# Some Iterative Approximation Results of F Iteration Process in Banach Spaces

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- (${a}_{1}$)
- the asymptotic radius r of the sequence $\left\{{x}_{k}\right\}$ at the point s is the real number $r(s,\left\{{x}_{k}\right\}):={lim\; sup}_{k\to \infty}\left|\right|s-{x}_{k}\left|\right|$;
- (${a}_{2}$)
- the asymptotic radius of the sequence $\left\{{x}_{k}\right\}$ in the connection with D is given by $inf\{r(s,\left\{{x}_{k}\right\}):s\in D\}=r(D,\left\{{x}_{k}\right\})$
- (${a}_{3}$)
- the asymptotic center A of the sequence $\left\{{x}_{k}\right\}$ in the connection with D is given by $\{s\in D:r(s,\left\{{x}_{k}\right\})=r(D,\left\{{x}_{k}\right\})\}=A(D,\left\{{x}_{k}\right\})$.

**Proposition**

**1.**

- (i)
- For every choice of $v\in D$ and $u\in F\left(T\right)$, follow that $\left|\right|Tv-Tu\left|\right|\le \left|\right|v-u\left|\right|$.
- (ii)
- The set $F\left(T\right)$ is always closed. However if X is strictly convex and the domain D is convex, then the set $F\left(T\right)$ also enjoys convexity.
- (iii)
- For every choice of $v,{v}^{\prime}$ in D, it follows that$$\left|\right|v-T{v}^{\prime}\left|\right|\le 9\left|\right|v-Tv\left|\right|+\left|\right|v-{v}^{\prime}\left|\right|.$$
- (iv)
- If we assume that X has Opial property, and $\left\{{x}_{k}\right\}$ is a weakly convergent to some element u with ${lim}_{k\to \infty}\left|\right|T{x}_{k}-{x}_{k}\left|\right|=0$, then u is the point of of the set $F\left(T\right)$.

**Lemma**

**1.**

## 3. Approximation Results

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Theorem**

**5.**

**Proof.**

## 4. Example

**Example**

**1.**

**Case(a):**when $v,{v}^{\prime}\in [3,6)$. Then $Tv=\frac{v+3}{2}$ and $T{v}^{\prime}=\frac{{v}^{\prime}+3}{2}$. Keeping triangle inequality in mind, we have

**Case(b):**when $v\in [3,6)$ and ${v}^{\prime}\in \left\{6\right\}$. Then $Tv=\frac{v+3}{2}$ and $T{v}^{\prime}=3$. Now

**Case(c):**finally, for $v={v}^{\prime}=6$. Then $Tv=T{v}^{\prime}=3$. Now

**Remark**

**1.**

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Convergence behaviors of F (red), M (brown), Thakur (green), Abbas (yellow), Agarwal (blue), Noor (cyan), Ishikawa (magenta) and Mann (black) iterative schemes.

**Table 1.**Some values generated by F, M, Thakur, Abbas, S, Noor, Ishikawa and Mann iterations for the mapping T of Example 1.

k | F | M | Thakur | Abbas | Agarwal | Noor | Ishikawa | Mann |
---|---|---|---|---|---|---|---|---|

1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |

2 | 3.0813 | 3.1625 | 3.1931 | 3.2456 | 3.3863 | 3.4851 | 3.5363 | 3.6500 |

3 | 3.0066 | 3.0264 | 3.0373 | 3.0603 | 3.1492 | 3.2353 | 3.2876 | 3.4225 |

4 | 3.0005 | 3.0043 | 3.0072 | 3.0148 | 3.0576 | 3.1141 | 3.1542 | 3.2746 |

5 | 3 | 3.0007 | 3.0014 | 3.0036 | 3.0223 | 3.0554 | 3.0827 | 3.1785 |

6 | 3 | 3.0001 | 3.0003 | 3.0009 | 3.0086 | 3.0269 | 3.0443 | 3.1160 |

7 | 3 | 3 | 3.0001 | 3.0002 | 3.0033 | 3.0130 | 3.0238 | 3.0754 |

8 | 3 | 3 | 3 | 3.0001 | 3.0013 | 3.0063 | 3.0123 | 3.0490 |

9 | 3 | 3 | 3 | 3 | 3.0005 | 3.0031 | 3.0068 | 3.0318 |

10 | 3 | 3 | 3 | 3 | 3.0002 | 3.0015 | 3.0037 | 3.0207 |

11 | 3 | 3 | 3 | 3 | 3.0001 | 3.0007 | 3.0020 | 3.0134 |

12 | 3 | 3 | 3 | 3 | 3 | 3.0003 | 3.0011 | 3.0088 |

13 | 3 | 3 | 3 | 3 | 3 | 3.0002 | 3.0006 | 3.0057 |

14 | 3 | 3 | 3 | 3 | 3 | 3.0001 | 3.0003 | 3.0037 |

15 | 3 | 3 | 3 | 3 | 3 | 3 | 3.0002 | 3.0024 |

16 | 3 | 3 | 3 | 3 | 3 | 3 | 3.0001 | 3.0016 |

17 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3.0010 |

18 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3.0007 |

19 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3.0004 |

20 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3.0003 |

21 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3.0002 |

22 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3.0001 |

23 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

**Table 2.**${\alpha}_{k}=\frac{k}{{(k+7)}^{\frac{11}{9}}}$ and ${\beta}_{k}=\frac{1}{{(k+2)}^{\frac{2}{5}}}$.

Number of Iterates for Obtaining Fixed Point. | |||
---|---|---|---|

Starting Points | Agarwal | M | F |

$3.2$ | 28 | 14 | $\mathbf{10}$ |

$3.7$ | 30 | 14 | $\mathbf{10}$ |

$4.3$ | 30 | 15 | $\mathbf{10}$ |

$4.8$ | 31 | 15 | $\mathbf{10}$ |

$5.3$ | 31 | 15 | $\mathbf{11}$ |

$5.8$ | 32 | 15 | $\mathbf{11}$ |

Number of Iterates for Obtaining Fixed Point. | |||
---|---|---|---|

Starting Points | Agarwal | M | F |

$3.2$ | 24 | 13 | $\mathbf{9}$ |

$3.7$ | 26 | 14 | $\mathbf{10}$ |

$4.3$ | 26 | 15 | $\mathbf{10}$ |

$4.8$ | 27 | 15 | $\mathbf{10}$ |

$5.3$ | 27 | 15 | $\mathbf{11}$ |

$5.8$ | 27 | 15 | $\mathbf{11}$ |

Number of Iterates for Obtaining Fixed Point. | |||
---|---|---|---|

Starting Points | Agarwal | M | F |

$3.2$ | 28 | 12 | $\mathbf{9}$ |

$3.7$ | 29 | 13 | $\mathbf{9}$ |

$4.3$ | 30 | 13 | $\mathbf{9}$ |

$4.8$ | 31 | 13 | $\mathbf{10}$ |

$5.3$ | 31 | 14 | $\mathbf{10}$ |

$5.8$ | 31 | 14 | $\mathbf{10}$ |

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**MDPI and ACS Style**

Ahmad, J.; Ullah, K.; Ahmad, I.; Arshad, M.; Jarasthitikulchai, N.; Sudsutad, W.
Some Iterative Approximation Results of *F* Iteration Process in Banach Spaces. *Axioms* **2022**, *11*, 153.
https://doi.org/10.3390/axioms11040153

**AMA Style**

Ahmad J, Ullah K, Ahmad I, Arshad M, Jarasthitikulchai N, Sudsutad W.
Some Iterative Approximation Results of *F* Iteration Process in Banach Spaces. *Axioms*. 2022; 11(4):153.
https://doi.org/10.3390/axioms11040153

**Chicago/Turabian Style**

Ahmad, Junaid, Kifayat Ullah, Imtiaz Ahmad, Muhammad Arshad, Nantapat Jarasthitikulchai, and Weerawat Sudsutad.
2022. "Some Iterative Approximation Results of *F* Iteration Process in Banach Spaces" *Axioms* 11, no. 4: 153.
https://doi.org/10.3390/axioms11040153