# Fixed-Discs in Rectangular Metric Spaces

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction and Preliminaries

**Definition**

**1.**

- (R
_{1}) $\theta =\vartheta $ if and only if ${d}_{R}(\theta ,\vartheta )=0$; - (R
_{2}) ${d}_{R}(\theta ,\vartheta )={d}_{R}(\vartheta ,\theta )$; - (R
_{3}) ${d}_{R}(\theta ,\vartheta )\le {d}_{R}(\theta ,\xi )+{d}_{R}(\xi ,\eta )+{d}_{R}(\eta ,\vartheta )$

**Definition**

**2.**

- 1.
- $\mathcal{S}(\xi ,\eta ,\theta )=0$ if and only if $\xi =\eta =\theta $,
- 2.
- $\mathcal{S}(\xi ,\eta ,\theta )\le \mathcal{S}(\xi ,\xi ,a)+\mathcal{S}(\eta ,\eta ,a)+\mathcal{S}(\theta ,\theta ,a)$.

**Lemma**

**1.**

- 1.
- the function given as ${\mathcal{S}}_{d}(\xi ,\eta ,\theta )=d(\xi ,\theta )+d(\eta ,\theta ),$ for all $\xi ,\eta ,\theta \in X$, is an S-metric on X.
- 2.
- ${\xi}_{n}\to \xi $ in $(X,d)$ if ${\xi}_{n}\to \xi $ in $(X,{\mathcal{S}}_{d})$.
- 3.
- $\left\{{\xi}_{n}\right\}$ is Cauchy in $(X,d)$ iff $\left\{{\xi}_{n}\right\}$ is Cauchy in $(X,{\mathcal{S}}_{d}).$
- 4.
- $(X,d)$ is complete iff $(X,{\mathcal{S}}_{d})$ is complete.

**Definition**

**3.**

**Proposition**

**1.**

**Corollary**

**1.**

**Proposition**

**2.**

**Corollary**

**2.**

**Example**

**1.**

**Example**

**2.**

## 2. Main Results

**Definition**

**4.**

#### 2.1. New Contractions via $\alpha $-${\xi}_{0}$-Admissible Maps

**Definition**

**5.**

**Theorem**

**1.**

**Proof.**

**Definition**

**6.**

**Definition**

**7.**

**Theorem**

**2.**

**Proof.**

**Definition**

**8.**

- (F
_{1}) F is strictly increasing; - (F
_{2}) For every positive sequence $\left\{{\lambda}_{n}\right\}$, we have$$\underset{n\to \infty}{lim}{\lambda}_{n}=0\phantom{\rule{4.pt}{0ex}}\mathit{iff}\phantom{\rule{4.pt}{0ex}}\underset{n\to \infty}{lim}F\left({\lambda}_{n}\right)=\infty ;$$ - (F
_{3}) There is $u\in (0,1)$ in order that $\underset{\lambda \to {0}^{+}}{lim}{\alpha}^{u}F\left(\lambda \right)=0$.

**Definition**

**9.**

**Theorem**

**3.**

**Proof.**

**Definition**

**10.**

**Proposition**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

#### 2.2. Branciari Type ${F}_{d}$-Contractions

**Definition**

**11.**

**Theorem**

**5.**

**Proof.**

**Definition**

**12.**

**Theorem**

**6.**

**Proof.**

#### 2.3. Some Remarks

#### 2.4. Illustrative Examples

**Example**

**3.**

**The**${\mathit{\xi}}_{\mathit{0}}$

**-contractive self-mapping**$\mathit{T}\mathit{:}$ The mapping T is an ${\xi}_{0}$-contraction with ${\xi}_{0}=0$ and $k=\frac{1}{2}$. Indeed, we get the following cases:

- Case 1:Let $\xi \in \left\{0\right\}\cup B$. Then, we have$${d}_{R}(\xi ,T\xi )=0\le \frac{1}{2}{d}_{R}(0,\xi ).$$
- Case 2: Let $\xi =2$. Then, we have$${d}_{R}(\xi ,T\xi )={d}_{R}\left(2,\frac{1}{2}\right)=\frac{1}{2}\le \frac{1}{2}{d}_{R}(0,2)=\frac{1}{2}.$$

**The**$\mathit{\alpha}$

**-**${\mathit{\xi}}_{\mathit{0}}$

**-contractive and**$\mathit{\alpha}$

**-**${\mathit{\xi}}_{\mathit{0}}$

**-admissible self-mapping**$\mathit{T}\mathbf{:}$ If we take ${\xi}_{0}=0$ and the function $\alpha :X\times X\to (0,\infty )$ defined as $\alpha (\xi ,\eta )=1$, then T verifies the condition of Theorem 2 similar to the above cases.

**The**${\mathit{F}}_{\mathit{d}}$

**-contractive and**$\mathit{\alpha}$

**-**${\mathit{\xi}}_{\mathit{0}}$

**-admissible self-mapping**$\mathit{T}\mathbf{:}$ If we take $F=ln\xi $, $t=ln4$, ${\xi}_{0}=0$ and $\alpha :X\times X\to (0,\infty )$ such that $\alpha (\xi ,\eta )=2$, then T satisfies the condition of Theorem 3. Indeed, we get

**The Ćirić type**${\mathit{F}}_{\mathit{d}}$

**-contractive and**$\mathit{\alpha}$

**-**${\mathit{\xi}}_{\mathit{0}}$

**-admissible self-mapping**$\mathit{T}\mathbf{:}$ If we take $F=ln\xi $, $t=ln4$, ${\xi}_{0}=0$ and $\alpha :X\times X\to (0,\infty )$ given as $\alpha (\xi ,\eta )=2$, then T verifies the conditions of Proposition 3 and Theorem 4. Indeed, we get

**The Branciari**${\mathit{F}}_{\mathit{d}}$

**-contractive self-mapping**$\mathit{T}\mathbf{:}$ If we take $F=ln\xi $, $t=ln2$ and ${\xi}_{0}=0$, then T verifies the condition of Theorem 5. Indeed, we get

**The Branciari**${\mathit{F}}_{\mathit{d}}$

**-rational contractive self-mapping**$\mathit{T}\mathbf{:}$ If we take $F=ln\xi $, $t=ln2$ and ${\xi}_{0}=0$, then T verifies the condition of Theorem 6. Indeed, we get

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

## 3. Conclusions and Perspectives

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Scarselli, F.; Gori, M.; Tsoi, A.C.; Hagenbuchner, M.; Monfardini, G. The graph neural network model. IEEE Trans. Neural Netw.
**2009**, 20, 61–80. [Google Scholar] [CrossRef] [PubMed] - Mandic, D.P. The use of Möbius transformations in neural networks and signal processing. In Proceedings of the Neural Networks for Signal Processing X, Sydney, NSW, Australia, 11–13 September 2000. [Google Scholar]
- Özdemir, N.; İskender, B.B.; Özgür, N.Y. Complex valued neural network with Möbius activation function, Commun. Nonlinear Sci. Numer. Simul.
**2011**, 16, 4698–4703. [Google Scholar] [CrossRef] - Özgür, N.Y.; Taş, N. Some fixed-circle theorems on metric spaces. Bull. Malays. Math. Sci. Soc.
**2017**, 1–17. [Google Scholar] [CrossRef] - Pant, R.P.; Özgür, N.Y.; Taş, N. On discontinuity problem at fixed point. Bull. Malays. Math. Sci. Soc.
**2018**, 1–19. [Google Scholar] [CrossRef] - Rashid, M.; Batool, I.; Mehmood, N. Discontinuous mappings at their fixed points and common fixed points with applications. J. Math. Anal.
**2018**, 9, 90–104. [Google Scholar] - Taş, N.; Özgür, N.Y. A new contribution to discontinuity at fixed point. Fixed Point Theory
**2019**, in press. [Google Scholar] - Clevert, D.A.; Unterthiner, T.; Hochreiter, S. Fast and accurate deep networks learning by exponential linear units (ELUs). In Proceedings of the International Conference on Learning Representations, San Juan, Puerto Rico, 2–4 May 2016. [Google Scholar]
- Jin, X.; Xu, C.; Feng, J.; Wei, Y.; Xiong, J.; Yan, S. Deep learning with S-shaped rectified linear activation units. AAAI
**2016**, 3, 1737–1743. [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] - Mlaiki, N.; Taş, N.; Özgür, N.Y. On the fixed-circle problem and Khan type contractions. Axioms
**2018**, 7, 80. [Google Scholar] [CrossRef] - Özgür, N.Y.; Taş, N. Some fixed-circle theorems and discontinuity at fixed circle. AIP Conf. Proc.
**2018**, 1926, 020048. [Google Scholar] [CrossRef] - Mlaiki, N.; Çelik, U.; Taş, N.; Özgür, N.Y.; Mukheimer, A. Wardowski type contractions and the fixed-circle problem on S-metric spaces. J. Math.
**2018**, 2018, 1–9. [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] - Taş, N. Various types of fixed-point theorems on S-metric spaces. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi
**2018**, 20, 211–223. [Google Scholar] - Taş, N. Suzuki-Berinde type fixed-point and fixed-circle results on S-metric spaces. J. Linear Topol. Algebra
**2018**, 7, 233–244. [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]
- Branciari, A. A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces. Publ. Math.
**2000**, 57, 31–37. [Google Scholar] - Alharbi, N.; Aydi, H.; Felhi, A.; Ozel, C.; Sahmim, S. α-Contractive mappings on rectangular b-metric spaces and an application to integral equations. J. Math. Anal.
**2018**, 9, 47–60. [Google Scholar] - Ansari, A.H.; Aydi, H.; Kumari, P.S.; Yildirim, I. New fixed point results via C-class functions in b-rectangular metric spaces. Commun. Math. Anal.
**2018**, 9, 109–126. [Google Scholar] - Aydi, H.; Karapınar, E.; Shatanawi, W. Tripled fixed point results in generalized metric spaces. J. Appl. Math.
**2012**, 2012, 1–10. [Google Scholar] [CrossRef] - Kadelburg, Z.; Radenović, S. Pata-type common fixed point results in b-metric and b-rectangular metric spaces. J. Nonlinear Sci. Appl.
**2015**, 8, 944–954. [Google Scholar] [CrossRef] - Aydi, H.; Karapınar, E.; Zhang, D. On common fixed points in the context of Brianciari metric spaces. Results Math.
**2017**, 71, 73–92. [Google Scholar] [CrossRef] - Karapinar, E. Discussion on (α,ψ)-contractions on generalized metric spaces. Abstr. Appl. Anal.
**2014**, 2014, 1–7. [Google Scholar] [CrossRef] - Kirk, W.A.; Shahzad, N. Generalized metrics and Caristi’s theorem. Fixed Point Theory Appl.
**2013**, 2013, 129. [Google Scholar] [CrossRef] [Green Version] - Aydi, H.; Chen, C.M.; Karapinar, E. Interpolative Ćirić-Reich-Rus type contractions via the Branciari distance. Mathematics
**2019**, 7, 84. [Google Scholar] [CrossRef] - Mlaiki, N.; Abodayeh, K.; Aydi, H.; Abdeljawad, T.; Abuloha, M. Rectangular metric-like type spaces and related fixed points. J. Math.
**2018**, 2018, 1–7. [Google Scholar] [CrossRef] - Shatanawi, W.; Al-Rawashdeh, A.; Aydi, H.; Nashine, H.K. On a fixed point for generalized contractions in generalized metric spaces. Abstr. Appl. Anal.
**2012**, 2012, 1–13. [Google Scholar] [CrossRef] - Suzuki, T. generalized metric spaces do not have the compatible topology. Abstr. Appl. Anal.
**2014**, 2014, 1–5. [Google Scholar] [CrossRef] - Souyah, N.; Aydi, H.; Abdeljawad, T.; Mlaiki, N. Best proximity point theorems on rectangular metric spaces endowed with a graph. Axioms
**2019**, 8, 17. [Google Scholar] [CrossRef] - Sedghi, S.; Shobe, N.; Aliouche, A. A generalization of fixed point theorems in S-metric spaces. Matematički Vesnik
**2012**, 64, 258–266. [Google Scholar] - Hieu, N.T.; Ly, N.T.; Dung, N.V. A generalization of Ciric quasi-contractions for maps on S-metric spaces. Thai J. Math.
**2015**, 13, 369–380. [Google Scholar] - Özgür, N.Y.; Taş, N. Some new contractive mappings on S-metric spaces and their relationships with the mapping (S25). Math. Sci.
**2017**, 11, 7–16. [Google Scholar] [CrossRef] - Gupta, A. Cyclic contraction on S-metric space. Int. J. Anal. Appl.
**2013**, 3, 119–130. [Google Scholar] - Roshan, J.R.; Hussain, N.; Parvaneh, V.; Kadelburg, Z. New fixed point results in rectangular b-metric spaces. Nonlinear Anal. Model. Control
**2016**, 21, 614–634. [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] - Banach, S. Sur les operations dans les ensembles abstraits et leur application aux equations integrals. Fundam. Math.
**1922**, 2, 133–181. [Google Scholar] [CrossRef]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Aydi, H.; Taş, N.; Özgür, N.Y.; Mlaiki, N.
Fixed-Discs in Rectangular Metric Spaces. *Symmetry* **2019**, *11*, 294.
https://doi.org/10.3390/sym11020294

**AMA Style**

Aydi H, Taş N, Özgür NY, Mlaiki N.
Fixed-Discs in Rectangular Metric Spaces. *Symmetry*. 2019; 11(2):294.
https://doi.org/10.3390/sym11020294

**Chicago/Turabian Style**

Aydi, Hassen, Nihal Taş, Nihal Yılmaz Özgür, and Nabil Mlaiki.
2019. "Fixed-Discs in Rectangular Metric Spaces" *Symmetry* 11, no. 2: 294.
https://doi.org/10.3390/sym11020294