# 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

- Özgür, N.Y.; Taş, N. Some fixed-circle theorems on metric spaces. Bull. Malays. Math. Sci. Soc.
**2017**. [Google Scholar] [CrossRef] - Özgür, N.Y.; Taş, N.; Çelik, U. New fixed-circle results on S-metric spaces. Bull. Math. Anal. Appl.
**2017**, 9, 10–23. [Google Scholar] - Özgür, N.Y.; Taş, N. Fixed-circle problem on S-metric spaces with a geometric viewpoint. arXiv, 2017; arXiv:1704.08838. [Google Scholar]
- Özgür, N.Y.; Taş, N. Some fixed-circle theorems and discontinuity at fixed circle. AIP Conf. Proc.
**2018**. [Google Scholar] [CrossRef] - Taş, N.; Özgür, N.Y.; Mlaiki, N. New fixed-circle results related to F
_{c}-contractive and F_{c}-expanding mappings on metric spaces. 2018; submitted for publication. [Google Scholar] - Taş, N.; Özgür, N.Y.; Mlaiki, N. New types of F
_{c}-contractions and the fixed-circle problem. Mathematics**2018**, 6, 188. [Google Scholar] [CrossRef] - Taş, N. Various types of fixed-point theorems on S-metric spaces. J. BAUN Inst. Sci. Technol.
**2018**. [Google Scholar] [CrossRef] - Özgür, N.Y.; Taş, N. Generalizations of metric spaces: From the fixed-point theory. In Applications of Nonlinear Analysis; Rassias, T., Ed.; Springer: Cham, Switzerland, 2018; Volume 134. [Google Scholar]
- Taş, N.; Özgür, N.Y. Some fixed-point results on parametric N
_{b}-metric spaces. Commun. Korean Math. Soc.**2018**, 33, 943–960. [Google Scholar] - Mlaiki, N.; Özgür, N.Y.; Mukheimer, A.; Taş, N. A new extension of the M
_{b}-metric spaces. J. Math. Anal.**2018**, 9, 118–133. [Google Scholar] - Taş, N.; Özgür, N.Y. A new contribution to discontinuity at fixed point. arXiv, 2017; arXiv:1705.03699. [Google Scholar]
- Fisher, B. On a theorem of Khan. Riv. Math. Univ. Parma
**1978**, 4, 135–137. [Google Scholar] - Khan, M.S. A fixed point theorem for metric spaces. Rend. Inst. Math. Univ. Trieste
**1976**, 8, 69–72. [Google Scholar] [CrossRef] - Piri, H.; Rahrovi, S.; Kumam, P. Generalization of Khan fixed point theorem. J. Math. Comput. Sci.
**2017**, 17, 76–83. [Google Scholar] [CrossRef] [Green Version] - Piri, H.; Rahrovi, S.; Marasi, H.; Kumam, P. A fixed point theorem for F-Khan-contractions on complete metric spaces and application to integral equations. J. Nonlinear Sci. Appl.
**2017**, 10, 4564–4573. [Google Scholar] [CrossRef] - Wardowski, D. Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl.
**2012**, 2012, 94. [Google Scholar] [CrossRef] [Green Version] - Tomar, A.; Sharma, R. Some coincidence and common fixed point theorems concerning F-contraction and applications. J. Int. Math. Virtual Inst.
**2018**, 8, 181–198. [Google Scholar] - Kadelburg, Z.; Radenovic, S. Notes on some recent papers concerning Fcontractions in b-metric spaces. Constr. Math. Anal.
**2018**, 1, 108–112. [Google Scholar] - Satish, S.; Stojan, R.; Zoran, K. Some fixed point theorems for F-generalized contractions in 0-orbitally complete partial metric spaces. Theory Appl. Math. Comput. Sci.
**2014**, 4, 87–98. [Google Scholar] - Lukacs, A.; Kajanto, S. Fixed point therorems for various types of F-contractions in complete b-metric spaces. Fixed Point Theory
**2018**, 19, 321–334. [Google Scholar] [CrossRef]

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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