# Common Fixed Point Theorems for Generalized Geraghty (α,ψ,ϕ)-Quasi Contraction Type Mapping in Partially Ordered Metric-Like Spaces

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## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

## 2. Preliminaries

**Definition**

**1.**

- (σ
_{1}) - $\sigma (u,v)=0\Rightarrow u=v,$
- (σ
_{2}) - $\sigma (u,v)=\sigma (v,u),$
- (σ
_{3}) - $\sigma (u,z)\le \sigma (u,v)+\sigma (v,z).$

**Example**

**1.**

**Remark**

**1.**

**Lemma**

**1.**

**Example**

**2.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

- (1)
- S is α-admissible,
- (2)
- $\alpha (u,z)\ge 1$ and $\alpha (z,v)\ge 1$ imply $\alpha (u,v)\ge 1$.

**Definition**

**5.**

- (1)
- The pair $(S,T)$ is α-admissible,
- (2)
- $\alpha (u,z)\ge 1$ and $\alpha (z,v)\ge 1$ imply $\alpha (u,v)\ge 1$.

- (1)
- $\psi $ is strictly continuous increasing,
- (2)
- $\psi \left(t\right)=0$⇔$t=0.$

**Definition**

**6.**

- (1)
- For all $x,y\in X$ are said to be comparable if $x\u2aafy$ or $y\u2aafx$ holds,
- (2)
- f is said to be nondecreasing if $x\u2aafy$ implies $fx\u2aaffy$,
- (3)
- $f,g$ are called weakly increasing if $fx\u2aafgfx$ and $gx\u2aaffgx$ for all $x\in X,$
- (4)
- f is called weakly increasing if f and I are weakly increasing, where I is denoted to the identity mapping on $X.$

## 3. Main Results

**Definition**

**7.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Theorem**

**4.**

- (1)
- $(S,T)$ is triangular α-admissible and there exists an ${x}_{0}\in X$ such that $\alpha ({x}_{0},S{x}_{0})\ge 1$,
- (2)
- the pair $(S,T)$ is weakly increasing,
- (3)
- the pair $(S,T)$ is a generalized Geraghty $(\alpha ,\psi ,\varphi )$-quasi contraction non-self mapping,
- (4)
- S and T are σ-continuous mappings.

**Proof.**

**Theorem**

**5.**

- (1)
- the pair $(S,T)$ is triangular α-admissible,
- (2)
- there exists an ${x}_{0}\in X$ such that $\alpha ({x}_{0},S{x}_{0})\ge 1$,
- (3)
- the pair $(S,T)$ is a generalized Geraghty $(\alpha ,\psi ,\varphi )$-quasi contraction non-self mapping,
- (4)
- the pair $(S,T)$ is weakly increasing,
- (5)
- $\left(\mathcal{C}\right)$ holds.

**Proof.**

**I**: Assume that ${lim}_{l\to \infty}{M}_{{x}_{2{n}_{l}},v}=\sigma (v,Tv).$

**II**: Assume that ${lim}_{l\to \infty}{M}_{{x}_{2{n}_{l}},v}=\sigma (v,Sv).$ Then, arguing like above, we get $v=Sv.$ Thus, $v=Sv=Tv.$ Uniqueness of the fixed point is follows from the Theorem 4. This completes the proof. □

**Corollary**

**1.**

- (1)
- there exists $\psi \in \Psi ,\beta \in \mathfrak{S}$ and a continuous function $\varphi :[0,\infty )\to [0,\infty )$ are continuous functions with $\varphi \left(t\right)<\psi \left(t\right)$ for all $t>0$ such that$$\alpha (x,y)\psi \left(\sigma (Sx,Sy)\right)\le \lambda \beta \left(\psi \left({M}_{x,y}\right)\right)\varphi \left({M}_{x,y}\right),$$
- (2)
- S is triangular α-admissible and there exists an ${x}_{0}\in X$ such that $\alpha ({x}_{0},S{x}_{0})\ge 1$,
- (3)
- $Sx\u2aafS\left(Sx\right)$ for all $x,y\in X,$
- (4)
- T is σ-continuous mappings.

**Corollary**

**2.**

- (1)
- there exists $\psi \in \Psi ,\beta \in \mathfrak{S}$ and a continuous function $\varphi :[0,\infty )\to [0,\infty )$ are continuous functions with $\varphi \left(t\right)<\psi \left(t\right)$ for all $t>0$ such that$$\alpha (x,y)\psi \left(\sigma (Sx,Sy)\right)\le \lambda \beta \left(\psi \left({M}_{x,y}\right)\right)\varphi \left({M}_{x,y}\right),$$
- (2)
- S is triangular α-admissible and there exists an ${x}_{0}\in X$ such that $\alpha ({x}_{0},S{x}_{0})\ge 1$,
- (3)
- $Sx\u2aafS\left(Sx\right)$ for all $x,y\in X,$
- (4)
- $\left(\mathcal{C}\right)$ holds.

**Corollary**

**3.**

- (1)
- there exists $\psi \in \Psi ,\beta \in \mathfrak{S}$ and a continuous function $\varphi :[0,\infty )\to [0,\infty )$ are continuous functions with $\varphi \left(t\right)<\psi \left(t\right)$ for all $t>0$ such that$$\psi \left(\sigma (Sx,Ty)\right)\le \lambda \beta \left(\psi \left({M}_{x,y}\right)\right)\varphi \left({M}_{x,y}\right),$$$${M}_{x,y}=max\{\sigma (x,y),\sigma (x,Sx),\sigma (y,Ty),\sigma (Sx,y),\sigma (x,Ty)\}.$$
- (2)
- the pair $(S,T)$ is weakly increasing,
- (3)
- S and T are σ-continuous mappings.

**Corollary**

**4.**

- (1)
- there exists $\psi \in \Psi ,\beta \in \mathfrak{S}$ and a continuous function $\varphi :[0,\infty )\to [0,\infty )$ are continuous functions with $\varphi \left(t\right)<\psi \left(t\right)$ for all $t>0$ such that$$\psi \left(\sigma (Tx,Ty)\right)\le \lambda \beta \left(\psi \left({M}_{x,y}\right)\right)\varphi \left({M}_{x,y}\right),$$$${M}_{x,y}=max\{\sigma (x,y),\sigma (x,Sx),\sigma (y,Ty),\sigma (Sx,y),\sigma (x,Ty)\},$$
- (2)
- the pair $(S,T)$ is weakly increasing,
- (3)
- the pair $(S,T)$ is a generalized $(\alpha ,\psi ,\varphi )$-quasi contraction non-self,
- (4)
- $\left(\mathcal{C}\right)$ holds.

## 4. Consequences

**Corollary**

**5.**

- (1)
- $(S,T)$ is triangular α-admissible and there exists an ${x}_{0}\in X$ such that $\alpha ({x}_{0},S{x}_{0})\ge 1$,
- (2)
- there exists $\psi \in \Psi ,\beta \in \mathfrak{S}$ and a continuous function $\varphi :[0,\infty )\to [0,\infty )$ are continuous functions with $\varphi \left(t\right)<\psi \left(t\right)$ for all $t>0$ in order that$$\psi \left(\sigma \right(Sx,Ty\left)\right)\le \lambda \beta \left(\psi \right(\sigma (x,y)\left)\varphi \right(\sigma (x,y)),$$
- (3)
- the pair $(S,T)$ is weakly increasing,
- (4)
- the pair $(S,T)$ is σ-continuous mappings.

**Corollary**

**6.**

- (1)
- $(S,T)$ is triangular α-admissible and there exists an ${x}_{0}\in X$ such that $\alpha ({x}_{0},S{x}_{0})\ge 1$,
- (2)
- there exists $\psi \in \Psi ,\beta \in \mathfrak{S}$ and a continuous function $\varphi :[0,\infty )\to [0,\infty )$ are continuous functions with $\varphi \left(t\right)\le \psi \left(t\right)$ for all $t>0$ in order that$$\psi \left(\sigma \right(Sx,Ty\left)\right)\le \lambda \beta \left(\psi \right(\sigma (x,y)\left)\varphi \right(\sigma (x,y)),$$
- (3)
- the pair $(S,T)$ is weakly increasing,
- (4)
- $\left(\mathcal{C}\right)$ holds.

**Corollary**

**7.**

- (1)
- S is triangular α-admissible and there exists an ${x}_{0}\in X$ such that $\alpha ({x}_{0},S{x}_{0})\ge 1$.
- (2)
- there exists $\psi \in \Psi ,\beta \in \mathfrak{S}$ and a continuous function $\varphi :[0,\infty )\to [0,\infty )$ are continuous functions with $\varphi \left(t\right)<\psi \left(t\right)$ for all $t>0$ in order that$$\alpha (x,y)\psi \left(\sigma \right(Sx,Sy\left)\right)\le \lambda \beta \left(\psi \right(\sigma (x,y)\left)\varphi \right(\sigma (x,y)),$$
- (3)
- $S\u2aafS\left(Sx\right)$,
- (4)
- the pair $(S,T)$ is σ-continuous mappings.

**Corollary**

**8.**

- (1)
- S is triangular α-admissible and there exists an ${x}_{0}\in X$ such that $\alpha ({x}_{0},S{x}_{0})\ge 1$,
- (2)
- there exists $\psi \in \Psi ,\beta \in \mathfrak{S}$ and a continuous function $\varphi :[0,\infty )\to [0,\infty )$ are continuous functions with $\varphi \left(t\right)<\psi \left(t\right)$ for all $t>0$ in order that$$\alpha (x,y)\psi \left(\sigma \right(Sx,Sy\left)\right)\le \lambda \beta \left(\psi \right(\sigma (x,y)\left)\varphi \right(\sigma (x,y)),$$
- (3)
- $S\u2aafS\left(Sx\right)$,
- (4)
- $\left(\mathcal{C}\right)$ holds.

**Example**

**3.**

## 5. Application

**Theorem**

**6.**

- (i)
- There exists $\zeta :X\times X\to [0,1)$ such that for all $r\in [0,1]$ and for all $x,y\in X$$$0\le \mid f(r,x(r\left)\right)-f(r,y(r\left)\right)\mid \le \zeta (x,y)\mid x\left(r\right)-y\left(r\right)\mid ,$$
- (ii)
- there exists $\beta :[0,\infty )\to [0,1)$ such that$$\underset{n\to \infty}{lim}\beta \left({t}_{n}\right)=1\phantom{\rule{0.222222em}{0ex}}\Rightarrow \phantom{\rule{0.222222em}{0ex}}\underset{n\to \infty}{lim}{t}_{n}=0,$$$$\parallel {\int}_{0}^{1}P(t,r)\zeta (x,y)dr{\parallel}_{\infty}\le \left(\frac{1}{4}\beta (\parallel x-y{\parallel}_{\infty})\right).$$

**Proof.**

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Qawaqneh, H.; Noorani, M.; Shatanawi, W.; Alsamir, H.
Common Fixed Point Theorems for Generalized Geraghty (*α*,*ψ*,*ϕ*)-Quasi Contraction Type Mapping in Partially Ordered Metric-Like Spaces. *Axioms* **2018**, *7*, 74.
https://doi.org/10.3390/axioms7040074

**AMA Style**

Qawaqneh H, Noorani M, Shatanawi W, Alsamir H.
Common Fixed Point Theorems for Generalized Geraghty (*α*,*ψ*,*ϕ*)-Quasi Contraction Type Mapping in Partially Ordered Metric-Like Spaces. *Axioms*. 2018; 7(4):74.
https://doi.org/10.3390/axioms7040074

**Chicago/Turabian Style**

Qawaqneh, Haitham, Mohd Noorani, Wasfi Shatanawi, and Habes Alsamir.
2018. "Common Fixed Point Theorems for Generalized Geraghty (*α*,*ψ*,*ϕ*)-Quasi Contraction Type Mapping in Partially Ordered Metric-Like Spaces" *Axioms* 7, no. 4: 74.
https://doi.org/10.3390/axioms7040074