Extension of an Eighth-Order Iterative Technique to Address Non-Linear Problems

Abstract

The convergence order of an iterative method used to solve equations is usually determined by using Taylor series expansions, which in turn require high-order derivatives, which are not necessarily present in the method. Therefore, such convergence analysis cannot guarantee the theoretical convergence of the method to a solution if these derivatives do not exist. However, the method can converge. This indicates that the most sufficient convergence conditions required by the Taylor approach can be replaced by weaker ones. Other drawbacks exist, such as information on the isolation of simple solutions or the number of iterations that must be performed to achieve the desired error tolerance. This paper positively addresses all these issues by considering a technique that uses only the operators on the method and Ω-generalized continuity to control the derivative. Moreover, both local and semi-local convergence analyses are presented for Banach space-valued operators. The technique can be used to extend the applicability of other methods along the same lines. A large number of concrete examples are shown in which the convergence conditions are fulfilled.

1. Introduction

In this study, our goal is to obtain a solution ${\varkappa }^{*}$ of the non-linear equation

$$P\left(x\right)=0,$$

(1)

where $P:\mathbb{D}\subset {\mathbb{G}}_1\to {\mathbb{G}}_2,$ is assumed to be a differentiable operator in the Fréchet sense with ${\mathbb{G}}_1$ and ${\mathbb{G}}_2$ Banach spaces, and $\mathbb{D}$ is an open and convex set. Only in certain cases is it possible to obtain exactly the solution ${\varkappa }^{*}$. As a result, given certain assumptions based on the initial estimate, researchers rely on the construction of iterative methods that produce a sequence converging to ${\varkappa }^{*}$. The fixed point method, successive substitutions method or Picard method is defined by

$$x_{\wp +1}=Q\left(x_{\wp }\right),$$

where $Q:{\mathbb{G}}_1\to {\mathbb{G}}_1$ is a continuous operator. This method is of convergence order one [1,2,3,4].

Newton’s method [1,2,3,4,5,6] is a well-known one-step iterative procedure, which is defined for $x_0\in \mathbb{D}$ and each $\wp =0,1,2,$ … by

$$x_{\wp +1}=x_{\wp }-P^{\prime }{\left(x_{\wp }\right)}^{-1}P\left(x_{\wp }\right).$$

This method has convergence of second order if $x_0$ is chosen close enough to the solution denoted by ${\varkappa }^{*}$. By adding substeps to a one-step method, higher-order convergence methods (of order greater than two) have been obtained, as found in the literature [7,8,9,10,11,12,13,14]. As an example, the Traub two-step method

$$\begin{array}{cc}\hfill y_{\wp }& =x_{\wp }-P^{\prime }{\left(x_{\wp }\right)}^{-1}P\left(x_{\wp }\right),\hfill \\ \hfill x_{\wp +1}& =y_{\wp }-P^{\prime }{\left(x_{\wp }\right)}^{-1}P\left(y_{\wp }\right)\hfill \end{array}$$

is of order three, whereas the two-step method

$$\begin{array}{cc}\hfill y_{\wp }& =x_{\wp }-P^{\prime }{\left(x_{\wp }\right)}^{-1}P\left(x_{\wp }\right),\hfill \\ \hfill x_{\wp +1}& =y_{\wp }-P^{\prime }{\left(y_{\wp }\right)}^{-1}P\left(y_{\wp }\right)\hfill \end{array}$$

is of order four. Furthermore, numerous one- and multi-step methods have been proposed to improve the convergence order and computational cost of Newton’s method [15,16,17,18,19,20,21,22,23].

For a given $x_0\in \mathbb{D}$, let us consider the following multi-step iterative approach to solving (1), given in [24]:

$$\begin{array}{cc}\hfill y_{\wp }& =x_{\wp }-P^{\prime }{\left(x_{\wp }\right)}^{-1}P\left(x_{\wp }\right),\hfill \\ \hfill A_{\wp }& =P^{\prime }{\left(x_{\wp }\right)}^{-1}\left(P^{\prime }\left(x_{\wp }\right)-P^{\prime }\left(y_{\wp }\right)\right),\hfill \\ \hfill z_{\wp }& =y_{\wp }-\left[I+\left(I+{\displaystyle \frac{5}{4}}{A}_{\wp }\right)A_{\wp }\right]P^{\prime }{\left(x_{\wp }\right)}^{-1}P\left(y_{\wp }\right),\hfill \\ \hfill x_{\wp +1}& =z_{\wp }-\left[I+\left(I+{\displaystyle \frac{3}{2}}{A}_{\wp }\right)A_{\wp }\right]P^{\prime }{\left(x_{\wp }\right)}^{-1}P\left(z_{\wp }\right),\wp =0,1,2,\dots .\hfill \end{array}$$

(2)

It was shown that this method presents eighth-order convergence, highlighting that it only uses one operator inversion per iteration. Its high efficiency was demonstrated by comparing it with other methods that have appeared in the literature. However, there are certain problems limiting the applicability of this method (2), particularly when Taylor series are used to establish convergence. These problems are as follows.

 (P₁) 

The convergence order eight was determined in [24] provided that ${\mathbb{G}}_1={\mathbb{G}}_2={\mathbb{R}}^k$ and assuming the existence and boundedness of higher-order derivatives $P^{\left(i\right)},i=2,3,\dots ,9,$ which do not appear in the formulation of the method. Let us see an example with ${\mathbb{G}}_1={\mathbb{G}}_2=\mathbb{R},\mathbb{D}=[-1.25,1.25]$. Define the function $p:\mathbb{D}\to \mathbb{R}$ by $p\left(t\right)={\theta }_1{t}^{2}logt+{\theta }_2{t}^3+{\theta }_3{t}^4$, if $t\ne 0$, and $p\left(t\right)=0,$ if $t=0$, where ${\theta }_1\ne 0,$ and ${\theta }_2+{\theta }_3=0$. It is clear that $t_{*}=1\in \mathbb{D}$ solves the equation $p\left(t\right)=0$. However, for $j\ge 2$, the derivatives $p^{\left(j\right)}\left(t\right)$ are not continuous at $t=0$. Hence, the results in [24] cannot ensure the convergence of the method to $t_{*}=1$. However, this method converges when taking, for example, $t_0=1.2\in \mathbb{D}$. Consequently, we can ensure that the conditions in [24] related to the convergence of the method are weakened.

In addition, there are other factors that restrict the applicability of this method.

 (P₂) 

No prior estimates are provided, and the number of iterations that must be performed to achieve a predetermined error tolerance is unknown.

 (P₃) 

Since the radius of convergence is not given in [24], selecting initial estimates that guarantee convergence to ${\varkappa }^{*}$ is difficult in general.

 (P₄) 

The uniqueness of ${\varkappa }^{*}$ in a neighborhood around it is not determined.

 (P₅) 

The study in [24] is restricted to ${\mathbb{R}}^k$.

 (P₆) 

The semi-local analysis of convergence, which is in fact the most interesting, has not been developed in [24].

Therefore, issues $\left(P_1\right)$–$\left(P_6\right)$ must be addressed in our study. It is also worth noting that Taylor series expansions constitute the dominant technique for the study of multi-step iterative methods, especially when it comes to showing the convergence order.

As a novelty Our study provides the following new insights.

$\left(P_1^{\prime }\right)$

The new local conditions are based on controlling the first derivative $P^{\prime }$ that is present in the method.

$\left(P_2^{\prime }\right)$

A prior estimate of $\parallel {\varkappa }^{*}-x_{\wp }\parallel$ is developed. Thus, the number of iterations to be performed can be known in advance.

$\left(P_3^{\prime }\right)$

A radius of convergence is provided.

$\left(P_4^{\prime }\right)$

A set is determined that contains only one solution.

$\left(P_5^{\prime }\right)$

The studies are provided for Banach space-valued operators.

$\left(P_6^{\prime }\right)$

A semi-local analysis based on majorizing sequences [2,4] is carried out.

Both types of analyses rely on generalized continuity conditions on $P^{\prime }$, which are employed to control it [1,2,3]. The same set of conditions is used in both analyses. It is also worth noting that the methodology of this study can be applied to other methods using Taylor series and inverses of linear operators, such as those in [6,15,16,17,18,19,20,21,22,23,25,26].

A detailed historical overview of convergence analysis methods can also be found in [2,4,5,21,26]. The flow chart of the proposed convergence technique is divided into two parts.

(1) Local convergence analysis

Sufficient local conditions are provided to establish the convergence of the method. The iterates are shown to exist on a ball centered at the solution and of a certain radius, which is well defined. Convergence is assured provided that the initial guess the initial point $x_0$ is chosen from the ball. It is also shown that the sequence ${\parallel x_{\wp }-{\varkappa }^{*}\parallel }_{p=0}^{\infty }$ converges to zero sequence. Furthermore, a certain ball is specified that contains only one solution of Equation (1).

(2) Semi-local convergence analysis

Scalar sequences majorize (control) the iterates, which are shown to exist in a ball centred at $x_0$ and of a certain computable radius. Convergence is established as long as $\parallel P^{\prime }{\left(x_0\right)}^{-1}P\left(x_0\right)\parallel$ is small enough. A priori estimates of $\parallel x_{\wp +1}-x_{\wp }\parallel$ and $\parallel {\varkappa }^{*}-x_{\wp }\parallel$ determine the number of iterations to that must be performed to reach a predetermined error tolerance. The uniqueness of the solution results in a finite radius ball centered at $x_0$, completing this type of analysis.

The rest of the study is structured as follows: local and semi-local analyses are developed in Section 2 and Section 3, respectively; Section 4 contains concrete numerical applications; and Section 5 presents some concluding remarks.

2. Analysis of the Local Convergence

Let $T=[0,+\infty ]$. The following abbreviations will be used in the analysis.

FCND: a function that is continuous as well as nondecreasing on an interval.

SPS: the smallest and positive solution.

The local analysis relies on the following conditions.

 (H₁) 

There exists an FCND ${\aleph }_0:T\to T$ such that the equation ${\aleph }_0\left(t\right)-1=0$ has the SPS denoted by $s_0$. Let us assume that $T_0=[0,s_0)$.

 (H₂) 

There exists an FCND $\aleph :T_0\to T$ such that, for $l_1:T_0\to T$ defined by

$$l_1\left(t\right)={\displaystyle \frac{{\int }_0^1\aleph \left((1-\theta )t\right)d\theta }{1-{\aleph }_0\left(t\right)}},$$

the equation $l_1\left(t\right)-1=0$ has the SPS in $T_0$. Let $r_1$ stand for this SPS.

 (H₃) 

The equation ${\aleph }_0\left(l_1\left(t\right)t\right)-1=0$ has solutions in $T_0$. Let us denote by $s_1$ the SPS on $T_0$. Set $T_1=[0,s_1)$.

Define the functions $\overline{\aleph }:T_1\to T$, $l_2:T_1\to T$ by

$$\overline{\aleph }\left(t\right)=\left\{\begin{array}{c}\aleph \left(\left(1+l_1\left(t\right)\right)t\right)\hfill \\ \mathrm{or}\hfill \\ {\aleph }_0\left(t\right)+{\aleph }_0\left(l_1\left(t\right)t\right)\hfill \end{array}\right.$$

and

$$\begin{array}{cc}\hfill l_2\left(t\right)=& [{\displaystyle \frac{{\int }_0^1\aleph \left((1-\theta )l_1\left(t\right)t\right)d\theta }{1-{\aleph }_0\left(l_1\left(t\right)t\right)}}+{\displaystyle \frac{\overline{\aleph }\left(t\right)\left(1+{\int }_0^1{\aleph }_0\left(\theta l_1\left(t\right)t\right)d\theta \right)}{(1-{\aleph }_0\left(t\right)\left(1-{\aleph }_0\left(l_1\left(t\right)t\right)\right)}}\hfill \\ \hfill & +\left(1+{\displaystyle \frac{5}{4}}{\displaystyle \frac{\overline{\aleph }\left(t\right)}{1-{\aleph }_0\left(t\right)}}\right){\displaystyle \frac{\overline{\aleph }\left(t\right)\left(1+{\int }_0^1{\aleph }_0\left(\theta l_1\left(t\right)t\right)d\theta \right)}{{\left(1-{\aleph }_0\left(t\right)\right)}^2}}]l_1\left(t\right).\hfill \end{array}$$

The choice of $\overline{\aleph }\left(t\right)$ depends on the functions ${\aleph }_0\left(t\right)$ and $\aleph \left(t\right)$. We will choose the one that provides the largest radius of convergence.

 (H₄) 

The equation $l_2\left(t\right)-1=0$ has the SPS on $T_1-\left\{0\right\}$. Let us denote by $r_2$ the SPS of this equation in $T_1-\left\{0\right\}$.

 (H₅) 

The equation ${\aleph }_0\left(l_2\left(t\right)t\right)-1=0$ has the SPS in $T_1$. Let us denote by $s_2$ the SPS in $T_1-\left\{0\right\}$. Set $T_2=[0,s_2)$ and define the function $l_3:T_2\to T$ by

$$\begin{array}{cc}\hfill l_3\left(t\right)=& [{\displaystyle \frac{{\int }_0^1\aleph \left((1-\theta )l_2\left(t\right)t\right)d\theta }{1-{\aleph }_0\left(l_2\left(t\right)t\right)}}+{\displaystyle \frac{\overline{\overline{\aleph }}\left(t\right)\left(1+{\int }_0^1{\aleph }_0\left(\theta l_2\left(t\right)t\right)d\theta \right)}{\left(1-{\aleph }_0\left(t\right)\right)\left(1-{\aleph }_0\left(l_2\left(t\right)t\right)\right)}}\hfill \\ \hfill & +\left(1+{\displaystyle \frac{3}{2}}{\displaystyle \frac{\overline{\aleph }\left(t\right)}{1-{\aleph }_0\left(t\right)}}\right){\displaystyle \frac{\overline{\aleph }\left(t\right)\left(1+{\int }_0^1{\aleph }_0\left(\theta l_2\left(t\right)t\right)d\theta \right)}{{\left(1-{\aleph }_0\left(t\right)\right)}^2}}]l_2\left(t\right).\hfill \end{array}$$

Define the function $\overline{\overline{\aleph }}:T_2\to T$ by

$$\overline{\overline{\aleph }}\left(t\right)=\left\{\begin{array}{c}\aleph \left(\left(1+l_2\left(t\right)\right)t\right)\hfill \\ \mathrm{or}\hfill \\ {\aleph }_0\left(t\right)+{\aleph }_0\left(l_2\left(t\right)t\right).\hfill \end{array}\right.$$

The choice of $\overline{\aleph }\left(t\right)$ depends on the functions ${\aleph }_0\left(t\right)$ and $\aleph \left(t\right)$. We will choose the one that provides the largest radius of convergence.

 (H₆) 

The equation $l_3\left(t\right)-1=0$ has a solution on $T_2$. Let us denote by $r_3$ the SPS in $T_2-\left\{0\right\}$. Let

$$r=min\left\{r_i\right\},i=1,2,3,\mathrm{and}{T}^{*}=[0,r).$$

(3)

These definition implies that for each $t\in T^{*}$, it holds that

$$0\le {\aleph }_0\left(t\right)<1,$$

(4)

$$0\le {\aleph }_0\left(l_1\left(t\right)t\right)<1,$$

(5)

$$0\le {\aleph }_0\left(l_2\left(t\right)t\right)<1,$$

(6)

and

$$0\le l_i\left(t\right)<1,i=1,2,3.$$

(7)

From now on, by $\mathbb{E}(x,\lambda )$, we mean the open ball with center $x\in \mathbb{D}$ and radius $\lambda >0$. Moreover, the closure of $\mathbb{E}(x,\lambda )$ is denoted by $\mathbb{E}[x,\lambda ]$.

Furthermore, we assume the existence of a linear operator related to the functions ${\aleph }_0$ and ℵ, as defined below:

 (H₇) 

There exists ${\varkappa }^{*}\in \mathbb{D}$ that solves the equation $P\left(x\right)=0$ and there also exists an invertible linear operator M such that, for each $x\in \mathbb{D}$,

$$\parallel M^{-1}\left(P^{\prime }\left(x\right)-M\right)\parallel \le {\aleph }_0(\parallel x-{\varkappa }^{*}\parallel ).$$

Set ${\mathbb{D}}_1=\mathbb{D}\cap \mathbb{E}({\varkappa }^{*},s_0)$.

 (H₈) 

For any $\tilde{x},\overline{x}\in {\mathbb{D}}_1$, we have that

$$\parallel M^{-1}\left(P^{\prime }\left(\tilde{x}\right)-P^{\prime }\left(\overline{x}\right)\right)\parallel \le \aleph (\parallel \tilde{x}-\overline{x}\parallel ).$$

 (H₉) 

$\mathbb{E}[{\varkappa }^{*},r]\subset \mathbb{D}$.

Remark 1.

The choice $M=P^{\prime }\left({\varkappa }^{*}\right)$ is popular, but not necessarily the most flexible one. However, in this case, ${\varkappa }^{*}$ is simple. We do not adopt such an assumption in conditions $\left(H_1\right)$–$\left(H_8\right)$. Consequently, our approach can be used to obtain solutions of multiplicity greater than one provided that $M\ne P^{\prime }\left({\varkappa }^{*}\right)$. Other choices can be $M=I$ (the identity operator) or $M=P^{\prime }\left(\tilde{x}\right)$, where $\tilde{x}\in \mathbb{D}$ is some auxiliary point.

The main local convergence analysis for method (2) is presented below.

Theorem 1.

Suppose that conditions $\left(H_1\right)$–$\left(H_9\right)$ hold. If $x_0\in \mathbb{E}({\varkappa }^{*},r)-\left\{{\varkappa }^{*}\right\}$, then the sequence ${\left\{x_{\wp }\right\}}_{p=0}^{\infty }$ generated by method (2) is well defined on the ball $\mathbb{E}({\varkappa }^{*},r)$. Moreover, for each $\wp =0,1,2,\dots$, it holds that

$$\left\{x_{\wp }\right\}\subset \mathbb{E}({\varkappa }^{*},r),$$

(8)

$$\parallel y_{\wp }-{\varkappa }^{*}\parallel \le l_1(\parallel x_{\wp }-{\varkappa }^{*}\parallel )\parallel x_{\wp }-{\varkappa }^{*}\parallel \le \parallel x_{\wp }-{\varkappa }^{*}\parallel ,$$

(9)

$$\parallel z_{\wp }-{\varkappa }^{*}\parallel \le l_2(\parallel x_{\wp }-{\varkappa }^{*}\parallel )\parallel x_{\wp }-{\varkappa }^{*}\parallel \le \parallel x_{\wp }-{\varkappa }^{*}\parallel ,$$

(10)

$$\parallel x_{\wp +1}-{\varkappa }^{*}\parallel \le l_3(\parallel x_{\wp }-{\varkappa }^{*}\parallel )\parallel x_{\wp }-{\varkappa }^{*}\parallel \le \parallel x_{\wp }-{\varkappa }^{*}\parallel ,$$

(11)

where the radius of convergence r is given in (1), and the functions $l_i$ are those given previously.

Proof. 

By the hypothesis $x_0\in \mathbb{E}({\varkappa }^{*},r)-\left\{{\varkappa }^{*}\right\}\subset \mathbb{E}({\varkappa }^{*},r)$. So, the expression (4) clearly holds if $\wp =0$. Let $u\in \mathbb{E}({\varkappa }^{*},r)$. The application of Condition $\left(H_7\right)$ and the Formula (3) give in return

$$∥M^{-1}\left(P^{\prime }\left(u\right)-M\right)∥\le {\aleph }_0(\parallel u-{\varkappa }^{*}\parallel )\le {\aleph }_0\left(r\right)<1.$$

(12)

The estimate (12), in combination with the perturbation lemma on linear and invertible operators by Banach [2], implies the existence of $P^{\prime }\left(u\right)$, which verifies

$$\parallel P^{\prime }{\left(u\right)}^{-1}M\parallel \le {\displaystyle \frac{1}{1-{\aleph }_0(\parallel u-{\varkappa }^{*}\parallel )}}.$$

(13)

In particular, for $u=x_0$, the linear operator $P^{\prime }{\left(x_0\right)}^{-1}$ exists. Thus, the iterate $y_0$ exists. Then, we can write

$$\begin{array}{cc}\hfill y_0-{\varkappa }^{*}=& x_0-{\varkappa }^{*}-P^{\prime }{\left(x_0\right)}^{-1}P\left(x_0\right)\hfill \\ \hfill =& {\int }_0^1\left[P^{\prime }{\left(x_0\right)}^{-1}{P}^{\prime }\left({\varkappa }^{*}\right)\right]\left[P^{\prime }\left({\varkappa }^{*}+\theta (x_0-{\varkappa }^{*})\right)-P^{\prime }\left(x_0\right)\right]d\theta (x_0-{\varkappa }^{*}).\hfill \end{array}$$

(14)

Using condition $\left(H_8\right)$, (13) (for $u=x_0$), (7) (for i = 1), and Formulas (3) and (14), we arrive at

$$\begin{array}{cc}\hfill \parallel y_0-{\varkappa }^{*}\parallel =& {\displaystyle \frac{{\int }_0^1\aleph \left((1-\theta )\parallel x_0-{\varkappa }^{*}\parallel \right)d\theta \parallel x_0-{\varkappa }^{*}\parallel }{1-{\aleph }_0(\parallel x_0-{\varkappa }^{*}\parallel )}}\hfill \\ \hfill \le & l_1(\parallel x_0-{\varkappa }^{*}\parallel )\parallel x_0-{\varkappa }^{*}\parallel \le \parallel x_0-{\varkappa }^{*}\parallel <r.\hfill \end{array}$$

(15)

Thus, the iterate $y_0\in \mathbb{E}(x_0,r)$, and (9) holds if $\wp =0$. Note that $z_0$ and $x_1$ are given in the second and third steps of (2), from which we can write

$$\begin{array}{cc}\hfill z_0-{\varkappa }^{*}=& y_0-{\varkappa }^{*}-P^{\prime }{\left(y_0\right)}^{-1}P\left(y_0\right)+P^{\prime }{\left(y_0\right)}^{-1}\left(P^{\prime }\left(y_0\right)-P^{\prime }\left(x_0\right)\right)P^{\prime }{\left(x_0\right)}^{-1}P\left(y_0\right)\hfill \\ \hfill & -\left(I+{\displaystyle \frac{5}{4}}{A}_0\right)A_0{P}^{\prime }{\left(x_0\right)}^{-1}P\left(y_0\right),\hfill \end{array}$$

leading to

$$\begin{array}{cc}\hfill \parallel z_0-{\varkappa }^{*}\parallel =& [{\displaystyle \frac{{\int }_0^1\aleph \left((1-\theta )\parallel y_0-{\varkappa }^{*}\parallel \right)d\theta }{1-{\aleph }_0(\parallel y_0-{\varkappa }^{*}\parallel )}}+{\displaystyle \frac{{\overline{\aleph }}_0\left(1+{\int }_0^1{\aleph }_0\left(\theta \parallel y_0-{\varkappa }^{*}\parallel \right)d\theta \right)}{\left(1-{\aleph }_0(\parallel x_0-{\varkappa }^{*}\parallel )\right)\left(1-{\aleph }_0(\parallel y_0-{\varkappa }^{*}\parallel )\right)}}\hfill \\ \hfill & +\left(1+{\displaystyle \frac{5}{4}}{\displaystyle \frac{{\aleph }_0}{\left(1-{\aleph }_0(\parallel x_0-{\varkappa }^{*}\parallel )\right)}}\right){\displaystyle \frac{{\overline{\aleph }}_0(1+{\int }_0^1{\aleph }_0\left(\theta \parallel y_0-{\varkappa }^{*}\parallel \right)d\theta )}{{\left(1-{\aleph }_0(\parallel x_0-{\varkappa }^{*}\parallel )\right)}^2}}]\parallel y_0-{\varkappa }^{*}\parallel \hfill \\ \hfill \le & l_1(\parallel x_0-{\varkappa }^{*}\parallel )\parallel x_0-{\varkappa }^{*}\parallel \le \parallel x_0-{\varkappa }^{*}\parallel ,\hfill \end{array}$$

(16)

where ${\overline{\aleph }}_0=\overline{\aleph }(\parallel x_0-{\varkappa }^{*}\parallel )$, and we also use the estimates

$$\begin{array}{cc}\hfill \parallel M^{-1}\left(P^{\prime }\left(y_0\right)-P^{\prime }\left(x_0\right)\right)\parallel & \le \aleph (\parallel y_0-x_0\parallel )\le \aleph (\parallel y_0-{\varkappa }^{*}\parallel +\parallel x_0-{\varkappa }^{*}\parallel )\hfill \\ \hfill & \le \aleph \left[\left(1+l_1(\parallel x_0-{\varkappa }^{*}\parallel )\right)\parallel x_0-{\varkappa }^{*}\parallel \right]\le {\overline{\aleph }}_0,\hfill \end{array}$$

(17)

or

$$\begin{array}{cc}\hfill \parallel M^{-1}\left(P^{\prime }\left(y_0\right)-P^{\prime }\left(x_0\right)\right)\parallel & \le \parallel M^{-1}\left(P^{\prime }\left(y_0\right)-M\right)\parallel +\parallel M^{-1}\left(P^{\prime }\left(x_0\right)-M\right)\parallel \hfill \\ \hfill & \le {\aleph }_0(\parallel y_0-{\varkappa }^{*}\parallel +{\aleph }_0\parallel x_0-{\varkappa }^{*}\parallel )\hfill \\ \hfill & \le {\aleph }_0\left(l_1(\parallel x_0-{\varkappa }^{*}\parallel )\right)\parallel x_0-{\varkappa }^{*}\parallel +{\aleph }_0(\parallel x_0-{\varkappa }^{*}\parallel )\le {\overline{\aleph }}_0\hfill \end{array}$$

(18)

and

$$P\left(y_0\right)=P\left(y_0\right)-P\left({\varkappa }^{*}\right)={\int }_0^1{P}^{\prime }\left({\varkappa }^{*}+\theta (y_0-{\varkappa }^{*})\right)d\theta (y_0-{\varkappa }^{*}),$$

leading to

$$\begin{array}{cc}\hfill \parallel M^{-1}{P}^{\prime }\left(y_0\right)\parallel & \le \parallel M^{-1}{\int }_0^1\left[P^{\prime }({\varkappa }^{*}+\theta (y_0-{\varkappa }^{*}))-M+M\right]d\theta (y_0-{\varkappa }^{*})\parallel \hfill \\ \hfill & \le \left(1+{\int }_0^1{\aleph }_0\left(\theta \parallel y_0-{\varkappa }^{*}\parallel \right)d\theta \right)\parallel y_0-{\varkappa }^{*}\parallel .\hfill \end{array}$$

(19)

Furthermore, from the last step of (2), we can write

$$\begin{array}{cc}\hfill x_1-{\varkappa }^{*}=& z_0-{\varkappa }^{*}-P^{\prime }{\left(z_0\right)}^{-1}P\left(z_0\right)+P^{\prime }{\left(z_0\right)}^{-1}\left(P^{\prime }\left(x_0\right)-P^{\prime }\left(z_0\right)\right)P^{\prime }{\left(z_0\right)}^{-1}P\left(z_0\right)\hfill \\ \hfill & -\left(I+{\displaystyle \frac{3}{2}}{A}_0\right)A_0{P}^{\prime }{\left(x_0\right)}^{-1}P\left(z_0\right).\hfill \end{array}$$

(20)

Thus, from (7) (for $i=3$), (13) $(\mathrm{for}u=x_0,z_0)$, (15)–(19) for $(y_0,z_0)$, and (20), we arrive at

$$\begin{array}{cc}\hfill \parallel x_1-{\varkappa }^{*}\parallel =& [{\displaystyle \frac{{\int }_0^1\aleph \left((1-\theta )\parallel z_0-{\varkappa }^{*}\parallel \right)d\theta }{1-{\aleph }_0(\parallel z_0-{\varkappa }^{*}\parallel )}}+{\displaystyle \frac{{\overline{\overline{\aleph }}}_0\left(1+{\int }_0^1{\aleph }_0\left(\theta \parallel z_0-{\varkappa }^{*}\parallel \right)d\theta \right)}{\left(1-{\aleph }_0(\parallel x_0-{\varkappa }^{*}\parallel )\right)\left(1-{\aleph }_0(\parallel z_0-{\varkappa }^{*}\parallel )\right)}}\hfill \\ \hfill & +\left(1+{\displaystyle \frac{3}{2}}{\displaystyle \frac{{\overline{\aleph }}_0}{\left(1-{\aleph }_0(\parallel x_0-{\varkappa }^{*}\parallel )\right)}}\right){\displaystyle \frac{\left(1+{\int }_0^1{\aleph }_0\left(\theta \parallel z_0-{\varkappa }^{*}\parallel \right)d\theta \right)}{(1-{\aleph }_0(\parallel x_0-{\varkappa }^{*}{\parallel \left)\right)}^2}}]\parallel z_0-{\varkappa }^{*}|\hfill \\ \hfill \le & l_3(\parallel x_0-{\varkappa }^{*}\parallel )\parallel x_0-{\varkappa }^{*}\parallel \le \parallel x_0-{\varkappa }^{*}\parallel ,\hfill \end{array}$$

(21)

where ${\overline{\overline{\aleph }}}_0=\overline{\overline{\aleph }}(\parallel x_0-{\varkappa }^{*}\parallel )$.

Hence, the iterate $x_1\in \mathbb{E}({\varkappa }^{*},r)$, and (8) holds for $\wp =1$ and $\wp =0$, respectively. Proceeding inductively, one can prove (8)–(11) provided that the iterates $x_m,y_m,x_{m+1},z_m$ replace $x_0,y_0,z_0,x_1$, respectively, in the preceding calculations. Finally, from the estimation

$$\begin{array}{cc}\hfill \parallel x_{m+1}-x_m\parallel & \le c\parallel x_m-{\varkappa }^{*}\parallel \hfill \\ \hfill & \le c^{m+1}\parallel x_m-{\varkappa }^{*}\parallel <r,\hfill \end{array}$$

(22)

where $c=l_3(\parallel x_0-{\varkappa }^{*}\parallel )\in [0,1]$, we conclude that the iterate $x_{m+1}\in \mathbb{E}({\varkappa }^{*},r)$ and ${\displaystyle \underset{m\to \infty }{lim}{x}_m={\varkappa }^{*}}$. The uniqueness of the solution ${\varkappa }^{*}$ in a certain neighborhood containing it is given in the next result. □

Proposition 1.

Let us suppose that there exists $s_3>0$ such that condition $\left(H_7\right)$ is satisfied in the ball $\mathbb{E}({\varkappa }^{*},s_3)$, and there exists $s_4\ge s_3$ such that, for the function ${\aleph }_0$ defined in $\left(C_1\right)$, it holds that

$${\int }_0^1{\aleph }_0\left(\theta s_4\right)d\theta <1.$$

(23)

Set ${\mathbb{D}}_3=\mathbb{D}\cap \mathbb{E}[{\varkappa }^{*},s_4]$. Then, the equation $P\left(x\right)=0$ has a unique solution ${\varkappa }^{*}$ in ${\mathbb{D}}_3$.

Proof. 

We proceed by contradiction. Suppose that there exists ${\varkappa }^{**}\in {\mathbb{D}}_3$ such that $P\left({\varkappa }^{**}\right)=0$. Define the linear operator $L_1$ by $L_1={\int }_0^1{P}^{\prime }\left({\varkappa }^{*}+\theta ({\varkappa }^{**}-{\varkappa }^{*})\right)d\theta$. It follows from (23) and $\left(H_7\right)$ that

$$\parallel M^{-1}(L_1-M)\parallel \le {\int }_0^1{\aleph }_0(\theta \parallel {\varkappa }^{**}-{\varkappa }^{*}\parallel )d\theta \le {\int }_0^1{\aleph }_0\left(\theta s_4\right)d\theta <1.$$

(24)

Thus, the operator $L_1$ is invertible. Consequently, from the identity

$${\varkappa }^{**}-{\varkappa }^{*}=L_1^{-1}\left(P\left({\varkappa }^{**}\right)-P\left({\varkappa }^{*}\right)\right)=L_1^{-1}\left(0\right)=0,$$

(25)

we conclude that ${\varkappa }^{**}={\varkappa }^{*}$. □

Remark 2.

Under all conditions $\left(H_1\right)$–$\left(H_9\right)$, we will obtain $s_3=r$ using Proposition 1.

3. Semi-Local Analysis for Method (2)

The conditions for this case, as well as the calculations, are the same as in the local case. In this case, the point ${\varkappa }^{*}$ and the functions ${\aleph }_0$ and ℵ are replaced by $x_0$ and the functions $v_0$ and v, respectively.

Now, we assume the following.

 (C₁) 

There exists a $FCND$$v_0:T\to T$ such that $v_0\left(t\right)-1=0$ has the SPS in $T_0$. Denote by ${\rho }_0$ the $SPS$ in $T_0$. Set $T_3=[0,{\rho }_0)$.

 (C₂) 

There exists a $FCND$$v:T_3\to T$ such that $v\left(t\right)-1=0$ has the SPS in $T_3$. Define the sequences $\left\{{\alpha }_{\wp }\right\}$, $\left\{{\beta }_{\wp }\right\}$, and $\left\{{\gamma }_{\wp }\right\}$ for ${\alpha }_0=0$, some ${\beta }_0\ge 0$, and each $\wp =0,1,2,\dots$ by

$${\lambda }_{\wp }={\int }_0^{1}v\left((1-\theta )({\beta }_{\wp }-{\alpha }_{\wp })\right)d\theta ({\beta }_{\wp }-{\alpha }_{\wp }),$$

$${\overline{v}}_{\wp }=\left\{\begin{array}{c}v({\beta }_{\wp }-{\alpha }_{\wp })\hfill \\ \mathrm{or}\hfill \\ v_0\left({\alpha }_{\wp }\right)+v_0\left({\beta }_{\wp }\right),\hfill \end{array}\right.$$

(26)

$${\gamma }_{\wp }={\beta }_{\wp }+\left(1+\left(1+{\displaystyle \frac{5}{4}}{\displaystyle \frac{{\overline{v}}_{\wp }}{1-v_0\left({\alpha }_{\wp }\right)}}\right){\displaystyle \frac{{\overline{v}}_{\wp }}{1-v_0\left({\alpha }_{\wp }\right)}}\right){\displaystyle \frac{{\lambda }_{\wp }}{1-v_0\left({\alpha }_{\wp }\right)}},$$

$${\mu }_{\wp }=\left(1+{\int }_0^1{v}_0\left({\beta }_{\wp }+\theta ({\gamma }_{\wp }-{\beta }_{\wp })\right)d\theta \right)({\gamma }_{\wp }-{\beta }_{\wp })+{\lambda }_{\wp },$$

$${\alpha }_{\wp +1}={\gamma }_{\wp }+\left[1+\left(1+{\displaystyle \frac{3}{2}}{\displaystyle \frac{{\overline{v}}_{\wp }}{1-v_0\left({\alpha }_{\wp }\right)}}\right){\displaystyle \frac{{\overline{v}}_{\wp }}{1-v_0\left({\alpha }_{\wp }\right)}}\right]{\displaystyle \frac{{\mu }_{\wp }}{1-v_0\left({\alpha }_{\wp }\right)}},$$

$${\delta }_{\wp +1}={\int }_0^{1}v\left((1-\theta )({\alpha }_{\wp +1}-{\alpha }_{\wp })\right)d\theta ({\alpha }_{\wp +1}-{\alpha }_{\wp })+\left(1+v_0\left({\alpha }_{\wp }\right)\right)({\alpha }_{\wp +1}-{\beta }_{\wp })$$

and

$${\beta }_{\wp +1}={\alpha }_{\wp +1}+{\displaystyle \frac{{\delta }_{\wp }+1}{1-v_0({\alpha }_{\wp }+1)}}.$$

Note that the sequences $\left\{{\alpha }_{\wp }\right\}$, $\left\{{\beta }_{\wp }\right\}$, and $\left\{{\gamma }_{\wp }\right\}$ are majorizing for $\left\{x_{\wp }\right\}$, $\left\{y_{\wp }\right\}$, and $\left\{z_{\wp }\right\}$, respectively (see Theorem 2).

 (C₃) 

There exists ${\rho }_1\in [0,{\rho }_0]$ such that, for each $\wp =0,1,2,\dots ,$

$$v_0\left({\alpha }_{\wp }\right)<1,\mathrm{and}{\alpha }_{\wp }\le {\rho }_1.$$

By adopting the above conditions in (26), we obtain

$$0\le {\alpha }_{\wp }\le {\beta }_{\wp }\le {\gamma }_{\wp }\le {\rho }_1$$

and there exists ${\alpha }^{*}\in [0,{\rho }_1)$ such that ${\displaystyle \underset{\wp \to \infty }{lim}{\alpha }_{\wp }={\alpha }^{*}}$.

 (C₄) 

There exist a point $x_0\in \mathbb{D}$ and an invertible operator K such that, for each $x\in \mathbb{D}$,

$$\parallel K^{-1}\left(P^{\prime }\left(x\right)-K\right)\parallel \le v_0(\parallel x-x_0\parallel ).$$

Set ${\mathbb{D}}_3=\mathbb{D}\cap \mathbb{E}(x_0,{\rho }_3)$. Notice that, if $x=x_0$, we have

$$\parallel K^{-1}\left(P^{\prime }\left(x_0\right)-M\right)\parallel \le v_0\left(0\right)<1.$$

Thus, $P^{\prime }\left(x_0\right)$ is invertible. Hence, we can take ${\beta }_0\ge \parallel P^{\prime }{\left(x_0\right)}^{-1}P\left(x_0\right)\parallel$.

 (C₅) 

For each $\overline{x},\tilde{x}\in \mathbb{D}$, we have

$$\parallel K^{-1}\left(P^{\prime }\left(\tilde{x}\right)-P^{\prime }\left(\overline{x}\right)\right)\parallel \le v(\parallel \tilde{x}-\overline{x}\parallel ).$$

 (C₆) 

$\mathbb{E}[x_0,a^{*}]\subset \mathbb{D}$.

Remark 3.

(i) We can take $K=P^{\prime }\left(x_0\right)$, although this is not the most flexible choice. Other choices can be $K=I$ or $K=P^{\prime }\left(\overline{x}\right),$ where $\overline{x}$ is an auxiliary point.

The main semi-local analysis of convergence follows for method (2).

Theorem 2.

Suppose that conditions $\left(C_1\right)$–$\left(C_6\right)$ hold. Then, the sequence ${\left\{x_{\wp }\right\}}_{p=0}^{\infty }$ is well defined and it holds that

$$\left\{x_{\wp }\right\}\subset \mathbb{E}(x_0,a^{*}),$$

(27)

$$\parallel y_{\wp }-x_{\wp }\parallel \le {\beta }_{\wp }-{\alpha }_{\wp },$$

(28)

$$\parallel z_{\wp }-y_{\wp }\parallel \le {\gamma }_{\wp }-{\beta }_{\wp }$$

(29)

and

$$\parallel x_{\wp +1}-z_{\wp }\parallel \le {\alpha }_{\wp +1}-{\gamma }_{\wp }.$$

(30)

Furthermore, there exists ${\varkappa }^{*}\in \mathbb{E}[x_0,a^{*}]$ solving Equation (1) such that

$$\parallel {\varkappa }^{*}-x_{\wp }\parallel \le a^{*}-a_{\wp }.$$

(31)

Proof. 

As in the local analysis, mathematical induction is employed to prove items (27)–(30). From the definition of ${\beta }_0$, (26), and the first substep of method (2), we obtain

$$\parallel y_0-x_0\parallel =\parallel P^{\prime }{\left(x_0\right)}^{-1}P\left(x_0\right)\parallel \le {\beta }_0={\beta }_0-{\alpha }_0<a_{*}.$$

Thus, we have that $y_0\in \mathbb{E}(x_0,a^{*})$, and (28) holds if $m=0$. Let $u\in \mathbb{E}(x_0,a^{*})$. From the definition of $a^{*}$, (26) and the condition $\left(C_4\right)$, we obtain

$$\parallel M^{-1}\left(P^{\prime }\left(u\right)-M\right)\parallel \le v_0(\parallel u-x_0\parallel )\le v_0\left(a^{*}\right)<1.$$

Therefore, the linear operator $P^{\prime }\left(u\right)$ is invertible, and

$$\parallel P^{\prime }{\left(u\right)}^{-1}M\parallel \le {\displaystyle \frac{1}{1-v_0(\parallel u-x_0\parallel )}}.$$

(32)

If $u=x_0$ in (32), then $P^{\prime }\left(x_0\right)$ is invertible. Consequently, the iterates $y_0$, $x_0$, and $x_1$ are well defined. We have

$$\begin{array}{cc}\hfill P\left(y_m\right)& =P\left(y_m\right)-P\left(x_m\right)-P^{\prime }\left(x_m\right)(y_m-x_m)\hfill \\ \hfill & ={\int }_0^1\left[P^{\prime }\left(x_m+\theta (y_m-x_m)\right)-P^{\prime }\left(x_m\right)\right]d\theta (y_m-x_m),\hfill \end{array}$$

Hence, we obtain

$$\begin{array}{cc}\hfill \parallel M^{-1}P\left(y_m\right)\parallel & \le {\int }_0^{1}v\left((1-\theta )\parallel y_m-x_m\parallel \right)d\theta \parallel y_m-x_m\parallel \hfill \\ \hfill & \le {\int }_0^{1}v\left((1-\theta )\parallel {\beta }_m-{\alpha }_m\parallel \right)d\theta \parallel {\beta }_m-{\alpha }_m\parallel ={\lambda }_m,\hfill \end{array}$$

(33)

$$\parallel M^{-1}\left(P^{\prime }\left(y_m\right)-P^{\prime }\left(x_m\right)\right)\parallel \le v(\parallel y_m-x_m\parallel )\le v(\parallel {\beta }_m-{\alpha }_m\parallel )\le {\overline{v}}_m$$

(34)

or

$$\begin{array}{cc}\hfill \parallel M^{-1}\left(P^{\prime }\left(y_m\right)-P^{\prime }\left(x_m\right)\right)\parallel & \le \parallel M^{-1}\left(P^{\prime }\left(y_m\right)-M\right)\parallel +\parallel M^{-1}\left(P^{\prime }\left(x_m\right)-M\right)\parallel \hfill \\ \hfill & \le v_0(\parallel y_m-x_0\parallel )+v_0(\parallel x_m-x_0\parallel )+v_0\left({\alpha }_m\right)+v_0\left({\beta }_m\right)\le {\overline{v}}_m.\hfill \end{array}$$

(35)

Then, from the second substep of (2), (26), (32) for $(u=x_0)$, and (33)–(35), we have that

$$\begin{array}{cc}\hfill \parallel z_m-y_m\parallel & \le \left[1+\left(1+{\displaystyle \frac{5}{4}}{\displaystyle \frac{{\overline{v}}_m}{1-v_0\left({\alpha }_m\right)}}\right){\displaystyle \frac{{\overline{v}}_m}{1-v_0\left({\alpha }_m\right)}}\right]{\displaystyle \frac{{\lambda }_m}{1-v_0\left({\alpha }_m\right)}}\hfill \\ \hfill & ={\gamma }_m-{\beta }_m\hfill \end{array}$$

(36)

and

$$\parallel z_m-x_0\parallel \le \parallel z_m-y_m\parallel +\parallel y_m-x_0\parallel \le {\gamma }_m-{\beta }_m+{\beta }_m-{\alpha }_0={\gamma }_m<a^{*}.$$

Thus, the iterate $z_m\in \mathbb{E}(x_0,a^{*})$, and (29) holds if $m=0$. We can write

$$P\left(z_m\right)=P\left(z_m\right)-P\left(y_m\right)+P\left(y_m\right).$$

Therefore, we obtain

$$\parallel L^{-1}P\left(z_m\right)\parallel \le \left(1+{\int }_0^1{v}_0\left({\beta }_m+\theta ({\gamma }_m-{\beta }_m)\right)d\theta \right)({\gamma }_m-{\beta }_m)+{\lambda }_m={\mu }_m,$$

(37)

where we have used

$$P\left(z_m\right)-P\left(y_m\right)={\int }_0^1{P}^{\prime }\left(y_m+\theta (z_m-y_m)\right)d\theta (z_m-y_m).$$

Then, we yield

$$\begin{array}{cc}\hfill \parallel M^{-1}\left(P\left(z_m\right)-P\left(y_m\right)\right)\parallel & \le \parallel {\int }_0^1{M}^{-1}\left[P^{\prime }\left(y_m+\theta (z_m-y_m)\right)-M+M\right]d\theta (z_m-y_m)\parallel \hfill \\ \hfill & \le \left(1+{\int }_0^1{v}_0\left(\parallel y_m-x_0\parallel +\theta \parallel z_m-y_m\parallel \right)d\theta \right)\parallel z_m-y_m\parallel \hfill \\ \hfill & \le \left(1+{\int }_0^1{v}_0\left({\beta }_m+\theta ({\gamma }_m-{\beta }_m)\right)d\theta \right)({\gamma }_m-{\beta }_m).\hfill \end{array}$$

Next, from the third substep of method (2), (26), and (37), we obtain in turn

$$\begin{array}{cc}\hfill \parallel x_{m+1}-z_m\parallel & \le \left[\parallel I\parallel +\left(\parallel I\parallel +{\displaystyle \frac{3}{2}}\parallel A_m\parallel \right)\parallel A_m\parallel \right]\parallel P^{\prime }{\left(x_m\right)}^{-1}M\parallel \parallel M^{-1}P\left(z_m\right)\parallel \hfill \\ \hfill & \le \left[1+\left(1+{\displaystyle \frac{3}{2}}{\displaystyle \frac{{\overline{v}}_m}{1-v_0\left({\alpha }_m\right)}}\right){\displaystyle \frac{{\overline{v}}_m}{1-v_0\left({\alpha }_m\right)}}\right]{\displaystyle \frac{{\mu }_m}{1-v_0\left({\alpha }_m\right)}}\hfill \\ \hfill & =x_{m+1}-{\gamma }_m\hfill \end{array}$$

(38)

and

$$\begin{array}{cc}\hfill \parallel x_{m+1}-x_0\parallel & \le \parallel x_{m+1}-z_m\parallel +\parallel z_m-x_0\parallel \hfill \\ \hfill & \le {\alpha }_{m+1}-{\gamma }_m+{\gamma }_m-{\alpha }_0={\alpha }_{m+1}<a^{*}.\hfill \end{array}$$

Thus, the iterates $x_{m+1}\in \mathbb{E}(x_0,a^{*})$, and (30) holds. We can write

$$P\left(x_{m+1}\right)=P\left(x_{m+1}\right)-P\left(x_m\right)-P^{\prime }\left(x_m\right)(x_{m+1}-x_m)+P^{\prime }\left(x_m\right)(x_{m+1}-y_m),$$

$$\parallel M^{-1}P\left(x_{m+1}\right)\parallel \le {\overline{\delta }}_{m+1}\le {\delta }_{m+1},$$

(39)

leading to

$$\begin{array}{cc}\hfill \parallel y_{m+1}-x_{m+1}\parallel & \le \parallel P^{\prime }{\left(x_{m+1}\right)}^{-1}M\parallel \parallel M^{-1}P\left(x_{m+1}\right)\parallel \hfill \\ \hfill & \le {\displaystyle \frac{{\overline{\delta }}_{m+1}}{1-v_0(\parallel x_{m+1}-x_0\parallel )}}\hfill \\ \hfill & \le {\displaystyle \frac{{\delta }_{m+1}}{1-v_0\left({\alpha }_{m+1}\right)}}={\beta }_{m+1}-{\alpha }_{m+1}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \parallel y_{m+1}-x_0\parallel & \le \parallel y_{m+1}-x_{m+1}\parallel +\parallel x_{m+1}-x_0\parallel \hfill \\ \hfill & \le {\beta }_{m+1}-{\alpha }_{m+1}+{\alpha }_{m+1}-{\alpha }_0\hfill \\ \hfill & ={\beta }_{m+1}<{\alpha }^{*}.\hfill \end{array}$$

Taking $m\to +\infty$ in (39) and using the continuity of P, we deduce that $P\left({\varkappa }^{*}\right)=0$. Finally, the estimate (31) follows from

$$\parallel x_{m+j}-x_m\parallel \le {\alpha }_{m+j}-{\alpha }_m$$

(40)

assuming that $j\to +\infty$. □

As in the local analysis, the uniqueness of the solution of equation $P\left(x\right)=0$ can be obtained as follows.

Proposition 2.

Let us assume that there exists a solution $y^{*}\in \mathbb{E}(x_0,{\rho }_2)$ of equation $P\left(x\right)=0$ for some ${\rho }_2>0$, the condition $\left(C_4\right)$ holds on the ball $\mathbb{E}(x_0,{\rho }_2)$, and there exists ${\rho }_3\ge {\rho }_2$ such that

$${\int }_0^1{v}_0\left((1-\theta ){\rho }_2+\theta {\rho }_3\right)d\theta <1.$$

(41)

Set ${\mathbb{D}}_4=\mathbb{D}\cap \mathbb{E}[x_0,{\rho }_3]$.

Then, the equation $P\left(x\right)=0$ has only one solution in ${\mathbb{D}}_4$.

Proof. 

Suppose that there also exists $z^{*}\in {\mathbb{D}}_4$ such that $P\left(z^{*}\right)=0$. Define the linear operator $L_2=P^{\prime }\left(y^{*}+\theta (z^{*}-y^{*})\right)d\theta$. It follows from $\left(C_4\right)$ and (41) that

$$\begin{array}{cc}\hfill \parallel M^{-1}(L_2-M)\parallel & \le {\int }_0^1{v}_0\left((1-\theta )\parallel y^{*}-x_0\parallel +\theta \parallel z^{*}-x_0\parallel \right)d\theta \hfill \\ \hfill & \le {\int }_0^1{v}_0\left((1-\theta ){\rho }_2+\theta {\rho }_3\right)d\theta <1,\hfill \end{array}$$

Therefore, the linear operator $L_2$ is invertible, and $z^{*}=y^{*}$. □

Remark 4.

(i) The limit point $a^{*}$ can be replaced by ${\rho }_0$ in condition $\left(C_6\right).$

(ii)

If all conditions $\left(C_1\right)$–$\left(C_6\right)$ hold, then set ${\rho }_2=a^{*}$ and $y^{*}={\varkappa }^{*}$ in Proposition 2.

4. Numerical Examples

In this section, we present the computational results, developed through a plethora of numerical examples. Among them, two examples are academic in nature, while the remaining four involve applied science problems, including the Hammerstein integral equation of the first kind, the Fisher’s equation, and boundary value problems (BVP). The corresponding results for these examples are collected in Table 1, Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7. In addition, we compute the computational order of convergence $\left(COC\right)$ based on the formula

$$\lambda ={\displaystyle \frac{ln\frac{\parallel x_{j+1}-{\varkappa }^{*}\parallel }{|x_j-{\varkappa }^{*}\parallel }}{ln\frac{\parallel x_j-{\varkappa }^{*}\parallel }{\parallel x_{j-1}-{\varkappa }^{*}\parallel }}},forj=1,2,\dots$$

(42)

or the approximate computational order of convergence $\left(ACOC\right)$ [25,26] given by

$${\lambda }^{*}={\displaystyle \frac{ln\frac{\parallel x_{j+1}-x_j\parallel }{\parallel x_j-x_{j-1}\parallel }}{ln\frac{\parallel x_j-x_{j-1}\parallel }{\parallel x_{j-1}-x_{j-2}\parallel }}},forj=2,3,\dots$$

(43)

The stopping criteria and error tolerance are established on the following basis:

Table 1. Example 1: Radii of convergence.

Method	${\mathit{s}}_0$	${\mathit{r}}_1$	${\mathit{s}}_1$	${\mathit{r}}_2$	${\mathit{r}}_3$	r
(2)	0.58198	0.38269	0.42236	0.21234	0.17579	0.17579

Table 2. Example 2: Radii of convergence.

Method	${\mathit{s}}_0$	${\mathit{r}}_1$	${\mathit{s}}_1$	${\mathit{r}}_2$	${\mathit{r}}_3$	r
(2)	0.16667	0.083333	0.103006	0.039488	0.031547	0.031547

Table 3. Computational results of Example 3.

Method	${\mathit{x}}_0$	$\parallel \mathit{P}\left({\mathit{x}}_{\mathit{\wp }}\right)\parallel$	$\parallel {\mathit{x}}_{\mathit{\wp }+1}-{\mathit{x}}_{\mathit{\wp }}\parallel$	℘	${\mathit{\lambda }}^{*}$	CPU
						Timing
(2)	$\left[\begin{array}{c}{\displaystyle \frac{1001}{1000}}\\ ⋮\\ {\displaystyle \frac{1001}{1000}}\end{array}\right]$	$1.6\times {10}^{-1494}$	$3.4\times {10}^{-1493}$	4	$6.0364$	$291.68$

Table 4. Computational results of Example 4.

Methods	${\mathit{x}}_0$	$\parallel \mathit{P}\left({\mathit{x}}_{\mathit{\wp }}\right)\parallel$	$\parallel {\mathit{x}}_{\mathit{\wp }+1}-{\mathit{x}}_{\mathit{\wp }}\parallel$	℘	${\mathit{\lambda }}^{*}$	CPU
						Timing
(2)	$\left[\begin{array}{c}1+{\displaystyle \frac{1}{121}}\\ 1+{\displaystyle \frac{2}{121}}\\ ⋮\\ 1+{\displaystyle \frac{121}{121}}.\end{array}\right]$	$3.3\times {10}^{-1197}$	$2.2\times {10}^{-1198}$	4	$5.9812$	$198.29$

Table 5. The values of abscissas $t_j$ and weights $w_j$.

j	${\mathit{t}}_{\mathit{j}}$	${\mathit{w}}_{\mathit{j}}$
1	$0.01304673574141413996101799\dots$	$0.03333567215434406879678440\dots$
2	$0.06746831665550774463395165\dots$	$0.07472567457529029657288816\dots$
3	$0.16029521585048779688283632\dots$	$0.10954318125799102199776746\dots$
4	$0.28330230293537640460036703\dots$	$0.13463335965499817754561346\dots$
5	$0.42556283050918439455758700\dots$	$0.14776211235737643508694649\dots$
6	$0.57443716949081560544241300\dots$	$0.14776211235737643508694649\dots$
7	$0.71669769706462359539963297\dots$	$0.13463335965499817754561346\dots$
8	$0.83970478414951220311716368\dots$	$0.10954318125799102199776746\dots$
9	$0.93253168334449225536604834\dots$	$0.07472567457529029657288816\dots$
10	$0.98695326425858586003898201\dots$	$0.03333567215434406879678440\dots$

Table 6. Computational results of Example 5.

Methods	${\mathit{x}}_0$	$\parallel \mathit{P}\left({\mathit{x}}_{\mathit{\wp }}\right)\parallel$	$\parallel {\mathit{x}}_{\mathit{\wp }+1}-{\mathit{x}}_{\mathit{\wp }}\parallel$	℘	${\mathit{\lambda }}^{*}$	CPU
						Timing
(2)	$\left[\begin{array}{c}{\displaystyle \frac{9}{10}}\\ ⋮\\ {\displaystyle \frac{9}{10}}.\end{array}\right]$	$4.7\times {10}^{-510}$	$9.9\times {10}^{-511}$	3	$5.9783$	$0.457194$

Table 7. Computational results of Example 6.

Method	${\mathit{x}}_0$	$\parallel \mathit{P}\left({\mathit{x}}_{\mathit{\wp }}\right)\parallel$	$\parallel {\mathit{x}}_{\mathit{\wp }+1}-{\mathit{x}}_{\mathit{\wp }}\parallel$	℘	$\mathit{\lambda }$	CPU
						Timing
(2)	$\left[\begin{array}{c}0.95\\ ⋮\\ 0.95\end{array}\right]$	$7.0\times {10}^{-468}$	$2.3\times {10}^{-468}$	3	$8.1734$	$698.052$

$\left(i\right)\parallel x_{j+1}-x_j\parallel <ϵ;$

$\left(ii\right)\parallel P\left(x_j\right)\parallel <ϵ$, where $ϵ={10}^{-400}$.

$Mathematica11$ with multiple precision arithmetic is used for the computational results.

The first two examples are used to validate local convergence through conditions $\left(H_1\right)$–$\left(H_9\right)$, provided that $M=P^{\prime }\left({\varkappa }^{*}\right)$.

Example 1.

Let us assume that ${\mathbb{G}}_1={\mathbb{G}}_2={\mathbb{R}}^3$ and $\mathbb{D}=\mathbb{E}(0,1)$. In addition, we choose P on $\mathbb{D}$ with $u=\left[\begin{array}{c}{u}_1\\ u_2\\ u_3\end{array}\right]$ as

$$P\left(u\right)=\left[\begin{array}{cc}\hfill & u_3\hfill \\ \hfill & {\displaystyle \frac{e-1}{2}}{u}_2^2+u_3\hfill \\ \hfill & e^{u_1}-1\hfill \end{array}\right].$$

(44)

It follows from (44) that

$$P^{\prime }\left(u\right)=\left[\begin{array}{ccc}0& 0& 1\\ 0& (e-1)u_2& 1\\ e^{u_1}& 0& 0\end{array}\right].$$

It is straightforward to see that ${\varkappa }^{*}=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$. Then, conditions $\left(H_1\right)$–$\left(H_9\right)$ hold, taking

$${\aleph }_0\left(t\right)=(e-1)t,\mathrm{and}\aleph \left(t\right)=e^{{\displaystyle \frac{1}{e-1}}}t.$$

In Table 1, we collect the obtained radii of convergence for Example 1.

Example 2.

We consider ${\mathbb{G}}_1={\mathbb{G}}_2=C[0,1]$ and $\mathbb{D}=\mathbb{E}(0,1)$. We consider the well-known Hammerstein first-kind non-linear integral operator P:

$$P\left(w\right)\left(x\right)=w\left(x\right)-2{\int }_0^{1}x\xi w{\left(\xi \right)}^{3}d\xi .$$

Then, we obtain

$$P^{\prime }\left(w\left(p\right)\right)\left(x\right)=p\left(x\right)-6{\int }_0^{1}x\xi w{\left(\xi \right)}^{2}p\left(\xi \right)d\xi ,$$

for each $p\in C[0,1]$.

By substituting these values in conditions $\left(H_1\right)$–$\left(H_9\right)$, we obtain

$${\aleph }_0\left(t\right)=6t,\mathrm{and}\aleph \left(t\right)=12t.$$

The radii of convergence when using (2) for Example 2 are depicted in Table 2.

The rest of the examples test the performance of method (2).

Example 3.

Let us consider the following boundary value problem [2]:

$$w^{″}+{\mu }^2{w}^{\prime 2}+1=0$$

(45)

with $w\left(0\right)=0$ and $w\left(1\right)=1$. The interval $[0,1]$ is uniformly divided into l subintervals, which yields

$${\gamma }_0=0<{\gamma }_1<{\gamma }_2<\dots <{\gamma }_{l-1}<{\gamma }_l=1,{\gamma }_{j+1}={\gamma }_j+h,h={\displaystyle \frac{1}{l}}.$$

Then, we can choose $w_0=w\left({\gamma }_0\right)=0,w_1=w\left({\gamma }_1\right),\dots ,w_{l-1}=w\left({\gamma }_{l-1}\right),w_l=w\left({\gamma }_l\right)=1$. Using the classical second-order approximations for the derivatives

$$w_{\theta }^{\prime }={\displaystyle \frac{w_{\theta +1}-w_{\theta -1}}{2h}},w_{\theta }^{\prime \prime }={\displaystyle \frac{w_{\theta -1}-2w_{\theta }+w_{\theta +1}}{h^2}},\theta \in \mathbb{N},$$

we discretize (45) and obtain the following non-linear system of equations:

$$w_{\theta -1}-2w_{\theta }+w_{\theta +1}+{\displaystyle \frac{{\mu }^2}{4}}{(w_{\theta +1}-w_{\theta -1})}^2+h^2=0,\theta =1,2,\dots ,l-1.$$

(46)

In Table 3, we present the computational results after solving the system (46) by taking $\ell =71$ and $\mu ={\displaystyle \frac{1}{2}}$ with method (2). The approximate root after five iterations is

$$\begin{array}{cc}\hfill x_5=(& 0.02371\dots ,0.04708\dots ,0.07012\dots ,0.09283\dots ,0.1152\dots ,0.1372\dots ,0.1590\dots ,\hfill \\ \hfill & 0.1804\dots ,0.2015\dots ,0.2223\dots ,0.2429\dots ,0.2631\dots ,0.2830\dots ,0.3026\dots ,\hfill \\ \hfill & 0.3219\dots ,0.3409\dots ,0.3597\dots ,0.3781\dots ,0.3963\dots ,0.4142\dots ,0.4318\dots ,\hfill \\ \hfill & 0.4491\dots ,0.4662\dots ,0.4830\dots ,0.4995\dots ,0.5158\dots ,0.5318\dots ,0.5475\dots ,\hfill \\ \hfill & 0.5630\dots ,0.5782\dots ,0.5932\dots ,0.6079\dots ,0.6224\dots ,0.6366\dots ,0.6506\dots ,\hfill \\ \hfill & 0.6643\dots ,0.6777\dots ,0.6910\dots ,0.7040\dots ,0.7167\dots ,0.7292\dots ,0.7415\dots ,\hfill \\ \hfill & 0.7535\dots ,0.7653\dots ,0.7769\dots ,0.7883\dots ,0.7994\dots ,0.8103\dots ,0.8209\dots ,\hfill \\ \hfill & 0.8313\dots ,0.8415\dots ,0.8515\dots ,0.8613\dots ,0.8708\dots ,0.8801\dots ,0.8892\dots ,\hfill \\ \hfill & 0.8981\dots ,0.9068\dots ,0.9152\dots ,0.9234\dots ,0.9315\dots ,0.9393\dots ,0.9468\dots ,\hfill \\ \hfill & 0.9542\dots ,0.9614\dots ,0.9683\dots ,0.9751\dots ,0.9816\dots ,0.9879\dots ,0.9940\dots {)}^T\hfill \end{array}$$

Example 4.

Let us consider a well-known Fisher’s problem [6], which is given by

$${\mu }_t=\eta {\mu }_{xx}+\mu (1-\mu )=0,$$

(47)

with homogeneous Neumann’s boundary conditions

$$\begin{array}{cc}\hfill & \mu (x,0)=1.5+0.5cos\left(\pi x\right),0\le x\le 1,\hfill \\ \hfill & {\mu }_x(0,t)=0,{\mu }_x(1,t)=0,0\le t\le 1,\hfill \end{array}$$

where η is a diffusion parameter. We apply the finite difference discretization approach to (47) and obtain a non-linear system, considering a mesh with $N_1$ points in the spatial direction and $N_2$ points in the temporal direction. At the grid points of a mesh, $w_{i,j}$ denotes the approximate values of the solution, i.e., $w_{i,j}=\mu (x_i,t_j)$. The corresponding step sizes are $h=1/N_1$ and $l=1/N_2$, respectively. By using the following centered, backward, and forward approximations of the derivatives

$$\begin{array}{cc}\hfill & {\mu }_{xx}(x_i,t_j)=(w_{i+1,j}-2w_{i,j}+w_{i-1,j})/h^2,\hfill \\ \hfill & {\mu }_t(x_i,t_j)=(w_{i,j}-w_{i,j-1})/l,\hfill \\ \hfill & \mathit{and}\hfill \\ \hfill & {\mu }_x(x_i,t_j)=(w_{i+1,j}-w_{i,j})/h,t\in [0,1],\hfill \end{array}$$

we obtain a discretization of problem (47) given by the system

$$\begin{array}{c}\hfill {\displaystyle \frac{w_{1,j}-w_{i,j-1}}{l}}-w_{i,j}\left(1-w_{i,j}\right)-\eta {\displaystyle \frac{w_{i+1,j}-2w_{i,j}+w_{i-1,j}}{h^2}},i=1,2,3,\dots ,N_1;j=1,2,3,\dots ,N_2.\end{array}$$

(48)

Choosing $N_1=N_2=11$, we obtain a system of 121 equations with 121 unknowns. For the numerical computations, we take $\eta =1.$ The approximate solution of this non-linear system after five iterations of method (2) is given below:

$$\begin{array}{cc}\hfill x_5=(& 1.645\dots ,1.473\dots ,1.375\dots ,1.312\dots ,1.269\dots ,1.236\dots ,1.210\dots ,1.188\dots ,\hfill \\ \hfill & 1.169\dots ,1.153\dots ,1.138\dots ,1.623\dots ,1.464\dots ,1.370\dots ,1.310\dots ,1.268\dots ,\hfill \\ \hfill & 1.236\dots ,1.210\dots ,1.188\dots ,1.169\dots ,1.153\dots ,1.138\dots ,1.583\dots ,1.445\dots ,\hfill \\ \hfill & 1.361\dots ,1.306\dots ,1.266\dots ,1.235\dots ,1.209\dots ,1.188\dots ,1.169\dots ,1.153\dots ,\hfill \\ \hfill & 1.138\dots ,1.528\dots ,1.419\dots ,1.349\dots ,1.300\dots ,1.263\dots ,1.233\dots ,1.208\dots ,\hfill \\ \hfill & 1.187\dots ,1.169\dots ,1.153\dots ,1.138\dots ,1.463\dots ,1.388\dots ,1.334\dots ,1.292\dots ,\hfill \\ \hfill & 1.259\dots ,1.231\dots ,1.208\dots ,1.187\dots ,1.169\dots ,1.153\dots ,1.138\dots ,1.395\dots ,\hfill \\ \hfill & 1.355\dots ,1.317\dots ,1.284\dots ,1.255\dots ,1.229\dots ,1.207\dots ,1.186\dots ,1.168\dots ,\hfill \\ \hfill & 1.152\dots ,1.138\dots ,1.328\dots ,1.322\dots ,1.301\dots ,1.276\dots ,1.251\dots ,1.227\dots ,\hfill \\ \hfill & 1.206\dots ,1.186\dots ,1.168\dots ,1.152\dots ,1.138\dots ,1.268\dots ,1.292\dots ,1.286\dots ,\hfill \\ \hfill & 1.269\dots ,1.248\dots ,1.226\dots ,1.205\dots ,1.186\dots ,1.168\dots ,1.152\dots ,1.138\dots ,\hfill \\ \hfill & 1.219\dots ,1.267\dots ,1.274\dots ,1.263\dots ,1.245\dots ,1.224\dots ,1.204\dots ,1.185\dots ,\hfill \\ \hfill & 1.168\dots ,1.152\dots ,1.138\dots ,1.185\dots ,1.250\dots ,1.265\dots ,1.258\dots ,1.242\dots ,\hfill \\ \hfill & 1.223\dots ,1.204\dots ,1.185\dots ,1.168\dots ,1.152\dots ,1.138\dots ,1.168\dots ,1.240\dots ,\hfill \\ \hfill & 1.261\dots ,1.256\dots ,1.241\dots ,1.223\dots ,1.203\dots ,1.185\dots ,1.168\dots ,1.152\dots ,\hfill \\ \hfill & 1.138\dots {)}^T.\hfill \end{array}$$

We illustrate the numerical results in Table 4.

Example 5.

We study the Hammerstein non-linear integral equation (details can be found in [2] (pp. 19–20)). The following Hammerstein integral equation is a standard applied science example of computational analysis:

$$x\left(s\right)=1+{\displaystyle \frac{1}{5}}{\int }_0^{1}G(s,t)x{\left(t\right)}^{3}dt,$$

where $x\in C[0,1],s,t\in [0,1]$ and the kernel G is

$$G(s,t)=\left\{\begin{array}{c}(1-s)t,t\le s,\hfill \\ s(1-t),s\le t.\hfill \end{array}\right.$$

The Gauss–Legendre quadrature formula can be used to transform the above equation into a finite-dimensional problem. We approximate the integral by ${\displaystyle {\int }_0^{1}g\left(t\right)dt\simeq \sum _{j=1}^{10}{w}_{j}g\left(t_j\right),}$ considering appropriate weights $w_j$ and abscissas $t_j$. For $i=j=10$, the $t_j$ and $w_j$ are depicted in Table 5. The $x_i(i=1,2,\dots ,10)$ are adopted to represent the approximations of $x\left(t_i\right)$. Thus, we obtain the following system of non-linear equations:

$${\displaystyle 5x_i-5-\sum _{j=1}^{10}{a}_{ij}{x}_j^3=0,}$$

where

$$a_{ij}=\left\{\begin{array}{c}\hfill w_j{t}_j(1-t_i),j\le i,\\ \hfill w_j{t}_i(1-t_j),i<j.\end{array}\right.$$

In Table 6, we collect the obtained data with method (2). The approximate root after four iterations is

$$\begin{array}{cc}\hfill x_4=(& 1.001\dots ,1.006\dots ,1.014\dots ,1.021\dots ,1.026\dots ,1.026\dots ,1.021\dots ,1.014\dots ,\hfill \\ \hfill & 1.006\dots ,1.0013\dots {)}^T.\hfill \end{array}$$

Example 6.

Finally, we examine a more complex system of non-linear equations, consisting of 300 equations with 300 unknowns. This analysis highlights the method’s ability to handle significant computational challenges and demonstrates its applicability for a wide range of practical applications involving large-scale non-linear systems. We consider the following system:

$$P\left(X\right)=\left\{\begin{array}{c}{x}_j^2{x}_{j+1}-1=0,1\le j\le 299,\hfill \\ x_{300}^2{x}_1-1=0,otherwise.\hfill \end{array}\right.$$

(49)

The desired zero of this problem is ${\varkappa }^{*}=\left[\begin{array}{c}1\\ 1\\ ⋮\\ 1\end{array}\right]$. The number of iterations, the CPU time, the absolute value of function at the corresponding point, the absolute residual error, and the COC of Example 6 are shown in Table 7.

5. Concluding Remarks

In this study, certain drawbacks are identified that prevent the applicability of iterative methods if the usual Taylor expansion series approach is utilized to demonstrate convergence. Motivated by these issues, and in order to extend the applicability of iterative methods, a different technique is developed that does not use the Taylor series. In this way, both local and semi-local convergence analyses are based solely on the operators that define the methods, which extends their applicability to more abstract spaces, such as Banach spaces. Although the technique has been demonstrated with method (2), it can also be used to analyze other methods, such as those in [6,15,16,17,18,19,20,21,22,23,25,26]. Numerous concrete examples have been included to demonstrate the presented approach. This will be the direction of our future studies. In addition, we will try to further weaken the sufficient convergence conditions and even consider the necessary ones.