On Continuity of the Invariant Torus for Countable System of Difference Equations Dependent on Parameters
By using the method of the Green – Samoilenko function, an invariant torus is constructed for a system of discrete equations which are defined on tori in the space of bounded number sequences. Sufficient conditions are established for continuous dependence of the invariant torus on the angular va...
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irk-123456789-1746912021-01-28T01:26:58Z On Continuity of the Invariant Torus for Countable System of Difference Equations Dependent on Parameters Marchuk, N.A. By using the method of the Green – Samoilenko function, an invariant torus is constructed for a system of discrete equations which are defined on tori in the space of bounded number sequences. Sufficient conditions are established for continuous dependence of the invariant torus on the angular variable and the parameter contained in this system. 2001 Article On Continuity of the Invariant Torus for Countable System of Difference Equations Dependent on Parameters / N.A. Marchuk // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 316-325 . — Бібліогр.: 6 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/174691 en Нелінійні коливання Інститут математики НАН України |
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By using the method of the Green – Samoilenko function, an invariant torus is constructed
for a system of discrete equations which are defined on tori in the space of bounded number
sequences. Sufficient conditions are established for continuous dependence of the invariant torus
on the angular variable and the parameter contained in this system. |
| format |
Article |
| author |
Marchuk, N.A. |
| spellingShingle |
Marchuk, N.A. On Continuity of the Invariant Torus for Countable System of Difference Equations Dependent on Parameters Нелінійні коливання |
| author_facet |
Marchuk, N.A. |
| author_sort |
Marchuk, N.A. |
| title |
On Continuity of the Invariant Torus for Countable System of Difference Equations Dependent on Parameters |
| title_short |
On Continuity of the Invariant Torus for Countable System of Difference Equations Dependent on Parameters |
| title_full |
On Continuity of the Invariant Torus for Countable System of Difference Equations Dependent on Parameters |
| title_fullStr |
On Continuity of the Invariant Torus for Countable System of Difference Equations Dependent on Parameters |
| title_full_unstemmed |
On Continuity of the Invariant Torus for Countable System of Difference Equations Dependent on Parameters |
| title_sort |
on continuity of the invariant torus for countable system of difference equations dependent on parameters |
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Інститут математики НАН України |
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2001 |
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http://dspace.nbuv.gov.ua/handle/123456789/174691 |
| citation_txt |
On Continuity of the Invariant Torus for Countable System of Difference Equations Dependent on Parameters / N.A. Marchuk // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 316-325 . — Бібліогр.: 6 назв. — англ. |
| series |
Нелінійні коливання |
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AT marchukna oncontinuityoftheinvarianttorusforcountablesystemofdifferenceequationsdependentonparameters |
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2025-07-15T11:44:32Z |
| last_indexed |
2025-07-15T11:44:32Z |
| _version_ |
1837713192381317120 |
| fulltext |
Nonlinear Oscillations, Vol. 4, No. 3, 2001
ON CONTINUITY OF THE INVARIANT TORUS FOR COUNTABLE
SYSTEM OF DIFFERENCE EQUATIONS DEPENDENT ON PARAMETERS
N. A. Marchuk
Kam’yanets’-Podil’s’ky Pedagogic University, Ukraine
By using the method of the Green – Samoilenko function, an invariant torus is constructed
for a system of discrete equations which are defined on tori in the space of bounded number
sequences. Sufficient conditions are established for continuous dependence of the invariant torus
on the angular variable and the parameter contained in this system.
AMS Subject Classification: 39A11
It is well known that in the theory of discrete dynamical systems, investigations of invariant
tori play an important role, particularly for their continuity and smoothness. In recent time
there were published some papers dedicated to an investigation of invariant tori for difference
equations in spaces of bounded number sequences (countable systems of difference equations).
Let us mention, for example, papers [1 – 5]. Closely related to the results obtained in them are
theorems on continuity of invariant tori proved in the present article.
Let us consider the system of equations
ϕn+1 = ϕn + a(ϕn, µ), xn+1 = P (ϕn+p, µ)xn + c(ϕn+g+1, µ), (1)
in which ϕ = (ϕ1, ϕ2, . . . , ϕm) ∈ Rm, x = (x1, x2, x3, . . . ) ∈ M, where M is the space of
bounded number sequences with the norm ‖x‖ = supi{|xi|}; the functions
a(ϕ, µ) = {a1(ϕ, µ), a2(ϕ, µ), . . . , am(ϕ, µ)}, c(ϕ, µ) = {c1(ϕ, µ), c2(ϕ, µ), . . . }
and the infinite matrix P (ϕ, µ) = [pij(ϕ, µ)]∞i,j=1 are real and periodic with respect to ϕi, i =
1, 2, . . . ,m, with period 2π; n ∈ Z, Z is the set of all integers; p and g are integer parameters
which determine deviations of the discreet argument; µ ∈ [µ1, µ2] ⊂ R1 is a real parameter.
With ϕi, i = 1,m, being interpreted as the angular coordinates, we will consider the system
of equations (1) as one which is defined on an m-dimensional torus, Tm.
Let ϕn(ϕ, µ) be a solution of the first equation from (1)such that ϕ0(ϕ, µ) = ϕ ∈ Tm for all
µ ∈ [µ1, µ2]. Let xn(p, g, µ) = xn(p, g, µ, ϕ, xk) stands for a solution of the equation
xn+1 = P (ϕn+p(ϕ, µ), µ)xn + c(ϕn+1+g(ϕ, µ), µ), n ∈ Z, (2)
such that xk(p, g, µ) = xk ∈ M, k ∈ Z.
The next conditions we will call “conditions A”:
1) for all µ ∈ [µ1, µ2] and ϕ ∈ Tm, the matrix P (ϕ, µ) is invertible and for every µ ∈ [µ1, µ2],
the mapping Φ(ϕ, µ) = ϕ+ a(ϕ, µ) : Rm → Rm is invertible as well;
2) ‖a(ϕ, µ)‖ ≤ A0, ‖c(ϕ, µ)‖ ≤ C0, ‖P (ϕ, µ)‖ = supi
∑∞
j=1 |pij(ϕ, µ)| ≤ P 0, ‖P−1(ϕ, µ)‖ ≤
P1, where A0, P 0, C0, P1 are positive constants independent of µ ∈ [µ1, µ2], ϕ ∈ Tm.
316 c© N. A. Marchuk, 2001
ON CONTINUITY OF THE INVARIANT TORUS FOR COUNTABLE SYSTEM .. . 317
It is easy to see that under conditions A for each xk ∈ M, ϕ ∈ Tm, {p, g} ⊂ Z, µ ∈ [µ1, µ2],
the solution xk(p, g, µ) for equation (2) exists, is unique and contained in M for all n ∈ Z.
The invariant torus T (p, g, µ) for the system of equations (1) stands for the set of points
x ∈ M such that
x = u(p, g, µ, ϕ) = (u1(p, g, µ, ϕ), u2(p, g, µ, ϕ), . . . ), ϕ ∈ Tm,
if the function u(p, g, µ, ϕ) is defined for each {p, g} ⊂ Z, µ ∈ [µ1, µ2], ϕ ∈ Rm, is 2π-periodic
with respect to ϕi, is bounded with respect to the norm ‖ · ‖, and for every ϕ ∈ Tm satisfies the
equality
u(p, g, µ, ϕn+1(ϕ, µ)) = P (ϕn+p(ϕ, µ), µ)u(p, g, µ, ϕn(ϕ, µ)) + c(ϕn+g+1(ϕ, µ), µ).
The torus T (p, g, µ) will called continuous with respect toϕ, µ, if this is true for its generating
function u(p, g, µ, ϕ).
Let us assume that the uniform equation
xn+1 = P (ϕn+p(ϕ, µ), µ)xn, n ∈ Z, (3)
for p = 0 has a GSF (Green – Samoilenko function),G0(l, µ, ϕ), for the invariant torus problem,
i.e., there exist an infinite matrixC(ϕ, µ), 2π-periodic with respect to ϕi, i = 1,m, and bounded
with respect to the norm for all ϕ ∈ Tm, µ ∈ [µ1, µ2], and constants M > 0 and 0 < λ < 1
such that the function
G0(l, µ, ϕ) =
Ω0
l (ϕ, µ)C(ϕl(ϕ, µ), µ) for l ≤ 0;
Ω0
l (ϕ, µ)[C(ϕl(ϕ, µ), µ)− E] for l > 0
satisfies the inequality ‖G0(l, µ, ϕ)‖ ≤ Mλ|l| uniformly with respect to ϕ and µ.
In paper [1] it is shown that the function
G0(l, p, µ, ϕ) =
Ω0
l (ϕp(ϕ, µ), µ)C(ϕl(ϕp(ϕ, µ)), µ) for l ≤ 0;
Ω0
l (ϕp(ϕ, µ), µ)[C(ϕl(ϕp(ϕ, µ)), µ)− E] for l > 0
is a GSF for equation (3) for every p ∈ Z. Note that, in the same paper, it is proved that a
GSF G0(l, µ, ϕ) exists in the case of Z-dichotomic equation (3) and necessary and sufficient
conditions of its Z-dichotomicity are given. Considering equation (3) as a linear extension for
discrete dynamical system on the torus and using the weighted shift operators it is possible to
tie together the conditions on existence and uniqueness of the GSF for this linear extension
with the conditions on its hyperbolicity [6].
The next proposition takes place.
Proposition 1. Let conditions A hold and for p = 0 there exists a GSF G0(l, µ, ϕ) for equati-
on (3). Then for every {p, g} ⊂ Z, µ ∈ [µ1, µ2], system of equations (1) has an invariant torus
318 N.A. MARCHUK
T (p, g, µ) defined by the function
u(p, g, µ, ϕ) =
+∞∑
l=−∞
G0(l, p, µ, ϕ)c(ϕl+p(ϕ, µ), µ). (4)
This torus is covered by the family of bounded solutions,
xn =
+∞∑
l=−∞
Gn(l, p, µ, ϕ)c(ϕl+p(ϕ, µ), µ)
for equation (2). This family depends on the parameters p, g, µ, ϕ.
The proof for this proposition does not present difficulties, so we omit it.
Let ω(z) be some scalar function, continuous and nondecreasing on the segment [0;µ2−µ1],
such that ω(0) = 0.
Theorem 1. Let the conditions of Proposition 1 hold and, moreover, for all {ϕ, ϕ̄} ⊂ Tm,
{µ, µ̄} ⊂ [µ1, µ2],
1) ‖a(ϕ, µ)− a(ϕ̄, µ̄)‖ ≤ α1‖ϕ− ϕ̄‖+ α2ω(|µ− µ̄|),
‖P (ϕ, µ)− P (ϕ̄, µ̄)‖ ≤ β1‖ϕ− ϕ̄‖+ β2ω(|µ− µ̄|),
‖c(ϕ, µ)− c(ϕ̄, µ̄)‖ ≤ γ1‖ϕ− ϕ̄‖+ γ2(|µ− µ̄|),
‖Φ−1(ϕ, µ)− Φ−1(ϕ̄, µ̄)‖ ≤ ξ1‖ϕ− ϕ̄‖+ ξ2ω(|µ− µ̄|),
where αi, βi, γi, ξi, i = 1, 2, are positive constants independent of ϕ, µ, ϕ̄, µ̄;
2) for all ϕ ∈ Tm, µ ∈ [µ1, µ2], and p = 0, equation (3) has a unique solution bounded on Z;
3) ξ1 ≥ 1, λ < min
{
1
1 + α1
;
1
ξ1
}
.
Then the function u(p, g, µ, ϕ), which determines the invariant torus for the system of equation
(1), is continuous with respect to the set of variables ϕ, µ and, moreover, since at some moment in
the process ‖ϕ− ϕ̄‖ → 0, |µ− µ̄| → 0, the inequality
‖u(p, g, µ, ϕ)− u(p, g, µ̄, ϕ̄)‖ ≤ M∗{‖ϕ− ϕ̄‖+ ω(|µ− µ̄|)}
1
2 (5)
holds, where M∗ is a positive constant independent of ϕ, µ, ϕ̄, µ̄.
Proof. Having in mind (4) and the condition 2) of the theorem, we can write the equality
G0(l, p, µ, ϕ)−G0(l, p, µ̄, ϕ̄) =
+∞∑
k=−∞
G0(k, p, µ, ϕ)
× {P (ϕk+p−1(ϕ, µ), µ)− P (ϕk+p−1(ϕ̄, µ̄), µ̄)}Gk−1(l, p, µ̄, ϕ̄). (6)
ON CONTINUITY OF THE INVARIANT TORUS FOR COUNTABLE SYSTEM .. . 319
It is easy to see that the next inequalities hold:
‖ϕn(ϕ, µ)− ϕn(ϕ̄, µ̄)‖ ≤ (1 + α1)
n{‖ϕ− ϕ̄‖+
α2
α1
ω(|µ− µ̄|)}, n ≥ 0,
(7)
‖ϕn(ϕ, µ)− ϕn(ϕ̄, µ̄)‖ ≤ ξ−n1 {‖ϕ− ϕ̄‖+
ξ2
ξ1 − 1
ω(|µ− µ̄|)}, n < 0.
Let us introduce the following notations for convenience:
ω(|µ− µ̄|) = ω, ‖G0(l, p, µ, ϕ)−G0(l, p, µ̄, ϕ̄)‖ = G,
‖c(ϕl+g(ϕ, µ), µ)− c(ϕl+g(ϕ̄, µ̄), µ̄)‖ = c̄,
‖u(p, g, µ, ϕ)− u(p, g, µ̄, ϕ̄)‖ = ū,
1
1− λξ1
+
λ
ξ1 − λ
= λξ,
1
1 + α1 − λ
+
1
1− λ(1 + α1)
= λα.
Relations (6) and (7) imply the inequality
G ≤ M2(I1 + I2), (8)
where
I1 =
−p∑
k=−∞
λ|k|
{
β1ξ
−(k+p−1)
1 ‖ϕ− ϕ̄‖+
[β1ξ2ξ−(k+p−1)1
ξ1 − 1
+ β2
]
ω
}
,
I2 =
+∞∑
k=−p+1
λ|k|
{
β1(1 + α1)
k+p−1‖ϕ− ϕ̄‖+
[β1(α2)(1 + α1)
k+p−1
α1
+ β2
]
ω
}
.
For I1 we have
I1 ≤ β1ξ
1−p
1 ‖ϕ− ϕ̄‖
+∞∑
k=−∞
λ|k|ξ−k1 +
β2(1 + λ)ω
1− λ
+
β1ξ2ξ
1−p
1
ξ1 − 1
ω
+∞∑
k=−∞
λ|k|ξ−k1 ,
320 N.A. MARCHUK
and, with condition 3) of the theorem, it implies the inequality
I1 ≤ K1‖ϕ− ϕ̄‖+ K̄1ω, (9)
where
K1 = β1ξ
1−p
1 λξ, K̄1 = β2
1 + λ
1− λ
+
β1ξ2ξ
1−p
1
ξ1 − 1
λξ.
The inequality for I2 is obtained in a similar way,
I2 ≤ K2‖ϕ− ϕ̄‖+ K̄2ω, (10)
where
K2 = β1(1 + α1)
p−1λα, K̄2 = β2
1 + λ
1− λ
+
β1α2(1 + α1)
p−1
α1
λα.
Using (8) – (10) we obtain the inequality
G ≤ ψ1‖ϕ− ϕ̄‖+ ψ2ω, (11)
where ψ1 = M2(K1 +K2), ψ2 = M2(K̄1 + K̄2).
Having in mind (11) it is easy to see that
G ≤ (2M)
1
2λ
|l|
2 (ψ1‖ϕ− ϕ̄‖+ ψ2ω)
1
2 ,
therefore,
ū ≤ M0{ψ1‖ϕ− ϕ̄‖+ ψ2ω}
1
2 +M
+∞∑
l=−∞
λ|l|c̄, (12)
where M0 = C0(2M)
1
2 (1 +
√
λ)/(1−
√
λ).
The relation
c̄ ≤
γ1ξ
−(l+g)
1
(
‖ϕ− ϕ̄‖+
ξ2ω
ξ1 − 1
)
+ γ2ω for l + g < 0;
γ1(1 + α1)
l+g
(
‖ϕ− ϕ̄‖+
α2
α1
ω
)
+ γ2ω for l + g ≥ 0
implies
M
+∞∑
l=−∞
λ|l|c̄ ≤ M(I3 + I4), (13)
ON CONTINUITY OF THE INVARIANT TORUS FOR COUNTABLE SYSTEM .. . 321
where
I3 =
−g−1∑
l=−∞
λ|l|
{
γ1ξ
−(l+g)
1
(
‖ϕ− ϕ̄‖+
ξ2ω
ξ1 − 1
)
+ γ2ω
}
,
I4 =
+∞∑
l=−g
λ|l|
{
γ1(1 + α1)
l+g
(
‖ϕ− ϕ̄‖+
α2
α1
ω
)
+ γ2ω
}
.
Similarly to inequalities (9), (10), we obtain the inequalities for I3 and I4,
I3 ≤ K3‖ϕ− ϕ̄‖+ K̄3ω, I4 ≤ K4‖ϕ− ϕ̄‖+ K̄4ω, (14)
where
K3 = γ1ξ
−g
1 λξ, K̄3 = γ2
1 + λ
1− λ
+
γ1ξ2ξ
−g
1
ξ1 − 1
λξ,
K4 = γ1(1 + α1)
gλα, K̄4 = γ2
1 + λ
1− λ
+
γ1α2(1 + α1)
g
α1
λα.
Having in mind (12) – (14) and denoting M(K3 + K4) and M(K̄3 + K̄4) by η1 and η2,
respectively, we obtain the inequality
ū ≤ M0{ψ1‖ϕ− ϕ̄‖+ ψ2ω(|µ− µ̄|)}
1
2 + η1‖ϕ− ϕ̄‖+ η2ω(|µ− µ̄|),
which, since at some moment in the process ‖ϕ− ϕ̄‖ → 0, |µ− µ̄| → 0, gives us inequality (5),
where M∗ stands for the expression max{M0(max{ψ1, ψ2})
1
2 , η1, η2}. The theorem is proved.
Now we prove a theorem that allows us to omit the condition 3) in Theorem 1.
Theorem 2. Let all the conditions of Theorem 1 hold, excluding the third one. Then the
function u(p, g, µ, ϕ) is continuous with respect to the set of variables ϕ, µ and, moreover, since at
some moment in the process ‖ϕ− ϕ̄‖ → 0, |µ− µ̄| → 0, the inequality
‖u(p, g, µ, ϕ)− u(p, g, µ̄, ϕ̄)‖ ≤ M∗{‖ϕ− ϕ̄‖+ ω(|µ− µ̄|)}
ν
2(ν+1) (15)
holds, where M∗ is a positive constant independent of {ϕ, ϕ̄} ⊂ Tm and {µ, µ̄} ⊂ [µ1, µ2], and ν
is an arbitrarily chosen positive real number satisfying the condition
ν
ν + 1
< min{− logξ1 λ;− log(1+α1) λ}, ξ1 > 1. (16)
322 N.A. MARCHUK
Proof. Let us put ξ1 > 1 and denote
‖P (ϕn(ϕ, µ), µ)− P (ϕn(ϕ̄, µ̄), µ̄)‖
by P̄ . Using (7) we get the inequality
P̄ ≤
P1ξ
− nν
ν+1
1 (‖ϕ− ϕ̄‖+ ω)
ν
ν+1 for n < 0;
P2(1 + α1)
nν
ν+1 (‖ϕ− ϕ̄‖+ ω)
ν
ν+1 for n ≥ 0,
(17)
where ν is an arbitrarily chosen positive number, and P1, P2 stand for
(2P 0)
1
ν+1
(
max{β1;
β1ξ2
ξ1 − 1
+ β2}
) ν
ν+1
,
(2P 0)
1
ν+1
(
max{β1;
β1α2
α1
+ β2}
) ν
ν+1
,
respectively.
Relations (6) and (17) give the inequality
G ≤ M2(I01 + I02 ), (18)
where
I01 =
−p∑
k=−∞
λ|k|P1ξ
− ν(k+p−1)
ν+1
1 (‖ϕ− ϕ̄‖+ ω)
ν
ν+1 ,
I02 =
+∞∑
k=−p+1
λ|k|P2(1 + α1)
ν(k+p−1)
ν+1 (‖ϕ− ϕ̄‖+ ω)
ν
ν+1 .
Because of condition (16), λξ
ν
ν+1
1 < 1 and λ(1 + α1)
ν
ν+1 < 1, which allows us to write the
inequalities
I0i ≤ P̄i(‖ϕ− ϕ̄‖+ ω)
ν
ν+1 , i = 1, 2, (19)
where
P̄1 = P1λ
(1)ξ
− ν(p−1)
ν+1
1 , P̄2 = P2λ
(2)(1 + α1)
ν(p−1)
ν+1 ,
λ(1) =
(
1− λξ
ν
ν+1
1
)−1
+ λξ
− ν
ν+1
1
(
1− λξ
− ν
ν+1
1
)−1
,
ON CONTINUITY OF THE INVARIANT TORUS FOR COUNTABLE SYSTEM .. . 323
λ(2) =
(
1− λ(1 + α1)
− ν
ν+1
)−1
+ λ(1 + α1)
ν
ν+1
(
1− λ(1 + α1)
− ν
ν+1
)−1
.
It is easy to see that (18) and (19) imply continuity of the GSF for equation (3) with respect
to the set of variables ϕ, µ, because
G ≤ M2(P̄1 + P̄2){‖ϕ− ϕ̄‖+ ω}
ν
ν+1 .
From the last inequality it, is easy to go to the next one,
G ≤ [2M3(P̄1 + P̄2]
1
2λ
|l|
2 {‖ϕ− ϕ̄‖+ ω}
ν
2(ν+1) . (20)
And, at last, let us write the relation
c̄ ≤
C1ξ
− ν(l+g)
ν+1
1 (‖ϕ− ϕ̄‖+ ω)
ν
ν+1 for l + g < 0;
C2(1 + α1)
ν(l+g)
ν+1 (‖ϕ− ϕ̄‖+ ω)
ν
ν+1 for l + g ≥ 0,
in which
C1 = (2C0)
1
ν+1
(
max
{
γ1;
γ1ξ2
ξ1 − 1
+ γ2
}) ν
ν+1
,
C2 = (2C0)
1
ν+1
(
max
{
γ1;
γ1α2
α1
+ γ2
}) ν
ν+1
.
Hence,
M
+∞∑
l=−∞
λ|l|c̄ ≤M(‖ϕ− ϕ̄‖+ ω)
ν
ν+1
{
C1ξ
− νg
ν+1
1
−g−1∑
l=−∞
λ|l|ξ
− νl
ν+1
1
+ C2(1 + α1)
νg
ν+1
+∞∑
l=−g
λ|l|(1 + α1)
νl
ν+1
}
≤ C3(‖ϕ− ϕ̄‖+ ω)
ν
ν+1 , (21)
where
C3 = M{C1ξ
− νg
ν+1
1 λ(1) + C2(1 + α1)
νg
ν+1λ(2)}.
From (20) and (21) we obtain the inequality
ū ≤
+∞∑
l=−∞
{C0G+Mλ|l|c̄} ≤ C4(‖ϕ− ϕ̄‖+ ω(|µ− µ̄|))
ν
2(ν+1)
+ C3(‖ϕ− ϕ̄‖+ ω(|µ− µ̄|))
ν
ν+1 ,
324 N.A. MARCHUK
where C4 = C0[2M3(P̄1 + P̄2)]
1
2 (1 +
√
λ)/(1−
√
λ).
Having denoted max{C3, C4} by M∗ and since at some moment in the process ‖ϕ − ϕ̄‖ →
0, ‖µ− µ̄‖ → 0, we obtain inequality (15) which finishes the proof of the theorem.
Remark 1. For any real ν ≥ 0 from inequality (15) it is not possible to obtain an inequality
of form (5) with the indicator of power equal to 1/2.
Remark 2. Under conditions of Theorems 1 and 2, the function u(p, g, µ, ϕ), with µ fixed,
satisfies the Gölder condition with respect to ϕ with indicators of power 1/2 and ν/(2(ν + 1)),
respectively.
In the case where the function a(ϕ, µ) does not depend on µ, i.e., a(ϕ, µ) = a(ϕ), the conti-
nuity conditions for the function u(p, g, µ, ϕ) (that defines the invariant torus for the system of
equations (1)) with respect to the parameter µ are much more simple.
Corollary 1. Let the conditions of Proposition 1 and condition 2) of Theorem 1 hold and
a(ϕ, µ) do not depend on µ. Then the inequalities
sup
ϕ
‖P (ϕ, µ)− P (ϕ, µ̄)‖ ≤ ω1(|µ− µ̄|),
sup
ϕ
‖c(ϕ, µ)− c(ϕ, µ̄)‖ ≤ ω2(|µ− µ̄|),
in which the functions ω1(z), ω2(z) have properties of ω(z), guarantee continuity of the function
u(p, g, µ, ϕ) with respect to the parameter µ.
Proof. The equality (6) implies the inequality
G ≤
+∞∑
k=−∞
M2λ|k|‖P (ϕk+p−1(ϕ), µ)− P (ϕk+p−1(ϕ), µ̄)‖ ≤ M2 1 + λ
1− λ
ω1(|µ− µ̄|),
which, in its turn, implies the inequality
G ≤
(
M3 1 + λ
1− λ
) 1
2
λ
|l|
2 ω
1
2
1 (|µ− µ̄|).
Similarly,
M
+∞∑
l=−∞
λ|l|‖c(ϕl+g(ϕ), µ)− c(ϕl+g(ϕ), µ̄)‖ ≤ M
1 + λ
1− λ
ω2(|µ− µ̄|).
ON CONTINUITY OF THE INVARIANT TORUS FOR COUNTABLE SYSTEM .. . 325
Then we have
‖u(p, g, µ, ϕ)− u(p, g, µ̄, ϕ)‖ ≤
≤
+∞∑
l=−∞
C0
(
M3 1 + λ
1− λ
) 1
2
λ
|l|
2 ω
1
2
1 (|µ− µ̄|) +M
1 + λ
1− λ
ω2(|µ− µ̄|)
≤ C0 1 +
√
λ
1−
√
λ
(
M3 1 + λ
1− λ
) 1
2
ω
1
2
1 (|µ− µ̄|) +M
1 + λ
1− λ
ω2(|µ− µ̄|).
The last inequality finishes the proof.
Finally, let us note that the results obtained here are preserved for the case where the torus, on
which the input system of equations (1) is being considered, is infinite-dimensional, i.e., when
ϕ = {ϕ1, ϕ2, ϕ3, . . . }.
REFERENCES
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[in Russian], Preprint, No. 98.3, Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv (1998).
3. Teplinsky Yu. V. and Marchuk N. A. “The method of reduction in the investigation of smoothness of the invari-
ant torus for a countable system of difference equations with parameters,” Zb. Nauk. Prats’ Kam’yanets’-
Podil’s’kogo Pedagog. Univ. Ser. Mat., No. 5, 117 – 126 (2000).
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systems,” Usp. Mat. Nauk, 48, No. 2, 85 – 143 (1991).
Received 10.12.99
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