# Fibonacci–Mann Iteration for Monotone Asymptotically Nonexpansive Mappings in Modular Spaces

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- (i)
- $\mathsf{\Omega}$ is a nonempty set;
- (ii)
- $\mathsf{\Sigma}$ is a nontrivial $\sigma $-algebra of subsets of $\mathsf{\Omega}$;
- (iii)
- $\mathcal{P}$ is a $\delta $-ring of subsets of $\mathsf{\Omega}$ such that $A\cap B\in \mathcal{P}$ for any $A\in \mathcal{P}$ and $B\in \mathsf{\Sigma}$;
- (iv)
- there exists an increasing sequence ${\left\{{\mathsf{\Omega}}_{n}\right\}}_{n\ge 1}$ in $\mathcal{P}$ such that $\mathsf{\Omega}={\displaystyle \bigcup _{n\ge 1}}{\mathsf{\Omega}}_{n}$.

**Definition**

**1.**

- (i)
- $\rho \left(f\right)=0$ implies $f=0$;
- (ii)
- $\left|f\right(t\left)\right|\le \left|g\right(t\left)\right|$ for all $t\in \mathsf{\Omega}$ implies $\rho \left(f\right)\le \rho \left(g\right)$, where $f,g\in {\mathcal{M}}_{\infty}$ (we will say that ρ is monotone);
- (iii)
- $|{f}_{n}\left(t\right)|\uparrow |f\left(t\right)|$ for all $t\in \mathsf{\Omega}$ implies $\rho \left({f}_{n}\right)\uparrow \rho \left(f\right)$, where $f\in {\mathcal{M}}_{\infty}$ (ρ has the Fatou property).

**Remark**

**1.**

**Theorem**

**1**

- (1)
- If $\rho \left(\beta {h}_{n}\right)\to 0$, for some $\beta >0$, then there exists a subsequence $\left\{{h}_{\psi \left(n\right)}\right\}$ such that ${h}_{\psi \left(n\right)}\to 0\phantom{\rule{4pt}{0ex}}\rho -a.e.$
- (2)
- If ${h}_{n}\to h\phantom{\rule{4pt}{0ex}}\rho -a.e.$, then $\rho \left(h\right)\le \underset{n\to \infty}{lim\; inf}\rho \left({h}_{n}\right)$.

**Definition**

**2**

- (1)
- $\left\{{g}_{n}\right\}$ is said to ρ-converge to g if $\underset{n\to \infty}{lim}\rho ({g}_{n}-g)=0.$
- (2)
- A sequence $\left\{{g}_{n}\right\}$ is called ρ-Cauchy if $\underset{n,m\to \infty}{lim}\rho ({g}_{n}-{g}_{m})=0$.
- (3)
- A subset C of ${L}_{\rho}$ is said to be ρ-closed if for any sequence $\left\{{g}_{n}\right\}$ in Cρ-convergent to g implies that $g\in C.$
- (4)
- A subset A of ${L}_{\rho}$ is called ρ-bounded if its ρ-diameter$${\delta}_{\rho}\left(A\right)=sup\{\rho (g-h);\phantom{\rule{4pt}{0ex}}g,\phantom{\rule{4pt}{0ex}}h\in A\}$$

**Theorem**

**2**

**.**Let ρ be convex regular modular. Let $\left\{{g}_{n}\right\}\subset {L}_{\rho}$ be a sequence which ρ-converges to g. The following hold:

- (i)
- if $\left\{{g}_{n}\right\}$ is monotone increasing, i.e., ${g}_{n}\le {g}_{n+1}$ ρ-a.e., for any $n\ge 1$, then ${g}_{n}\le g$ ρ-a.e., for any $n\ge 1$.;
- (ii)
- if $\left\{{g}_{n}\right\}$ is monotone decreasing, i.e., ${g}_{n+1}\le {g}_{n}$ ρ-a.e., for any $n\ge 1$, then $g\le {g}_{n}$ ρ-a.e., for any $n\ge 1$.

**Definition**

**3**

**.**Let ρ be convex regular modular. Let $r>0$ and $\epsilon >0$. Consider the following set:

- (i)
- ρ is said to be uniformly convex $\left(UC\right)$ if for every $r>0$ and $\epsilon >0$, we have ${\delta}_{\rho}(r,\epsilon )>0$.
- (ii)
- ρ is said to be $\left(UUC\right)$ if for every $s\ge 0,\epsilon >0$ there exists $\eta (s,\epsilon )>0$ such that ${\delta}_{\rho}(r,\epsilon )>\eta (s,\epsilon )>0$, for $r>s$.

**Example**

**1.**

**Theorem**

**3**

**.**Let ρ be $\left(UUC\right)$ convex regular modular. Then ${L}_{\rho}$ has the property $\left(R\right)$, i.e., every sequence $\left\{{C}_{n}\right\}$ of nonempty, ρ-bounded, ρ-closed, convex subsets of ${L}_{\rho}$ such that ${C}_{n+1}\subset {C}_{n}$, for any $n\in \mathbb{N}$, has a nonempty intersection, i.e., $\bigcap _{n\in \mathbb{N}}}{C}_{n}\ne \varnothing $.

**Remark**

**2.**

**Lemma**

**1**

**.**Let ρ be $\left(UUC\right)$ convex regular modular. Let $R>0$ and $\left\{{\alpha}_{n}\right\}\subset [a,b]$ with $0<a\le b<1$. Let $\left\{{f}_{n}\right\}$ and $\left\{{g}_{n}\right\}$ be in ${L}_{\rho}$. Assume that

**Definition**

**4**

**.**Let ρ be convex regular modular. Let K be a nonempty subset of ${L}_{\rho}$. The function $\phi :K\to [0,\infty ]$ is said to be a ρ-type if there exists a sequence $\left\{{h}_{m}\right\}$ in ${L}_{\rho}$ such that

**Lemma**

**2**

**.**Let ρ be $\left(UUC\right)$ convex regular modular. Then any minimizing sequence of any ρ-type defined on a ρ-bounded ρ-closed convex nonempty subset C of ${L}_{\rho}$ is ρ-convergent. Its ρ-limit is independent of the minimizing sequence.

**Definition**

**5.**

- (i)
- T is said to be monotone if $f\le g$ρ-a.e. implies $T\left(f\right)\le T\left(g\right)$ρ-a.e., for any $f,g\in C$.
- (ii)
- T is called monotone asymptotically nonexpansive (in short M-A-N) if T is monotone and there exists $\left\{{k}_{n}\right\}$, with ${k}_{n}\ge 1$ for any $n\ge 1$ such that $\underset{n\to \infty}{lim}{k}_{n}=1$ and$$\rho ({T}^{n}\left(g\right)-{T}^{n}\left(h\right))\le {k}_{n}\phantom{\rule{4pt}{0ex}}\rho (g-h),$$
- (iii)
- g is a fixed point of T whenever $T\left(g\right)=g$.

**Theorem**

**4**

**.**Let ρ be $\left(UUC\right)$ convex regular modular. Let C be a ρ-bounded ρ-closed convex nonempty subset of ${L}_{\rho}$. Let $T:C\to C$ be a ρ-continuous M-A-N mapping. Assume there exists ${f}_{0}\in K$ such that ${f}_{0}\le T\left({f}_{0}\right)$ (resp. $T\left({f}_{0}\right)\le {f}_{0}$) ρ-a.e. Then T has a fixed point f such that ${f}_{0}\le f$ (resp. $f\le {f}_{0}$) ρ-a.e.

**Lemma**

**3.**

- (i)
- ${h}_{0}\le {h}_{n}\le {h}_{n+1}\le {T}^{\varphi \left(n\right)}\left({h}_{n}\right)\le f$ (resp. $f\le {T}^{\varphi \left(n\right)}\left({h}_{n}\right)\le {h}_{n+1}\le {h}_{n}\le {h}_{0}$) ρ-a.e.;
- (ii)
- ${h}_{0}\le {T}^{\varphi \left(n\right)}\left({h}_{0}\right)\le {T}^{\varphi \left(n\right)}\left({h}_{n}\right)\le f$ (resp. $f\le {T}^{\varphi \left(n\right)}\left({h}_{n}\right)\le {T}^{\varphi \left(n\right)}\left({h}_{0}\right)\le {h}_{0}$) ρ-a.e.;

**Lemma**

**4.**

**Proof.**

## 3. Main Results

**Proposition**

**1.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Theorem**

**6.**

**Proof.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Bin Dehaish, B.A.; Khamsi, M.A.
Fibonacci–Mann Iteration for Monotone Asymptotically Nonexpansive Mappings in Modular Spaces. *Symmetry* **2018**, *10*, 481.
https://doi.org/10.3390/sym10100481

**AMA Style**

Bin Dehaish BA, Khamsi MA.
Fibonacci–Mann Iteration for Monotone Asymptotically Nonexpansive Mappings in Modular Spaces. *Symmetry*. 2018; 10(10):481.
https://doi.org/10.3390/sym10100481

**Chicago/Turabian Style**

Bin Dehaish, Buthinah A., and Mohamed A Khamsi.
2018. "Fibonacci–Mann Iteration for Monotone Asymptotically Nonexpansive Mappings in Modular Spaces" *Symmetry* 10, no. 10: 481.
https://doi.org/10.3390/sym10100481