1. Introduction
Finding solutions to nonlinear equations of the form (where is a differentiable nonlinear operator, are complete normed linear spaces, and D is a non-empty, open, and convex set) is a fundamental and widely studied problem in applied mathematics. As it is often difficult to obtain an exact solution to the previous equation, we typically seek a numerical approximation instead. In such cases, we rely on approximation methods, which are typically iterative.
There are different types of convergence results used to approximate solutions to nonlinear equations. The first type, referred to as local convergence, depends on the existence of a presumed solution and requires the initial value to be sufficiently close to it. The second type, known as semilocal convergence, does not assume the existence of a specific solution, but the initial values must still satisfy certain conditions to guarantee convergence. Analyzing the semilocal convergence of iterative methods in Banach spaces is of great theoretical interest, as it enables the derivation of key results such as the existence and uniqueness of solutions, the convergence rate, a priori error estimates, and convergence regions, based solely on assumptions about the initial approximation rather than the solution itself. Such results have practical relevance in solving equations arising from differential and integral models. A semilocal convergence study can be developed through two distinct approaches, namely recurrent relations and recurrent functions.
For the classical Newton’s method [
1], the semilocal convergence using recurrence relations was obtained by Kantorovich in [
2]. Polyak [
3] affirms that Kantorovich established in [
4] a rigorous theoretical framework for the method of successive approximations applied to solving functional equations and introduced a semilocal analysis of Newton’s method by leveraging Banach’s contraction mapping principle. He focused on proving the conditions under which the method converges to a solution, thereby laying the groundwork for subsequent developments in numerical analysis and optimization. This was later refined to semilocal quadratic convergence in 1948, known as the Newton–Kantorovich theorem [
5]. Kantorovich developed two distinct approaches to proving the aforementioned theorem, with one involving recurrence relations and the other employing majorant functions. The first proof [
5] used recurrence relations, while the second [
6] was founded on the majorant function framework. Mysovskikh [
7] presented a more straightforward proof of semilocal quadratic convergence, based on slightly altered theoretical premises. Since the work carried out by Kantorovich, extensive research has explored the convergence and error bounds of Newton’s method based on the theorem’s assumptions or related conditions. Among these contributions, the convergence theorems by Ortega and Rheinboldt [
8] stand out. Rall (see [
9]) proposed an alternative approach for studying semilocal convergence, which relied on recurrence relations and was simpler to analyze. Furthermore, numerous variations of the Newton–Kantorovich theorem have been proposed, each with distinct assumptions and conclusions.
Several studies have explored the semilocal convergence of third-order methods, including the Chebyshev method, addressed in [
10,
11], and the Halley scheme, discussed in [
12]. Many of these works build on Kantorovich’s framework for analyzing the Newton method (see [
13,
14]). Gutierrez and Hernández [
15] provide a comprehensive theory that unifies these third-order methods. All of these papers established that the convergence of sequences in Banach spaces can be traced back to the convergence of a majorant sequence. Then, Hernández [
16] introduced a novel form of recurrence relations for the Chebyshev method, composed of two sequences of positive real numbers.
Recently, scholars have carried out studies on semilocal convergence for higher-order iterative methods. For example, Chen et al. [
17] studied the semilocal convergence for a modified Newton’s method using recurrence relations. Using that approach, Wang et al. [
18] analyzed the semilocal convergence for a sixth-order variant of the Jarratt method. Hueso and Martínez [
19] proved a convergence result for a third- and fourth-order family of methods. Jaiswal [
20] examined the semilocal convergence of an existing eighth-order method for solving nonlinear equations in Banach spaces under relaxed conditions, utilizing Rall’s straightforward techniques. Later, Cordero et al. [
21] used recurrence relations to prove the semilocal convergence in Banach spaces for the multidimensional extension of Chun’s scheme. Argyros et al. [
22] provided the semilocal convergence of a two-step Jarratt-type method by using restricted convergence regions combined with majorizing scalar sequences.
Wang et al. [
23] established a semilocal convergence study of an iterative method using a recursive relation without requiring higher-order derivatives. They determined the domains of existence and uniqueness by appropriately choosing the initial point and applying the Lipschitz condition to the first-order Fréchet derivative across the region. In [
24], a fourth-order family of methods for solving nonlinear equations was proposed and the stability of the class was analyzed using complex dynamical tools. Subsequently, ref. [
25] provided local convergence results. Now, the purpose of this paper is to establish the semilocal convergence of this iterative scheme in Banach spaces and obtain error estimates through a system of recurrence relations.
This paper is organized as follows:
Section 2 presents preliminary results and introduces the auxiliary functions.
Section 3 outlines the development of the recurrence relations. In
Section 4, the semilocal convergence is established.
Section 5 provides numerical examples to support our theoretical findings.
Section 6 concludes the paper and provides final remarks.
2. Preliminary Results
Let
be Banach spaces and
a nonlinear twice continuously Fréchet differentiable operator in an open convex domain of
D. Consider the family of iterative schemes defined for
by
where
is a parameter and
is the initial estimate. This family was studied in [
24], and its local convergence analysis was carried out in [
25]. Let us denote
and
.
Let such that exists and
- (C1)
- (C2)
- (C3)
- (C4)
Let us define
, and
, where
. From the first step in (
1), we have
Prior to performing the semilocal convergence analysis, we apply the following lemmas:
Lemma 1 ([
26])
. If operator has continuous derivatives up to order in an open convex subset D and for all , then if , the following is obtained: where the remainder can be expressed asand The previous lemma provides an expression and a bound for the remainder of Taylor’s expansion of the operator , and the next lemma (Banach Lemma) will allow us to guarantee the existence and boundedness of an inverse matrix. Both lemmas will be used in the upcoming proofs.
Lemma 2 (Banach Lemma [
27])
. Let , where is the space of all continuous linear operators on X into X. For such T, the normis defined. If , then is an invertible matrix and Now, in order to set the semilocal study, we analyze the iterative method (
1) step by step. Let us find a bound for
and
. We have
and using the triangular inequality, we can write
where from (
1), we obtain the following:
and
By applying Banach’s Lemma, we aim to prove the existence of the inverse of the operator
and then the existence of the inverse of
. Thus, using (
C1), (
C3) and the definition of
, the following condition must be met:
By the Banach Lemma,
exists and
Next, we prove the existence of the inverse of the operator
. Hence, the next condition must be satisfied
if
, which leads to
and by the Banach Lemma,
exists and
By taking the norm in (
3) and using (
4)–(
6), the following is obtained:
where
replacing (
2) and (
7) in (
2), we have
where by definition
The result obtained from Lemma 2 is applied to
in order to prove the existence of the inverse matrix of
and the inverse of
. Therefore, the following condition must hold:
by the Banach Lemma,
exists and
Therefore,
where by definition
Taylor’s expansion of
around
can be described as follows:
Multiplying by
, the first step in (
1), we have
Then, replacing (
15) in (
14), we obtain
Now, Taylor’s expansion of
around
is
From the first step in (
1), we have
, where
Then,
and taking the norm
where
and
Using (
2) and conditions (
C1)–(
C4), we have
Now, using (
8), (
9), and the definition for
, we obtain
Then, taking (
11) and (
16), we obtain
where by definition
In the following section, we will establish the recurrence relations and present some technical lemmas, which will be crucial for the demonstration of the convergence properties of the iterative scheme (
1).
3. Recurrence Relations for the Scheme
Consider the following sequences:
for
Now, we present the following lemmas:
Lemma 3. Let be the functions defined by (8), (10), (13) and (17), respectively. Suppose that and . Then, the following can be stated: - (i)
, and are increasing functions and .
- (ii)
is increasing as a function of t and increasing as a function of and .
- (iii)
The sequences are decreasing; and .
Proof of Lemma 3. Let
, and
be the functions defined by (
8), (
10), (
13) and (
17), respectively.
- (i)
Considering
, we determine that the derivative of
is
We have determined that
for all
t in the interval
. Therefore,
is increasing for
. Also, the following can be determined:
and
.
Consequently,
is increasing;
and
. Thus,
since
, then
. Therefore,
is increasing;
and
.
- (ii)
From (
17), we have determined that
then
Consequently,
is increasing as a function of
t and
. In addition,
Therefore, is increasing as a function of u.
It is concluded that is increasing as a function of t and also increasing as a function of u, , and .
- (iii)
Then,
and
as
and
implies that
.
Furthermore, we have determined that
By induction, suppose that
, we have
For the above reasons and the fact that
and
are increasing, we have
Consequently, and are decreasing sequences.
Since is increasing and , then by induction and are obtained. Hence, and .
□
Lemma 4. Let be the functions defined by (8), (10), (13) and (17), respectively. Suppose that . Then, , and . Proof of Lemma 4. For the proof of this lemma, we have considered the definitions of the functions.
Suppose that
. Then,
□
Lemma 5. Under the hypothesis of Lemma 3, if we define and , the following are obtained:
- (i)
.
- (ii)
,
.
- (iii)
.
- (iv)
.
- (v)
.
Proof of Lemma 5. Suppose that and . Then,
- (i)
, because .
- (ii)
,
Using mathematical induction, we can ascertain that
Similarly,
and using mathematical induction, we determine that
- (iii)
- (iv)
As a direct consequence of the previous statement, we have
and the product expansion is
- (v)
Since
and
, we have determined that
; therefore,
We then normalize the terms of the geometric progression to obtain
□
Furthermore, as a consequence of the last statement from the aforementioned Lemma 5, since
and
, we obtain a convergent series which establishes that
Lemma 6. Under the hypothesis of Lemma 3 and conditions (C1)–(C4), it is verified, for all , that
- (i)
exists and .
- (ii)
.
- (iii)
.
- (iv)
.
- (v)
, and
- (vi)
Proof of Lemma 6. By following the previously outlined development and employing an inductive method, the proof of this lemma can be obtained.
- (i)
Let
; then,
exists by the Banach Lemma and
with
. Moreover, there exists
, and from (
16), we have
By induction,
exists and
- (ii)
From (
C2), we have determined that
, with
. Thus,
Using induction, the following is obtained:
- (iii)
We have determined that
, with
and
. Therefore,
Using this method,
, and by induction,
- (iv)
Similarly to the previous item,
Using this method,
, and by induction,
- (v)
For this proof, we proceed as follows:
with
.
For the sequence
, we have
Therefore, .
- (vi)
From the definition of
, and
, the following can be obtained:
where
and
by the hypothesis of Lemma 3 and the definitions of
, and
.
□
5. Numerical Examples
In this section, we will apply the theoretical results to applied problems, some of which have already been cited in earlier studies.
Example 1 (see [
17,
19])
. Consider the nonlinear integral equation , where with and . This is a Chandrasekhar integral equation. These equations represent a category of integral equations used to tackle problems related to radiative transfer theory in a plane-parallel atmosphere. Additionally, these equations are applicable in other areas of research, such as traffic modeling, queuing theory, neutron transport, and the kinetic theory of gases. Let , the space of continuous functions defined on with the norm The first and second derivatives of
are, respectively,
and
for
.
The second derivative verifies
and using the mean value theorem,
with
Taking the norm in (
24),
we consider the Lipschitz condition
Thus, .
Choosing an initial estimate of the solution,
, we obtain
and
Therefore, by the Banach Lemma,
exists and
The auxiliary functions and recurrence relations of the scheme for some chosen values of parameter
are shown in
Table 1 and
Table 2.
Next, let us discretize the integral equation using Simpson’s quadrature, with the aim of transforming it into a large finite-dimensional problem. Let us denote the nodes as
for
and the weights as
, where
is the step size and
n is an even number of subintervals. Expressing
, we obtain the (nonlinear) system of equations as follows:
Table 3 displays the results of a comparative study between a selected member of the family of methods (
1) and three other iterative methods, including the Newton [
1], Chun [
28], and Jarratt [
29] methods. The solution obtained by the schemes is
for
, where the number of iterations (iter) required to converge to the solution is presented, as well as the execution time (ex-time) in seconds required for convergence to the solution, which is determined as the average over 10 consecutive runs for each scheme. The stopping criterion is
, where
represents the error estimate between two consecutive iterations,
denotes the residual error, and the error tolerance is set to
. The approximated computational order of convergence (ACOC) (see [
30]) quantifies the rate at which successive iterates approach the solution, providing an empirical estimate of the method’s order of convergence. It is employed to verify the validity of the theoretical convergence order
p.
Example 2. Consider the system of equations of size with the following expression:with and with the norm The first derivative of
is
Choosing an initial estimate of the solution,
and denoting
, we obtain
Therefore, the system fails to meet the conditions established in Banach’s Lemma. For this example, the results obtained for
are presented in
Table 4.
This iterative scheme is not limited to systems with standard algebraic or differential structures; rather, it extends naturally to a broader class of nonlinear operator equations, including those that include coefficients involving domain integrals, such as terms of the form , provided that certain conditions are met. These include that the operator is well defined in a suitable Banach space, the existence and continuity of its Fréchet derivative, and the satisfaction of regularity conditions such as the Lipschitz continuity. Integral terms are common in nonlinear and functional analysis, and their presence does not rule out the use of iterative methods, especially when the operator remains differentiable and bounded. Therefore, with appropriate analytical justification, such operators are compatible with standard semilocal and local convergence theories.