# Inertial Mann-Type Algorithm for a Nonexpansive Mapping to Solve Monotone Inclusion and Image Restoration Problems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**MTA**) to solve the fixed point problem for a nonexpansive mapping and proved strong convergence of the iterate without using viscosity and projection method under some control conditions of parameters sequences. Their algorithm is defined by

## 2. Preliminaries

**Lemma**

**1.**

- 1.
- ${\parallel x-y\parallel}^{2}={\parallel x\parallel}^{2}-{\parallel y\parallel}^{2}-2\langle x-y,y\rangle $ for all $x,y\in \mathcal{H}$,
- 2.
- ${\parallel x+y\parallel}^{2}\le {\parallel x\parallel}^{2}+2\langle x+y,y\rangle $ for all $x,y\in \mathcal{H}$,
- 3.
- ${\parallel rx+(1-r)y\parallel}^{2}={r\parallel x\parallel}^{2}+{(1-r)\parallel y\parallel}^{2}-r(1-r){\parallel x-y\parallel}^{2}$ for all $r\in [0,1]$ and $x,y\in \mathcal{H}.$

**Lemma**

**2.**

- 1.
- If ${\mu}_{n}\le c{\delta}_{n}$ (where $c\ge 0$), then ${\left({a}_{n}\right)}_{n\ge 1}$ is bounded.
- 2.
- If ${\sum}_{n\ge 0}{\delta}_{n}=\infty $ and ${lim\; sup}_{n\to +\infty}\frac{{\mu}_{n}}{{\delta}_{n}}\le 0$, then the sequence ${\left({a}_{n}\right)}_{n\ge 0}$ converges to 0.

**Lemma**

**3.**

**Assumption**

**1.**

- 1.
- $\underset{n\to +\infty}{lim\; inf}{\alpha}_{n}>0$ and ${\sum}_{n\ge 1}|{\alpha}_{n}-{\alpha}_{n-1}|<+\infty $,
- 2.
- $\underset{n\to +\infty}{lim}{\delta}_{n}=1$, ${\sum}_{n\ge 0}(1-{\delta}_{n})=+\infty $ and ${\sum}_{n\ge 1}|{\delta}_{n}-{\delta}_{n-1}|<+\infty $,
- 3.
- ${\sum}_{n\ge 0}\parallel {\epsilon}_{n}\parallel <+\infty .$

**Remark**

**1.**

## 3. Main Results

**Lemma**

**4.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Remark**

**2.**

## 4. Applications

**Proposition**

**1.**

**Lemma**

**5.**

**Remark**

**3.**

- 1.
- If we set $Cx=0$ for all $x\in \mathcal{H}$ in Lemma 5, $\mathbf{zer}(A+B)={J}_{\mu B}\left(\mathbf{Fix}\left(T\right)\right),$ where $T:={J}_{\mu A}\circ (2{J}_{\mu B}-Id)+Id-{J}_{\mu B}$ with $\mu >0.$
- 2.
- If we set $Bx=0$ for all $x\in \mathcal{H}$ in Lemma 5, $\mathbf{zer}(A+C)=\mathbf{Fix}\left(T\right),$ where $T:={J}_{\mu A}\circ (Id-\mu C)$ with $\mu >0.$

**Theorem**

**2.**

- 1.
- ${\left({x}_{n}\right)}_{n\ge 0}$ strongly converges to ${x}^{*}:={\mathbf{proj}}_{\mathbf{Fix}\left(T\right)}\left(0\right)$, where$T:={J}_{\mu A}\circ (2{J}_{\mu B}-Id-\mu C\circ {J}_{\mu B})+Id-{J}_{\mu B}$ for some $\mu >0$.
- 2.
- ${\left({y}_{n}\right)}_{n\ge 1}$ and ${\left({z}_{n}\right)}_{n\ge 1}$ strongly converge to ${J}_{\mu B}\left({x}^{*}\right)\in \mathbf{zer}(A+B+C)$.

**Proof.**

**Corollary**

**1.**

- 1.
- ${\left({x}_{n}\right)}_{n\ge 0}$ strongly converges to ${x}^{*}:={\mathbf{proj}}_{\mathbf{Fix}({J}_{\mu A}\circ (2{J}_{\mu B}-Id)+Id-{J}_{\mu B})}\left(0\right)$ for some $\mu >0$.
- 2.
- ${\left({y}_{n}\right)}_{n\ge 1}$ and ${\left({z}_{n}\right)}_{n\ge 1}$ strongly converge to ${J}_{\mu B}\left({x}^{*}\right)\in \mathbf{zer}(A+B)$.

**Proof.**

**Corollary**

**2.**

## 5. Numerical Experiments

#### 5.1. Convex Minimization Problems

**Example**

**1.**

#### 5.2. Image Restoration Problems

**SNR**) for monochrome images, which is defined by

**NCD**) [37] which is defined by

**WF**) [38,39]. Figure 6 presents the comparative results of two degradation images ’Artsawang’ and ’Mandril’ corrupted by motion blur and different salt and pepper noise from $0\%$ to $10\%$.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Illustration the behavior of $\parallel {y}_{n}-{y}_{n-1}\parallel $ for Algorithm 2, MTA, and Shehu et al.’s algorithm in Equation (3).

**Figure 2.**(

**a**) The original image ‘camera man’; (

**b**) the images degraded by average blur and random noise (Gaussian noise); and (

**c**–

**e**) the reconstructed image by using Weiner filter, Kitkuan et al.’s algorithm, and our algorithm in Equation (19), respectively.

**Figure 3.**(

**a**) The original image ‘Artsawang’; (

**b**) the images degraded by Gaussian blur and random noise (Gaussian noise); and (

**c**–

**e**) the reconstructed image by using Weiner filter, Kitkuan et al.’s algorithm, and our algorithm in Equation (19), respectively.

**Figure 4.**(

**a**) The original image ‘Mandril’; (

**b**) the images degraded by motion blur and random noise (Gaussian noise); and (

**c**–

**e**) the reconstructed image by usingWeiner filter, Kitkuan et al.’s algorithm, and our algorithm in Equation (19), respectively.

**Figure 6.**(

**a**,

**b**) The behavior of

**NCD**in motion blur and different different salt and pepper noise from 0% to 10%.

**Table 1.**Comparison: Algorithm 2, MTA and Shehu et al.’s algorithm in Equation (3).

$(\mathit{l},\mathit{s})$ | Algorithm 2 | MTA | Shehu et al.’s Algorithm Equation (3) | |||
---|---|---|---|---|---|---|

CPU Time (s) | Iterations | CPU Time (s) | Iterations | CPU Time (s) | Iterations | |

(20,700) | 0.0218 | 7 | 0.0428 | 278 | 0.0756 | 626 |

(20,800) | 0.0189 | 7 | 0.0914 | 350 | 0.1745 | 796 |

(20,7000) | 0.0302 | 7 | 1.7751 | 1273 | 0.0977 | 53 |

(20,8000) | 0.0308 | 6 | 1.2419 | 1290 | 0.0671 | 54 |

(200,7000) | 0.0365 | 8 | 1.9452 | 858 | 4.6538 | 2028 |

(200,8000) | 0.0406 | 7 | 2.5115 | 977 | 0.1425 | 53 |

(500,7000) | 0.0403 | 7 | 4.1647 | 892 | 8.3620 | 1956 |

(500,8000) | 0.0548 | 8 | 4.3239 | 813 | 9.0929 | 1835 |

(1000,7000) | 0.0703 | 7 | 6.7954 | 786 | 14.1693 | 1751 |

(1000,8000) | 0.0728 | 7 | 7.8302 | 825 | 16.3752 | 1784 |

(3000,7000) | 0.1597 | 7 | 18.0559 | 779 | 44.8129 | 1940 |

(3000,8000) | 0.1763 | 7 | 22.3514 | 841 | 49.6872 | 1891 |

(100,80,000) | 0.1376 | 8 | 26.6863 | 1489 | 1.5926 | 94 |

(1000,80,000) | 0.6949 | 8 | 344.7048 | 3289 | 9.4181 | 93 |

The Normalized Color Difference (NCD). | ||||
---|---|---|---|---|

Kitkuan et al.’s Algorithm | Our Algorithm in Equation (19) | |||

$\mathit{n}$ | Artsawang Image | Mandril Image | Artsawang Image | Mandril Image |

1 | 0.99803 | 0.99842 | 0.99663 | 0.99731 |

50 | 0.99660 | 0.99730 | 0.99659 | 0.99727 |

100 | 0.99661 | 0.99729 | 0.99658 | 0.99726 |

200 | 0.99660 | 0.99728 | 0.99658 | 0.99726 |

300 | 0.99659 | 0.99727 | 0.99658 | 0.99726 |

400 | 0.99659 | 0.99727 | 0.99658 | 0.99726 |

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**MDPI and ACS Style**

Artsawang, N.; Ungchittrakool, K.
Inertial Mann-Type Algorithm for a Nonexpansive Mapping to Solve Monotone Inclusion and Image Restoration Problems. *Symmetry* **2020**, *12*, 750.
https://doi.org/10.3390/sym12050750

**AMA Style**

Artsawang N, Ungchittrakool K.
Inertial Mann-Type Algorithm for a Nonexpansive Mapping to Solve Monotone Inclusion and Image Restoration Problems. *Symmetry*. 2020; 12(5):750.
https://doi.org/10.3390/sym12050750

**Chicago/Turabian Style**

Artsawang, Natthaphon, and Kasamsuk Ungchittrakool.
2020. "Inertial Mann-Type Algorithm for a Nonexpansive Mapping to Solve Monotone Inclusion and Image Restoration Problems" *Symmetry* 12, no. 5: 750.
https://doi.org/10.3390/sym12050750