# An Inertial Forward–Backward Splitting Method for Solving Modified Variational Inclusion Problems and Its Application

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Lemma**

**1**

- (i)
- ${\parallel u\parallel}^{2}-{\parallel v\parallel}^{2}-2\langle u-v,v\rangle \ge {\parallel u-v\parallel}^{2}$;
- (ii)
- ${\parallel u\parallel}^{2}+2\langle v,u+v\rangle \ge {\parallel u+v\parallel}^{2}$;
- (iii)
- ${t\parallel u\parallel}^{2}+{(1-t)\parallel v\parallel}^{2}{-t(1-t)\parallel u-v\parallel}^{2}={\parallel tu+(1-t)v\parallel}^{2}$, $\forall t\in [0,1]$.

**Remark**

**1.**

**Remark**

**2**

**.**Z is firmly nonexpansive iff $I-Z$ is firmly nonexpansive. Obviously, the projection ${P}_{K}$ is firmly nonexpansive.

**Theorem**

**1**

**.**Assume that $Z:K\to K$ is a nonexpansive mapping with a fixed point. For $t\in (0,1)$ and fixed $p\in K$, the unique fixed point ${u}_{t}\in K$ of the contraction $u\u27fctp+(1-t)Zu$ converges strongly as $t\to 0$ to a fixed point of Z.

**Lemma**

**2**

**.**Assume that $\left\{{x}_{n}\right\}$ and $\left\{{y}_{n}\right\}$ are sequences of nonnegative real numbers such that

- (i)
- $\left\{{x}_{n}\right\}$ is a bounded sequence, if ${y}_{n}\le {\omega}_{n}M$ for some $M\ge 0;$
- (ii)
- $\underset{n\to \infty}{lim}{x}_{n}=0$, if $\sum _{n=1}^{\infty}{\omega}_{n}=\infty \phantom{\rule{0.166667em}{0ex}}$ and $\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\underset{n\to \infty}{lim\; sup}\frac{{y}_{n}}{{\omega}_{n}}\le 0$.

**Lemma**

**3**

**.**Assume $\left\{{z}_{n}\right\}$ is a sequence of nonnegative real numbers such that

- (i)
- $\sum _{n=1}^{\infty}{\omega}_{n}=\infty$;
- (ii)
- $\underset{n\to \infty}{lim}{\rho}_{n}=0$;
- (iii)
- $\underset{k\to \infty}{lim}{\eta}_{{n}_{k}}=0$ implies $\underset{k\to \infty}{lim\; sup}{t}_{n}\le 0$ for each subsequence of real number $\left\{{n}_{k}\right\}$ of $\left\{n\right\}$.

**Proposition**

**1**

**.**Let H be a real Hilbert space. Let $j\in N$ be fixed. Let ${\left\{{x}_{i}\right\}}_{i=1}^{j}\subset H$ and ${t}_{i}\ge 0$ for all $i=1,2,...,j$ with $\sum _{i=1}^{j}{t}_{i}\le 1$. Then, we have

## 3. Main Result

**Lemma**

**4.**

**Proof.**

- (ii)
- Since ${G}_{\lambda}u={(I+\lambda T)}^{-1}(u-\lambda \sum _{i=1}^{N}{\delta}_{i}{S}_{i}u),$ then ${\displaystyle \frac{u-{G}_{\lambda}u}{\lambda}}-\sum _{i=1}^{N}{\delta}_{i}{S}_{i}u\in T({G}_{\lambda}u).$ Let $0<r\le \lambda $; by the monotone of T, we obtain that$$\langle {\displaystyle \frac{u-{G}_{\lambda}u}{r}}-{\displaystyle \frac{u-{G}_{\lambda}u}{\lambda}},{G}_{r}u-{G}_{\lambda}u\rangle \ge 0.$$It follows that$$\begin{array}{cc}\hfill 0\phantom{\rule{1.em}{0ex}}& \le r\langle {\displaystyle \frac{u-{G}_{\lambda}u}{r}}-{\displaystyle \frac{u-{G}_{\lambda}u}{\lambda}},{G}_{r}u-{G}_{\lambda}u\rangle \hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\langle {\displaystyle \frac{\lambda -r}{\lambda}}u-{\displaystyle \frac{\lambda -r}{\lambda}}{G}_{\lambda}u,{G}_{r}u-{G}_{\lambda}u\rangle -\langle {G}_{r}u-{G}_{\lambda}u,{G}_{r}u-{G}_{\lambda}u\rangle .\hfill \end{array}$$Since $0<r\le \lambda ,$ we obtain that$$\begin{array}{cc}\hfill \parallel {G}_{r}u-{G}_{\lambda}{u\parallel}^{2}\phantom{\rule{1.em}{0ex}}& \le 1-{\displaystyle \frac{r}{\lambda}}\langle u-{G}_{\lambda}u,{G}_{r}u-{G}_{\lambda}u\rangle \hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le \parallel u-{G}_{\lambda}u\parallel \parallel {G}_{r}u-{G}_{\lambda}u\parallel .\hfill \end{array}$$Hence, $\parallel u-{G}_{r}u\parallel \le \parallel u-{G}_{\lambda}u\parallel +\parallel {G}_{r}u-{G}_{\lambda}u\parallel \le 2\parallel u-{G}_{\lambda}u\parallel .$
- (iii)
- Since T is maximal monotone, it is know that ${J}_{\lambda}^{T}$ is firmly nonexpaxsive and so we have$$\begin{array}{cc}\hfill \parallel {G}_{\lambda}u-{G}_{\lambda}{v\parallel}^{2}\phantom{\rule{1.em}{0ex}}& {\displaystyle {=\parallel (I+\lambda T)}^{-1}(I-\lambda \sum _{i=1}^{N}{\delta}_{i}{S}_{i})u-{(I+\lambda T)}^{-1}(I-\lambda \sum _{i=1}^{N}{\delta}_{i}{S}_{i}){v\parallel}^{2}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\parallel {J}_{\lambda}^{T}{u}^{\prime}-{J}_{\lambda}^{T}{v}^{\prime}{\parallel}^{2}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le \parallel {u}^{\prime}-{v}^{\prime}{\parallel}^{2}-{\parallel (I-{J}_{\lambda}^{T}){u}^{\prime}-(I-{J}_{\lambda}^{T}){v}^{\prime}\parallel}^{2}\hfill \end{array}$$

**Theorem**

**2.**

- (i)
- $\sum _{n=1}^{\infty}{\theta}_{n}|{u}_{n}-{u}_{n-1}|<\infty$,
- (ii)
- $\underset{n\to \infty}{lim}{\omega}_{n}=0$ and $\sum _{n=0}^{\infty}{\omega}_{n}=\infty ,\sum _{n=0}^{\infty}|{\omega}_{n+1}-{\omega}_{n}|<\infty$,
- (iii)
- $0<c\u2a7d{\xi}_{n}\u2a7dd<1$ for all $n\ge 1$ and $\sum _{n=0}^{\infty}|{\xi}_{n+1}-{\xi}_{n}|<\infty$,
- (iv)
- $\sum _{n=1}^{N}{\delta}_{i}=1.$

**Proof.**

## 4. Numerical Result

**Example**

**1**

**Remark**

**3.**

**Figure 2.**Original grayscale images. (

**a**) X-ray film of the brain image and (

**b**) X-ray film of the right shoulder image.

**Figure 3.**Blurred X-ray film of the brain image with filtering ${M}_{i}x$ by (

**a**) ${M}_{1}x$, (

**b**) ${M}_{2}x$, (

**c**) ${M}_{3}x$ and (

**d**) ${M}_{4}x$.

**Figure 4.**(

**a**) X-ray film of the brain image obtained via Theorem 2 and (

**b**) X-ray film of the brain image obtained via Theorem 3.1 in [16].

**Figure 6.**Blurred X-ray film of the right shoulder image with filtering ${M}_{i}x$ by (

**a**) ${M}_{1}x$, (

**b**) ${M}_{2}x$, (

**c**) ${M}_{3}x$ and (

**d**) ${M}_{4}x$.

**Figure 7.**(

**a**) X-ray film of the right shoulder image obtained via Theorem 2 and (

**b**) X-ray film of the right shoulder image obtained via Theorem 3.1 in [16].

**Figure 9.**X-ray film of the brain image obtained via Theorem 2 when Example 1 was tuned for the parameter $\lambda $ by setting (

**a**) $\lambda $ = 0.25, (

**b**) $\lambda $ = 0.5, (

**c**) $\lambda $ = 0.75, (

**d**) $\lambda $ = 1.

**Figure 10.**The SNR of Figure 9a–d.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Sombut, K.; Sitthithakerngkiet, K.; Arunchai, A.; Seangwattana, T.
An Inertial Forward–Backward Splitting Method for Solving Modified Variational Inclusion Problems and Its Application. *Mathematics* **2023**, *11*, 2107.
https://doi.org/10.3390/math11092107

**AMA Style**

Sombut K, Sitthithakerngkiet K, Arunchai A, Seangwattana T.
An Inertial Forward–Backward Splitting Method for Solving Modified Variational Inclusion Problems and Its Application. *Mathematics*. 2023; 11(9):2107.
https://doi.org/10.3390/math11092107

**Chicago/Turabian Style**

Sombut, Kamonrat, Kanokwan Sitthithakerngkiet, Areerat Arunchai, and Thidaporn Seangwattana.
2023. "An Inertial Forward–Backward Splitting Method for Solving Modified Variational Inclusion Problems and Its Application" *Mathematics* 11, no. 9: 2107.
https://doi.org/10.3390/math11092107