# On Pata–Suzuki-Type Contractions

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction and Preliminaries

**Definition**

**1**

**.**A simulation function is a mapping $\zeta :[0,\infty )\times [0,\infty )\to \mathbb{R}$ satisfying the following conditions:

- $\left({\zeta}_{1}\right)$
- $\zeta (t,s)<s-t$ for all $t,s>0$;
- $\left({\zeta}_{2}\right)$
- if $\{{t}_{n}\},\{{s}_{n}\}$ are sequences in $(0,\infty )$ such that $\underset{n\to \infty}{lim}{t}_{n}=\underset{n\to \infty}{lim}{s}_{n}>0$, then$$\underset{n\to \infty}{lim\; sup}\zeta ({t}_{n},{s}_{n})<0.$$

**Theorem**

**1.**

**Definition**

**2**

**.**We say that f, defined on a $(X,d)$, satisfies C-condition if

**Theorem**

**2**

**.**f, defined on $({X}^{\ast},d)$, possesses a unique fixed point if

## 2. Main Results

**Definition**

**3.**

**Theorem**

**3.**

**Proof.**

**Definition**

**4.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Example**

**1.**

**Example**

**2.**

## 3. Application to Ordinary Differential Equations

**Theorem**

**6.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Khojasteh, F.; Shukla, S.; Radenović, S. A new approach to the study of fixed point theorems via simulation functions. Filomat
**2015**, 29, 1189–1194. [Google Scholar] [CrossRef] - Alsulami, H.H.; Karapınar, E.; Khojasteh, F.; Roldán-López-de-Hierro, A.F. A proposal to the study of contractions in quasi-metric spaces. Discr. Dyn. Nat. Soc.
**2014**, 2014, 10. [Google Scholar] [CrossRef] - Alharbi, A.S.S.; Alsulami, H.H.; Karapinar, E. On the Power of Simulation and Admissible Functions in Metric Fixed Point Theory. J. Funct. Spaces
**2017**, 2017, 7. [Google Scholar] [CrossRef] - Alqahtani, B.; Fulga, A.; Karapinar, E. Fixed Point Results On Δ-Symmetric Quasi-Metric Space Via Simulation Function With An Application To Ulam Stability. Mathematics
**2018**, 6, 208. [Google Scholar] [CrossRef] [Green Version] - Aydi, H.; Felhi, A.; Karapinar, E.; Alojail, F.A. Fixed points on quasi-metric spaces via simulation functions and consequences. J. Math. Anal. (Ilirias)
**2018**, 9, 10–24. [Google Scholar] - Roldán-López-de-Hierro, A.F.; Karapınar, E.; Roldán-López-de-Hierro, C.; Martínez-Moreno, J. Coincidence point theorems on metric spaces via simulation functions. J. Comput. Appl. Math.
**2015**, 275, 345–355. [Google Scholar] [CrossRef] - Suzuki, T. A generalized Banach contraction principle which characterizes metric completeness. Proc. Am. Math. Soc.
**2008**, 136, 1861–1869. [Google Scholar] [CrossRef] - Karapinar, E.; Erhan, I.M.; Aksoy, U. Weak ψ-contractions on partially ordered metric spaces and applications to boundary value problems. Bound. Value Probl.
**2014**, 2014, 149. [Google Scholar] [CrossRef] [Green Version] - Pata, V. A fixed point theorem in metric spaces. J. Fixed Point Theory Appl.
**2011**, 10, 299–305. [Google Scholar] [CrossRef] - Balasubramanian, S. A Pata-type fixed point theorem. Math. Sci.
**2014**, 8, 65–69. [Google Scholar] [CrossRef] [Green Version] - Choudhury, B.S.; Metiya, N.; Bandyopadhyay, C.; Maity, P. Fixed points of multivalued mappings satisfying hybrid rational Pata-type inequalities. J. Anal.
**2019**, 27, 813–828. [Google Scholar] [CrossRef] - Choudhury, B.S.; Kadelburg, Z.; Metiya, N.; Radenović, S. A Survey of Fixed Point Theorems Under Pata-Type Conditions. Bull. Malays. Math. Soc.
**2019**, 43, 1289–1309. [Google Scholar] [CrossRef] - Choudhury, B.S.; Metiya, N.; Kundu, S. End point theorems of multivalued operators without continuity satisfying hybrid inequality under two different sets of conditions. Rend. Circ. Mat. Palermo
**2019**, 68, 65–81. [Google Scholar] - Geno, K.J.; Khan, M.S.; Park, C.; Sungsik, Y. On Generalized Pata Type Contractions. Mathematics
**2018**, 6, 25. [Google Scholar] [CrossRef] [Green Version] - Kadelburg, Z.; Radenovic, S. Fixed point theorems under Pata-type conditions in metric spaces. J. Egypt. Math. Soc.
**2016**, 24, 77–82. [Google Scholar] [CrossRef] [Green Version] - Kadelburg, Z.; Radenovic, S. A note on Pata-type cyclic contractions. Sarajevo J. Math.
**2015**, 11, 235–245. [Google Scholar] - Kadelburg, Z.; Radenovic, 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] - Kadelburg, Z.; Radenovic, S. Fixed point and tripled fixed point theprems under Pata-type conditions in ordered metric spaces. Int. J. Anal. Appl.
**2014**, 6, 113–122. [Google Scholar] - Kolagar, S.M.; Ramezani, M.; Eshaghi, M. Pata type fixed point theorems of multivalued operators in ordered metric spaces with applications to hyperbolic differential inclusions. Proc. Am. Math. Soc.
**2016**, 6, 21–34. [Google Scholar] - Ramezani, M.; Ramezani, H. A new generalized contraction and its application in dynamic programming. Cogent Math.
**2018**, 5, 1559456. [Google Scholar] [CrossRef] - Ćirić, L. Some Recent Results in Metrical Fixed Point Theory; University of Belgrade: Belgrade, Serbia, 2003. [Google Scholar]
- Todorčević, V. Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics; Springer Nature Switzerland AG: Basel, Switzerland, 2019. [Google Scholar]
- Chanda, A.; Dey, L.K.; Radenović, S. Simulation functions: A Survey of recent results. RACSAM
**2019**, 113, 2923–2957. [Google Scholar] [CrossRef] - Radenović, S.; Vetro, F.; Vujaković, J. An alternative and easy approach to fixed point results via simulation functions. Demonstr. Math.
**2017**, 50, 224–231. [Google Scholar] [CrossRef] [Green Version] - Radenović, S.; Chandok, S. Simulation type functions and coincidence point results. Filomat
**2018**, 32, 141–147. [Google Scholar] [CrossRef] [Green Version]

© 2020 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**

Karapınar, E.; Hima Bindu, V.M.L.
On Pata–Suzuki-Type Contractions. *Mathematics* **2020**, *8*, 389.
https://doi.org/10.3390/math8030389

**AMA Style**

Karapınar E, Hima Bindu VML.
On Pata–Suzuki-Type Contractions. *Mathematics*. 2020; 8(3):389.
https://doi.org/10.3390/math8030389

**Chicago/Turabian Style**

Karapınar, Erdal, and V. M. L. Hima Bindu.
2020. "On Pata–Suzuki-Type Contractions" *Mathematics* 8, no. 3: 389.
https://doi.org/10.3390/math8030389