# On the Fixed-Circle Problem and Khan Type Contractions

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## Abstract

**:**

## 1. Introduction

**Open Problem $CC$:**What is (are) the condition(s) to make any circle ${C}_{{x}_{0},r}$ as the common fixed circle for two (or more than two) self-mappings?

## 2. New Fixed-Circle Theorems

**Definition**

**1**

**Definition**

**2**

**.**Let $(X,d)$ be a metric space. A mapping $T:X\to X$ is said to be an F-contraction on $(X,d)$, if there exist $F\in \mathbb{F}$ and $\tau \in (0,\infty )$ such that

**Definition**

**3**

**.**Let ${\mathbb{F}}_{k}$ be the family of all increasing functions $F:(0,\infty )\to \mathbb{R}$, that is, for all $x,y\in (0,\infty )$, if $x<y$ then $F\left(x\right)\le F\left(y\right)$.

**Definition**

**4**

**.**Let $(X,d)$ be a metric space and $T:X\to X$ be a self-mapping. T is said to be an F-Khan-contraction if there exist $F\in {\mathbb{F}}_{k}$ and $t>0$ such that for all $x,y\in X$ if $max\left\{d(Ty,x),d(Tx,y)\right\}\ne 0$ then $Tx\ne Ty$ and

**Definition**

**5.**

**Proposition**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**2**

**.**Let $(X,d)$ be a metric space and $T:X\to X$ satisfy

**Definition**

**6.**

**Proposition**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**2.**

**Theorem**

**4.**

**Proof.**

**Remark**

**1.**

**Example**

**1.**

**Definition**

**7.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Example**

**2.**

## 3. Common Fixed-Circle Results

**Definition**

**8.**

**Proposition**

**3.**

**Proof.**

**Definition**

**9.**

**Proposition**

**4.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Corollary**

**3.**

**Example**

**3.**

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Mlaiki, N.; Taş, N.; Özgür, N.Y.
On the Fixed-Circle Problem and Khan Type Contractions. *Axioms* **2018**, *7*, 80.
https://doi.org/10.3390/axioms7040080

**AMA Style**

Mlaiki N, Taş N, Özgür NY.
On the Fixed-Circle Problem and Khan Type Contractions. *Axioms*. 2018; 7(4):80.
https://doi.org/10.3390/axioms7040080

**Chicago/Turabian Style**

Mlaiki, Nabil, Nihal Taş, and Nihal Yılmaz Özgür.
2018. "On the Fixed-Circle Problem and Khan Type Contractions" *Axioms* 7, no. 4: 80.
https://doi.org/10.3390/axioms7040080