We develop the Newton-Kantorovich method to solve the system of 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.

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: where , , and the functions , , , , and . In 2010, Eshkuvatov et al. [19] used the Newton-Kantorovich hypothesis to solve the system of nonlinear Volterra integral equation of the formwhere and are unknown functions defined on , , and , . In 2010, Eshkuvatov et al. [20] developed the modified Newton-Kantorovich to obtain an approximate solution of system with the form where , , and the functions , , and the unknown functions , , .

In this paper, we consider the systems of nonlinear integral equation of the form where , , , and the unknown functions , to be determined, and .

The paper is organized as follows, in Section 2, 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 formwhere and To solve (5) we use initial iteration of Newton-Kantorovich method which is of the form where is the initial guess and and can be any continuous functions provided that and .

The Frechet derivative of at the point is defined as Hence, From (7) and (9) it follows that where , , and is the initial given functions. To solve (10) with respect to and we need to compute all partial derivatives: and in the same manner we obtain So that from (10)–(12) it follows that Equation (13) is a linear, and, by solving it for and , we obtain . By continuing this process, a sequence of approximate solution can be evaluated from which is equivalent to the system where and , .

Thus, one should solve a system of two linear Volterra integral equations to find each successive approximation. Let us eliminate from the system (13) by finding the expression of from the first equation of this system and substitute it in the second equation to yield where and , and the second equation of (16) yields where In an analogous way, and can be written in the form where

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 . To do this the grid () of points , , is introduced, and by the collocation method with mixed rule we require that the approximate solution satisfies (18) and (20). Hence On the grid () we set , suct that Consequently, the system (23) can be written in the form 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 , , then where

Case 2. When , , then  where Also, to compute on the grid , (18) can be represented in the form Let us set and and Then (30) can be written as and by applying mixed formula for (32) we obtain the following four cases.

Case 1. When and , we have

Case 2. If and , then

Case 3. When and , we get

Case 4. If and , then 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) the set of all continuous functions defined on the interval ,(ii) the set of all continuous functions defined on the region ,(iii),(iv).

And define the following normsLet Let us consider real valued function where and are nonnegative real coefficients.

Theorem 1. Assume that the operator in (5) is defined in and has continuous second derivative in closed ball where . Suppose the following conditions are satisfied: (1),(2), when ,
where and as in (39). Then the function defined by (39) majorizes the operator .

Proof. Let us rewrite (5) and (39) in the form where and .
Let us show that (40) and (41) satisfy the majorizing conditions [21, Theorem 1, page 525]. In fact and for the with the Remark in [21, Remark 1, page 504] we have Hence is a majorant function of .

Theorem 2. Let the functions , , , , and the kernels and ; then (1)the system (7) has unique solution in the interval ; that is, there exists , and ,(2),(3),(4) and ,
where and as in (39). Then the system (4) has unique solution in the closed ball and the sequence , of successive approximations where and , , and converge to the solution . The rate of convergence is given by

Proof. It is shown that (7) is reduced to (17). Since (17) is a linear Volterra integral equation of 2nd kind with respect to and since , which implies that the kernel 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 is uniquely determined from (16). Hence the existence of is archived.
To verify that is bounded we need to establish the resolvent kernel of (17), so we assume the integral operator from is given by where , and is defined in (18).
Due to (46), (17) can be written as The solution of (47) is expressed in terms of by means of the formula where is an integral operator and can be expanded as a series in powers of [21, Theorem 1, page 378]: and it is known that the powers of are also integral operators. In fact where is the iterated kernel.
Substituting (50) into (48) we obtain an expression for the solution of (47): Next, we show that the series in (51) is convergent uniformly for all . Since Let ; then by mathematical induction we get then Therefore the th root test of the sequence yields Hence 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): where Since the series in (57) is convergent we obtain To establish the validity of second condition, let us represent operator equation as in (41) and its the successive approximations is For initial guess we have From second condition of (Theorem 1) we have In addition, we need to show that for all where is defined in (38). It is known that the second derivative of the nonlinear operator is described by 3-dimensional array , which is called bilinear operator; that is, where