# Fixed Points of Eventually Δ-Restrictive and Δ(ϵ)-Restrictive Set-Valued Maps in Metric Spaces

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Preliminaries

**Definition**

**1.**

**Remark**

**1.**

- 1.
- It is to be noted that in the MS$\left(\mathsf{\Gamma}\right(X),\mathcal{PH})$,$\mu \in X$is a fixed point of R if and only if$\Delta (\mu ,R\mu )=0$. The function Δ plays a special role in the current paper.
- 2.
- It is well known that the metric function$\delta :X\times X\to [0,\infty )$is continuous in the sense that if$\left\{{\mu}_{n}\right\},\left\{{\nu}_{n}\right\}$are two sequences in X with$({\mu}_{n},{\nu}_{n})\to (\mu ,\nu )$for some$\mu ,\nu \in X$, as$n\to \infty $, then$\delta ({\mu}_{n},{\nu}_{n})\to \delta (\mu ,\nu )$as$n\to \infty $. It follows that the function Δ is continuous in the sense that if ${\mu}_{n}\to \mu $ as $n\to \infty $, then $\Delta ({\mu}_{n},U)\to \Delta (\mu ,U)$ as $n\to \infty $ for any $U\subseteq X$.

**Definition**

**2.**

**Definition**

**3.**

**Remark**

**2.**

- 1.
- R is$\mathcal{PH}$-continuous on a subset S of X if it is continuous on every point of S.
- 2.
- If R is a set-valued contraction, then it is$\mathcal{PH}$-continuous.

**Example**

**1.**

**Definition**

**4.**

## 2. Eventually $\Delta $-Restrictive Set-Valued Map

**Definition**

**5.**

**Definition**

**6.**

**Theorem**

**1.**

**Proof.**

**Example**

**2.**

**Theorem**

**2.**

**Proof.**

## 3. $\Delta \left(\u03f5\right)$-Restrictive Set-Valued Map

**Definition**

**7.**

**Theorem**

**3.**

**Proof.**

**Example**

**3.**

**Theorem**

**4.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Debnath, P.; de La Sen, M.
Fixed Points of Eventually Δ-Restrictive and Δ(*ϵ*)-Restrictive Set-Valued Maps in Metric Spaces. *Symmetry* **2020**, *12*, 127.
https://doi.org/10.3390/sym12010127

**AMA Style**

Debnath P, de La Sen M.
Fixed Points of Eventually Δ-Restrictive and Δ(*ϵ*)-Restrictive Set-Valued Maps in Metric Spaces. *Symmetry*. 2020; 12(1):127.
https://doi.org/10.3390/sym12010127

**Chicago/Turabian Style**

Debnath, Pradip, and Manuel de La Sen.
2020. "Fixed Points of Eventually Δ-Restrictive and Δ(*ϵ*)-Restrictive Set-Valued Maps in Metric Spaces" *Symmetry* 12, no. 1: 127.
https://doi.org/10.3390/sym12010127