# Fixed-Point Convergence of Multi-Valued Non-Expansive Mappings with Applications

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- $N\left(A\right)$: all of the non-empty subsets of A;
- $CK\left(A\right)$: all of the non-empty, convex, and compact subsets of A;
- $C\left(A\right)$: all of the non-empty, closed subsets of A;
- $CB\left(A\right)$: all of the non-empty, closed, and bounded subsets of A;
- $CC\left(A\right)$: all of the non-empty, closed, and convex subsets of A;
- $CCB\left(A\right)$: all of the non-empty, closed, convex, and bounded subsets of A.

**Definition**

**1.**

**Definition**

**2.**

**Remark**

**1.**

**Definition**

**3.**

**Definition**

**4.**

**Theorem**

**1**

- 1.
- X has the property $\left(R\right)$, i.e., any decreasing sequence of non-empty, convex, bounded, and closed sets that have a non-empty intersection.
- 2.
- If $Z\in CC\left(X\right),$ then any type function $\eta :X\to [0,\infty )$ attains a minimal point u in Z that is unique, thereby satisfying$$\begin{array}{c}\hfill \eta \left(u\right)=inf\left\{\eta \right(x);x\in Z\}.\end{array}$$Furthermore, any minimizing sequence $\left\{{u}_{m}\right\}$ in Z is convergent, that is, ${lim}_{m\to \infty}\eta \left({u}_{m}\right)=\eta \left(u\right)$.
- 3.
- Let $\Omega >0$ and $z\in X$. Suppose $\left\{{x}_{m}\right\}$ and $\left\{{y}_{m}\right\}$ are any two arbitrary sequences in X satisfying$$\underset{m\to \infty}{lim\; sup}\varrho ({x}_{m},z)\le \Omega ,\phantom{\rule{4pt}{0ex}}\underset{m\to \infty}{lim\; sup}\varrho ({y}_{m},z)\le \Omega ,$$and$$\begin{array}{c}\hfill \underset{m\to \infty}{lim}\varrho (\alpha {x}_{m}\oplus (1-\alpha ){y}_{m},z)=\Omega ,\end{array}$$then ${lim}_{m\to \infty}\varrho ({x}_{m},{y}_{m})=0$.

**Definition**

**5.**

**Remark**

**2.**

- 1.
- In the case of a compact valued operator T, H-continuity coincides with the lower and upper semi-continuity.
- 2.
- An asymptotically non-expansive map $T:C\to N\left(C\right)$ always fulfills the criterion of H-continuity.

**Theorem**

**2.**

**Definition**

**6.**

- 1.
- The edges of $\mathfrak{G}$ under $\mathfrak{f}$ are preserved, that is, for all elements $\nu ,\omega $ in X, such that$$\left(\mathfrak{f}\nu ,\mathfrak{f}\omega \right)\in \mathcal{E}\left(\mathfrak{G}\right)whenever\left(\nu ,\omega \right)\in \mathcal{E}\left(\mathfrak{G}\right).$$
- 2.
- The corresponding weights of edges of $\mathfrak{G}$ under $\mathfrak{f}$ decrease in a subsequent manner, that is, an element $k\in (0,1)$ exists by satisfying$$\varrho (\mathfrak{f}\nu ,\mathfrak{f}\omega )\le k\varrho (\nu ,\omega )\phantom{\rule{4pt}{0ex}}whenever\phantom{\rule{4pt}{0ex}}(\nu ,\omega )\in \mathcal{E}\left(\mathfrak{G}\right).$$

## 3. Convergence Results for Multi-Valued $\mathfrak{G}$-Asymptotically Non-Expansive Mappings

**Definition**

**7**

**(Multi-valued $\mathfrak{G}$-Asymptotically Non-expansive Mapping).**Let $\mathfrak{G}$ represent a directed graph on X. Then, a mapping $\mathcal{P}:X\to C\left(X\right)$ is said to be a multi-valued $\mathfrak{G}$-asymptotically non-expansive mapping if the following conditions hold:

- 1.
- There exists $\left\{{b}_{m}\right\}$ with ${lim}_{m\to \infty}{b}_{m}=1$;
- 2.
- $\mathcal{P}$ preserves the edges, that is,$$\begin{array}{c}\hfill (q,r)\in \mathcal{E}\left(\mathfrak{G}\right)\phantom{\rule{4pt}{0ex}}implies\phantom{\rule{4pt}{0ex}}(\stackrel{\xb4}{q},\stackrel{\xb4}{r})\in \mathcal{E}\left(\mathfrak{G}\right),\end{array}$$where $\stackrel{\xb4}{q}$ is an element of $\mathcal{P}\left(q\right)$ and $\stackrel{\xb4}{r}$ belongs to $\mathcal{P}\left(r\right).$
- 3.
- Let q and r be any two elements of X. Then, for any generalized orbit $\left\{{q}_{m}\right\}$ of q, there exists a generalized orbit $\left\{{r}_{m}\right\}$ of r such that $({q}_{m},r)\in \mathcal{E}\left(\mathfrak{G}\right),$ and$$\begin{array}{c}\hfill \varrho ({q}_{m+h},{r}_{h})\le {b}_{h}\varrho ({q}_{m},r),\phantom{\rule{4pt}{0ex}}for\phantom{\rule{4pt}{0ex}}m,h\in \mathbb{N}.\end{array}$$

**Condition $(\mathcal{A}$)**. Let $\mathfrak{G}$ represent a directed graph on X. Let $Z\in CC\left(X\right)$ and $\left\{{q}_{m}\right\}$ be a generalized orbit of q in X. Then, the type function

**Condition ($\mathcal{B}$)**. Let $\mathfrak{G}$ represent a directed graph on X and $Z\in CCB\left(X\right)$. Let $q\in Z$ and $\left\{{q}_{m}\right\}$ be the generalized orbit of q. Then, for $r=\alpha q\oplus (1-\alpha ){q}_{1}$, we have

- (i)
- $(q,r)\in \mathcal{E}\left(\mathfrak{G}\right)$,
- (ii)
- $({q}_{m},r)\in \mathcal{E}\left(\mathfrak{G}\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}for\phantom{\rule{4.pt}{0ex}}all\phantom{\rule{4pt}{0ex}}m\in \mathbb{N}$.

**Theorem**

**3.**

**Proof.**

- (i)
- $({q}_{2},{q}_{1})\in \mathcal{E}\left(\mathfrak{G}\right)$;
- (ii)
- $({q}_{m}^{1},{q}_{2})\in \mathcal{E}\left(\mathfrak{G}\right)$.

- (i)
- $({q}_{2},{q}_{3})\in \mathcal{E}\left(\mathfrak{G}\right);$
- (ii)
- $({q}_{m}^{2},{q}_{3})\in \mathcal{E}\left(\mathfrak{G}\right);$

**Theorem**

**4.**

- (i)
- ${\left\{{b}_{m}\right\}}_{m\in \mathbb{N}}$ is the Lipschitz sequence associated with $\mathcal{P}$ and
- (ii)
- the series ${\sum}_{m\in \mathbb{N}}({b}_{m}-1)$ is convergent.

**Proof.**

## 4. Some Consequences of the Convergence Results

**Corollary**

**1.**

- (i)
- ${\left\{{b}_{m}\right\}}_{m\in \mathbb{N}}$ is a Lipschitz sequence associated with $\mathcal{P}$, and that
- (ii)
- the sequence ${\sum}_{m\in \mathbb{N}}({b}_{m}-1)$ converges.

**Proof.**

**Remark**

**3.**

**Example**

**1.**

**Case 1:**For $h=0$, (7) becomes

**Case 2:**For $h=1$, (7) reduces to

**Case 3:**For $h=2$, on similar lines of the abovementioned case, we have

**Open Problem**: On similar lines, one can also define the idea of $\mathfrak{G}$-total asymptotically non-expansive mappings and prove convergence theorems in Hadamard spaces, as well as in convex hyperbolic metric spaces.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Graphical representation of the unique FPs of the mapping, as defined by (1).

**Figure 2.**Visualization of the numerous FPs of the mapping, as defined by (2).

**Figure 3.**Illustration of the infinitely many FPs of the mapping, as defined by (3).

**Figure 4.**Representation of multi-valued $\mathfrak{G}$-asymptotically non-expansive mapping, as defined by (6).

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**MDPI and ACS Style**

Azam, A.; Rashid, M.; Kalsoom, A.; Ali, F.
Fixed-Point Convergence of Multi-Valued Non-Expansive Mappings with Applications. *Axioms* **2023**, *12*, 1020.
https://doi.org/10.3390/axioms12111020

**AMA Style**

Azam A, Rashid M, Kalsoom A, Ali F.
Fixed-Point Convergence of Multi-Valued Non-Expansive Mappings with Applications. *Axioms*. 2023; 12(11):1020.
https://doi.org/10.3390/axioms12111020

**Chicago/Turabian Style**

Azam, Akbar, Maliha Rashid, Amna Kalsoom, and Faryad Ali.
2023. "Fixed-Point Convergence of Multi-Valued Non-Expansive Mappings with Applications" *Axioms* 12, no. 11: 1020.
https://doi.org/10.3390/axioms12111020