# On the Fixed Circle Problem on Metric Spaces and Related Results

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## Abstract

**:**

## 1. Introduction

**Theorem**

**1**

**.**Consider the metric space $(\mathrm{Y},\mathrm{d})$ and let the mapping

**Theorem**

**2**

**Theorem**

**3**

**Theorem**

**4**

- 1.
- $\mathrm{d}(fy,{y}_{0})=\varrho $ for every $y\in {C}_{{y}_{0},\varrho}$,
- 2.
- $\mathrm{d}(fy,fz)>\rho $ for every $y,z\in {C}_{{y}_{0},\varrho}$ and $y\ne z$,
- 3.
- $\mathrm{d}(fy,fz)\le \mathrm{d}(y,z)-{\varphi}_{\varrho}\left(\mathrm{d}(y,fy)\right)$ for every $y,z\in {C}_{{y}_{0},\varrho}$,

## 2. New Fixed-Circle Theorems for Some Generalized Contractive Mappings

**Theorem**

**5.**

- 1.
- $\mathrm{d}(fy,{y}_{0})\le {\theta}_{\varrho}\left(\mathrm{d}(y,{y}_{0})\right)+L\mathrm{d}(y,fy)$ for some $L\in \left(-\infty ,0\right]$ and every $y\in \mathrm{Y}$,
- 2.
- $\varrho \le \mathrm{d}(fy,{y}_{0})$ for every $y\in {C}_{{y}_{0},\varrho}$,
- 3.
- $\mathrm{d}(fy,fz)\ge 2\varrho $ for every $y,z\in {C}_{{y}_{0},\varrho}$ and $y\ne z$,
- 4.
- $\mathrm{d}(fy,fz)<\varrho +\mathrm{d}(z,fy)$ for every $y,z\in {C}_{{y}_{0},\varrho}$ and $y\ne z$,

**Proof.**

**Case 1.**If $L=0$, then we find $\mathrm{d}(fy,{y}_{0})=\varrho $ by (4), that is, we have $fy\in {C}_{{y}_{0},\varrho}$. Assume that $\mathrm{d}(y,fy)\ne 0$ for $y\in {C}_{{y}_{0},\varrho}$. Since $y\ne fy$, by using condition (3), we obtain

**Case 2.**Let $L\in \left(-\infty ,0\right)$. If $\mathrm{d}(y,fy)\ne 0$, we obtain a contradiction by (4). Hence, it should be $\mathrm{d}(y,fy)=0$.

**Remark**

**1.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Theorem**

**6.**

- 1.
- $2\mathrm{d}(y,{y}_{0})-\mathrm{d}(fy,{y}_{0})\le {\theta}_{\rho}\left(\mathrm{d}(y,{y}_{0})\right)+L\mathrm{d}(y,fy)$ for some $L\in \left(-\infty ,0\right]$ and each $y\in \mathrm{Y}$,
- 2.
- $\mathrm{d}(fy,{y}_{0})\le \rho $ for each $y\in {C}_{{y}_{0},\rho}$,
- 3.
- $\mathrm{d}(fy,fz)\ge 2\rho $ for every $y,z\in {C}_{{y}_{0},\rho}$ and $y\ne z$,
- 4.
- $\mathrm{d}(fy,fz)<\rho +\mathrm{d}(z,fy)$ for each $y,z\in {C}_{{y}_{0},\rho}$ and $y\ne z$,

**Proof.**

**Remark**

**2.**

**Example**

**5.**

**Example**

**6.**

**Theorem**

**7.**

**Proof.**

**Example**

**7.**

**Example**

**8.**

**Theorem**

**8.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**9**

**Theorem**

**10.**

**Proof.**

## 3. New Classes of Contractive and Expanding Mappings in Metric Spaces

**Definition**

**1**

**Definition**

**2.**

**Proposition**

**1.**

**Theorem**

**11.**

**Proof.**

**Remark**

**3.**

**Example**

**9.**

**Example**

**10.**

**Example**

**11.**

**Definition**

**3.**

**Theorem**

**12.**

**Proof.**

**Example**

**12.**

**Example**

**13.**

**Remark**

**4.**

## 4. Fixed Point Sets of Activation Functions

**Example**

**14.**

## 5. Conclusions and Prospective Initiatives

**Question**

**13.**

**Notation**

**14.**

- 1.
- ${m}_{{s}_{a,b,c}}:=min\{{m}_{s}(a,a,a),{m}_{s}(b,b,b),{m}_{s}(c,c,c)\}$
- 2.
- ${M}_{{s}_{a,b,c}}:=max\{{m}_{s}(a,a,a),{m}_{s}(b,b,b),{m}_{s}(c,c,c)\}$

**Definition**

**4.**

- 1.
- ${m}_{s}(a,a,a)={m}_{s}(b,b,b)={m}_{s}(c,c,c)={m}_{s}(a,b,c)\u27faa=b=c,$
- 2.
- ${m}_{{s}_{a,b,c}}\le {m}_{s}(a,b,c),$
- 3.
- ${m}_{s}(a,a,b)={m}_{s}(b,b,a),$
- 4.
- $$\begin{array}{cc}\hfill ({m}_{s}(a,b,c)-{m}_{{s}_{a,b,c}})\le & ({m}_{s}(a,a,t)-{m}_{{s}_{a,a,t}})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& +({m}_{s}(b,b,t)-{m}_{{s}_{b,b,t}})+({m}_{s}(c,c,t)-{m}_{{s}_{c,c,t}}).\hfill \end{array}$$

**Question**

**15.**

**Question**

**16.**

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Mlaiki, N.; Özgür, N.; Taş, N.; Santina, D.
On the Fixed Circle Problem on Metric Spaces and Related Results. *Axioms* **2023**, *12*, 401.
https://doi.org/10.3390/axioms12040401

**AMA Style**

Mlaiki N, Özgür N, Taş N, Santina D.
On the Fixed Circle Problem on Metric Spaces and Related Results. *Axioms*. 2023; 12(4):401.
https://doi.org/10.3390/axioms12040401

**Chicago/Turabian Style**

Mlaiki, Nabil, Nihal Özgür, Nihal Taş, and Dania Santina.
2023. "On the Fixed Circle Problem on Metric Spaces and Related Results" *Axioms* 12, no. 4: 401.
https://doi.org/10.3390/axioms12040401