# Viscosity Methods and Split Common Fixed Point Problems for Demicontractive Mappings

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

**Algorithm**

**1.**

## 2. Preliminaries

**Definition**

**1.**

- (i)
- contractive if there exists $\kappa \in (0,1)$ such that$$\parallel Sy-Sz\parallel \le \kappa \parallel y-z\parallel ,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{4pt}{0ex}}y,z\in \mathcal{H};$$
- (ii)
- nonexpansive if$$\parallel Sy-Sz\parallel \le \parallel y-z\parallel ,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{4pt}{0ex}}y,z\in \mathcal{H};$$
- (iii)
- quasi-nonexpansive if $F(S)\ne \varnothing $ and$$\parallel Sy-p\parallel \le \parallel y-p\parallel ,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{4pt}{0ex}}y\in \mathcal{H},p\in F(S);$$
- (iv)
- firmly nonexpansive if$${\parallel Sy-Sz\parallel}^{2}\le \langle y-z,Sy-Sz\rangle ,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{4pt}{0ex}}y,z\in \mathcal{H};$$
- (v)
- directed if $F(S)\ne \varnothing $ and$${\parallel Sy-p\parallel}^{2}\le {\parallel y-p\parallel}^{2}-{\parallel y-Sy\parallel}^{2},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{4pt}{0ex}}y\in \mathcal{H},p\in F(S);$$
- (vi)
- τ-demicontractive if $F(S)\ne \varnothing $ and there exists $\tau \in (-\infty ,1)$, such that$${\parallel Sy-p\parallel}^{2}\le {\parallel y-p\parallel}^{2}+\tau {\parallel y-Sy\parallel}^{2},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{4pt}{0ex}}y\in \mathcal{H},p\in F(S),$$$$\begin{array}{c}\hfill {\displaystyle \langle y-Sy,y-p\rangle \ge \frac{1-\tau}{2}{\parallel y-Sy\parallel}^{2}.}\end{array}$$

**Remark**

**1.**

**Example**

**1**

**.**The mapping $S:[0,1]\to [0,1]$ is defined by

**Remark**

**2.**

**Lemma**

**1**

**.**Assume that $\{{c}_{n}\}$ is a sequence of non-negative real numbers, such that

- (i)
- ${\sum}_{n=0}^{\infty}{\gamma}_{n}=\infty ;$
- (ii)
- ${\sum}_{n=0}^{\infty}{\u03f5}_{n}<\infty ;$
- (iii)
- ${lim\; sup}_{n\to \infty}{\delta}_{n}\le 0$ or ${\Sigma}_{n=1}^{\infty}{\gamma}_{n}|{\delta}_{n}|<\infty .$

**Lemma**

**2**

**Lemma**

**3**

**.**The demiclosedness principle of nonexpansive mappings. If $V:\mathcal{H}\to \mathcal{H}$ is a nonexpansive mapping, then $I-V$ is demiclosed at 0.

## 3. Main Results

**Algorithm**

**2.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Theorem**

**1.**

- (a)
- $I-U$ and $I-T$ are demiclosed at 0.
- (b)
- ${\sum}_{n=0}^{\infty}{\rho}_{n}^{2}<\infty $.
- (c)
- ${lim}_{n\to \infty}{\beta}_{n}=0$ and ${\sum}_{n=0}^{\infty}{\beta}_{n}=\infty $.
- (d)
- ${lim}_{n\to \infty}\frac{{\beta}_{n}}{{\rho}_{n}}=0.$

**Proof.**

**Remark**

**3.**

**Corollary**

**1.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Wang, Y.; Fang, X.; Kim, T.-H.
Viscosity Methods and Split Common Fixed Point Problems for Demicontractive Mappings. *Mathematics* **2019**, *7*, 844.
https://doi.org/10.3390/math7090844

**AMA Style**

Wang Y, Fang X, Kim T-H.
Viscosity Methods and Split Common Fixed Point Problems for Demicontractive Mappings. *Mathematics*. 2019; 7(9):844.
https://doi.org/10.3390/math7090844

**Chicago/Turabian Style**

Wang, Yaqin, Xiaoli Fang, and Tae-Hwa Kim.
2019. "Viscosity Methods and Split Common Fixed Point Problems for Demicontractive Mappings" *Mathematics* 7, no. 9: 844.
https://doi.org/10.3390/math7090844