# On Newton-Kantorovich Method for Solving the Nonlinear Operator Equation

Authors: Hameed Husam Hameed; Z. K. Eshkuvatov; Anvarjon Ahmedov; N. M. A. Nik Long
Journal: Abstract and Applied Analysis
 (2015)

---

## Abstract

We develop the Newton-Kantorovich  method to solve the system of2
×
2 nonlinear Volterra integral equations where the unknown function is in logarithmic form. A new majorant function is introduced which  leads to the increment of the convergence interval. The existence and uniqueness of approximate solution are proved and a numerical example is provided to show the validation of the method.


---

## Body

## 1. Introduction

Nonlinear phenomenon appears in many scientific areas such as physics, fluid mechanics, population models, chemical kinetics, economic systems, and medicine and can be modeled by system of nonlinear integral equations. The difficulty lies in finding the exact solution for such system. Alternatively, the approximate or numerical solutions can be sought. One of the well known approximate method is Newton-Kantorovich method which reduces the nonlinear into sequence of linear integral equations. The the approximate solution is then obtained by processing the convergent sequence. In 1939, Kantorovich [1] presented an iterative method for functional equation in Banach space and derived the convergence theorem for Newton method. In 1948, Kantorovich [2] proved a semilocal convergence theorem for Newton method in Banach space, later known as the Newton-Kantorovich method. Uko and Argyros [3] proved a weak Kantorovich-type theorem which gives the same conclusion under the weaker conditions. Shen and Li [4] have established the Kantorovich-type convergence criterion for inexact Newton methods, assuming that the first derivative of an operator satisfies the Lipschitz condition. Argyros [5] provided a sufficient condition for the semilocal convergence of Newton’s method to a locally unique solution of a nonlinear operator equation. Saberi-Nadjafi and Heidari [6] introduced a combination of the Newton-Kantorovich and quadrature methods to solve the nonlinear integral equation of Urysohn type in the systematic procedure. Ezquerro et al. [7] studied the nonlinear integral equation of mixed Hammerstein type using Newton-Kantorovich method with majorant principle. Ezquerro et al. [8] provided the semilocal convergence of Newton method in Banach space under a modification of the classic conditions of Kantorovich. There are many methods of solving the system of nonlinear integral equations, for example, product integration method [9], Adomian method [10], RBF network method [11], biorthogonal system method [12], Chebyshev wavelets method [13], analytical method [14], reproducing kernel method [15], step method [16], and single term Wlash series [17]. In 2003, Boikov and Tynda [18] implemented the Newton-Kantorovich method to the following system: (1)
x
(
t
)
-
∫
y
(
t
)
t
‍
h
(
t
,
τ
)
g
(
τ
)
x
(
τ
)
d
τ
=
0
,
∫
y
(
t
)
t
‍
k
(
t
,
τ
)
[
1
-
g
(
τ
)
]
x
(
τ
)
d
τ
=
f
(
t
)
,where 0
<
t
0
≤
t
≤
T, y
(
t
)
<
t, and the functions h
(
t
,
τ
), k
(
t
,
τ
)
∈
C
[
t
0
,
T
]
×
[
t
0
,
T
], f
(
t
), g
(
t
)
∈
C
[
t
0
,
T
], and (
0
<
g
(
t
)
<
1
). In 2010, Eshkuvatov et al. [19] used the Newton-Kantorovich hypothesis to solve the system of nonlinear Volterra integral equation of the form(2)
x
(
t
)
-
∫
y
(
t
)
t
h
(
t
,
τ
)
x
2
(
τ
)
d
τ
=
0
,
∫
y
(
t
)
t
‍
k
(
t
,
τ
)
x
2
(
τ
)
d
τ
=
f
(
t
)
,where x
(
t
) and y
(
t
) are unknown functions defined on [
t
0
,
∞
), t
0
>
0, and h
(
t
,
τ
)
,
k
(
t
,
τ
)
∈
C
[
t
0
,
∞
]
×
[
t
0
,
∞
], f
(
t
)
∈
C
[
t
0
,
∞
]. In 2010, Eshkuvatov et al. [20] developed the modified Newton-Kantorovich to obtain an approximate solution of system with the form (3)
x
(
t
)
-
∫
y
(
t
)
t
‍
H
(
t
,
τ
)
x
n
(
τ
)
d
τ
=
0
,
∫
y
(
t
)
t
‍
K
(
t
,
τ
)
x
n
(
τ
)
d
τ
=
f
(
t
)
,where 0
<
t
0
≤
t
≤
T, y
(
t
)
<
t, and the functions H
(
t
,
τ
)
,
K
(
t
,
τ
)
∈
C
[
t
0
,
∞
]
×
[
t
0
,
∞
], f
(
t
)
∈
C
[
t
0
,
∞
], and the unknown functions x
(
t
)
∈
C
[
t
0
,
∞
], y
(
t
)
∈
C
[
t
0
,
∞
]
1, y
(
t
)
<
t.In this paper, we consider the systems of nonlinear integral equation of the form(4)
x
(
t
)
-
∫
y
t
t
‍
h
t
,
τ
log
⁡
x
τ
d
τ
=
g
(
t
)
,
∫
y
t
t
‍
k
t
,
τ
log
⁡
x
τ
d
τ
=
f
(
t
)
,where 0
<
t
0
≤
t
≤
T, y
(
t
)
<
t, x
(
t
)
≠
0, h
(
t
,
τ
)
,
h
τ
(
t
,
τ
)
,
k
(
t
,
τ
)
,
k
τ
(
t
,
τ
)
∈
C
(
D
) and the unknown functions x
(
t
)
∈
C
[
t
0
,
T
], y
(
t
)
∈
C
1
[
t
0
,
T
] to be determined, and D
=
[
t
0
,
T
]
×
[
t
0
,
T
].The paper is organized as follows, in Section2, Newton-Kantorovich method for the system of integral equations (4) is presented. Section 3 deals with mixed method followed by discretizations. In Section 4, the rate of convergence of the method is investigated. Lastly, Section 5 demonstrates the numerical example to verify the validity and accuracy of the proposed method, followed by the conclusion in Section 6.


## 2. Newton-Kantorovich Method for the System

Let us rewrite the system of nonlinear Volterra integral equation (4) in the operator form(5)
P
(
X
)
=
(
P
1
(
X
)
,
P
2
(
X
)
)
=
0
,where X
=
(
x
(
t
)
,
y
(
t
)
) and (6)
P
1
X
=
x
(
t
)
-
∫
y
t
t
‍
h
t
,
τ
log
⁡
x
τ
d
τ
-
g
(
t
)
,
P
2
X
=
∫
y
t
t
‍
k
t
,
τ
log
⁡
x
τ
d
τ
-
f
(
t
)
.To solve (5) we use initial iteration of Newton-Kantorovich method which is of the form (7)
P
′
X
0
X
-
X
0
+
P
X
0
=
0
,where X
0
=
(
x
0
(
t
)
,
y
0
(
t
)
) is the initial guess and x
0
(
t
) and y
0
(
t
) can be any continuous functions provided that t
0
<
y
(
t
)
<
t and x
(
t
)
≠
0.The Frechet derivative ofP
(
X
) at the point X
0 is defined as (8)
P
′
X
0
X
=
lim
s
→
0
1
s
P
1
X
0
+
s
X
-
P
1
X
,
vvvvvv
lim
s
→
0
1
s
P
2
X
0
+
s
X
-
P
2
X
=
lim
s
→
0
1
s
P
1
x
0
+
s
x
,
y
0
+
s
y
-
P
1
x
0
,
y
0
,
vvvvvv
lim
s
→
0
1
s
P
2
x
0
+
s
x
,
y
0
+
s
y
-
P
2
x
0
,
y
0
=
∑
lim
s
→
0
∂
2
P
1
∂
y
2
∂
P
1
(
x
0
,
y
0
)
∂
x
s
x
+
∂
P
1
(
x
0
,
y
0
)
∂
y
s
y
vvvvvvvvvv
+
1
2
∂
2
P
1
∂
x
2
(
x
0
+
θ
s
x
,
y
0
+
δ
s
y
)
s
2
x
2
vvvvvvvvivvvvvv
+
2
∂
2
P
1
∂
x
∂
y
x
0
+
θ
s
x
,
y
0
+
δ
s
y
s
2
x
y
vvvvvvvvvivvvvv
+
∂
2
P
1
∂
y
2
x
0
+
θ
s
x
,
y
0
+
δ
s
y
s
y
2
,
vvvvv
lim
s
→
0
1
s
∂
2
P
1
∂
y
2
∂
P
2
∂
x
x
0
,
y
0
s
x
+
∂
P
2
∂
y
x
0
,
y
0
s
y
vvvvvvvvvv
+
1
2
∂
2
P
2
∂
x
2
x
0
+
θ
s
x
,
y
0
+
δ
s
y
s
2
x
2
vvvvvvivvvvvvvv
+
2
∂
2
P
2
∂
x
∂
y
x
0
+
θ
s
x
,
y
0
+
δ
s
y
s
2
x
y
vvvvvvvivvvvvvv
+
∂
2
P
2
∂
y
2
x
0
+
θ
s
x
,
y
0
+
δ
s
y
s
y
2
∑

=
∂
P
1
(
x
0
,
y
0
)
∂
x
x
+
∂
P
1
(
x
0
,
y
0
)
∂
y
y
,
000000
∂
P
2
x
0
,
y
0
∂
x
x
+
∂
P
2
x
0
,
y
0
∂
y
y
.Hence, (9)
P
′
(
X
0
)
X
=
∂
P
1
∂
x
x
0
,
y
0
∂
P
1
∂
y
x
0
,
y
0
∂
P
2
∂
x
x
0
,
y
0
∂
P
2
∂
y
x
0
,
y
0
x
y
.From (7) and (9) it follows that (10)
∂
P
1
∂
x
x
0
,
y
0
Δ
x
t
+
∂
P
1
∂
y
x
0
,
y
0
Δ
y
t
=
-
P
1
x
0
t
,
y
0
t
,
∂
P
2
∂
x
(
x
0
,
y
0
)
Δ
x
t
+
∂
P
2
∂
y
(
x
0
,
y
0
)
Δ
y
(
t
)
=
-
P
2
x
0
t
,
y
0
t
,where Δ
x
(
t
)
=
x
1
(
t
)
-
x
0
(
t
), Δ
y
(
t
)
=
y
1
(
t
)
-
y
0
(
t
), and (
x
0
(
t
)
,
y
0
(
t
)
) is the initial given functions. To solve (10) with respect to Δ
x and Δ
y we need to compute all partial derivatives: (11)
∂
P
1
∂
x
(
x
0
,
y
0
)
=
lim
s
→
0
1
s
P
1
x
0
+
s
x
,
y
0
-
P
1
x
0
,
y
0
=
lim
s
→
0
1
s
∫
y
0
(
t
)
t
‍
h
(
t
,
τ
)
s
x
(
t
)
vvvvvvvv
-
∫
y
0
(
t
)
t
‍
h
(
t
,
τ
)
log
⁡
x
0
τ
+
s
x
τ
vvvvvvvvvvvvvvvvvvvvv
-
log
⁡
x
0
τ
d
τ
∫
y
0
t
t
‍
=
x
(
t
)
-
∫
y
0
(
t
)
t
‍
h
(
t
,
τ
)
x
(
τ
)
x
0
(
τ
)
d
τ
,
∂
P
1
∂
y
(
x
0
,
y
0
)
=
lim
s
→
0
1
s
P
1
(
x
0
,
y
0
+
s
y
)
-
P
1
(
x
0
,
y
0
)
=
lim
s
→
0
1
s
∫
y
0
t
y
0
t
+
s
y
t
‍
h
t
,
τ
log
⁡
x
0
τ
d
τ
=
h
t
,
y
0
t
log
⁡
x
0
y
0
t
y
(
t
)
,and in the same manner we obtain (12)
∂
P
2
∂
x
(
x
0
,
y
0
)
=
∫
y
0
(
t
)
t
‍
k
(
t
,
τ
)
x
(
τ
)
x
0
(
τ
)
d
τ
,
∂
P
2
∂
y
(
x
0
,
y
0
)
=
-
k
t
,
y
0
t
log
⁡
x
0
y
0
t
y
t
.So that from (10)–(12) it follows that (13)
Δ
x
t
-
∫
y
0
t
t
‍
h
t
,
τ
Δ
x
(
τ
)
x
0
(
τ
)
d
τ
+
h
t
,
y
0
t
log
⁡
x
0
y
0
t
Δ
y
(
t
)
=
∫
y
0
t
t
‍
h
t
,
τ
log
⁡
x
0
τ
d
τ
-
x
0
t
+
g
t
,
∫
y
0
t
t
‍
k
(
t
,
τ
)
Δ
x
τ
x
0
τ
d
τ
-
k
t
,
y
0
t
log
⁡
x
0
y
0
t
Δ
y
t
=
-
∫
y
0
t
t
‍
k
t
,
τ
log
⁡
x
0
τ
d
τ
+
f
t
.Equation (13) is a linear, and, by solving it for Δ
x and Δ
y, we obtain (
x
1
(
t
)
,
y
1
(
t
)
). By continuing this process, a sequence of approximate solution x
m
(
t
)
,
y
m
(
t
) can be evaluated from (14)
P
′
(
X
0
)
Δ
X
m
+
P
(
X
m
)
=
0
,which is equivalent to the system (15)
Δ
x
m
t
-
∫
y
0
t
t
‍
h
t
,
τ
Δ
x
m
τ
x
0
τ
d
τ
+
h
t
,
y
0
t
log
⁡
x
0
y
0
t
Δ
y
m
(
t
)
=
∫
y
0
t
t
‍
h
t
,
τ
log
⁡
x
0
τ
d
τ
-
x
0
(
t
)
+
g
(
t
)
,
∫
y
0
t
t
‍
k
t
,
τ
Δ
x
m
τ
x
0
τ
d
τ
-
k
t
,
y
0
t
log
⁡
x
0
y
0
t
Δ
y
m
t
=
-
∫
y
0
t
t
‍
k
t
,
τ
log
⁡
x
0
τ
d
τ
+
f
(
t
)
,where Δ
x
m
(
t
)
=
x
m
(
t
)
-
x
m
-
1
(
t
) and Δ
y
m
(
t
)
=
y
m
(
t
)
-
y
m
-
1
(
t
), m
=
1,2
,
3
,
….Thus, one should solve a system of two linear Volterra integral equations to find each successive approximation. Let us eliminateΔ
y
(
t
) from the system (13) by finding the expression of Δ
y
(
t
) from the first equation of this system and substitute it in the second equation to yield (16)
Δ
y
t
=
1
H
(
t
)
∫
y
0
t
t
‍
h
t
,
τ
Δ
x
τ
x
0
τ
+
log
⁡
x
0
τ
d
τ
vvvvvvvvvvvvv
-
Δ
x
t
+
x
0
t
-
g
t

∫
y
0
t
t
‍
,
G
(
t
)
∫
y
0
t
t
‍
h
t
,
τ
Δ
x
τ
x
0
τ
+
log
⁡
x
0
τ
d
τ
vvvvvv
-
Δ
x
t
+
x
0
t
-
g
t

∫
y
0
t
t
‍
=
∫
y
0
t
t
‍
k
t
,
τ
Δ
x
τ
x
0
τ
d
τ
-
∫
y
0
t
t
‍
k
t
,
τ
log
⁡
x
0
τ
d
τ
+
f
t
,where G
(
t
)
=
k
(
t
,
y
0
(
t
)
)
/
h
(
t
,
y
0
(
t
)
) and H
(
t
)
=
1
/
h
t
,
y
0
t
log
⁡
x
0
y
0
t, and the second equation of (16) yields (17)
Δ
x
(
t
)
-
∫
y
0
(
t
)
t
‍
k
1
(
t
,
τ
)
Δ
x
(
τ
)
x
0
(
τ
)
τ
=
F
0
(
t
)
,where (18)
k
1
t
,
τ
=
h
t
,
τ
-
k
(
t
,
τ
)
G
(
t
)
,
G
t
=
k
t
,
y
0
t
h
t
,
y
0
t
,
k
t
,
y
0
t
≠
0
∀
t
∈
t
0
,
T
,
F
0
t
=
∫
y
0
t
t
‍
k
1
t
,
τ
log
⁡
x
0
τ
d
τ
-
x
0
t
+
g
t
+
f
t
G
t
.In an analogous way, Δ
y
m
(
t
) and Δ
x
m
(
t
) can be written in the form (19)
Δ
y
m
t
=
1
H
t
∫
y
0
t
t
‍
h
t
,
τ
Δ
x
m
τ
x
0
τ
d
τ
vvvvvvvvvv
+
∫
y
m
-
1
t
t
‍
h
t
,
τ
log
⁡
x
m
-
1
τ
d
τ
vvvvvvvvvv
-
Δ
x
m
t
-
x
m
-
1
t
+
g
t

∫
y
0
(
t
)
t
‍
,
(20)
Δ
x
m
t
-
∫
y
0
t
t
k
1
t
,
τ
Δ
x
m
τ
x
0
τ
d
τ
=
F
m
-
1
t
,where (21)
F
m
-
1
t
=
∫
y
m
-
1
t
t
‍
k
1
t
,
τ
log
⁡
x
m
-
1
τ
d
τ
-
x
m
-
1
(
t
)
+
g
(
t
)
+
f
(
t
)
G
(
t
)
.


## 3. The Mixed Method (Simpson and Trapezoidal) for Approximate Solution

At each step of the iterative process we have to find the solution of (18) and (20) on the closed interval [
t
0
,
T
]. To do this the grid (ω) of points t
i
=
t
0
+
i
h, i
=
1,2
,
3
,
…
,
2
N, h
=
T
-
t
0
/
2
N is introduced, and by the collocation method with mixed rule we require that the approximate solution satisfies (18) and (20). Hence (22)
Δ
x
m
(
t
0
)
=
-
x
m
-
1
(
t
0
)
+
g
(
t
0
)
+
f
(
t
0
)
G
(
t
0
)
,
(23)
Δ
x
m
(
t
2
i
)
-
∫
y
0
(
t
2
i
)
t
2
i
‍
k
1
(
t
2
i
,
τ
)
Δ
x
m
(
τ
)
x
0
(
τ
)
d
τ
=
F
m
-
1
t
2
i
,
i
=
1,2
,
…
,
N
.On the grid (ω) we set v
2
i
=
y
0
(
t
2
i
), suct that (24)
t
v
2
i
=
t
v
2
i
,
t
0
≤
y
0
t
2
i
<
t
2
i
-
2
,
t
2
i
,
t
2
i
-
2
≤
y
0
t
2
i
<
t
2
i
.Consequently, the system (23) can be written in the form (25)
Δ
x
m
t
2
i
-
∫
y
0
t
2
i
t
v
2
i
‍
k
1
t
2
i
,
τ
Δ
x
m
τ
x
0
τ
d
τ
-
∑
j
=
v
2
i
i
-
1
‍
∫
t
2
j
t
2
j
+
2
‍
k
1
t
2
i
,
τ
Δ
x
m
(
τ
)
x
0
τ
d
τ
=
F
m
-
1
t
2
i
,
i
=
1,2
,
…
,
N
.By computing the integral in (26) using tapezoidal formula on the first integrals and Simpson formula on the second integral, we consider two cases.Case 1. When v
2
i
≠
2
i, i
=
1,2
,
…
,
N, then (26)
Δ
x
m
(
t
2
i
)
=
F
m
-
1
(
t
2
i
)
+
A
(
i
)
+
B
(
i
)
+
C
(
i
)
1
-
(
t
2
i
-
t
2
i
-
2
/
6
·
x
0
t
2
i
)
k
1
(
t
2
i
,
t
2
i
)
,where (27)
A
i
=
0.5
t
v
2
i
-
y
0
t
2
i
×
Δ
x
m
t
v
2
i
-
2
y
0
t
2
i
-
t
v
2
i
-
2
t
v
2
i
-
t
v
2
i
-
2
x
0
y
0
t
2
i
k
1
t
2
i
,
t
v
2
i
Δ
x
m
t
v
2
i
x
0
t
v
2
i
+
k
1
t
2
i
,
y
0
t
2
i
vvvvv
×
Δ
x
m
t
v
2
i
t
v
2
i
-
y
0
t
2
i
t
v
2
i
-
t
v
2
i
-
2
x
0
y
0
t
2
i
vvvvv
+
k
1
t
2
i
,
y
0
t
2
i
vvvvv
×
Δ
x
m
t
v
2
i
-
2
y
0
t
2
i
-
t
v
2
i
-
2
t
v
2
i
-
t
v
2
i
-
2
x
0
y
0
t
2
i
,
B
i
=
∑
j
=
v
2
i
i
-
2
‍
(
t
2
j
+
2
-
t
2
j
)
6
×
k
1
(
t
2
i
,
t
2
j
)
Δ
x
m
t
2
j
x
0
t
2
j
vvvvv
+
4
k
1
(
t
2
i
,
t
2
j
+
1
)
Δ
x
m
t
2
j
+
1
x
0
t
2
j
+
1
vvvvv
+
k
1
(
t
2
i
,
t
2
j
+
2
)
Δ
x
m
(
t
2
j
+
2
)
x
0
(
t
2
j
+
2
)
,
C
i
=
t
2
i
-
t
2
i
-
2
6
k
1
t
2
i
,
t
2
i
-
2
Δ
x
m
t
2
i
-
2
x
0
t
2
i
-
2
vvvvvvvvvvvvv
+
4
k
1
t
2
i
,
t
2
i
-
1
Δ
x
m
t
2
i
-
1
x
0
t
2
i
-
1
.Case 2. When v
2
i
=
2
i, i
=
1,2
,
…
,
N, then  (28)
Δ
x
m
(
t
2
i
)
=
D
1
i
D
2
i
,where (29)
D
1
i
=
F
m
-
1
(
t
2
i
)
+
0.5
k
1
t
2
i
,
y
0
t
2
i
×
Δ
x
m
t
2
i
-
2
x
0
y
0
t
2
i
t
2
i
-
y
0
t
2
i
y
0
t
2
i
-
t
2
i
-
2
t
2
i
-
t
2
i
-
2
,
D
2
i
=
t
2
i
-
y
0
t
2
i
2
x
0
y
0
t
2
i
t
2
i
-
t
2
i
-
2
∑
1
-
0.5
t
2
i
-
y
0
t
2
k
1
t
2
i
,
t
2
i
x
0
t
2
i
vvvv
-
0.5
k
1
t
2
i
,
y
0
t
2
i
t
2
i
-
y
0
t
2
i
2
x
0
y
0
t
2
i
t
2
i
-
t
2
i
-
2
k
1
t
2
i
,
t
2
i
x
0
t
2
i
∑
.Also, to compute Δ
y
m
t on the grid ω, (18) can be represented in the form (30)
Δ
y
m
t
2
i
=
1
H
t
2
i
×
∫
y
0
(
t
2
i
)
t
2
i
‍
h
(
t
2
i
,
τ
)
Δ
x
m
τ
x
0
τ
d
τ
vvvvv
+
∫
y
m
-
1
(
t
2
i
)
t
2
i
‍
h
t
2
i
,
τ
log
⁡
x
m
-
1
τ
d
τ
vvvvv
-
Δ
x
m
t
2
i
-
x
m
-
1
t
2
i
+
g
t
2
i
∫
y
0
t
2
i
t
2
i
x
‍
.Let us set v
2
i
=
y
0
(
t
2
i
) and u
2
i
=
y
m
-
1
(
t
2
i
) and (31)
t
v
2
i
=
t
2
i
,
t
2
i
-
2
≤
y
0
(
t
2
i
)
<
t
2
i
,
t
v
2
i
,
t
0
≤
y
0
(
t
2
i
)
<
t
2
i
-
2
,
t
u
2
i
=
t
2
i
,
t
2
i
-
2
≤
y
m
-
1
(
t
2
i
)
<
t
2
i
,
t
u
2
i
,
t
0
≤
y
m
-
1
(
t
2
i
)
<
t
2
i
-
2
.Then (30) can be written as (32)
Δ
y
m
t
2
i
=
1
H
(
t
2
i
)
×
∫
y
0
(
t
2
i
)
t
v
2
i
‍
h
t
2
i
,
τ
Δ
x
m
(
τ
)
x
0
(
τ
)
d
τ
vvvvv
+
∑
j
=
v
2
i
i
-
1
‍
∫
t
2
j
t
2
j
+
2
‍
h
t
2
i
,
τ
Δ
x
m
(
τ
)
x
0
(
τ
)
d
τ
vvvvv
+
∫
y
m
-
1
(
t
2
i
)
t
u
2
i
‍
h
t
2
i
,
τ
log
⁡
x
m
-
1
τ
d
τ
vvvvv
+
∑
j
=
u
2
i
i
-
1
‍
∫
t
2
j
t
2
j
+
2
‍
h
t
2
i
,
τ
log
⁡
x
m
-
1
τ
d
τ
vvvvv
-
Δ
x
m
(
t
2
i
)
-
x
m
-
1
(
t
2
i
)
+
g
(
t
2
i
)
∫
y
0
t
2
i
t
v
2
i
‍
h
t
2
i
,
τ
Δ
x
m
(
τ
)
x
0
(
τ
)
d
τ
,and by applying mixed formula for (32) we obtain the following four cases.Case 1. When v
2
i
≠
2
i and u
2
i
≠
2
i, we have (33)
Δ
y
m
t
2
i
=
1
H
t
2
i
×
∑
j
=
v
2
i
i
-
1
‍
0.5
t
v
2
i
-
y
0
t
2
i
vvvvv
×
h
t
2
i
,
t
v
2
i
Δ
x
m
t
v
2
i
x
0
t
v
2
i
vvvvvvvv
+
h
t
2
i
,
y
0
t
2
i
Δ
x
m
y
0
t
2
i
x
0
y
0
t
2
i
Δ
x
m
t
v
2
i
x
0
t
v
2
i
vvvvv
+
∑
j
=
v
2
i
i
-
1
‍
(
t
2
j
+
2
-
t
2
j
)
6
vvvvvvvvvv
×
h
t
2
i
,
t
2
j
Δ
x
m
t
2
j
x
0
t
2
j
vvvvvvvvvvvvv
+
4
h
t
2
i
,
t
2
j
+
1
Δ
x
m
t
2
j
+
1
x
0
t
2
j
+
1
vvvvvvvvvvvvv
+
h
t
2
i
,
t
2
j
+
2
Δ
x
m
t
2
j
+
2
x
0
t
2
j
+
2
vvvvv
+
0.5
t
u
2
i
-
y
m
-
1
t
2
i
vvvvv
×
1
H
t
2
i
h
t
2
i
,
t
u
2
i
log
⁡
x
m
-
1
t
u
2
i
vvvvvvvv
+
h
t
2
i
,
y
m
-
1
t
2
i
log
⁡
x
m
-
1
y
m
-
1
t
2
i
1
H
t
2
i
vvvvv
+
∑
j
=
u
2
i
i
-
1
‍
(
t
2
j
+
2
-
t
2
j
)
6
vvvvvvvvvv
×
1
H
t
2
i
h
(
t
2
i
,
t
2
j
)
log
⁡
(
x
m
-
1
(
t
2
j
)
)
vvvvvvvvvvvvv
+
4
h
t
2
i
,
t
2
j
+
1
log
⁡
x
m
-
1
t
2
j
+
1
vvvvvvvvvvvvv
+
h
t
2
i
,
t
2
j
+
2
log
⁡
x
m
-
1
t
2
j
+
2
1
H
t
2
i
vvvvv
-
Δ
x
m
t
2
i
-
x
m
-
1
t
2
i
+
g
t
2
i
∑
j
=
v
2
i
i
-
1
‍
.Case 2. If v
2
i
=
2
i and u
2
i
≠
2
i, then (34)
Δ
y
m
(
t
2
i
)
=
1
H
(
t
2
i
)
×
∑
j
=
u
2
i
i
-
1
‍
0.5
(
t
2
i
-
y
0
(
t
2
i
)
)
vvvvv
×
h
(
t
2
i
,
t
2
i
)
Δ
x
m
t
v
2
i
x
0
t
2
i
vvvvvvvv
+
h
(
t
2
i
,
y
0
(
t
2
i
)
)
Δ
x
m
(
y
0
(
t
2
i
)
)
x
0
(
y
0
(
t
2
i
)
)
Δ
x
m
t
v
2
i
x
0
t
2
i
vvvvv
+
0.5
(
t
u
2
i
-
y
m
-
1
(
t
2
i
)
)
vvvvv
×
1
H
(
t
2
i
)
h
t
2
i
,
t
u
2
i
log
⁡
x
m
-
1
t
u
2
i
vvvvvvvv
+
h
t
2
i
,
y
m
-
1
t
2
i
log
⁡
x
m
-
1
y
m
-
1
t
2
i
1
H
(
t
2
i
)
vvvvv
+
∑
j
=
u
2
i
i
-
1
‍
(
t
2
j
+
2
-
t
2
j
)
6
vvvvvvvvvv
×
1
H
t
2
i
h
t
2
i
,
t
2
j
log
⁡
⁡
x
m
-
1
t
2
j
vvvvvvvvvvvvv
+
4
h
t
2
i
,
t
2
j
+
1
log
⁡
x
m
-
1
t
2
j
+
1
vvvvvvvvvvvvv
+
h
t
2
i
,
t
2
j
+
2
log
⁡
x
m
-
1
t
2
j
+
2
1
H
(
t
2
i
)
vvvvv
-
Δ
x
m
(
t
2
i
)
-
x
m
-
1
(
t
2
i
)
+
g
(
t
2
i
)
∑
j
=
u
2
i
i
-
1
‍
.Case 3. When v
2
i
≠
2
i and u
2
i
=
2
i, we get (35)
Δ
y
m
(
t
2
i
)
=
1
H
(
t
2
i
)
×
∑
j
=
v
2
i
i
-
1
‍
0.5
(
t
v
2
i
-
y
0
(
t
2
i
)
)
vvvvv
×
h
t
2
i
,
t
v
2
i
Δ
x
m
t
v
2
i
x
0
t
v
2
i
vvvvvvvv
+
h
(
t
2
i
,
y
0
(
t
2
i
)
)
Δ
x
m
(
y
0
(
t
2
i
)
)
x
0
(
y
0
(
t
2
i
)
)
vvvvv
+
∑
j
=
v
2
i
i
-
1
‍
(
t
2
j
+
2
-
t
2
j
)
6
vvvvvvvvvv
×
h
(
t
2
i
,
t
2
j
)
Δ
x
m
t
2
j
x
0
t
2
j
vvvvvvvvvvvvv
+
4
h
(
t
2
i
,
t
2
j
+
1
)
Δ
x
m
(
t
2
j
+
1
)
x
0
(
t
2
j
+
1
)
vvvvvvvvvvvvv
+
h
(
t
2
i
,
t
2
j
+
2
)
Δ
x
m
t
2
j
+
2
x
0
t
2
j
+
2
vvvvv
+
0.5
t
2
i
-
y
m
-
1
t
2
i
vvvvv
×
1
H
(
t
2
i
)
h
t
2
i
,
t
2
i
log
⁡
x
m
-
1
t
2
i
vvvvvvvv
+
h
t
2
i
,
y
m
-
1
t
2
i
log
⁡
x
m
-
1
y
m
-
1
t
2
i
1
H
(
t
2
i
)
-
Δ
x
m
(
t
2
i
)
-
x
m
-
1
(
t
2
i
)
+
g
(
t
2
i
)
∑
j
=
v
2
i
i
-
1
‍
.Case 4. If v
2
i
=
2
i and u
2
i
=
2
i, then (36)
Δ
y
m
(
t
2
i
)
=
1
H
(
t
2
i
)
×
∑
j
=
v
2
i
i
-
1
‍
0.5
t
2
i
-
y
0
t
2
i
vvvvv
×
h
t
2
i
,
t
2
i
Δ
x
m
t
2
i
x
0
t
2
i
vvvvvvvv
+
h
t
2
i
,
y
0
t
2
i
Δ
x
m
y
0
t
2
i
x
0
y
0
t
2
i
vvvvv
+
0.5
t
2
i
-
y
m
-
1
t
2
i
vvvvv
×
1
H
(
t
2
i
)
h
t
2
i
,
t
2
i
log
⁡
x
m
-
1
t
2
i
vvvvvvvv
+
h
t
2
i
,
y
m
-
1
t
2
i
log
⁡
x
m
-
1
y
m
-
1
t
2
i
1
H
(
t
2
i
)
vvvvv
-
Δ
x
m
t
2
i
-
x
m
-
1
t
2
i
+
g
t
2
i
∑
j
=
v
2
i
i
-
1
‍
.Thus, (32) can be computed by one of (33)–(36) according to the cases.


## 4. The Convergence Analysis of the Method

On the basis of general theorems of Newton-Kantorovich method [21, Chapter XVIII] for the convergence, we state the following theorem regarding the successive approximations described by (18)–(20).First, consider the following classes of functions:(i)
C
[
t
0
,
T
] the set of all continuous functions f
(
t
) defined on the interval [
t
0
,
T
],
(ii)
C
[
t
0
,
T
]
×
[
t
0
,
T
] the set of all continuous functions ψ
(
t
,
τ
) defined on the region [
t
0
,
T
]
×
[
t
0
,
T
],
(iii)
C
¯
=
{
X
:
X
=
(
x
(
t
)
,
y
(
t
)
)
:
x
(
t
)
,
y
(
t
)
∈
C
[
t
0
,
T
]
},
(iv)
C
[
t
0
,
T
]
<
=
{
y
(
t
)
∈
C
[
t
0
,
T
]
1
:
y
(
t
)
<
t
}.And define the following norms(37)
x
=
max
⁡
t
∈
[
t
0
,
T
]
x
t
,
Δ
X
C
¯
=
max
⁡
Δ
x
C
t
0
,
T
,
Δ
y
C
t
0
,
T
,
X
C
1
=
max
⁡
x
C
t
0
,
T
,
x
′
C
t
0
,
T
,
X
¯
C
¯
=
max
⁡
x
¯
C
t
0
,
T
,
y
¯
C
t
0
,
T
h
t
,
τ
=
H
1
,
h
τ
′
t
,
τ
=
H
1
′
,
k
(
t
,
τ
)
=
H
2
,
k
τ
′
(
t
,
τ
)
=
H
2
′
,
1
x
0
=
max
⁡
t
∈
[
t
0
,
T
]
1
x
0
t
=
c
1
,
1
x
0
2
=
max
⁡
t
∈
[
t
0
,
T
]
1
x
0
2
(
t
)
=
c
2
,
1
G
(
t
)
=
max
⁡
t
∈
[
t
0
,
T
]
1
G
(
t
)
=
c
3
,
x
0
=
max
⁡
t
∈
[
t
0
,
T
]
x
0
t
=
H
3
,
x
0
′
=
max
⁡
t
∈
[
t
0
,
T
]
x
0
′
t
=
H
3
′
,
min
⁡
t
∈
[
t
0
,
T
]
y
0
t
=
H
4
,
log
⁡
=
max
⁡
t
∈
[
t
0
,
T
]
log
⁡
x
t
=
H
5
,
g
=
max
⁡
t
∈
[
t
0
,
T
]
g
t
=
H
6
,
f
=
max
⁡
t
∈
[
t
0
,
T
]
f
t
=
H
7
.Let (38)
η
1
=
max
⁡
H
1
c
2
T
-
H
4
,
H
1
c
1
,
H
1
′
H
5
+
H
1
H
3
′
c
1
,
vvvvvv
H
2
c
2
(
T
-
H
4
)
,
H
2
c
1
,
H
2
′
H
5
+
H
2
H
3
′
c
1
.Let us consider real valued function (39)
ψ
(
t
)
=
K
t
-
t
0
2
-
1
+
K
η
t
-
t
0
+
η
,where K
>
0 and η are nonnegative real coefficients.Theorem 1.
Assume that the operatorP
(
X
)
=
0 in (5) is defined in Ω
=
{
X
∈
C
(
[
t
0
,
T
]
)
:
∥
X
-
X
0
∥
≤
R
} and has continuous second derivative in closed ball Ω
0
=
{
X
∈
C
(
[
t
0
,
T
]
)
:
∥
X
-
X
0
∥
≤
r
} where T
=
t
0
+
r
≤
t
0
+
R. Suppose the following conditions are satisfied: (1)
∥
Γ
0
P
(
X
0
)
∥
≤
η
/
1
+
K
η,
(2)
∥
Γ
0
P
′′
(
X
)
∥
≤
2
K
/
1
+
K
η, when ∥
X
-
X
0
∥
≤
t
-
t
0
≤
r,
whereK and η as in (39). Then the function ψ
(
t
) defined by (39) majorizes the operator P
(
X
).Proof.
Let us rewrite (5) and (39) in the form (40)
t
=
ϕ
(
t
)
,
ϕ
(
t
)
=
t
+
c
0
ψ
(
t
)
,
(41)
X
=
S
(
X
)
,
S
(
X
)
=
X
-
Γ
0
P
(
X
)
,where c
0
=
-
1
/
ψ
′
(
t
0
)
=
1
/
1
+
K
η and Γ
0
=
[
P
′
(
X
0
)
]
-
1.
Let us show that (40) and (41) satisfy the majorizing conditions [21, Theorem 1, page 525]. In fact (42)
S
(
X
0
)
-
X
0
=
-
Γ
0
P
(
X
0
)
≤
η
1
+
K
η
=
ϕ
(
t
0
)
-
t
0
,and for the X
-
X
0
≤
t
-
t
0 with the Remark in [21, Remark 1, page 504] we have (43)
S
′
(
X
)
=
S
′
(
X
)
-
S
′
(
X
0
)
≤
∫
X
0
X
‍
S
′′
(
X
)
d
X
=
∫
X
0
X
‍
Γ
0
P
′′
(
X
)
d
X
≤
∫
t
0
t
‍
c
0
ψ
′′
(
τ
)
d
τ
=
∫
t
0
t
‍
2
K
1
+
K
η
d
τ
=
2
K
1
+
K
η
(
t
-
t
0
)
=
ϕ
′
(
t
)
.Hence ψ
(
t
)
=
0 is a majorant function of P
(
X
)
=
0.Theorem 2.
Let the functionsf
(
t
)
,
g
(
t
)
∈
C
[
t
0
,
T
], x
0
(
t
)
∈
C
1
[
t
0
,
T
], x
0
(
y
0
(
t
)
)
≠
0, x
0
2
(
t
)
≠
0, and the kernels h
(
t
,
τ
)
,
k
(
t
,
τ
)
∈
C
[
t
0
,
T
]
×
[
t
0
,
T
]
1 and (
x
0
(
t
)
,
y
0
(
t
)
)
∈
Ω
0; then (1)
the system (7) has unique solution in the interval [
t
0
,
T
]; that is, there exists Γ
0, and ∥
Γ
0
∥
≤
∑
j
=
1
∞
‍
c
1
H
1
+
c
1
c
3
H
2
j
(
T
-
H
4
)
j
-
1
/
(
j
-
1
)
!
=
η
2,
(2)
Δ
X
≤
η
/
1
+
K
η,
(3)
P
′′
(
X
)
≤
η
1,
(4)
η
>
1
/
K and r
<
η
+
t
0,
whereK and η as in (39). Then the system (4) has unique solution X
* in the closed ball Ω
0 and the sequence X
m
(
t
)
=
(
x
m
(
t
)
,
y
m
(
t
)
), m
≥
0 of successive approximations (44)
Δ
y
m
t
=
1
H
(
t
)
∫
y
0
t
t
‍
h
t
,
τ
Δ
x
m
τ
x
0
τ
d
τ
vvvvvvvvv
+
∫
y
m
-
1
t
t
‍
h
t
,
τ
log
⁡
x
m
-
1
τ
d
τ
vvvvvvvvv
-
Δ
x
m
t
-
x
m
-
1
t
+
g
t
∫
y
0
(
t
)
t
‍
,
Δ
x
m
(
t
)
-
∫
y
0
(
t
)
t
‍
k
1
(
t
,
τ
)
Δ
x
m
(
τ
)
x
0
(
τ
)
d
τ
=
F
m
-
1
(
t
)
,where Δ
x
m
(
t
)
=
x
m
(
t
)
-
x
m
-
1
(
t
) and Δ
y
m
(
t
)
=
y
m
(
t
)
-
y
m
-
1
(
t
), m
=
2,3
,
…, and X
m converge to the solution X
*. The rate of convergence is given by (45)
X
*
-
X
m
≤
2
1
+
K
η
m
1
K
.Proof.
It is shown that (7) is reduced to (17). Since (17) is a linear Volterra integral equation of 2nd kind with respect to Δ
x
(
t
) and since k
(
t
,
y
0
(
t
)
)
≠
0, ∀
t
∈
[
t
0
,
T
] which implies that the kernel k
1
(
t
,
τ
) defined by (18) is continues it follows that (17) has a unique solution which can be obtained by the method of successive approximations. Then the function Δ
y
(
t
) is uniquely determined from (16). Hence the existence of Γ
0 is archived.
To verify thatΓ
0 is bounded we need to establish the resolvent kernel Γ
0
(
t
,
τ
) of (17), so we assume the integral operator U from C
[
t
0
,
T
]
→
C
[
t
0
,
T
] is given by (46)
Z
=
U
(
Δ
x
)
,
Z
(
t
)
=
∫
y
0
(
t
)
t
‍
k
2
(
t
,
τ
)
Δ
x
(
τ
)
d
τ
,where k
2
(
t
,
τ
)
=
k
1
(
t
,
τ
)
/
x
0
(
τ
), and k
1
(
t
,
τ
) is defined in (18).
Due to (46), (17) can be written as (47)
Δ
x
-
U
(
Δ
x
)
=
F
0
.The solution Δ
x
* of (47) is expressed in terms of F
0 by means of the formula (48)
Δ
x
*
=
F
0
+
B
(
F
0
)
,where B is an integral operator and can be expanded as a series in powers of U [21, Theorem 1, page 378]: (49)
B
(
F
0
)
=
U
(
F
0
)
+
U
2
(
F
0
)
+
⋯
+
U
n
(
F
0
)
+
⋯
,and it is known that the powers of U are also integral operators. In fact (50)
Z
n
=
U
n
,
Z
n
t
=
∫
y
0
t
t
‍
k
2
n
t
,
τ
Δ
x
τ
d
τ
,
000000000000000000000000
n
=
1,2
,
…
,where k
2
(
n
) is the iterated kernel.
Substituting (50) into (48) we obtain an expression for the solution of (47): (51)
Δ
x
*
=
F
0
(
t
)
+
∑
j
=
1
∞
‍
∫
y
0
(
t
)
t
‍
k
2
(
j
)
(
t
,
τ
)
F
0
(
τ
)
d
τ
.Next, we show that the series in (51) is convergent uniformly for all t
∈
[
t
0
,
T
]. Since (52)
k
2
t
,
τ
=
k
1
t
,
τ
x
0
τ
≤
h
t
,
τ
x
0
τ
+
k
t
,
τ
x
0
τ
G
t
≤
c
1
H
1
+
c
1
c
3
H
2
.Let M
=
c
1
H
1
+
c
1
c
3
H
2; then by mathematical induction we get (53)
k
2
2
t
,
τ
≤
∫
y
0
t
t
‍
k
2
t
,
u
k
2
u
,
τ
d
u
≤
M
2
t
-
H
4
1
!
,
k
2
3
t
,
τ
≤
∫
y
0
t
t
‍
k
2
t
,
u
k
2
2
u
,
τ
d
u
≤
M
3
t
-
H
4
2
2
!
,
⋮
k
2
n
t
,
τ
≤
∫
y
0
t
t
‍
k
2
t
,
u
k
2
n
-
1
u
,
τ
d
u
≤
M
n
t
-
H
4
n
-
1
n
-
1
!
,
vvvvvvvvvvvvvvvvvvvvvv
n
=
1,2
,
…
;then (54)
U
n
=
max
⁡
t
∈
[
t
0
,
T
]
∫
y
0
t
t
‍
k
2
n
t
,
τ
d
τ
≤
M
n
T
-
H
4
(
n
-
1
)
n
-
1
!
.Therefore the nth root test of the sequence yields (55)
U
n
n
≤
M
T
-
H
4
1
-
1
/
n
n
-
1
!
n
⟶
n
→
∞
0
.Hence ρ
=
1
/
lim
n
→
∞
∥
U
n
∥
n
=
∞ and a Volterra integral equations (17) has no characteristic values. Since the series in (51) converges uniformly (48) can be written in terms of resolvent kernel of (17): (56)
Δ
x
*
=
F
0
+
∫
y
0
t
t
‍
Γ
0
t
,
τ
F
0
τ
d
τ
,where (57)
Γ
0
t
,
τ
=
∑
j
=
1
∞
‍
k
2
j
t
,
τ
.Since the series in (57) is convergent we obtain (58)
Γ
0
=
B
F
0
≤
∑
j
=
1
∞
‍
U
j
≤
∑
j
=
1
∞
‍
M
j
T
-
H
j
-
1
j
-
1
!
≤
η
2
.To establish the validity of second condition, let us represent operator equation (59)
P
(
X
)
=
0
,as in (41) and its the successive approximations is (60)
X
n
+
1
=
S
X
n
,
n
=
0,1
,
2
,
…
.For initial guess X
0 we have (61)
S
X
0
=
X
0
-
Γ
0
P
X
0
.From second condition of (Theorem 1) we have (62)
Γ
0
P
(
X
0
)
=
S
(
X
0
)
-
X
0
=
X
1
-
X
0
=
Δ
X
≤
η
1
+
K
η
.
In addition, we need to show thatP
′′
(
X
)
≤
η
1 for all X
∈
Ω
0 where η
1 is defined in (38). It is known that the second derivative P
′′
X
0
X
,
X
¯ of the nonlinear operator P
X is described by 3-dimensional array P
′′
X
0
X
X
¯
=
D
1
,
D
2
X
,
X
¯, which is called bilinear operator; that is, P
′′
X
0
X
X
¯
=
B
X
0
,
X
,
X
¯ where (63)
P
′′
(
X
0
)
(
X
,
X
¯
)
=
lim
s
→
0
1
s
[
P
′
(
x
0
+
s
X
¯
)
-
P
′
(
X
0
)
]
=
lim
s
→
0
1
s
∂
P
1
∂
x
x
0
+
s
x
¯
,
y
0
+
s
y
¯
-
∂
P
1
∂
x
x
0
,
y
0
x
vvvvvvvvv
+
∂
P
1
∂
y
x
0
+
s
x
¯
,
y
0
+
s
y
¯
-
∂
P
1
∂
y
x
0
,
y
0
y
,
vvvv
lim
s
→
0
1
s
∂
P
2
∂
x
x
0
+
s
x
¯
,
y
0
+
s
y
¯
-
∂
P
2
∂
x
x
0
,
y
0
x
vvvvvvvvv
+
∂
P
2
∂
y
x
0
+
s
x
¯
,
y
0
+
s
y
¯
-
∂
P
2
∂
y
x
0
,
y
0
y
=
∑
P
P
lim
s
→
0
1
s
∑
∂
2
P
1
∂
x
2
x
0
,
y
0
s
x
¯
+
∂
2
P
1
∂
y
∂
x
(
x
0
,
y
0
)
s
y
¯
vvvvvvvvvvv
+
1
2
∂
3
P
1
∂
x
3
x
0
+
θ
s
x
¯
,
y
0
+
δ
s
y
¯
s
2
x
¯
2
vvvvvvvvvvvvvvvv
+
2
∂
3
P
1
∂
x
2
∂
y
x
0
+
θ
s
x
¯
,
y
0
+
δ
s
y
¯
s
2
x
¯
y
¯
vvvvvvvvvvvvvvvv
+
∂
3
P
1
∂
y
2
∂
x
x
0
+
θ
s
x
¯
,
y
0
+
δ
s
y
¯
s
2
y
¯

∑
x
vvvvvvvvv
+
∑
∂
2
P
1
∂
x
∂
y
x
0
,
y
0
s
x
¯
+
∂
2
P
1
∂
y
2
x
0
,
y
0
s
y
¯
vvvvvvvvvvvv
+
1
2
∂
3
P
1
∂
x
2
∂
y
x
0
+
θ
s
x
¯
,
y
0
+
δ
s
y
¯
s
2
x
¯
2
vvvvvvvvvvvvvvvvv
+
2
∂
3
P
1
∂
x
∂
y
2
x
0
+
θ
s
x
¯
,
y
0
+
δ
s
y
¯
s
2
x
¯
y
¯
vvvvvvvvvvvvvvvvv
+
∂
3
P
1
∂
y
3
x
0
+
θ
s
x
¯
,
δ
s
y
¯
s
2
y
¯
2

∑
y
,
vv
v
lim
s
→
0
1
s
∑
∂
2
P
2
∂
x
2
(
x
0
,
y
0
)
s
x
¯
+
∂
2
P
2
∂
y
∂
x
(
x
0
,
y
0
)
s
y
¯
vvvvvvvvvvv
+
1
2
∂
3
P
2
∂
x
3
x
0
+
θ
s
x
¯
,
y
0
+
δ
s
y
¯
s
2
x
¯
2
vvvvvvvvvvvvvvvv
+
2
∂
3
P
2
∂
x
2
∂
y
x
0
+
θ
s
x
¯
,
y
0
+
δ
s
y
¯
s
2
x
¯
y
¯
vvvvvvvvvvvvvvvv
+
∂
3
P
2
∂
y
2
∂
x
x
0
+
θ
s
x
¯
,
y
0
+
δ
s
y
¯
s
2
y
¯

∑
x
vvvvvvvv
+
∑
∂
2
P
2
∂
x
∂
y
(
x
0
,
y
0
)
s
x
¯
+
∂
2
P
2
∂
y
2
(
x
0
,
y
0
)
s
y
¯
vvvvvvvvvvvv
+
1
2
∂
3
P
2
∂
x
2
∂
y
(
x
0
+
θ
s
x
¯
,
y
0
+
δ
s
y
¯
)
s
2
x
¯
2
vvvvvvvvvvvvvvvvv
+
2
∂
3
P
2
∂
x
∂
y
2
x
0
+
θ
s
x
¯
,
y
0
+
δ
s
y
¯
s
2
x
¯
y
¯
vvvvvvvvvvvvvvvvv
+
∂
3
P
2
∂
y
3
(
x
0
+
θ
s
x
¯
,
δ
s
y
¯
)
s
2
y
¯
2

∑
y
=
∂
2
P
1
∂
x
2
(
x
0
,
y
0
)
x
¯
x
+
∂
2
P
1
∂
y
∂
x
(
x
0
,
y
0
)
y
¯
x
vvvv
+
∂
2
P
1
∂
x
∂
y
x
0
,
y
0
x
¯
y
+
∂
2
P
1
∂
y
2
x
0
,
y
0
y
¯
x
,
vvvv
∂
2
P
2
∂
x
2
x
0
,
y
0
x
¯
x
+
∂
2
P
2
∂
y
∂
x
x
0
,
y
0
y
¯
x
vvvv
+
∂
2
P
2
∂
x
∂
y
x
0
,
y
0
x
¯
y
+
∂
2
P
2
∂
y
2
x
0
,
y
0
y
¯
x
,where θ
,
δ
∈
(
0,1
), so we have (64)
P
′′
X
0
X
,
X
¯
=
D
1
D
2
x
¯
y
¯
x
y
,where (65)
D
1
=
∂
2
P
1
∂
x
2
x
0
,
y
0
∂
2
P
1
∂
y
∂
x
x
0
,
y
0
∂
2
P
1
∂
x
∂
y
x
0
,
y
0
∂
2
P
1
∂
y
2
x
0
,
y
0
,
D
2
=
∂
2
P
2
∂
x
2
(
x
0
,
y
0
)
∂
2
P
2
∂
y
∂
x
(
x
0
,
y
0
)
∂
2
P
2
∂
x
∂
y
(
x
0
,
y
0
)
∂
2
P
2
∂
y
2
(
x
0
,
y
0
)
.Then the norms of every components of D
1 and D
2 have the estimate (66)
∂
2
P
1
∂
x
2
=
max
⁡
X
≤
1
,
‖
X
¯
‖
≤
1
∫
y
0
t
t
‍
h
t
,
τ
x
τ
x
0
2
τ
x
¯
τ
d
τ
≤
H
1
c
2
(
T
-
H
4
)
,
∂
2
P
1
∂
x
∂
y
=
max
⁡
X
≤
1
,
‖
X
¯
‖
≤
1
h
t
,
y
0
t
x
y
0
t
x
0
y
0
t
y
¯
t
≤
H
1
c
1
,
∂
2
P
1
∂
y
∂
x
=
max
⁡
X
≤
1
,
‖
X
¯
‖
≤
1
h
t
,
y
0
t
x
¯
y
0
t
x
0
y
0
t
y
t
≤
H
1
c
1
,
∂
2
P
1
∂
y
2
=
max
⁡
X
≤
1
,
‖
X
¯
‖
≤
1
∑
x
0
′
y
0
t
x
0
y
0
t
h
τ
′
t
,
y
0
t
log
⁡
x
0
y
0
t
vvvvvvvvvvvv
+
h
t
,
y
0
t
x
0
′
y
0
t
x
0
y
0
t
vvvvvvvvvvv
×
y
t
y
¯
t
∑
x
0
′
y
0
t
x
0
y
0
t
≤
H
1
′
H
5
+
H
1
H
3
′
c
1
,
∂
2
P
2
∂
x
2
=
max
⁡
X
≤
1
,
‖
X
¯
‖
≤
1
∫
y
0
t
t
‍
k
t
,
τ
x
τ
x
0
2
τ
x
¯
τ
d
τ
≤
H
2
c
2
(
T
-
H
4
)
,
∂
2
P
2
∂
x
∂
y
=
max
⁡
X
≤
1
,
‖
X
¯
‖
≤
1
-
k
t
,
y
0
t
x
y
0
t
x
0
y
0
t
y
¯
t
≤
H
2
c
1
,
∂
2
P
2
∂
y
∂
x
=
max
⁡
X
≤
1
,
‖
X
¯
‖
≤
1
-
k
t
,
y
0
t
x
¯
y
0
t
x
0
y
0
t
y
t
≤
H
2
c
1
,
∂
2
P
2
∂
y
2
=
max
⁡
X
≤
1
,
‖
X
¯
‖
≤
1
∑
x
0
′
(
y
0
(
t
)
)
x
0
(
y
0
(
t
)
)
k
τ
′
t
,
y
0
t
log
⁡
x
0
y
0
t
vvvvvvvvvvvv
+
k
t
,
y
0
t
x
0
′
y
0
t
x
0
y
0
t
vvvvvvvvvvv
×
y
t
y
¯
t
∑
x
0
′
y
0
t
x
0
y
0
t
≤
H
2
′
H
5
+
H
2
H
3
′
c
1
.Therefore, all the second derivatives exist and are bounded: (67)
P
′′
X
≤
η
1
.Since ψ
(
t
) majorizes operator P
(
X
) and utilizing the second condition of (Theorem 1) we get (68)
Γ
0
P
′′
X
≤
2
K
1
+
K
η
.Let us consider the discriminant of equation ψ
(
t
)
=
0: (69)
D
=
K
2
η
2
-
2
K
η
+
1
=
k
η
-
1
2
,and the two roots of ψ
(
t
)
=
0 are r
1
=
1
/
K
+
t
0 and r
2
=
η
+
t
0; therefore, when r
1
<
r
<
r
2 implies (70)
ψ
r
≤
0
,then under the assumption of the fourth condition, that is, 1
/
K
+
t
0 is the unique solution of ψ
t
=
0 in t
0
,
T and the condition in (70) [21, Theorem 4, page 530] implies that X
* is the unique solution of operator equation (5) [21, Theorem 6, page 532] and (71)
X
*
-
X
0
≤
t
*
-
t
0
,where t
* is the unique solution of ψ
(
t
)
=
0 in t
0
,
r.
To show the rate of convergence let us write the equationψ
(
t
)
=
0 in a same form as in (40) then its successive approximation is (72)
t
m
+
1
=
ϕ
t
m
,
m
=
0,1
,
2
,
…
.To estimate the difference between t
* and successive approximation t
m: (73)
t
*
-
t
m
=
ϕ
t
*
-
ϕ
t
m
-
1
=
ϕ
′
t
m
~
t
*
-
t
m
-
1
,where t
m
~
∈
(
t
m
-
1
,
t
*
) and (74)
ϕ
′
t
=
1
+
c
0
ψ
′
t
=
2
K
1
+
K
η
t
-
t
0
;therefore (75)
ϕ
′
t
m
~
=
2
K
1
+
K
η
t
m
~
-
t
0
≤
2
K
1
+
K
η
t
*
-
t
0
=
2
1
+
K
η
;then (76)
t
*
-
t
m
≤
2
1
+
K
η
t
*
-
t
m
-
1
,
t
*
-
t
m
-
1
≤
2
1
+
K
η
t
*
-
t
m
-
2
;
⋮
t
*
-
t
1
≤
2
1
+
K
η
t
*
-
t
0
,consequently, (77)
t
*
-
t
m
≤
2
1
+
K
η
m
1
K
;it implies (78)
X
*
-
X
m
≤
t
*
-
t
m
≤
2
1
+
K
η
m
1
K
.


## 5. Numerical Example

Consider the system of nonlinear equation(79)
x
t
-
∫
y
t
t
‍
t
τ
log
⁡
x
τ
d
τ
=
e
t
-
t
2
3
,
∫
y
t
t
‍
τ
log
⁡
x
τ
d
τ
=
t
3
,
t
∈
10,15
.The exact solution is (80)
x
*
t
=
e
t
,
y
*
t
=
t
3
-
t
3
,and the initial guesses are (81)
x
0
t
=
e
10
t
-
9
,
y
0
t
=
0.6
t
+
4
.Table1 shows that x
m
(
t
) coincides with the exact x
*
(
t
) from the first iteration whereas only six iterations are needed for y
m
(
t
) to be very close to y
*
(
t
). Notations used here are as follows: N is the number of nodes, m is the number of iterations, and ϵ
x
=
max
⁡
t
∈
[
10,15
]
x
m
t
-
x
*
t and ϵ
y
=
max
⁡
t
∈
[
10,15
]
y
m
t
-
y
*
t.Table 1
Numerical results for (79).
N
=
20
,
h
=
0.25
m
ϵ
x
ϵ
y
1
0.00
0.0029
2
0.00
4.3597
E
-
006
3
0.00
3.1061
E
-
008
4
0.00
1.0140
E
-
009
5
0.00
1.2541
E
-
010
6
0.00
3.9968
E
-
011


## 6. Conclusion

In this paper, the Newton-Kantorovich method is developed to solve the system of nonlinear Volterra integral equations which contains logarithmic function. We have introduced a new majorant function that leads to the increment of range of convergence of successive approximation process. A new theorem is stated based on the general theorems of Kantorovich. Numerical example is given to show the validation of the method. Table1 shows that the proposed method is in good agreement with the theoretical findings.



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*Source: 219616-2015-04-14.xml*