# On the Study of Fixed Points for a New Class of α-Admissible Mappings

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## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

- (a)
- F is nondecreasing.
- (b)
- For every sequence $\left\{{t}_{n}\right\}\subset (0,+\infty )$, we have$$\underset{n\to +\infty}{lim}F\left({t}_{n}\right)=-\infty \u27fa\underset{n\to +\infty}{lim}{t}_{n}=0.$$
- (c)
- There exists $k\in (0,1)$ such that $\underset{t\to {0}^{+}}{lim}{t}^{k}F\left(t\right)=0$.

**Theorem**

**2.**

## 2. The Class of Generalized Ćirić-Contractions

- (${\Phi}_{1}$)
- $\phi $ is non-decreasing, i.e., $0<t<s\Rightarrow \phi \left(t\right)\le \phi \left(s\right)$.
- (${\Phi}_{2}$)
- For every sequence $\left\{{t}_{n}\right\}\subset (0,+\infty )$,$$\underset{n\to +\infty}{lim}\phi \left({t}_{n}\right)=-\infty $$$$\underset{n\to +\infty}{lim}{t}_{n}=0.$$
- (${\Phi}_{3}$)
- There exists $k\in (0,1)$ such that $\underset{t\to {0}^{+}}{lim}{t}^{k}\phi \left(t\right)=0$.

**Definition**

**1.**

**Theorem**

**3.**

**Proof.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 3. A Larger Class of Mappings

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

- (i)
- $\left\{{x}_{n}\right\}$ is α-regular.
- (ii)
- $\left\{{x}_{n}\right\}$ is a Cauchy sequence.

**Definition**

**6.**

- (${\mathcal{T}}_{1}$)
- T is $\alpha $-continuous.
- (${\mathcal{T}}_{2}$)
- There exists $(\phi ,\psi )\in \Phi \times \Psi $ such that for all $(x,y)\in X\times X$ with $d(Tx,Ty)>0$,$$\alpha (x,y)exp\left(\phi \left(d\right(Tx,Ty\left)\right)\right)\le exp\left(\phi \left(d\right(x,y\left)\right)+\psi \left(d\right(x,y\left)\right)\right).$$

**Proposition**

**1**(The class of generalized Ćirić-contractions)

**.**

**Proof.**

**Proposition**

**2**(The class of F-contractions)

**.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Remark**

**1.**

**Proposition**

**4.**

**Proof.**

**Remark**

**2.**

**Proposition**

**5**(The class of almost F-contractions)

**.**

**Proof.**

**Remark**

**3.**

**Theorem**

**4.**

- (i)
- There exists $\alpha :X\times X\to \mathbb{R}$ such that $(X,d)$ is α-complete.
- (ii)
- There exists $(\phi ,\psi )\in \Phi \times \Psi $ such that $T\in {\mathcal{T}}_{\alpha}$.
- (iii)
- T is α-admissible.
- (iv)
- There exists some ${x}_{0}\in X$ such that $\alpha ({x}_{0},T{x}_{0})\ge 1$.

**Proof.**

**Remark**

**4.**

**Lemma**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Remark**

**5.**

**Corollary**

**4.**

**Proof.**

**Corollary**

**5.**

**Proof.**

**Corollary**

**6.**

- (i)
- T is non-decreasing with respect to ⪯, i.e.,$$Tx\u2aafTy,$$
- (ii)
- There exists ${x}_{0}\in X$ such that ${x}_{0}\u2aafT{x}_{0}$.
- (iii)
- There exist $F\in \Phi $ and $\tau >0$ such that$$\tau +F\left(d\right(Tx,Ty\left)\right)\le F\left(d\right(x,y\left)\right),$$

**Proof.**

**Corollary**

**7**

**.**Let $(X,d)$ be a complete metric space, and let $T:X\to X$ be continuous mapping. Suppose that X is partially ordered by a certain binary relation ⪯. Suppose that

- (i)
- T is non-decreasing with respect to ⪯.
- (ii)
- There exists ${x}_{0}\in X$ such that ${x}_{0}\u2aafT{x}_{0}$.
- (iii)
- There exists $0<q<1$ such that for all $(x,y)\in X\times X$ with $x\u2aafy$,$$d(Tx,Ty)\le qd(x,y).$$

**Proof.**

**Remark**

**6.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Darwish, M.A.; Jleli, M.; O’Regan, D.; Samet, B. On the Study of Fixed Points for a New Class of *α*-Admissible Mappings. *Mathematics* **2019**, *7*, 1240.
https://doi.org/10.3390/math7121240

**AMA Style**

Darwish MA, Jleli M, O’Regan D, Samet B. On the Study of Fixed Points for a New Class of *α*-Admissible Mappings. *Mathematics*. 2019; 7(12):1240.
https://doi.org/10.3390/math7121240

**Chicago/Turabian Style**

Darwish, Mohamed Abdalla, Mohamed Jleli, Donal O’Regan, and Bessem Samet. 2019. "On the Study of Fixed Points for a New Class of *α*-Admissible Mappings" *Mathematics* 7, no. 12: 1240.
https://doi.org/10.3390/math7121240