Lemmas

18.1–18.3, it follows from (18.11) for j = 0 that

_Φ1(y, ψ, τ, ε)_ ≤ sup

G

___

∂b

∂x

___

∞ _

τ

_yξ,0

τ

_dξ ≤ n

γ1

c1K_y_ ≤ d1_y_,

___

∂

∂y

Φ1(y, ψ, τ, ε)

___

≤ sup

G

___

∂b

∂x

___

∞ _

τ

___∂

∂y

yξ,0

τ

__ _

dξ ≤

n

γ1

c1K ≤ d2.

To estimate

___

∂

∂ψ

Φ1(y, ψ, τ, ε)

__ _

, we use inequalities (

18.22), the first of which

yields

___

∂

∂ψ

ψξ

τ (ψ, ε)

___

≤ m +

___

∂

∂ψ

(ψξ

τ (ψ, ε) − ψ)

___

≤ m + c4εα(1 + ξ − τ )ec4εα(ξ−τ)

≤ m + e2c4εα(ξ−τ) ≤ (m + 1)e

1

4 γ(ξ−τ)

Section 19 Proof of Theorem 18.1 219

for ξ ≥ τ and 2c4εα0

≤ 1

4γ1. Then

___

∂

∂ψ

Φ1(y, ψ, τ, ε)

___

≤

_

2nc1c5

1

γ2

+ n2c1c1K

1

γ1

+ mnc1(m + 1)K

4

3γ1

_

_y_

≤ d3_y_,

provided that hd3 ≤ 1. The constants d1, d2, and d3 in the inequalities presented

above will be fixed in what follows. Consider the integral obtained from

(18.11) for j = 0 by differentiation with respect to τ under the integral sign.

Estimates (18.22) and (18.24) yield

∞ _

τ

___

∂

∂τ

(b0 −_b)

__ _

dξ

≤ c5_y_

__ω(τ )_

ε

+ 1

_

nc1

∞ _

τ

e

−γ2(ξ−τ)dξ

+ (nc1εα0

+ m)nc1c4K_y_

__ω(τ )_

ε

+ 1

_ ∞ _

τ

e

−3

4 γ1(ξ−τ)dξ.

The estimates presented above guarantee the uniform convergence of the improper

integrals

∞ _

τ

(b0 −_b)dξ,

∞ _

τ

∂

∂y

(b0 −_b)dξ,

∞ _

τ

∂

∂ψ

(b0 −_b)dξ,

∞ _

τ

∂

∂τ

(b0 −_b)dξ

on the set

y ∈ Ph, ψ∈ Rm, τ ∈ [−T,T], ε∈ [Δ, ε0], (19.1)

where T > 0 and Δ > 0 (Δ< ε0) are arbitrary. Then, for every ε ∈ [Δ, ε0],

the function Φ1(y, ψ, τ, ε) is continuously differentiable with respect to y, ψ,

and τ from set (19.1) and satisfies Eq. (18.14) with j = 0 for these values of

y, ψ, τ, and ε. Since Δ and T are arbitrary, we obtain relation (18.14) for all

(y, ψ, τ, ε) ∈ G.

Now assume that, for all j = 1, s, s > 1, the functions Φj(y, ψ, τ, ε) are

continuously differentiable with respect to (y, ψ, τ) ∈ Ph × Rm × R for every

value of the parameter ε and satisfy the inequalities

220 Integral Manifolds Chapter 3

_Φj_ ≤ d1_y_,

___

∂

∂y

Φj

___

≤ d2,

___

∂

∂ψ

Φj

___

≤ d3_y_ (19.2)

and Eq. (18.14) ∀(y, ψ, τ, ε) ∈ G. Using Lemmas 18.1–18.3 for j = s and

estimates (18.15) and (18.22), we deduce from (18.11) that

_Φs+1(y, ψ, τ, ε)_ ≤

_

nc1

1

γ1

K(1 + c1εα0

d1) +

1

γ1

K sup

G

___

∂b

∂ϕ

__ _

d1

_

_y_.

Since γ1 >

γ

2

and

K

γ1

sup

G

___

∂b

∂ϕ

___

< 

2

γ

K sup

G

___

∂b

∂ϕ

___

= σ0 < 1,

for d1εα0

≤ 1 we get

_Φs+1(y, ψ, τ, ε)_ ≤

_

nc1

1

γ1

K(1 + c1) + σ0d1

_

_y_ ≤ d1_y_,

where

d1 = nc1(1 + c1)K

γ1(1 − σ0) .

By analogy, for d2εα0

≤ 1 we obtain

___

∂

∂y

Φs+1(y, ψ, τ, ε)

___

≤ 2nc1(1 + c1)

γ1(1 − σ0) K ≡ d2.

We now represent

∂

∂ψ

Φs+1(y, ψ, τ, ε) in the form

∂

∂ψ

Φs+1(y, ψ, τ, ε)

= −

∞ _

τ

_∂bs

∂y

+

_∂bs

∂y

∂Xs

∂ϕ

+ ∂bs

∂ϕ

_∂Φs

∂y

_∂yξ,s

τ

∂ψ

dξ

−

∞ _

τ

_∂(bs −_b)

∂y

∂Xs

∂ϕ

+ ∂_b

∂y

_∂Xs

∂ϕ

− ∂X

∂ϕ

_

+ ∂(bs −_b)

∂ϕ

_∂ψξ

τ

∂ψ

dξ

−

∞ _

τ

_∂bs

∂y

∂Xs

∂ϕ

∂Φs

∂ϕ

∂ψξ

τ

∂ψ

+ ∂bs

∂ϕ

∂Φs

∂ϕ

∂(ψξ

τ − ψ)

∂ψ

+ ∂bs

∂ϕ

∂Φs

∂ϕ

_

dξ

Section 19 Proof of Theorem 18.1 221

and use the inequalities

_Φs(yξ,s

τ , ψξ

τ, ξ, ε)_ ≤ d1K_y_e

−γ1(ξ−τ),

___

∂

∂ϕ

Φs(yξ,s

τ , ψξ

τ, ξ, ε)

___

≤ d3K_y_e

−γ1(ξ−τ), ξ≥ τ.

Then, for c1εα0

≤ 1, d3εα0

≤ 1, and hd3 ≤ 1, we get

___

∂

∂ψ

Φs+1(y, ψ, τ, ε)

___

≤ (c10 + σ0d3)_y_ ≤ c10

1 − σ0

_y_ ≡ d3_y_,

where

c10 =

2

γ2

(n + (n + m)d2)c1c5 +

4

3γ1

c1K(m + 1)[(n + m)2(1 + 2d1) + nd1]

+

4

3γ1

n(m + 1)c1c1K +

4

9γ2

1

(4 + 3γ1)mc1c4K.

Thus, by the method of mathematical induction, we establish that, for sufficiently

small h > 0 and ε0 > 0, inequalities (19.2) with the constants d1, d2,

and d3 defined above hold for all j ≥ 0 and (y, ψ, τ, ε) ∈ G. Moreover, the

fact that the norms of the matrices

yξ,j

τ ,

∂

∂y

yξ,j

τ ,

∂

∂ψ

yξ,j

τ ,

∂

∂τ

yξ,j

τ

tend exponentially to zero as ξ → ∞ guarantees the uniform convergence [on

set (19.1)] of the improper integral (18.11) and the integrals obtained from it by

differentiation with respect to y, ψ, and τ under the integral sign. Therefore,

the functions Φj(y, ψ, τ, ε), j ≥ 0, are continuously differentiable with respect

to y, ψ, and τ for every fixed ε from set (19.1) and satisfy Eq. (18.14). Since

Δ and T are arbitrary, equality (18.14) holds for all (y, ψ, τ, ε) ∈ G, and the

functions Φj(y, ψ, τ, ε) have continuous partial derivatives of the first order with

respect to (y, ψ, τ) ∈ Ph × Rm × R for every ε ∈ (0, ε0].

We now prove that the matrices

∂

∂y

Φj(y, ψ, τ, ε) and

∂

∂ψ

Φj(y, ψ, τ, ε) satisfy

the Lipschitz condition with respect to the variables y and ψ. Using Lemma

18.4 for j = 0 and the inequalities

_ψξ

τ (ψ, ε) − ψξ

τ (ψ, ε)_ ≤ [1 + c4εα(1 + ξ − τ )ec4εα(ξ−τ)]_ψ − ψ_,

_yξ,j

τ (y, ψ, ε) − yξ,j

τ (y, ψ, ε)_

≤ Ke

−γ1(ξ−τ)]_y − y_ + c5h(1 + hd3)e

−γ2(ξ−τ)_ψ − ψ_, ξ≥ τ, (19.3)

222 Integral Manifolds Chapter 3

which follow from relations (18.22) and (18.23) and Lemma 18.2, we deduce the

following estimate from (18.11):

___

∂

∂y

Φ1(y, ψ, τ, ε) − ∂

∂y

Φ1(y, ψ, τ, ε)

___

≤ Knc1

_ c6

γ3

+ n

2γ1

K + nc5h

1

γ1 + γ2

(1 + hd3)

+ (m + nc1εα0

)

_ 1

γ2

+

4

4γ2 − γ1

__

(_y − y_ + _ψ − ψ_)

≤ μ(_y − y_ + _ψ − ψ_).

Here, μ > 0 is a constant, which will be fixed in what follows. Assume that

inequalities (18.25) are satisfied for all j = 1, s, s > 1. Then

___

∂

∂y

Φs+1(y, ψ, τ, ε) − ∂

∂y

Φs+1(y, ψ, τ, ε)

___

≤ c1(n + d2(n + m))

∞ _

τ

___

∂

∂y

yξ,s

τ (y, ψ, ε) − ∂

∂y

yξ,s

τ (y, ψ, ε)

__ _

dξ

+

_

c1(1 + d2 + d3h)K[(1 + d2)(n + m)2 + nd2]

+

_

nc1c1εα0

+ sup

G

___

∂b

∂ϕ

___

_

Kμ

_ ∞ _

τ

(_yξ,s

τ (y, ψ, ε) − yξ,s

τ (y, ψ, ε)_

+ _ψξ

τ (ψ, ε) − ψξ

τ (ψ, ε)_)e

−γ1(ξ−τ)dξ.

Choosing h > 0 and ε0 > 0 so small that hμ ≤ 1 and εα0

μ ≤ 1 and using

Lemma 18.4 and estimates (19.3), we get

___

∂

∂y

Φs+1(y, ψ, τ, ε) − ∂

∂y

Φs+1(y, ψ, τ, ε)

___

≤ c11(_y − y_ + _ψ − ψ_) + sup

G

___

∂b

∂ϕ

___

1

2γ1

K2μ_y − y_

+ sup

G

___

∂b

∂ϕ

___

1

γ1

Kμ_ψ − ψ_, (19.4)

Section 19 Proof of Theorem 18.1 223

where

c11 = c1[n + d2(n + m)]

2

γ3

c6

+ c1K[(2 + d2)(nd2 + (1 + d2)(n + m)2 + nc1)]

_ 2c5

(γ1 + γ2)d3

+ K

2γ1

_

+ mc1K

_ 2c5

γ1 + γ2

+

4

9γ2

1

c4(4 + 3γ1)

_

.

Since γ1 >

1

2γ and, according to condition (18.15), the constant

σ0 = K

γ

sup

G

___

∂b

∂ϕ

___

max{2;K}

is less than 1, it follows from inequality (19.4) that

___

∂

∂y

Φs+1(y, ψ, τ, ε) − ∂

∂y

Φs+1(y, ψ, τ, ε)

___

≤ (c11 + μσ0)(_y − y_ + _ψ − ψ_) ≤ μ(_y − y_ + _ψ − ψ_),

μ = c11

1 − σ0

.

Thus, we have established inequalities (18.25) for all j ≥ 0.

By analogy, using Lemma 18.5, for sufficiently small h > 0 and ε0 > 0 one

can prove inequalities (18.26) for all j ≥ 0 with the constant ν independent of

ε and h.

Consider the problem of the convergence of the sequence {Φj(y, ψ, τ, ε)} on

the set G. In view of inequalities (18.27) and (18.31), relation (18.11) yields

_Φj+1(y, ψ, τ, ε) − Φj(y, ψ, τ, ε)_

≤

_

(n + (m + nc1εα0

)d2) c1

γ5

_y_2 + sup

G

___

∂b

∂ϕ

___

1

γ1

K_y_

_

sup

G

vj−1

or

sup

G

vj ≤

_

σ0 + c1(n + m)d2

1

γ5

h

_

sup

G

vj−1. (19.5)

For small h > 0, the expression in the square brackets on the right-hand side of

inequality (19.5) can be estimated by a constant c12 less than 1. Consequently,

the inequalities

224 Integral Manifolds Chapter 3

sup

G

vj ≤ c12 sup

G

vj−1, c12 < 1, sup

G

v0 ≤ d1

imply that the numerical series

∞,

j=0

sup

G

vj is convergent, and, hence, the functional

sequence {Φj(y, ψ, τ, ε)} converges uniformly on the set G to the limit

function

Φ(y, ψ, τ, ε) = lim

j→∞

Φj(y, ψ, τ, ε)

continuous in y, ψ, and τ.

By analogy, using estimates (18.28) and (18.29) we prove the uniform convergence

of the sequences

_ ∂

∂ψ

Φj(y, ψ, τ, ε)

_

and

_ ∂

∂y

Φj(y, ψ, τ, ε)

_

on the

set G.

Consider the sequence

_ ∂

∂τ

Φj(y, ψ, τ, ε)

_

each element of which is determined

by equality (18.14). The smoothness conditions for the right-hand side of

system (18.1) and the uniform convergence of the sequences {Φj},

_ ∂

∂ψ

Φj

_

,

and

_ ∂

∂y

Φj

_

on the set G yield the uniform convergence of the sequence

_ ∂

∂τ

Φj

_

on set (19.1). Therefore, the limit function Φ(y, ψ, τ, ε) is continuous

in y, ψ, and τ for every fixed ε from set (19.1). Passing to the limit as

j → ∞ in Eq. (18.4), we obtain Eq. (18.10) for all y, ψ, τ, and ε from set

(19.1). Since T and Δ are arbitrary, we get Eq. (18.10) for any (y, ψ, τ, ε) ∈ G.

The Lipschitz conditions (18.17) follow from inequalities (18.25) and (18.26).

Theorem 18.1 is proved.

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