# Semilocal Convergence of the Extension of Chun’s Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Recurrence Relations

- (C
_{1}) - $\parallel {\Gamma}_{0}\parallel \le \beta $,
- (C
_{2}) - $\parallel {\Gamma}_{0}F\left({x}_{0}\right)\parallel \le \eta $,
- (C
_{3}) - $\parallel {F}^{\prime}\left(x\right)-{F}^{\prime}\left(y\right)\parallel \le K\parallel x-y\parallel $,

#### Preliminary Results

- (I
_{n}) - $\parallel {\Gamma}_{n}\parallel \le f\left({a}_{n-1}\right)\parallel {\Gamma}_{n-1}\parallel $,
- (II
_{n}) - $\parallel {y}_{n}-{x}_{n}\parallel =\parallel {\Gamma}_{n}F\left({x}_{n}\right)\parallel \le f\left({a}_{n-1}\right)g\left({a}_{n-1}\right)\parallel {y}_{n-1}-{x}_{n-1}\parallel $,
- (III
_{n}) - $K\parallel {\Gamma}_{n}\parallel \parallel {y}_{n}-{x}_{n}\parallel \le {a}_{n}$,
- (IV
_{n}) - $\parallel {x}_{n}-{x}_{n-1}\parallel \le \left(1+{\displaystyle \frac{1}{2}}h\left({a}_{n-1}\right)\right)\parallel {y}_{n-1}-{x}_{n-1}\parallel $.

- (II
_{1}): - By means of the Taylor’s expansion of $F\left({x}_{1}\right)$ around ${y}_{0}$, we get

- (III
_{1}): - using (${I}_{1}$) and ($I{I}_{1}$),

- (IV
_{1}): - for $n=1$ it has been proven in (9).

## 3. Convergence Analysis

**Lemma**

**1.**

- (i)
- $f\left(x\right)$ is increasing and $f\left(x\right)>1$ for $x\in (0,0.650629)$,
- (ii)
- $h\left(x\right)$ and $g\left(x\right)$ are increasing for $x\in (0,0.650629)$.

**Proof.**

**Lemma**

**2.**

- (i)
- $f\left({a}_{0}\right)g\left({a}_{0}\right)<1$ for ${a}_{0}<0.367826$,
- (ii)
- $f{\left({a}_{0}\right)}^{2}g\left({a}_{0}\right)<1$ for ${a}_{0}<0.300637$,
- (iii)
- the sequence $\left\{{a}_{n}\right\}$ is decreasing and ${a}_{n}<0.300637$ for $n\ge 0.$

**Proof.**

**Theorem**

**1.**

**Proof.**

## 4. Numerical Experiments

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Parameters of (16) for different initial estimations.

${{\mathit{x}}_{0}}_{\mathit{i}}$ | $\mathit{\beta}$ | $\mathit{\eta}$ | ${\mathit{a}}_{0}$ | ${\mathit{R}}_{\mathit{e}}$ | ${\mathit{R}}_{\mathit{u}}$ |
---|---|---|---|---|---|

0 | 1.0000 | 1.0000 | 0.2471 | 2.0825 | 6.0108 |

0.2 | 1.0516 | 0.8465 | 0.2200 | 1.5346 | 6.1614 |

0.4 | 1.1080 | 0.6864 | 0.1879 | 1.0901 | 6.2142 |

0.6 | 1.1699 | 0.5189 | 0.1500 | 0.7256 | 6.1925 |

0.8 | 1.2380 | 0.3428 | 0.1049 | 0.4238 | 6.1134 |

1.0 | 1.3134 | 0.1567 | 0.0509 | 0.1720 | 5.9899 |

1.2 | 1.3973 | 0.1879 | 0.0649 | 0.2123 | 5.5796 |

1.4 | 1.4912 | 0.3889 | 0.1433 | 0.5380 | 4.8892 |

1.6 | 1.5969 | 0.5911 | 0.2333 | 1.1986 | 3.8694 |

**Table 2.**Numerical solution of (16).

i | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

${x}_{i}^{*}$ | 1.0122… | 1.0584… | 1.1181… | 1.1598… | 1.1598… | 1.1181… | 1.0584… | 1.0122… |

${{\mathit{x}}_{0}}_{\mathit{i}}$ | iter | $\parallel {\mathit{x}}_{\mathit{n}}-{\mathit{x}}_{\mathit{n}-1}{\parallel}_{\mathit{\infty}}$ | $\parallel \mathit{F}\left({\mathit{x}}_{\mathit{n}}\right){\parallel}_{\mathit{\infty}}$ | $\mathit{\rho}$ |
---|---|---|---|---|

0.2 | 5 | $5.2749\times {10}^{-189}$ | $9.396\times {10}^{-757}$ | 4.0 |

0.4 | 5 | $2.9207\times {10}^{-211}$ | $8.8316\times {10}^{-846}$ | 4.0 |

0.6 | 5 | $9.5406\times {10}^{-242}$ | $1.0055\times {10}^{-967}$ | 4.0 |

0.8 | 5 | $2.2172\times {10}^{-288}$ | $1.7003\times {10}^{-1008}$ | 4.0 |

1.0 | 5 | $4.7315\times {10}^{-381}$ | $1.8738\times {10}^{-1008}$ | 4.0 |

1.2 | 5 | $1.28\times {10}^{-455}$ | $3.258\times {10}^{-1823}$ | 4.0 |

1.4 | 5 | $9.6618\times {10}^{-300}$ | $1.0576\times {10}^{-1199}$ | 4.0 |

1.6 | 5 | $4.943\times {10}^{-231}$ | $7.2453\times {10}^{-925}$ | 4.0 |

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Cordero, A.; Maimó, J.G.; Martínez, E.; Torregrosa, J.R.; Vassileva, M.P. Semilocal Convergence of the Extension of Chun’s Method. *Axioms* **2021**, *10*, 161.
https://doi.org/10.3390/axioms10030161

**AMA Style**

Cordero A, Maimó JG, Martínez E, Torregrosa JR, Vassileva MP. Semilocal Convergence of the Extension of Chun’s Method. *Axioms*. 2021; 10(3):161.
https://doi.org/10.3390/axioms10030161

**Chicago/Turabian Style**

Cordero, Alicia, Javier G. Maimó, Eulalia Martínez, Juan R. Torregrosa, and María P. Vassileva. 2021. "Semilocal Convergence of the Extension of Chun’s Method" *Axioms* 10, no. 3: 161.
https://doi.org/10.3390/axioms10030161