Volterra functional integral equation of the first kind with nonlinear right-hand side and variable limits of integration
We prove a theorem on the existence and uniqueness of a solution of a Volterra functional integral equation of the first kind with nonlinear right-hand side and nonlinear deviation. We use the method of successive approximations combined with the method of contracting mappings.
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| author | Artykova, J. A. Yuldashev, T. K. Артикова, Й. А. Юлдашев, Т. К. |
| author_facet | Artykova, J. A. Yuldashev, T. K. Артикова, Й. А. Юлдашев, Т. К. |
| author_sort | Artykova, J. A. |
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| datestamp_date | 2020-03-18T19:56:51Z |
| description | We prove a theorem on the existence and uniqueness of a solution of a Volterra functional integral equation of the first kind with nonlinear right-hand side and nonlinear deviation. We use the method of successive approximations combined with the method of contracting mappings. |
| first_indexed | 2026-03-24T02:44:15Z |
| format | Article |
| fulltext |
UDC 517.91
T. K. Yuldashev (Kyrgyz Law Acad., Osh),
J. A. Artykova (Osh Univ., Kyrgyzstan)
VOLTERRA FUNCTIONAL INTEGRAL EQUATION
OF THE FIRST KIND WITH NONLINEAR RIGHT-HAND SIDE
AND VARIABLE LIMITS OF INTEGRATION
ФУНКЦIОНАЛЬНО-IНТЕГРАЛЬНЕ РIВНЯННЯ ВОЛЬТЕРРИ
ПЕРШОГО РОДУ З НЕЛIНIЙНОЮ ПРАВОЮ ЧАСТИНОЮ
I ЗМIННИМИ МЕЖАМИ IНТЕГРУВАННЯ
We prove a theorem on the existence and uniqueness of a solution on a Volterra functional integral equation
of the first kind with nonlinear right-hand side and nonlinear deviation. We use the method of successive
approximations in combination with the method of compressing mapping.
Доведено теорему про iснування та єдинiсть розв’язку функцiонально-iнтегрального рiвняння Воль-
терри першого роду з нелiнiйним вiдхиленням. При цьому використано метод послiдовних набли-
жень у поєднаннi з методом стискаючих вiдображень.
In this paper, we consider a Volterra functional integral equation of the form
β(t)∫
α(t)
K(t, s)u
[
u(s)
]
ds = f
(
t, u
[
δ(t, u(t))
])
, t ∈ T1, (1)
with initial value condition
u(t) = g(t), t ∈ E0 ≡ [t0; t1], (2)
where K(t, s) ∈ C(T 2
0 ), 0 ≤ K(t) ≡ K(t, t), f(t, u) ∈ C(T1 × X), T 2
0 ≡ T0 × T0,
T1 ≡ [t1;T ], T0 ≡ [t0;T ], 0 < t0 < T < ∞, t0 < t1, X is a bounded closed set in
R ≡ (−∞;∞), t0 ≤ α(t) < β(t) ≤ T, α(t), β(t) ∈ C(T0), δ(t, u) ∈ C(T1 × X),
t0 ≤ δ(t, u) ≤ t, g(t) ∈ C(E0).
We study the existence and uniqueness of a solution of the Volterra functional integral
equation (1) with initial value condition (2) on the segment T1. Here, we use the method
of successive approximations in combination with the method of compressing mapping.
We note that the Volterra integral equation of the first kind, in which the right-
hand side is presented by f(t)-known continuous function studied by many authors (see
bibliography in [1]).
The Volterra functional integral equations with such right-hand side were considered
in our works [2 – 4]. The present paper is the further development of these works.
We rewrite the Volterra functional integral equation (1) in the following form:
β(t)∫
α(t)
K(t, s)u(s)ds =
=
β(t)∫
α(t)
K(t, s)
[
u(s)− u[u(s)]
]
ds + f
(
t, u
[
δ(t, u(t))
])
, t ∈ T1,
c© T. K. YULDASHEV, J. A. ARTYKOVA, 2006
ISSN 1027-3190. Укр. мат. журн., 2006, т. 58, № 9 1285
1286 T. K. YULDASHEV, J. A. ARTYKOVA
or
u(t) +
t∫
t0
K(t, s)u(s)ds =
= u(t) +
β(t)∫
α(t)
K(t, s)
[
u(s)− u[u(s)]
]
ds +
α(t)∫
t0
K(t, s)u(s)ds+
+
t∫
β(t)
K(t, s)u(s)ds + f
(
t, u
[
δ(t, u(t))
])
, t ∈ T1. (3)
We change equation (3) as follows:
u(t) +
t∫
t0
K(s)u(s)ds =
= −
t∫
t0
[
K(t, s)−K(s)
]
u(s)ds + f0(t, u), t ∈ T1,
where we denote the right-hand side of (2) by f0(t, u), i.e.,
f0(t, u) = u(t) +
β(t)∫
α(t)
K(t, s)
[
u(s)− u[u(s)]
]
ds+
+
α(t)∫
t0
K(t, s)u(s)ds +
t∫
β(t)
K(t, s)u(s)ds + f
(
t, u
[
δ(t, u(t))
])
, t ∈ T1. (4)
Hence, using the resolvent method for
[
−K(s)
]
, we obtain
u(t) = −
t∫
t0
[
K(t, s)−K(s)
]
u(s)ds + f0(t, u)+
+
t∫
t0
K(s) exp
{
−ϕ(t, s)
}{
−f0(s, u) +
s∫
t0
[
K(s, τ)−K(τ)
]
u(τ)dτ
}
ds, t ∈ T1,
(5)
where ϕ(t, s) =
∫ t
s
K(τ)dτ, ϕ(t, t0) = ϕ(t), ϕ(t1) 6= 0.
Applying Direchlet’s formulation to (5), we derive
u(t) =
t∫
t0
H(t, s)u(s)ds + f0
(
(t, u(t)
)
exp
{
−ϕ(t)
}
+
+
t∫
t0
K(s) exp
{
−ϕ(t, s)
}[
f0
(
t, u(t)
)
− f0
(
s, u(s)
)]
ds, t ∈ T1, (6)
ISSN 1027-3190. Укр. мат. журн., 2006, т. 58, № 9
VOLTERRA FUNCTIONAL INTEGRAL EQUATION OF THE FIRST KIND ... 1287
where
H(t, s) ≡ −
[
K(t, s)−K(s)
]
exp
{
−ϕ(t, s)
}
−
−
t∫
s
K(τ) exp
{
−ϕ(t, τ)
}[
K(t, s)−K(τ, s)
]
dτ
and f0(t, u) is defined from (4).
The Volterra functional integral equations (1) and (6) are equivalent.
Theorem. Assume that the following conditions are satisfied:
1)
∣∣K(τ, s)−K(η, t)
∣∣ ≤ L1(s)ϕ(τ, η), 0 ≤ L1(s);
2) f(t, u) ∈ Bnd(M) ∩ Lip(L3|u), 0 ≤ M, L3 = const;
3)
∣∣ϕ(t, s)
∣∣ ≤ L4|t− s|, 0 ≤ L4 = const;
4) δ(t, u) ∈ Lip(L5|u), 0 ≥ L5 = const;
5) for all t ∈ T1 there holds ρ = ρ(t) < 1,
ρ =
t∫
t0
∥∥L1(s)
∥∥ds +
t∫
t0
∥∥L1(s)
∥∥ds+
+
(
1 + ∆1 + L3 + (2 + L2L4)∆2 + L2L3L4L5
)
×
×
exp
{
−ϕ(t)
}
+ 2
t∫
t0
∥∥K(s)
∥∥ exp
{
−ϕ(t, s)
}
ds
,
∆1 =
α(t)∫
t0
∥∥K(t, s)
∥∥ds +
t∫
β(t)
∥∥K(t, s)
∥∥ds,
∆2 =
β(t)∫
α(t)
∥∥K(t, s)
∥∥ds, 0 ≤ L2 = const.
Then the Volterra functional integral equation (1) with initial value condition (2) has
the unique solution on T1.
The theorem is proved by the method of successive approximations, which is defined
by the following relations:
u0(t) =
g(t), t ∈ E0,
f(t, 0) exp
{
−ϕ(t)
}
+
+
t∫
t0
K(s) exp
{
−ϕ(t, s)
}[
f(t, 0)− f(s, 0)
]
ds, t ∈ T1,
ISSN 1027-3190. Укр. мат. журн., 2006, т. 58, № 9
1288 T. K. YULDASHEV, J. A. ARTYKOVA
uk+1(t) =
g(t), t ∈ E0,
t∫
t0
H(t, s)uk(s)ds + f0
(
t, uk(t)
)
exp
{
−ϕ(t)
}
+
+
t∫
t0
K(s) exp
{
−ϕ(t, s)
}[
f0
(
t, uk(t)
)
− f0
(
s, uk(s)
)]
ds,
k = 0, 1, . . . , t ∈ T1.
1. Асанов А. Устойчивость решений систем линейных интегральных уравнений Вольтерра вто-
рого рода на полуинтервале // Исслед. по интегро-дифференц. уравнениям. – 1989. – Вып. 22.
– С. 123 – 129.
2. Юлдашев Т. К., Артыкова Ж. А. Интегральное уравнение Вольтерра первого рода с нелиней-
ной правой частью // Складнi системи i процеси. – 2005. – № 1, 2. – С. 3 – 5.
3. Юлдашев Т. К., Артыкова Ж. А. Случайные интегральные уравнения Вольтерра первого
рода с нелинейной правой частью // Материалы V междунар. Ферганской конф. „Предельные
теоремы теории вероятностей и их приложения” (Фергана, 10 – 12 мая 2005 г.). – Ташкент:
Ин-т математики АН Узбекистана, 2005. – С. 204 – 206.
4. Юлдашев Т. К., Артыкова Ж. А. Интегральное уравнение Вольтерра первого рода с нели-
нейной правой частью и сложным отклонением // Тез. междунар. семинара „Геометрия в
Одессе-2005. Дифференц. геометрия и ее прил.” (Одесса, 23 – 29 мая 2005 г.). – Одесса, 2005.
– С. 112 – 113.
Received 30.03.2005,
after revision — 10.05.2006
ISSN 1027-3190. Укр. мат. журн., 2006, т. 58, № 9
|
| id | umjimathkievua-article-3530 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:44:15Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/cc/2fe85d0a4cb1a865a1273fcc3eb349cc.pdf |
| spelling | umjimathkievua-article-35302020-03-18T19:56:51Z Volterra functional integral equation of the first kind with nonlinear right-hand side and variable limits of integration Функціонально-інтегральне рівняння Вольтерри першого роду з нелінійною правою частиною i змінними межами iнтегрування Artykova, J. A. Yuldashev, T. K. Артикова, Й. А. Юлдашев, Т. К. We prove a theorem on the existence and uniqueness of a solution of a Volterra functional integral equation of the first kind with nonlinear right-hand side and nonlinear deviation. We use the method of successive approximations combined with the method of contracting mappings. Доведено теорему про існування та єдиність розв'язку функціонально-інтегрального рівняння Вольтерри першого роду з нелінійним відхиленням. При цьому використано метод послідовних наближень у поєднанні з методом стискаючих відображень. Institute of Mathematics, NAS of Ukraine 2006-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3530 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 9 (2006); 1285–1288 Український математичний журнал; Том 58 № 9 (2006); 1285–1288 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3530/3797 https://umj.imath.kiev.ua/index.php/umj/article/view/3530/3798 Copyright (c) 2006 Artykova J. A.; Yuldashev T. K. |
| spellingShingle | Artykova, J. A. Yuldashev, T. K. Артикова, Й. А. Юлдашев, Т. К. Volterra functional integral equation of the first kind with nonlinear right-hand side and variable limits of integration |
| title | Volterra functional integral equation of the first kind with nonlinear right-hand side and variable limits of integration |
| title_alt | Функціонально-інтегральне рівняння Вольтерри першого роду з нелінійною правою частиною i змінними межами iнтегрування |
| title_full | Volterra functional integral equation of the first kind with nonlinear right-hand side and variable limits of integration |
| title_fullStr | Volterra functional integral equation of the first kind with nonlinear right-hand side and variable limits of integration |
| title_full_unstemmed | Volterra functional integral equation of the first kind with nonlinear right-hand side and variable limits of integration |
| title_short | Volterra functional integral equation of the first kind with nonlinear right-hand side and variable limits of integration |
| title_sort | volterra functional integral equation of the first kind with nonlinear right-hand side and variable limits of integration |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3530 |
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