# The Meir–Keeler Fixed Point Theorem for Quasi-Metric Spaces and Some Consequences

^{*}

## Abstract

**:**

**2016**, 9, 5245–5251. We also give an application to the study of existence of solution for a type of recurrence equations associated to certain nonlinear difference equations.

## 1. Introduction

**Example**

**1.**

## 2. Results

**Definition**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Proposition**

**1.**

**Proof**

**of**

**Proposition**

**1.**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

**Corollary**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Definition**

**2.**

**Remark**

**3.**

**Proposition**

**2.**

**Theorem**

**5.**

**Theorem**

**6.**

**Proof**

**of**

**Theorem**

**6.**

**Theorem**

**7.**

**Theorem**

**8.**

**Corollary**

**2.**

**Remark**

**4.**

**Example**

**6.**

## 3. An Application

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Alegre, C.; Dağ, H.; Romaguera, S.; Tirado, P. On the fixed point theory in bicomplete quasi-metric spaces. J. Nonlinear Sci. Appl.
**2016**, 9, 5245–5251. [Google Scholar] [CrossRef] - Boyd, D.W.; Wong, J.S.W. On nonlinear contractions. Proc. Am. Math. Soc.
**1969**, 20, 458–464. [Google Scholar] [CrossRef] - Meir, A.; Keeler, E. A theorem on contraction mappings. J. Math. Anal. Appl.
**1969**, 28, 326–329. [Google Scholar] [CrossRef] [Green Version] - Aydi, H.; Karapinar, E. A Meir–Keeler common type fixed point theorem on partial metric spaces. Fixed Point Theory Appl.
**2012**, 2012, 26. [Google Scholar] [CrossRef] - Chen, C.M. Fixed point theory for the cyclic weaker Meir–Keeler function in complete metric spaces. Fixed Point Theory Appl.
**2012**, 2012, 17. [Google Scholar] [CrossRef] - Chen, C.M. Fixed point theorems for cyclic Meir–Keeler type mappings in complete metric spaces. Fixed Point Theory Appl.
**2012**, 2012, 41. [Google Scholar] [CrossRef] - Chen, C.M.; Karapinar, E. Fixed point results for the α-Meir–Keeler contraction on partial Hausdorff metric spaces. J. Inequalities Appl.
**2013**, 2013, 410. [Google Scholar] [CrossRef] - Choban, M.; Verinde, B. Multiple fixed point theorems for contractive and Meir–Keeler type mappings defined on partially ordered spaces with a distance. Appl. Gen. Top.
**2017**, 18, 317–330. [Google Scholar] [CrossRef] - Di Bari, C.; Suzuki, T.; Vetro, C. Best proximity points for cyclic Meir–Keeler contractions. Nonlinear Anal.
**2008**, 69, 3790–3794. [Google Scholar] [CrossRef] - Jachymski, J. Equivalent conditions and the Meir–Keeler type theorems. J. Math. Anal. Appl.
**1995**, 194, 293–303. [Google Scholar] [CrossRef] - Karapinar, E.; Czerwik, S.; Aydi, H. (α,ψ)-Meir–Keeler contraction mappings in generalized b-metric spaces. J. Funct. Spaces
**2018**, 2018, 3264620. [Google Scholar] [CrossRef] - Mustafa, Z.; Aydi, H.; Karapinar, E. Generalized Meir–Keeler type contractions on G-metric spaces. Appl. Math. Comput.
**2013**, 219, 10441–10447. [Google Scholar] [CrossRef] - Nashine, H.K.; Romaguera, S. Fixed point theorems for cyclic self-maps involving weaker Meir–Keeler functions in complete metric spaces and applications. Fixed Point Theory Appl.
**2013**, 2013, 224. [Google Scholar] [CrossRef] - Pant, R.P.; Joshi, P.C.; Gupta, V. A Meir–Keeler fixed point theorem. Indian J. Pure Appl. Math.
**2001**, 32, 779–787. [Google Scholar] - Park, S.; Bae, J.S. Extensions of a fixed point theorem of Meir and Keeler. Arkiv Mat.
**1981**, 19, 223–228. [Google Scholar] [CrossRef] - Piatek, B. On cyclic Meir–Keeler contractions in metric spaces. Nonlinear Anal.
**2011**, 74, 35–40. [Google Scholar] [CrossRef] - Rhoades, B.E.; Park, S.; Moon, K.B. On generalizations of the Meir–Keeler type contraction maps. J. Math. Anal. Appl.
**1990**, 146, 482–494. [Google Scholar] [CrossRef] - Samet, B. Coupled fixed point theorems for a generalized MeirKeeler contraction in partially ordered metric spaces. Nonlinear Anal.
**2010**, 72, 4508–4517. [Google Scholar] [CrossRef] - Samet, B.; Vetro, C.; Yazidic, H. A fixed point theorem for a Meir–Keeler type contraction through rational expression. J. Nonlinear Sci. Appl.
**2013**, 6, 162–169. [Google Scholar] [CrossRef] - Cobzaş, S. Functional Analysis in Asymmetric Normed Spaces; Frontiers in Mathematics; Birkhäuser/Springer Basel AG: Basel, Switzerland, 2013. [Google Scholar]
- Schellekens, M. The Smyth completion: A common foundation for denonational semantics and complexity analysis. Electron. Notes Theor. Comput. Sci.
**1995**, 1, 535–556. [Google Scholar] [CrossRef] - Romaguera, S.; Schellekens, M. Quasi-metric properties of complexity spaces. Top. Appl.
**1999**, 98, 311–322. [Google Scholar] [CrossRef] [Green Version] - García-Raffi, L.M.; Romaguera, S.; Schellekens, M.P. Applications of the complexity space to the General Probabilistic Divide and Conquer Algorithms. J. Math. Anal. Appl.
**2008**, 348, 346–355. [Google Scholar] [CrossRef] [Green Version] - Mohammadi, Z.; Valero, O. A new contribution to the fixed point theory in partial quasi-metric spaces and its applications to asymptotic complexity analysis of algorithms. Top. Appl.
**2016**, 203, 42–56. [Google Scholar] [CrossRef] - Romaguera, S.; Tirado, P. The complexity probabilistic quasi-metric space. J. Math. Anal. Appl.
**2011**, 376, 732–740. [Google Scholar] [CrossRef] [Green Version] - Romaguera, S.; Tirado, P. A characterization of Smyth complete quasi-metric spaces via Caristi’s fixed point theorem. Fixed Point Theory Appl.
**2015**, 2015, 183. [Google Scholar] [CrossRef] - Stević, S. More on a rational recurrence relation. Appl. Math. E-Notes
**2004**, 4, 80–84. [Google Scholar] - Stević, S. On the recursive sequence x
_{n+1}= g(x_{n},x_{n+1})/(A + x_{n}). Appl. Math. Lett.**2002**, 15, 305–308. [Google Scholar] [CrossRef] - Stević, S. On the recursive sequence x
_{n+1}= x_{n−1}/g(x_{n}). Taiwan. J. Math.**2002**, 6, 405–414. [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**

Romaguera, S.; Tirado, P.
The Meir–Keeler Fixed Point Theorem for Quasi-Metric Spaces and Some Consequences. *Symmetry* **2019**, *11*, 741.
https://doi.org/10.3390/sym11060741

**AMA Style**

Romaguera S, Tirado P.
The Meir–Keeler Fixed Point Theorem for Quasi-Metric Spaces and Some Consequences. *Symmetry*. 2019; 11(6):741.
https://doi.org/10.3390/sym11060741

**Chicago/Turabian Style**

Romaguera, Salvador, and Pedro Tirado.
2019. "The Meir–Keeler Fixed Point Theorem for Quasi-Metric Spaces and Some Consequences" *Symmetry* 11, no. 6: 741.
https://doi.org/10.3390/sym11060741