# A New Multi-Step Iterative Algorithm for Approximating Common Fixed Points of a Finite Family of Multi-Valued Bregman Relatively Nonexpansive Mappings

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## Abstract

**:**

## 1. Introduction

**Definition 1.**

- (1)
- Essentially smooth if f is both locally bounded and single-valued on its domain.
- (2)
- Essentially strictly convex if ${(\partial f)}^{-1}$ is locally bounded on its domain and f is strictly convex on every convex subset of $domf$.
- (3)
- Legendre if it is both essentially smooth and essentially strictly convex.

**Remark 1.**

- (a)
- f is essentially smooth if and only if ${f}^{*}$ is essentially strictly convex (see [21], Theorem 5.4).
- (b)
- ${(\partial )}^{-1}=\partial {f}^{*}$ (see [22]).
- (c)
- (d)
- If f is Legendre, then $\nabla f$ is a bijection satisfying:$$\nabla f={(\nabla {f}^{*})}^{-1},ran\nabla f=dom\nabla {f}^{*}=int(dom{f}^{*})\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\text{and}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\nabla {f}^{*}=dom\nabla f=int(domf)$$

**Definition 2.**

- (1)
- (The three point identity): for each $x\in domf$ and $y,z\in int(domf)$,$$\begin{array}{c}\hfill {D}_{f}(x,y)+{D}_{f}(y,z)-{D}_{f}(z,x)=\langle \nabla f(z)-\nabla f(y),x-y\rangle \end{array}$$
- (2)
- (The four point identity): for each $y,\omega \in domf$ and $x,z\in int(domf)$.$$\begin{array}{c}\hfill {D}_{f}(y,x)-{D}_{f}(y,z)-{D}_{f}(\omega ,x)+{D}_{f}(\omega ,z)=\langle \nabla f(z)-\nabla f(x),y-\omega \rangle \end{array}$$

**Definition 3.**

**Definition 4.**

- (1)
- Relatively quasi-nonexpansive if$$\begin{array}{c}\hfill \varphi (p,Tx)\le \varphi (p,x)\phantom{\rule{3.33333pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{3.33333pt}{0ex}}x\in C,\phantom{\rule{3.33333pt}{0ex}}p\in F(T)\end{array}$$
- (2)
- Relatively nonexpansive if $\widehat{F}(T)=F(T)$,$$\begin{array}{c}\hfill \varphi (p,Tx)\le \varphi (p,x)\phantom{\rule{3.33333pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{3.33333pt}{0ex}}x\in C,\phantom{\rule{3.33333pt}{0ex}}p\in F(T)\end{array}$$
- (3)
- Bregman relatively quasi-nonexpansive if,$$\begin{array}{c}\hfill {D}_{f}(p,Tx)\le {D}_{f}(p,x)\phantom{\rule{3.33333pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{3.33333pt}{0ex}}x\in C,\phantom{\rule{3.33333pt}{0ex}}p\in F(T)\end{array}$$
- (4)
- Bregman relatively nonexpansive if, $\widehat{F}(T)=F(T)$,$$\begin{array}{c}\hfill {D}_{f}(p,Tx)\le {D}_{f}(p,x)\phantom{\rule{3.33333pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{3.33333pt}{0ex}}x\in C,\phantom{\rule{3.33333pt}{0ex}}p\in F(T)\end{array}$$

**Remark 2.**

**Definition 5.**

- (1)
- Relatively quasi-nonexpansive if,$$\begin{array}{c}\hfill \varphi (p,u)\le \varphi (p,x)\phantom{\rule{3.33333pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{3.33333pt}{0ex}}u\in Tx,\phantom{\rule{3.33333pt}{0ex}}x\in C\phantom{\rule{3.33333pt}{0ex}}\text{and}\phantom{\rule{3.33333pt}{0ex}}p\in F(T)\end{array}$$
- (2)
- Relatively nonexpansive if T is relatively quasi-nonexpansive and $\widehat{F}(T)=F(T)$;
- (3)
- Bregman relatively quasi-nonexpansive if,$$\begin{array}{c}\hfill {D}_{f}(p,u)\le {D}_{f}(p,x)\phantom{\rule{3.33333pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{3.33333pt}{0ex}}u\in Tx,\phantom{\rule{3.33333pt}{0ex}}x\in C,\phantom{\rule{3.33333pt}{0ex}}p\in F(T)\end{array}$$
- (4)
- Bregman relatively nonexpansive if T is Bregman relatively quasi-nonexpansive and $\widehat{F}(T)=F(T)$.

**Example 1.**

**Lemma 1.**

**Lemma 2.**

- (1)
- f is strongly coercive and uniformly convex on bounded sets;
- (2)
- ${f}^{*}$ is Fr$\stackrel{\xb4}{e}$chet differentiable, and $\nabla {f}^{*}$ is uniformly norm-to-norm continuous on bounded sets of $dom({f}^{*})={E}^{*}$.

**Lemma 3.**

**Lemma 4.**

- (1)
- $z={P}_{C}^{f}(x)$ if and only if $\langle \nabla f(x)-\nabla f(z),y-z\rangle \le 0$, $\forall y\in C$.
- (2)
- ${D}_{f}(y,{P}_{C}^{f}(x))+{D}_{f}({P}_{C}^{f}(x),x)\le {D}_{f}(y,x)$, $\forall y\in C$.

**Lemma 5.**

**Lemma 6.**

**Lemma 7.**

**Lemma 8.**

**Lemma 9.**

- (1)
- ${D}_{f}(x,\nabla {f}^{*}({x}^{*}))={V}_{f}(x,{x}^{*}),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall x\in E,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{x}^{*},{y}^{*}\in {E}^{*}$.
- (2)
- ${V}_{f}(x,{x}^{*})+\langle {y}^{*},\nabla {f}^{*}({x}^{*})-x\rangle \le {V}_{f}(x,{x}^{*}+{y}^{*}\rangle ),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall x\in E,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{x}^{*},{y}^{*}\in {E}^{*}$.

**Lemma 10.**

**Lemma 11.**

**Lemma 12.**

## 2. Main Results

- (C1)
- ${lim}_{n\u27f6\infty}{\alpha}_{n}=0$ and ${\sum}_{n=1}^{\infty}{\alpha}_{n}=\infty $;
- (C2)
- ${\left\{{\beta}_{n,i}\right\}}_{i=1}^{N}\subset [a,b]\subset (0,1)$.

**Proof.**

**Case 1.**

**Case 2.**

**Corollary 1.**

**Corollary 2.**

**Corollary 3.**

**Corollary 4.**

**Corollary 5.**

## 3. Some Applications

#### 3.1. Variational Inequality Problems

**Definition 6.**

**Definition 7.**

**Lemma 13.**

- (1)
- ${P}_{C}^{f}\circ {A}^{f}$ is Bregman relatively nonexpansive mapping, where ${A}^{f}=\nabla {f}^{*}\circ (\nabla f-A)$;
- (2)
- $F({P}_{C}^{f}\circ {A}^{f})=VI(C,A)$.

#### 3.2. Zeros of Maximal Monotone Operators

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Kumam, W.; Sunthrayuth, P.; Phunchongharn, P.; Akkarajitsakul, K.; Ngiamsunthorn, P.S.; Kumam, P.
A New Multi-Step Iterative Algorithm for Approximating Common Fixed Points of a Finite Family of Multi-Valued Bregman Relatively Nonexpansive Mappings. *Algorithms* **2016**, *9*, 37.
https://doi.org/10.3390/a9020037

**AMA Style**

Kumam W, Sunthrayuth P, Phunchongharn P, Akkarajitsakul K, Ngiamsunthorn PS, Kumam P.
A New Multi-Step Iterative Algorithm for Approximating Common Fixed Points of a Finite Family of Multi-Valued Bregman Relatively Nonexpansive Mappings. *Algorithms*. 2016; 9(2):37.
https://doi.org/10.3390/a9020037

**Chicago/Turabian Style**

Kumam, Wiyada, Pongsakorn Sunthrayuth, Phond Phunchongharn, Khajonpong Akkarajitsakul, Parinya Sa Ngiamsunthorn, and Poom Kumam.
2016. "A New Multi-Step Iterative Algorithm for Approximating Common Fixed Points of a Finite Family of Multi-Valued Bregman Relatively Nonexpansive Mappings" *Algorithms* 9, no. 2: 37.
https://doi.org/10.3390/a9020037