Conditions of Invertibility for Functional Operators with Shift in Weighted Hölder Spaces
We consider functional operators with shift in weighted Hölder spaces. The main result of the work is the proof of the conditions of invertibility for these operators. We also indicate the forms of the inverse operators. As an application, we propose to use these results for the solution of equation...
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irk-123456789-1659152020-02-18T01:27:29Z Conditions of Invertibility for Functional Operators with Shift in Weighted Hölder Spaces Tarasenko, G. Karelin, O. Статті We consider functional operators with shift in weighted Hölder spaces. The main result of the work is the proof of the conditions of invertibility for these operators. We also indicate the forms of the inverse operators. As an application, we propose to use these results for the solution of equations with shift encountered in the study of cyclic models for natural systems with renewable resources. Розглядаються функціональні оператори із зсувом у просторах Гельдера з вагою. Основним результатом роботи є встановлення умов оборотності для цих операторів. Вказано види оберненого оператора. Як застосування запропоновано використовувати отримані результати для розв'язання рівнянь із зсувом, які виникають при дослідженні циклічних моделей природних систем з ресурсами, що відновлюються. 2015 Article Conditions of Invertibility for Functional Operators with Shift in Weighted Hölder Spaces / G. Tarasenko, O. Karelin // Український математичний журнал. — 2015. — Т. 67, № 11. — С. 1557–1568. — Бібліогр.: 6 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165915 517.9 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Tarasenko, G. Karelin, O. Conditions of Invertibility for Functional Operators with Shift in Weighted Hölder Spaces Український математичний журнал |
| description |
We consider functional operators with shift in weighted Hölder spaces. The main result of the work is the proof of the conditions of invertibility for these operators. We also indicate the forms of the inverse operators. As an application, we propose to use these results for the solution of equations with shift encountered in the study of cyclic models for natural systems with renewable resources. |
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Article |
| author |
Tarasenko, G. Karelin, O. |
| author_facet |
Tarasenko, G. Karelin, O. |
| author_sort |
Tarasenko, G. |
| title |
Conditions of Invertibility for Functional Operators with Shift in Weighted Hölder Spaces |
| title_short |
Conditions of Invertibility for Functional Operators with Shift in Weighted Hölder Spaces |
| title_full |
Conditions of Invertibility for Functional Operators with Shift in Weighted Hölder Spaces |
| title_fullStr |
Conditions of Invertibility for Functional Operators with Shift in Weighted Hölder Spaces |
| title_full_unstemmed |
Conditions of Invertibility for Functional Operators with Shift in Weighted Hölder Spaces |
| title_sort |
conditions of invertibility for functional operators with shift in weighted hölder spaces |
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Інститут математики НАН України |
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2015 |
| topic_facet |
Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/165915 |
| citation_txt |
Conditions of Invertibility for Functional Operators with Shift in Weighted Hölder Spaces / G. Tarasenko, O. Karelin // Український математичний журнал. — 2015. — Т. 67, № 11. — С. 1557–1568. — Бібліогр.: 6 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT tarasenkog conditionsofinvertibilityforfunctionaloperatorswithshiftinweightedholderspaces AT karelino conditionsofinvertibilityforfunctionaloperatorswithshiftinweightedholderspaces |
| first_indexed |
2025-07-14T20:22:19Z |
| last_indexed |
2025-07-14T20:22:19Z |
| _version_ |
1837655176908898304 |
| fulltext |
UDC 517.9
G. Tarasenko, O. Karelin (Univ. Autonoma del Estado de Hidalgo, México)
CONDITIONS OF INVERTIBILITY FOR FUNCTIONAL OPERATORS
WITH SHIFT IN WEIGHTED HÖLDER SPACES
УМОВИ ОБОРОТНОСТI ДЛЯ ФУНКЦIОНАЛЬНИХ ОПЕРАТОРIВ
IЗ ЗСУВОМ У ПРОСТОРАХ ГЕЛЬДЕРА З ВАГОЮ
We consider functional operators with shift in weighted Hölder spaces. The main result of this work is the proof of the
conditions of invertibility for these operators. We also indicate the forms of the inverse operator. As an application, we
propose to use these results for solution of equations with shift which arise in the study of cyclic models for natural systems
with renewable resources.
Розглядаються функцiональнi оператори iз зсувом у просторах Гельдера з вагою. Основним результатом роботи є
встановлення умов оборотностi для цих операторiв. Вказано види оберненого оператора. Як застосування запро-
поновано використовувати отриманi результати для розв’язання рiвнянь iз зсувом, якi виникають при дослiдженнi
циклiчних моделей природних систем з ресурсами, що вiдновлюються.
1. Introduction. The interest towards the study of functional operators with shift was stipulated by
the development of the solvability theory and Fredholm theory for some classes of linear operators,
in particular, singular integral operators with Carleman and non-Carleman shift [1 – 3]. Conditions of
invertibility for functional operators with shift in weighted Lebesgue spaces were obtained in [1].
Our study of functional operators with shift in the weighted Hölder spaces has an additional
motivation: on modeling systems with renewable resources, equations with shift arise in [4, 5], and
the theory of linear functional operators with shift is the adequate mathematical instrument for the
investigation of such systems.
In Section 2, the boundedness of functional operators with shift in the Hölder spaces and in the
weighted Hölder spaces is proved.
In Section 3, some auxiliary lemmas are proved. They will be used in the proof of invertibility
conditions.
In Section 4, forms of the inverse operator are given.
In Section 5, conditions of invertibility for functional operators with shift in the Hölder spaces
with power wight are obtained. At the end of the article, an application to modeling systems with
renewable resources is given.
2. Boundedness of shift operators in the weighted Hölder spaces. We introduce [6] the
weighted Hölder spaces H0
µ(J, ρ).
A function ϕ(x) that satisfies the following condition on J = [0, 1],
|ϕ(x1)− ϕ(x2)| ≤ C|x1 − x2|µ, x1 ∈ J, x2 ∈ J, µ ∈ (0, 1),
is called a Hölder’s function with exponent µ and constant C on J .
Let ρ be a power function which has zeros at the endpoints x = 0, x = 1:
ρ(x) = (x− 0)µ0(1− x)µ1 , µ < µ0 < 1 + µ, µ < µ1 < 1 + µ.
The functions that become Hölder functions and valued zero at the points x = 0, x = 1, after being
multiplied by ρ(x), form a Banach space:
c© G. TARASENKO, O. KARELIN, 2015
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11 1557
1558 G. TARASENKO, O. KARELIN
H0
µ(J, ρ), J = [0, 1].
The norm in the space H0
µ(J, ρ) is defined by
‖f(x)‖H0
µ(J,ρ)
= ‖ρ(x)f(x)‖Hµ(J),
where
‖ρ(x)f(x)‖Hµ(J) = ‖ρ(x)f(x)‖C + ‖ρ(x)f(x)‖µ,
and
‖ρ(x)f(x)‖C = max
x∈J
|ρ(x)f(x)|,
‖ρ(x)f(x)‖µ = sup
x1,x2∈J,x1 6=x2
|ρ(x)f(x)|µ,
|ρ(x)f(x)|µ =
|ρ(x1)f(x1)− ρ(x2)f(x2)|
|x1 − x2|µ
.
Let β(x) be a bijective orientation-preserving shift on J :
if x1 < x2, then β(x1) < β(x2) for any x1 ∈ J, x2 ∈ J ; and let β(x) have only two fixed points:
β(0) = 0, β(1) = 1, β(x) 6= x, when x 6= 0, x 6= 1.
In addition, let β(x) be a differentiable function with
d
dx
β(x) 6= 0 and
d
dx
β(x) ∈ Hµ(J).
Let us begin with the shift operator (Bβϕ)(x) = ϕ[β(x)].
Theorem 1. Operator Bβ is bounded on the space Hµ(J),
‖Bβ‖B(Hµ(J)) ≤ ‖β
′‖µC .
Operator Bβ is bounded on the space H0
µ(J, ρ),
‖Bβ‖B(H0
µ(J,ρ))
≤
∥∥∥∥ ρ
ρ[β]
∥∥∥∥
Hµ(J)
‖Bβ‖B(Hµ(J)).
Proof. Let ϕ ∈ Hµ(J),
‖Bβϕ‖Hµ(J)=‖Bβϕ‖C + ‖Bβϕ‖µ =
= ‖ϕ‖C + sup
x1 6=x2
|ϕ[β(x2)]− ϕ[β(x1)]| |β(x2)− β(x1)|µ
|x2 − x1|µ |β(x2)− β(x1)|µ
≤
≤ ‖ϕ‖C + sup
x1 6=x2
∣∣∣∣β(x2)− β(x1)
x2 − x1
∣∣∣∣µ ‖ϕ‖µ.
From here, it follows that
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
CONDITIONS OF INVERTIBILITY FOR FUNCTIONAL OPERATORS WITH SHIFT . . . 1559
‖Bβ‖B(Hµ(J)) ≤ max
{
1, sup
x1 6=x2
∣∣∣∣β(x2)− β(x1)
x2 − x1
∣∣∣∣µ
}
=
= sup
x1 6=x2
∣∣∣∣β(x2)− β(x1)
x2 − x1
∣∣∣∣µ = ‖β′‖µC .
Let ϕ ∈ H0
µ(J, ρ); from
∥∥∥∥ ρ
ρ[β]
Bβ(ρϕ)
∥∥∥∥
µ
= sup
x1 6=x2
∣∣∣∣∣∣∣∣
ρ(x1)
ρ[β(x1)]
(Bβ(ρϕ)) (x1)−
ρ(x2)
ρ[β(x2)]
(Bβ(ρϕ)) (x2)
(x1 − x2)µ
∣∣∣∣∣∣∣∣ =
= sup
x1 6=x2
∣∣∣∣∣∣∣∣
(
ρ(x1)
ρ[β(x1)]
− ρ(x2)
ρ[β(x2)]
)
(Bβ(ρϕ))(x1)+((Bβ(ρϕ))(x1)− (Bβ(ρϕ))(x2))
(
ρ(x2)
ρ[β(x2)]
)
(x1 − x2)µ
∣∣∣∣∣∣∣∣≤
≤
∥∥∥∥ ρ
ρ[β]
∥∥∥∥
µ
‖Bβ(ρϕ)‖C +
∥∥∥∥ ρ
ρ[β]
∥∥∥∥
C
∥∥Bβ(ρϕ)
∥∥
µ
and ∥∥∥∥ ρ
ρ[β]
∥∥∥∥
C
≤
∥∥∥∥ ρ
ρ[β]
∥∥∥∥
Hµ(J)
,
it follows that
‖Bβϕ‖H0
µ(J,ρ)
= ‖ρBβϕ‖Hµ(J) =
∥∥∥∥ ρ
ρ[β]
Bβ(ρϕ)
∥∥∥∥
C
+
∥∥∥∥ ρ
ρ[β]
Bβ(ρϕ)
∥∥∥∥
µ
≤
≤
∥∥∥∥ ρ
ρ[β]
∥∥∥∥
C
‖Bβ(ρϕ)‖C +
∥∥∥∥ ρ
ρ[β]
∥∥∥∥
µ
‖Bβ(ρϕ)‖C +
∥∥∥∥ ρ
ρ[β]
∥∥∥∥
C
‖Bβ(ρϕ)‖µ ≤
≤
∥∥∥∥ ρ
ρ[β]
∥∥∥∥
Hµ(J)
∥∥Bβ(ρϕ)
∥∥
C
+
∥∥∥∥ ρ
ρ[β]
∥∥∥∥
C
‖Bβ(ρϕ)‖µ ≤
∥∥∥∥ ρ
ρ[β]
∥∥∥∥
Hµ(J)
‖Bβ(ρϕ)‖Hµ(J) ≤
≤
∥∥∥∥ ρ
ρ[β]
∥∥∥∥
Hµ(J)
‖Bβ‖B(Hµ(J))‖ρϕ‖Hµ(J) =
∥∥∥∥ ρ
ρ[β]
∥∥∥∥
Hµ(J)
‖Bβ‖B(Hµ(J))‖ϕ‖H0
µ(J,ρ)
.
Since
ρ(x)
ρ[β(x)]
=
∣∣∣∣ x
β(x)
∣∣∣∣µ0 ∣∣∣∣ 1− x
1− β(x)
∣∣∣∣µ1 ∈ Hµ(J), we complete the proof.
Thus the operator A = aI − bBβ, with coefficients a ∈ Hµ(J), b ∈ Hµ(J), is bounded on the
space H0
µ(J, ρ).
3. Auxiliary lemmas. We keep the conditions on the shift β given in Section 2. Without loss
of generality, we assume also that for any fixed x ∈ (0, 1),
lim
m→+∞
βm(x) = 0, lim
m→+∞
β−m(x) = 1;
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
1560 G. TARASENKO, O. KARELIN
which implies that β
′
(0) ≤ 1 and β
′
(1) ≥ 1.
We will use the following notation:
r = µ0 − µ, s = µ1 − µ, ρr,s(x) = xr(1− x)s, ρµ,µ(x) = ρµ(x) = xµ(1− x)µ,
ρr,s;j(x)=ρr,s[βj(x)], ρµ;j(x)=ρµ,µ;j(x), β(x1, x2) =
β(x1)− β(x2)
x1 − x2
.
Lemma 1. We have
(∀β(x), x ∈ J )(∀ε > 0)(∃n0 ∈ N)(∀x ∈ J)(∃n1, n2 ∈ N, n1 < n2, n0 = n2 − n1)[
βn(x) ∈ [0, ε]
⋃
[1− ε, 1], n ∈ N \ [n1, n2]
]
.
An essential point here is that n0 = n2 − n1 is independent of x.
Proof. Follows directly from the properties of β(x).
Lemma 2. Under the conditions
a(x) 6= 0;
∣∣β′(0)
∣∣−µ0+µ ∣∣∣∣ b(0)
a(0)
∣∣∣∣ < 1,
∣∣β′(1)
∣∣−µ1+µ ∣∣∣∣ b(1)
a(1)
∣∣∣∣ < 1, (1)
the following inequalities hold in some one-sided ε1-neighborhoods of the endpoints x = 0, x = 1:∣∣∣∣u(x)
ρr,s(x)
ρr,s;1(x)
∣∣∣∣ ≤ q1 < 1, x ∈ [0, ε1]
⋃
[1− ε1, 1]. (2)
Proof. Follows from (1) and from the properties of β(x), a(x), b(x).
Lemma 3. Under the condition (1), there is ε2 > 0 such that the following inequality holds:∣∣∣∣∣∣∣∣β(x1)− β(x2)
x1 − x2
∣∣∣∣µ u(x2)
ρ(x2)
ρ[β(x2)]
∣∣∣∣ ≤ q2 < 1, (3)
for x1, x2 ∈ [0, ε2] or x1, x2 ∈ [1− ε2, 1], or x1 ∈ [0, ε2], x2 ∈ [1− ε2, 1].
Proof. It is easy to see that the following identity:∣∣∣∣∣∣∣∣β(x1)−β(x2)
x1 − x2
∣∣∣∣µu(x2)
ρ(x2)
ρ[β(x2)]
∣∣∣∣ =
=
∣∣∣∣β(x1)− β(x2)
x1 − x2
∣∣∣∣µ∣∣∣∣ x2
β(x2)
∣∣∣∣µ∣∣∣∣ 1− x2
1− β(x2)
∣∣∣∣µ∣∣∣∣u(x2)
ρr,s(x2)
ρr,s;1(x2)
∣∣∣∣ (4)
holds. We estimate then the expression
∣∣∣∣β(x1)− β(x2)
x1 − x2
∣∣∣∣µ ∣∣∣∣ x2
β(x2)
∣∣∣∣µ ∣∣∣∣ 1− x2
1− β(x2)
∣∣∣∣µ .
By (2) of Lemma 2 and limx1,x2→0
∣∣∣∣β(x1)− β(x2)
x1 − x2
∣∣∣∣µ ∣∣∣∣ x2
β(x2)
∣∣∣∣µ = 1, limx2→0
∣∣∣∣ x2 − 1
β(x2)− 1
∣∣∣∣µ = 1,
we can choose ε3 > 0 such that the inequality
∣∣∣∣β(x1)− β(x2)
x1 − x2
∣∣∣∣µ ∣∣∣∣ x2
β(x2)
∣∣∣∣µ ∣∣∣∣ 1− x2
1− β(x2)
∣∣∣∣µ q1 ≤ q3 < 1
holds for x1, x2 ∈ [0, ε3].
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
CONDITIONS OF INVERTIBILITY FOR FUNCTIONAL OPERATORS WITH SHIFT . . . 1561
By (2) of Lemma 2 and limx1,x2→1
∣∣∣∣β(x1)− β(x2)
x1 − x2
∣∣∣∣µ ∣∣∣∣ 1− x2
1− β(x2)
∣∣∣∣µ = 1, limx2→1
∣∣∣∣ x2
β(x2)
∣∣∣∣µ = 1,
we can choose ε4 > 0 such that the inequality
∣∣∣∣β(x1)− β(x2)
x1 − x2
∣∣∣∣µ ∣∣∣∣ x2
β(x2)
∣∣∣∣µ ∣∣∣∣ 1− x2
1− β(x2)
∣∣∣∣µ q1 ≤ q4 < 1
holds for x1, x2 ∈ [1− ε4, 1].
As
limx1→0,x2→1
∣∣∣∣β(x1)− β(x2)
x1 − x2
∣∣∣∣µ = 1, limx2→1
∣∣∣∣ x2
β(x2)
∣∣∣∣µ = 1,
limx2→1
∣∣∣∣ 1− x2
1− β(x2)
∣∣∣∣µ =
∣∣β′(1)
∣∣−µ ≤ 1
we can choose ε5 > 0 such that the inequality∣∣∣∣β(x1)− β(x2)
x1 − x2
∣∣∣∣µ ∣∣∣∣ x2
β(x2)
∣∣∣∣µ ∣∣∣∣ 1− x2
1− β(x2)
∣∣∣∣µ q1 ≤ q5 < 1
holds for x1 ∈ [0, ε5], x2 ∈ [1− ε5, 1].
To prove inequality (3), it is sufficient to choose ε2 = min (ε3, ε4, ε5), take q2 = max (q3, q4, q5)
and apply the obtained estimates to expression (4).
By Lemma 1, for ε = min(ε1, ε2) there exists a positive integer n0 such that for each x ∈ [0, 1]
at most n0 values of βn(x) is outside of [0, ε]
⋃
[1− ε, 1]. Let q = max(q1, q2).
Lemma 4. If {w(x)}|x=0 = 0, then ‖w(x)‖Hµ(J) ≥ sup0<x<1
|w(x)|
xµ
.
If {w(x)}|x=1 = 0, then ‖w(x)‖Hµ(J) ≥ sup0<x<1
|w(x)|
(1− x)µ
.
If {w(x)}|x=0 = {w(x)}|x=1 = 0, then ‖w(x)‖Hµ(J) ≥
(
1
2
)µ
sup0<x<1
|w(x)|
xµ(1− x)µ
.
For ϕ ∈ H0
µ(J, ρ), the inequality∥∥∥∥ ρϕρµ,µ
∥∥∥∥
C
≤ 2µ ‖ϕ‖H0
µ(J,ρ)
(5)
holds.
Proof. The proof follows from the inequalities
‖w(x)‖C(J)+ sup
x1,x2∈J
x1 6=x2
|w(x1)− w(x2)|
|x1 − x2|µ
≥
≥ ‖w(x)‖C(J)+ sup
0<x1<1
|w(x1)− 0|
|x1 − 0|µ
≥ sup
0<x<1
|w(x)|
xµ
,
‖w(x)‖C(J)+ sup
x1,x2∈J
x1 6=x2
|w(x1)− w(x2)|
|x1 − x2|µ
≥
≥ ‖w(x)‖C(J)+ sup
0<x2<1
|0− w(x2)|
|1− x2|µ
≥ sup
0<x<1
|w(x)|
(1− x)µ
,
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
1562 G. TARASENKO, O. KARELIN
‖w(x)‖Hµ(J) =
1
2
(
‖w(x)‖Hµ(J) + ‖w(x)‖Hµ(J)
)
≥
≥ 1
2
(
sup
0<x<1
|w(x)|
xµ
+ sup
0<x<1
|w(x)|
(1− x)µ
)
≥
≥ 1
2
sup
0<x<1
((1− x)µ + xµ) |w(x)|
xµ(1− x)µ
≥
(
1
2
)µ
sup
0<x<1
|w(x)|
xµ(1− x)µ
.
It remains to prove (5): ∥∥∥∥ ρϕρµ,µ
∥∥∥∥
C
≤ 2µ−1
∥∥∥∥xµ + (1− x)µ
xµ(1− x)µ
ρ(x)ϕ(x)
∥∥∥∥
C
≤
≤ 2µ−1
[ ∥∥∥∥ρ(x)ϕ(x)
(1− x)µ
∥∥∥∥
C
+
∥∥∥∥ρ(x)ϕ(x)
(x)µ
∥∥∥∥
C
]
≤
≤ 2µ−1
[ ∥∥∥∥ρ(1)ϕ(1)− ρ(x2)ϕ(x2)
(1− x2)µ
∥∥∥∥
C
+
∥∥∥∥ρ(x1)ϕ(x1)− ρ(0)ϕ(0)
(x1 − 0)µ
∥∥∥∥
C
]
≤
≤ 2µ−1
[ ∥∥∥∥ρ(x1)ϕ(x1)− ρ(x2)ϕ(x2)
(x1 − x2)µ
∥∥∥∥
C
+
∥∥∥∥ρ(x1)ϕ(x1)− ρ(x2)ϕ(x2)
(x1 − x2)µ
∥∥∥∥
C
]
=
= 2µ‖ρϕ‖µ ≤ 2µ‖ρϕ‖Hµ(J) = 2µ‖ϕ‖H0
µ(J,ρ)
.
In the above, we use that 1 ≤ 2µ−1
∣∣xµ + (1− x)µ
∣∣, ρ(0)ϕ(0) = ρ(1)ϕ(1) = 0.
We will use these lemmas in the proof of invertibility conditions in Section 5.
4. Structure of the inverse operator. The operators
A = aI − bBβ,
where a ∈ Hµ, b ∈ Hµ, a 6= 0, and
U = I − uBβ,
where u = b/a, are invertible simultaneously on the weighted Hölder space H0
µ(J, ρ).
If there exists a natural number n such that∥∥∥∥∥∥
n−1∏
j=0
uj(x)
Bn
β
∥∥∥∥∥∥
B(H0
µ(J,ρ))
< 1,
where
uj(x) = u
[
βj(x)
]
,
then the operator U is invertible on H0
µ(J, ρ) and
U−1 =
I + uBβ + . . .+
n−2∏
j=0
uj(x)
Bn−1
β
I −
n−1∏
j=0
uj(x)
Bn
β
−1 .
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CONDITIONS OF INVERTIBILITY FOR FUNCTIONAL OPERATORS WITH SHIFT . . . 1563
This statement was proved in [1] for weighted Lebesgue spaces. The proof for the weighted Hölder
spaces literally follows the above one as an application of algebraic operations does not depend on
the specific properties of the spaces.
We note that the inverse operator U−1 can be written in the form
U−1 =
I + uBβ + . . .+
m−2∏
j=0
uj(x)
Bm−1
β
I −
m−1∏
j=0
uj(x)
Bm
β
−1 ,
with any another m, m 6= n, subject to the condition
∥∥∥∏m−1
j=0
uj(x)Bm
β
∥∥∥
B(H0
µ(J,ρ))
< 1.
Analogously, if b 6= 0 and there exists a natural number n such that∥∥∥∥∥∥
n−1∏
j=0
vj(x)
B−nβ
∥∥∥∥∥∥
B(H0
µ(J,ρ))
< 1,
where
v(x) =
a
[
β−1(x)
]
b
[
β−1(x)
] , vj(x) = v
[
β−1j (x)
]
,
then the operator
V = I − vB−1β
is invertible on the space B(H0
µ(J, ρ)) and its inverse operator is given by
V −1 =
I + vB−1β + . . .+
n−2∏
j=0
vj(x)
B−n+1
β
I −
n−1∏
j=0
vj(x)
B−nβ
−1 .
It is obvious that A = −bBβ
[
I −
(
B−1β
a
b
)
B−1β
]
, A−1 = −V −1B−1β
(
1
b
)
I.
5. Invertibility conditions for the operator A on the weighted Hölder spaces. We will use
the following notation:
fn(x) =
(
Bn
βf
)
(x), βµn(x1, x2) =
∣∣∣∣βn+1(x1)− βn+1(x2)
βn(x1)− βn(x2)
∣∣∣∣µ , ũj(x) =
ρj(x)
ρj+1(x)
uj(x).
Theorem 2. Conditions (1) implies that there exists a natural number n for which∥∥∥∥∥∥
n−1∏
j=0
uj
Bn
β
∥∥∥∥∥∥
H0
µ(J,ρ)
< 1.
Proof. To prove that∥∥∥∥∥∥ρ
n−1∏
j=0
uj
Bn
βϕ
∥∥∥∥∥∥
C
+
∥∥∥∥∥∥ρ
n−1∏
j=0
uj
Bn
βϕ
∥∥∥∥∥∥
µ
< ‖ϕ‖H0
µ(J,ρ)
, (6)
we estimate each summand separately. For the first one we have
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1564 G. TARASENKO, O. KARELIN
∥∥∥∥∥∥ρ
n−1∏
j=0
uj
Bn
βϕ
∥∥∥∥∥∥
C
=
∥∥∥∥∥∥ρρr,sρr,s
n−1∏
j=0
uj
1
ρr,s;n
Bn
β (ρr,sϕ)
∥∥∥∥∥∥
C
≤
≤
∥∥∥∥∥∥ρµ,µ
n−1∏
j=0
uj
ρr,s;j
ρr,s;j+1
∥∥∥∥∥∥
C
‖ρr,sϕ‖C ≤
∥∥∥∥∥∥ρµ,µ
n−1∏
j=0
uj
ρr,s;j
ρr,s;j+1
∥∥∥∥∥∥
C
2µ ‖ϕ‖H0
µ(J,ρ)
. (7)
We took into account
ρ
ρr,s
= ρµ,µ,
ρr,s
ρr,s;n
=
n−1∏
j=0
ρr,s;j
ρr,s;j+1
,
∥∥Bn
β (ρr,sϕ)
∥∥
C
= ‖(ρr,sϕ)‖C
and inequality (5) from Lemma 4.
By (2) of Lemma 2, it follows that the first factor on the right-hand side of inequality (7)∥∥∥∥ρµ,µ∏n−1
j=0
uj
ρr,s;j
ρr,s;j+1
∥∥∥∥
C
tends to zero when n→∞.
Now, we estimate the second summand of (6). We use the following notation:
fn(x) =
(
Bn
βf
)
(x), βµn(x1, x2) =
∣∣∣∣βn+1(x1)− βn+1(x2)
βn(x1)− βn(x2)
∣∣∣∣µ , ũj(x) =
ρj(x)
ρj+1(x)
uj(x).
We obtain ∥∥∥∥∥∥ρ
n−1∏
j=0
uj
Bn
βϕ
∥∥∥∥∥∥
µ
=
∥∥∥∥∥∥
n−1∏
j=0
ũjρnϕn
∥∥∥∥∥∥
µ
≤
≤ sup
x1<x2
∣∣∣∏n−1
j=0
ũj(x1)ρn(x1)ϕn(x1)−
∏n−1
j=0
ũj(x2)ρn(x2)ϕn(x2)
∣∣∣
|x1 − x2|µ
=
= sup
x1<x2
∣∣∣(ρnϕn)(x1)
(∏n−1
j=0
ũj(x1)−
∏n−1
j=0
ũj(x2)
)
+
∏n−1
j=0
ũj(x2) ((ρnϕn)(x1)−(ρnϕn)(x2))
∣∣∣
|x1 − x2|µ
≤
≤ sup
x1<x2
∣∣∣(ρnϕn)(x1) (ũ(x1)−ũ(x2))
∏n−2
j=0
ũj+1(x1)+(ũn−1(x1)−ũn−1(x2))
∏n−2
j=0
ũj(x2)
∣∣∣
|x1 − x2|µ
+
+ sup
x1<x2
∣∣∣(ρnϕn)(x1)
∑n−3
j=0
(
(ũj+1(x1)−ũj+1(x2))
∏n−3
i=j
ũi+2(x1)
∏j
k=0
ũk(x2)
)∣∣∣
|x1 − x2|µ
+
+ sup
x1<x2
|(ρnϕn)(x1)− (ρnϕn)(x2)|
|βn(x1)− βn(x2)|µ
sup
x1<x2
∣∣∣∣∣∣
n−1∏
j=0
ũj(x2)
|βn(x1)− βn(x2)|µ
|x1 − x2|µ
∣∣∣∣∣∣ .
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CONDITIONS OF INVERTIBILITY FOR FUNCTIONAL OPERATORS WITH SHIFT . . . 1565
Here, we used the identities
n−1∏
j=0
ũj(x1)−
n−1∏
j=0
ũj(x2) =
= (ũ(x1)− ũ(x2))
n−2∏
j=0
ũj+1(x1) + (ũn−1(x1)− ũn−1(x2))
n−2∏
j=0
ũj(x2)+
+ũ(x2)
n−2∏
j=0
ũj+1(x1)− ũn−1(x1)
n−2∏
j=0
ũj(x2)
and
ũ(x2)
n−2∏
j=0
ũj+1(x1)− ũn−1(x1))
n−2∏
j=0
ũj(x2) =
=
n−3∑
j=0
(ũj+1(x1)− ũj+1(x2))
n−3∏
i=j
ũi+2(x1)
j∏
k=0
ũk(x2)
.
Finally, taking into account the identities
ρµ;n(x1)
ρµ;j+2(x1)
=
n−3∏
i=j
ρµ;i+3(x1)
ρµ;i+2(x1)
,
∣∣∣∣βj+1(x1)− βj+1(x2)
x1 − x2
∣∣∣∣µ =
j∏
k=0
βµk (x1, x2),
we get ∥∥∥∥∥∥ρ
n−1∏
j=0
uj
Bn
βϕ
∥∥∥∥∥∥
µ
≤‖ũ‖µ sup
x1<x2
∣∣∣∣∣∣ρn(x1)ϕn(x1)
ρµ;n(x1)
ρµ;1(x1)
n−2∏
j=0
ũj+1(x1)
ρµ;n(x1)
ρµ;1(x1)
∣∣∣∣∣∣+
+ sup
x1<x2
∣∣∣∣∣∣ ρn(x1)ϕn(x1)
(ũn−1(x1)−ũn−1(x2))
(βn−1(x1)−βn−1(x2))µ
n−2∏
j=0
ũj(x2)
(βn−1(x1)−βn−1(x2))µ
|x1 − x2|µ
∣∣∣∣∣∣+
+ sup
x1<x2
∣∣∣∣∣∣ρn(x1)ϕn(x1)
ρµ;n(x1)
n−3∑
j=0
(
(ũj+1(x1)− ũj+1(x2))
(βj+1(x1)− βj+1(x2))µ
(βn−1(x1)−βn−1(x2))µ
|x1 − x2|µ
×
×ρµ;j+2(x1)
n−3∏
i=j
ũi+2(x1)
ρµ;n(x1)
ρµ;j+2(x1)
j∏
k=0
ũk(x2)
)∣∣∣∣∣∣+
+
∥∥∥∥(ρϕ)(x1)−(ρϕ)(x2))
(βn(x1)−βn(x2))
µ
∥∥∥∥
C
sup
x1<x2
∣∣∣∣∣∣
n−1∏
j=0
ũj(x2)
(βn(x1)− βn(x2))
µ
(x1−x2)µ
∣∣∣∣∣∣ ≤
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
1566 G. TARASENKO, O. KARELIN
≤ ‖ũ‖µ
∥∥∥∥ρnϕnρµ;n
∥∥∥∥
C
sup
x1<x2
∣∣∣∣∣∣ρµ;1(x1)
n−2∏
j=0
ũj+1(x1)
ρµ;j+2(x1)
ρµ;j+1(x1)
∣∣∣∣∣∣+
+ ‖ũ‖µ ‖ρnϕn‖C sup
x1<x2
∣∣∣∣∣∣
n−2∏
j=0
ũj(x2)β
µ
j (x1, x2)
∣∣∣∣∣∣+
+ ‖ũ‖µ
∥∥∥∥ρnϕnρµ;n
∥∥∥∥
C
sup
x1<x2
∣∣∣∣∣∣
n−3∑
j=0
ρµ;j+2
n−3∏
i=j
ũi+2(x1)
ρµ;i+3(x1)
ρµ;i+2(x1)
j∏
k=0
ũk(x2)β
µ
k (x1, x2)
∣∣∣∣∣∣+
+ ‖ρϕ‖µ sup
x1<x2
∣∣∣∣∣∣
n−1∏
j=0
ũj(x2)β
µ
j (x1, x2)
∣∣∣∣∣∣ .
By (2) of Lemma 2, the inequality∣∣∣∣ũl+1(x1)
ρµ;l(x1)
ρµ;l+1(x1)
∣∣∣∣ ≤ q < 1 (8)
holds for every fixed x1 with a possible exception of n0 values of l.
From Lemma 1 it follows that only n0 values of βl(x1) may be outside of the set [0, ε]
⋃
[1−ε, 1],
where inequality (8) holds. Here the number n0 is from Lemma 1.
By (3) of Lemma 3, the inequality∣∣ũl(x2)βµl (x1, x2)
∣∣ ≤ q < 1 (9)
holds for all fixed x1, x2, x1 < x2 with a possible exception of 2n0 values of l. In fact, under
x1 < x2, we have only two failures of the condition βl(x1), βl(x2) ∈ [0, ε]
⋃
[1− ε, 1], and βl(x1) ∈
∈ [0, ε], βl(x2) ∈ [1 − ε, 1], where inequality (9) holds. It means that the failures may occur when
βl(x1) ∈ (ε, 1− ε), or βl(x2) ∈ (ε, 1− ε). According to Lemma 1, there are no more than n0 values
of βl(x1) in (ε, 1− ε) and there are no more than n0 values of βl(x2) in (ε, 1− ε).
We have∥∥∥∥∥∥ρ
n−1∏
j=0
uj
Bn
βϕ
∥∥∥∥∥∥
µ
≤
‖ũ‖µ 2µ ‖ρµ,µ‖C q
n−1−m0Mm0 + ‖ũ‖µ q
n−1−2m0M2m0 +
+‖ũ‖µ2µ ‖ρµ,µ‖C
n−3∑
j=0
qn−2−j−m0Mm0qj+1−2m0M2m0 +qn−2m0M2m0
‖ϕ‖H0
µ(J,ρ)
, (10)
where the constant M is given by
M = max
(∥∥∥∥ũ(x)
ρµ;1(x)
ρµµ(x)
∥∥∥∥
C
, ‖ũ(x2)β
µ(x1, x2)‖C
)
.
Here, inequalities (5), (8) and (9) were used. The factor in the brackets of (10) tends to zero when
n→∞.
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CONDITIONS OF INVERTIBILITY FOR FUNCTIONAL OPERATORS WITH SHIFT . . . 1567
Thus, there exists n such that∥∥∥∥∥∥
n−1∏
j=0
uj
Bn
βϕ
∥∥∥∥∥∥
H0
µ(J,ρ)
< ‖ϕ‖H0
µ(J,ρ)
,
which means that the operator U = I − uBβ is invertible on the space H0
µ(J, ρ).
Theorem 2 is proved.
Theorem 3. The operator A acting on Banach space H0
µ(J, ρ), is invertible if the following
condition holds:
σβ[a(x), b(x)] 6= 0, x ∈ J,
where the function σβ is defined by
σβ[a(x), b(x)] =
a(x), when |a(i)| > [β′(i)]−µi+µ |b(i)|, i = 0, 1,
b(x), when |a(i)| < [β′(i)]−µi+µ
∣∣b(i)∣∣, i = 0, 1,
0, otherwise.
Proof. We consider only the case
a(x) 6= 0, x ∈ J,
|a(i)| > |β′(i)|−µi+µ|b(i)|, i = 0, 1.
(11)
The case
b(x) 6= 0, x ∈ J,
|a(i)| < |β′(i)|−µi+µ|b(i)|, i = 0, 1,
can be considered analogously.
Recall that the operators aI−bBβ and U = I−uBβ,where u = b/a, are invertible simultaneously
on H0
µ(J, ρ).
Thus, there exists n such that∥∥∥∥∥∥
n−1∏
j=0
uj
Bn
βϕ
∥∥∥∥∥∥
H0
µ(J,ρ)
< ||ϕ||H0
µ(J,ρ)
,
which means that operator U = I − uBβ is invertible in space H0
µ(J, ρ).
Theorem 3 is proved.
Now, we will focus on the application of the above results to a modeling of systems with renewable
resources. For the study of such systems, cyclic models based on functional operators with shift were
elaborated in [4]. The Balance relation describing the state of cyclic equilibrium is the equation
aIν − bBβν = g for the unknown distribution function ν ∈ H0
µ(J, ρ).
In [5], a reproductive summand has been added for a more accurate description of the process of
reproduction; this term has been expressed by integrals with degenerate kernels.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
1568 G. TARASENKO, O. KARELIN
If we model the behavior of a system with two resources, taking into account the interaction
between them, by integrals with degenerate kernels and follow the principles of modeling from [4],
we will obtain two equations with two unknowns, ν1 and ν2:
a1(x)ν1(x)− b1(x)ν1[β1(x)] + Σ1(x) + Γ1(x) = g1(x), (12)
a2(x)ν2(x)− b2(x)ν2[β2(x)] + Σ2(x) + Γ2(x) = g2(x), (13)
where ν1 and ν2 are the densities of the distributions of the first and second resources by their
respective individual parameters (such as weight or length), and
Σ1(x) =
m1∑
i=1
∫
J
ζ1,i(x)ξ1,i(t)ν1(t)dt, Γ1(x) =
n1∑
i=1
∫
J
%1,i(x)δ1,i(t)ν2(t)dt,
Σ2(x) =
m2∑
i=1
∫
J
ζ2,i(x)ξ2,i(t)ν2(t)dt, Γ2(x) =
n2∑
i=1
∫
J
%2,i(x)δ2,i(t)ν1(t)dt,
are the terms of reproduction and interaction process respectively.
We consider our model on the space H0
µ(J, ρ). Suppose that for
A1 = a1(x)ν1(x)− b1(x)ν1[β1(x)], A2 = a2(x)ν2(x)− b2(x)ν2[β2(x)]
on H0
µ(J, ρ) the invertibility conditions of Theorem 3 hold. Thus, the inverse operators A−11 and A−12
for A1 and A2 exist. We apply these inverse operators to the left-hand side of equations (12), (13)
and obtain Fredholm equations of the second type with degenerate kernels. Using a known method
of solving such equations, we can find densities ν1 and ν2 of the cyclic equilibrium of the system.
1. Karlovich Yu. I., Kravchenko V. G. Singular integral equations with non-Carleman shift on an open contour //
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3. Kravchenko V. G., Litvinchuk G. S. Introduction to the theory of singular integral operators with shift. – Dordrecht
etc.: Kluwer Acad. Publ., 1994. – 288 p.
4. Tarasenko A., Karelin A., Lechuga G. P., Hernández M. G. Modelling systems with renewable resources based on
functional operators with shift // Appl. Math. and Comput. – 2010. – 216, № 7. – P. 1938 – 1944.
5. Karelin O., Tarasenko A., Hernández M. G. Application of functional operators with shift to the study of renewable
systems when the reproductive processed is describedby integrals with degenerate kernels // Appl. Math. – 2013. –
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Received 13.10.14
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