# Strong Convergence of Mann’s Iteration Process in Banach Spaces

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## Abstract

**:**

## 1. Introduction

- “${x}_{n}\rightharpoonup x$” stands for the weak convergence of $\left({x}_{n}\right)$ to x,
- “${x}_{n}\to x$” stands for the strong convergence of $\left({x}_{n}\right)$ to x,
- ${\omega}_{w}\left({x}_{n}\right):=\{x:\exists {x}_{{n}_{k}}\rightharpoonup x\}$ is the set of all weak accumulation points of the sequence $\left({x}_{n}\right)$.

## 2. Preliminaries

#### 2.1. Uniform Convexity

**Proposition**

**1**

**.**Let X be a uniformly convex Banach space. Then for each fixed real number $r>0$, there exists a strictly increasing continuous function $h:[0,\infty )\to [0,\infty )$, $h\left(0\right)=0$, satisfying the property:

#### 2.2. Duality Maps

- (i)
- $\mu \left(0\right)=0$,
- (ii)
- $\mu $ is continuous and strictly increasing, and
- (iii)
- $\underset{t\to \infty}{lim}\mu \left(t\right)=\infty$.

**Proposition**

**2.**

#### 2.3. Demiclosedness Principle for Nonexpansive Mappings

#### 2.4. Accretive Operators

**Definition**

**1**

**.**Let $\phi :[0,\infty )\to [0,\infty )$ be a continuous, strictly increasing function with $\phi \left(0\right)=0$ and ${lim}_{t\to \infty}\phi \left(t\right)=\infty $. Then we say that an operator $S:C\to X$ is φ-accretive (or uniformly φ-accretive, respectively) if

#### 2.5. A Useful Lemma

**Lemma**

**1**

**.**Assume $\left({s}_{n}\right)$ is a sequence of nonnegative real numbers satisfying the condition:

- (i)
- ${\sum}_{n=1}^{\infty}{\lambda}_{n}=\infty $,
- (ii)
- ${lim\; sup}_{n\to \infty}{\beta}_{n}\le 0$ (or ${\sum}_{n=1}^{\infty}{\lambda}_{n}\left|{\beta}_{n}\right|<\infty $),
- (iii)
- ${\sum}_{n=1}^{\infty}{\delta}_{n}<\infty $.

## 3. Strong Convergence Analysis of Mann’s Iteration Process

#### 3.1. Strong Convergence of Mann’s Algorithm

**Lemma**

**2.**

- (i)
- for each $z\in \mathrm{Fix}\left(T\right)$, the sequence $\{\parallel {x}_{n}-z\parallel \}$ is nonincreasing, and
- (ii)
- the sequence $\{\parallel {x}_{n}-T{x}_{n}\parallel \}$ is nonincreasing.

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Theorem**

**2.**

**Proof.**

**Remark**

**2.**

#### 3.2. Regularization

**Theorem**

**3**

**.**Let X be a uniformly convex Banach space, C a nonempty closed convex subset of X, and T a nonexpansive self-mapping of C with $\mathrm{Fix}\left(T\right)\ne \varnothing $. Assume $R:C\to X$ is a continuous, bounded, uniformly φ-accretive operator. Assume the normalized duality map $J:X\to {X}^{*}$ is weakly sequentially continuous. Choose positive real numbers ${\delta}_{n}$ and ${\epsilon}_{n}\in (0,1)$ such that

- (i)
- ${lim}_{n\to \infty}{\epsilon}_{n}=0$,
- (ii)
- ${lim}_{n\to \infty}({\delta}_{n}/{\epsilon}_{n})=0$.

**Remark**

**3.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**1.**

**Remark**

**4.**

**Theorem**

**5.**

- (C1)
- ${\sum}_{n=0}^{\infty}{\alpha}_{n}{\epsilon}_{n}=\infty $,
- (C2)
- ${lim}_{n\to \infty}\frac{|{\epsilon}_{n}-{\epsilon}_{n-1}|}{{\alpha}_{n}{\epsilon}_{n}^{2}}=0$,
- (C3)
- ${lim}_{n\to \infty}({\delta}_{n}/{\epsilon}_{n})=0$,
- (C4)
- ${lim}_{n\to \infty}\frac{{\delta}_{n}+{\delta}_{n-1}}{{\alpha}_{n}{\epsilon}_{n}^{2}}=0$.

**Proof.**

**Remark**

**5.**

**Remark**

**6.**

**Corollary**

**2.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Xu, H.-K.; Altwaijry, N.; Chebbi, S.
Strong Convergence of Mann’s Iteration Process in Banach Spaces. *Mathematics* **2020**, *8*, 954.
https://doi.org/10.3390/math8060954

**AMA Style**

Xu H-K, Altwaijry N, Chebbi S.
Strong Convergence of Mann’s Iteration Process in Banach Spaces. *Mathematics*. 2020; 8(6):954.
https://doi.org/10.3390/math8060954

**Chicago/Turabian Style**

Xu, Hong-Kun, Najla Altwaijry, and Souhail Chebbi.
2020. "Strong Convergence of Mann’s Iteration Process in Banach Spaces" *Mathematics* 8, no. 6: 954.
https://doi.org/10.3390/math8060954