# On Pata–Suzuki-Type Contractions

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## 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

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**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