# Generalized Iterative Method of Order Four with Divided Differences

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## Abstract

**:**

## 1. Introduction

**Motivation**

- $\left({\mathcal{C}}_{1}\right)$
- A priori upper bounds on $\parallel {x}_{m}-\lambda \parallel $ are not given, $\lambda \in \mathcal{D}$ being a solution of the Equation (1). The number of iterations to be performed to reach a predecided error tolerance is not known.
- $\left({\mathcal{C}}_{2}\right)$
- The initial guess ${x}_{0}$ is “shot in dark”, and no information is available on the uniqueness of the solution.
- $\left({\mathcal{C}}_{3}\right)$
- There convergence of the method is not assured (although it may converge to $\lambda $) if at least ${H}^{\left(5\right)}$ does not exist.
- $\left({\mathcal{C}}_{4}\right)$
- The results are limited to the case only when $\mathcal{S}={\mathbb{R}}^{k}$.
- $\left({\mathcal{C}}_{5}\right)$
- The semi-local convergence, more interesting than the local convergence, is not given in [19].
- $\left({\mathcal{C}}_{6}\right)$

**Novelty**

## 2. Local Analysis

- $\left({E}_{1}\right)$
- There exist functions ${g}_{1}:\mathcal{Q}\to \mathcal{Q}$, ${\phi}_{0}:\mathcal{Q}\times \mathcal{Q}\to \mathcal{Q}$ which are continuous as well as non-decreasing (FCN) such that the equation ${\phi}_{0}(t,{g}_{1}\left(t\right))-1=0$ has a minimal positive solution called $P.$ Define the set ${\mathcal{Q}}_{0}=[0,P)$.
- $\left({E}_{2}\right)$
- There exist FCN ${g}_{2}:{\mathcal{Q}}_{0}\to \mathcal{Q}$, ${\phi}_{3}:{\mathcal{Q}}_{0}\to \mathcal{Q}$, $\phi :{\mathcal{Q}}_{0}\times {\mathcal{Q}}_{0}\to \mathcal{Q}$, ${\phi}_{1}:{\mathcal{Q}}_{0}\times {\mathcal{Q}}_{0}\times {\mathcal{Q}}_{0}\to \mathcal{Q}$ and ${\phi}_{2}:{\mathcal{Q}}_{0}\times {\mathcal{Q}}_{0}\times {\mathcal{Q}}_{0}\times {\mathcal{Q}}_{0}\to \mathcal{Q}$ such that the equations ${h}_{i}\left(t\right)-1=0$, $i=1,2$ have minimal positive solutions in the interval $(0,P)$ denoted by ${P}_{i}$, respectively, where the functions ${h}_{i}:{\mathcal{Q}}_{0}\to \mathcal{Q}$ are given as$$\begin{array}{cc}\hfill {h}_{1}\left(t\right)& =\frac{\phi (t,{g}_{1}\left(t\right))}{1-{\phi}_{0}(t,{g}_{1}\left(t\right))},\hfill \\ \hfill a\left(t\right)& =\frac{{\phi}_{2}(t,{h}_{1}\left(t\right)t,{g}_{1}\left(t\right),{g}_{2}\left(t\right))}{1-{\phi}_{0}(t,{g}_{1}\left(t\right))}\hfill \end{array}$$$$\begin{array}{c}\hfill {h}_{2}\left(t\right)=\left[\frac{{\phi}_{1}(t,{h}_{1}\left(t\right)t,{g}_{1}\left(t\right))+a\left(t\right)(1+2a\left(t\right))(1+{\phi}_{3}\left({h}_{1}\left(t\right)t\right))}{1-{\phi}_{0}(t,{g}_{1}\left(t\right))}\right]{h}_{1}\left(t\right).\end{array}$$$$\begin{array}{c}\hfill \mathrm{Define}{P}^{\ast}=min\left\{{P}_{i}\right\}.\end{array}$$
- $\left({E}_{3}\right)$
- There exist an invertible operator M and a solution $\lambda \in \mathcal{D}$ of the Equation (1) such that for all $x\in \mathcal{D}$, $u=x+H\left(x\right)$,$$\begin{array}{c}\hfill \parallel u-\lambda \parallel \le {g}_{1}(\parallel x-\lambda \parallel )\end{array}$$$$\begin{array}{c}\hfill \parallel {M}^{-1}([x,u,;H]-M)\parallel \le {\phi}_{0}(\parallel x-\lambda \parallel ,\parallel u-\lambda \parallel ).\end{array}$$Define the set ${\mathcal{Q}}_{0}=\mathcal{D}\cap \mathcal{B}(\lambda ,P)$.
- $\left({E}_{4}\right)$
- $$\begin{array}{cc}\hfill \parallel {M}^{-1}([x,u;H]-[x,\lambda ;H])\parallel & \le \phi (\parallel x-\lambda \parallel ,\parallel u-\lambda \parallel ),\hfill \\ \hfill \parallel {M}^{-1}([x,u;H]-[y,\lambda ;H])\parallel & \le {\phi}_{1}(\parallel x-\lambda \parallel ,\parallel y-\lambda \parallel ,\parallel u-\lambda \parallel ),\hfill \\ \hfill \parallel {M}^{-1}([x,u;H]-[y,v;H])\parallel & \le {\phi}_{2}(\parallel x-\lambda \parallel ,\parallel y-\lambda \parallel ,\parallel u-\lambda \parallel ,\parallel v-\lambda \parallel ),\hfill \\ \hfill \parallel v-\lambda \parallel \le {g}_{2}(\parallel y-\lambda \parallel ),& v=y+H\left(y\right),\hspace{1em}y=x-{[x,u;H]}^{-1}H\left(x\right),\hfill \end{array}$$$$\begin{array}{c}\hfill \parallel {M}^{-1}([y,\lambda ;H]-M)\parallel \le {\phi}_{3}(\parallel y-\lambda \parallel ).\end{array}$$Notice that by the definition of P, $\left({E}_{1}\right)$, and $\left({E}_{3}\right)$,$$\begin{array}{c}\hfill \parallel {M}^{-1}([x,u;H]-M)\parallel \le {\phi}_{0}(\parallel x-\lambda \parallel ,{g}_{1}(\parallel x-\lambda \parallel \left)\right)<1.\end{array}$$
- $\left({E}_{5}\right)$
- $\mathcal{B}[\lambda ,{\overline{P}}^{\ast}]\subset \mathcal{D}$, where ${\overline{P}}^{\ast}=max\{{P}^{\ast},{g}_{1}\left({P}^{\ast}\right),{g}_{2}\left({P}^{\ast}\right)\}$.

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Proposition**

**1.**

**Proof.**

## 3. Semi-Local Analysis

- $\left({T}_{1}\right)$
- There exists FCN ${g}_{3}:\mathcal{Q}\to \mathcal{Q}$, ${\psi}_{0}:\mathcal{Q}\times \mathcal{Q}\to \mathcal{Q}$ such that the equation ${\psi}_{0}(t,{g}_{3}\left(t\right))-1=0$ has a minimal positive solution denoted by q. Let ${\mathcal{Q}}_{1}=[0,q)$. Consider FCN ${g}_{4}:{\mathcal{Q}}_{1}\to M$, $\psi :{\mathcal{Q}}_{1}\times {\mathcal{Q}}_{1}\to M$, ${\psi}_{1}:{\mathcal{Q}}_{1}\times {\mathcal{Q}}_{1}\times {\mathcal{Q}}_{1}\to M$ and ${\psi}_{2}:{\mathcal{Q}}_{1}\times {\mathcal{Q}}_{1}\times {\mathcal{Q}}_{1}\times {\mathcal{Q}}_{1}\to M$. Define for ${\alpha}_{0}=0$, ${\beta}_{0}\ge 0$ the sequence $\left\{{\alpha}_{m}\right\}$ as$$\begin{array}{cc}\hfill {b}_{m}& =\frac{\psi ({\alpha}_{m},{\beta}_{m},{g}_{3}\left({\alpha}_{m}\right),{g}_{4}\left({\beta}_{m}\right))}{1-{\psi}_{0}({\alpha}_{m},{g}_{3}\left({\alpha}_{m}\right))},\hfill \\ \hfill {c}_{m}& =\psi ({\alpha}_{m},{\beta}_{m},{g}_{3}\left({\alpha}_{m}\right))({\beta}_{m}-{\alpha}_{m}),\hfill \\ \hfill {\alpha}_{m+1}& ={\beta}_{m}+\frac{(1+{b}_{m}+2{b}_{m}^{2}){c}_{m}}{1-{\psi}_{0}({\alpha}_{m},{g}_{3}\left({\alpha}_{m}\right))},\hfill \\ \hfill {d}_{m+1}& =(1+\psi ({\alpha}_{m},{\alpha}_{m+1}))({\alpha}_{m+1}-{\alpha}_{m})\hfill \\ \hfill & +(1+{\psi}_{0}({\alpha}_{m},{g}_{3}\left({\alpha}_{m}\right))({\beta}_{m}-{\alpha}_{m})\hfill \end{array}$$$$\begin{array}{c}\hfill {b}_{m+1}={\alpha}_{m+1}+\frac{{d}_{m+1}}{1-{\psi}_{0}({\alpha}_{m},{g}_{3}\left({\alpha}_{m}\right))}.\end{array}$$
- $\left({T}_{2}\right)$
- There exists ${q}_{0}\in [0,q)$ such that for all $m=0,1,2,\cdots $$$\begin{array}{c}\hfill {\psi}_{0}({\alpha}_{m},{g}_{3}\left({\alpha}_{m}\right)<1\mathrm{and}{\alpha}_{m}\le {q}_{0}.\end{array}$$It follows via (12) and $\left({T}_{2}\right)$ that$$\begin{array}{c}\hfill 0\le {\alpha}_{m}\le {\beta}_{m}\le {\alpha}_{m+1}<{q}_{0},\end{array}$$
- $\left({T}_{3}\right)$
- There exist an invertible linear operator M and ${x}_{0}\in \mathcal{D}$ such that$$\begin{array}{cc}\hfill \parallel {M}^{-1}([x,y,;H]-M)\parallel & \le \psi (\parallel x-{x}_{0}\parallel ,\parallel y-{x}_{0}\parallel ),\hfill \\ \hfill \parallel {M}^{-1}([y,x,;H]-[x,u;H])\parallel & \le {\psi}_{1}(\parallel x-{x}_{0}\parallel ,\parallel y-{x}_{0}\parallel ,\parallel u-{x}_{0}\parallel ),\hfill \\ \hfill \parallel {M}^{-1}([x,u,;H]-[y,v;H])\parallel & \le {\psi}_{2}(\parallel x-{x}_{0}\parallel ,\parallel y-{x}_{0}\parallel ,\parallel u-{x}_{0}\parallel ,\parallel v-{x}_{0}\parallel )\hfill \end{array}$$$$\begin{array}{c}\hfill \parallel v-{x}_{0}\parallel \le {g}_{4}(\parallel y-{x}_{0}\parallel )\end{array}$$It follows via conditions $\left({T}_{1}\right)$ and $\left({T}_{3}\right)$ that$$\begin{array}{cc}\hfill \parallel {M}^{-1}([{x}_{0},{u}_{0};H]-M)\parallel & \le {\psi}_{0}(\parallel {x}_{0}-{x}_{0}\parallel ,\parallel u-{x}_{0}\parallel )\hfill \\ \hfill & \le {\psi}_{0}(0,{g}_{3}(\parallel {x}_{0}-{x}_{0}\parallel )={\psi}_{0}(0,{g}_{3}\left(0\right))1.\hfill \end{array}$$Thus, ${\mathcal{G}}_{0}^{-1}$ exists. Set $\parallel {\mathcal{G}}_{0}^{-1}G\left({x}_{0}\right)\parallel \le {\beta}_{0}$.

- $\left({T}_{4}\right)$
- $\mathcal{B}[{x}_{0},{\overline{q}}^{\ast}]\subset \mathcal{D}$, where ${\overline{q}}^{\ast}=max\{{q}^{\ast},{g}_{3}\left({q}^{\ast}\right),{g}_{4}\left({q}^{\ast}\right)\}$.

**Theorem**

**2.**

**Proof.**

**Remark**

**2.**

**Proposition 2.**

**Proof.**

**Remark 3.**

## 4. Numerical Examples

**Example 1.**

**Example**

**2.**

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Correction Statement

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**MDPI and ACS Style**

Regmi, S.; Argyros, I.K.; Deep, G.
Generalized Iterative Method of Order Four with Divided Differences. *Foundations* **2023**, *3*, 561-572.
https://doi.org/10.3390/foundations3030033

**AMA Style**

Regmi S, Argyros IK, Deep G.
Generalized Iterative Method of Order Four with Divided Differences. *Foundations*. 2023; 3(3):561-572.
https://doi.org/10.3390/foundations3030033

**Chicago/Turabian Style**

Regmi, Samundra, Ioannis K. Argyros, and Gagan Deep.
2023. "Generalized Iterative Method of Order Four with Divided Differences" *Foundations* 3, no. 3: 561-572.
https://doi.org/10.3390/foundations3030033