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# A note on recent fixed point results involving g-quasicontractive type mappings in partially ordered metric spaces

Fixed Point Theory and Applications20142014:126

https://doi.org/10.1186/1687-1812-2014-126

• Accepted: 13 May 2014
• Published:

## Abstract

In this note, we establish the equivalence between recent fixed point theorems involving quasicontractive type mappings in metric spaces endowed with a partial order.

MSC:47H10.

## Keywords

• g-quasicontraction
• coincidence point
• fixed point
• partial order
• metric space

## 1 Introduction

Let $\left(X,d\right)$ be a metric space and let $f,g:X\to X$ be two self-maps on X. Let
Suppose that X is endowed with a partial order . We say that f is an ordered g-quasicontraction (see [1, 2]) if

for some constant $\lambda \in \left(0,1\right)$. If $g=i{d}_{X}$ (the identity map on X), then f is said to be an ordered quasicontraction.

In , the authors established the following result.

Theorem 1.1 Let $\left(X,d\right)$ be a metric space endowed with a certain partial order . Let $f,g:X\to X$ be two self-maps on X satisfying the following conditions:
1. (i)

$fX\subseteq gX$;

2. (ii)

gX is complete;

3. (iii)

f is g-nondecreasing, i.e., $gx⪯gy⟹fx⪯fy$;

4. (iv)

f is an ordered g-quasicontraction;

5. (v)

there exists ${x}_{0}\in X$ such that $g{x}_{0}⪯f{x}_{0}$;

6. (vi)

if $\left\{g{x}_{n}\right\}$ is a nondecreasing sequence (w.r.t. ) that converges to some $gz\in gX$, then $g{x}_{n}⪯gz$ for each $n\in \mathbb{N}$.

Then f and g have a coincidence point, i.e., there exists $z\in X$ such that $fz=gz$.

Taking $g=i{d}_{X}$ in Theorem 1.1, we obtain immediately the following result.

Theorem 1.2 Let $\left(X,d\right)$ be a complete metric space endowed with a certain partial order . Let $f:X\to X$ be a self-map on X satisfying the following conditions:
1. (iii)

f is nondecreasing, i.e., $x⪯y⟹fx⪯fy$;

2. (iv)

f is an ordered quasicontraction;

3. (v)

there exists ${x}_{0}\in X$ such that ${x}_{0}⪯f{x}_{0}$;

4. (vi)

if $\left\{{x}_{n}\right\}$ is a nondecreasing sequence (w.r.t. ) that converges to some $z\in X$, then ${x}_{n}⪯z$ for each $n\in \mathbb{N}$.

Then f has a fixed point.

Let us denote by Ψ the set of functions $\psi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ satisfying the following conditions:

(${\mathrm{\Psi }}_{1}$) ψ is nondecreasing;

(${\mathrm{\Psi }}_{2}$) ψ is subadditive, i.e., $\psi \left(s+t\right)\le \psi \left(s\right)+\psi \left(t\right)$, for every $s,t\ge 0$;

(${\mathrm{\Psi }}_{3}$) ψ is continuous;

(${\mathrm{\Psi }}_{4}$) $\psi \left(t\right)=0⟺t=0$.

In , the authors established the following result.

Theorem 1.3 Let $\left(X,d\right)$ be a metric space endowed with a certain partial order . Let $f,g:X\to X$ be two self-maps on X satisfying the following conditions:
1. (i)

$fX\subseteq gX$;

2. (ii)

gX is complete;

3. (iii)

f is g-nondecreasing;

4. (iv)
there exists $\psi \in \mathrm{\Psi }$ such that
$\begin{array}{rl}\psi \left(d\left(fx,fy\right)\right)\le & \lambda max\left\{\psi \left(d\left(gx,gy\right)\right),\psi \left(d\left(gx,fx\right)\right),\psi \left(d\left(gy,fy\right)\right),\\ \psi \left(d\left(gx,fy\right)\right),\psi \left(d\left(gy,fx\right)\right)\right\}\end{array}$

for all $x,y\in X$ such that $gy⪯gx$;
1. (v)

there exists ${x}_{0}\in X$ such that $g{x}_{0}⪯f{x}_{0}$;

2. (vi)

if $\left\{g{x}_{n}\right\}$ is a nondecreasing sequence that converges to some $gz\in gX$, then $g{x}_{n}⪯gz$ for each $n\in \mathbb{N}$.

Then f and g have a coincidence point.

The aim of this note is to prove that Theorems 1.1, 1.2 and 1.3 are equivalent.

## 2 Main result

Our main result in this note is the following.

Theorem 2.1 We have the following equivalence:
$\mathit{\text{Theorem}}\phantom{\rule{0.25em}{0ex}}\text{1.2}\phantom{\rule{1em}{0ex}}⟺\phantom{\rule{1em}{0ex}}\mathit{\text{Theorem}}\phantom{\rule{0.25em}{0ex}}\text{1.1}\phantom{\rule{1em}{0ex}}⟺\phantom{\rule{1em}{0ex}}\mathit{\text{Theorem}}\phantom{\rule{0.25em}{0ex}}\text{1.3}.$

Proof We consider three steps in the proof.

Step 1. Theorem 1.2 Theorem 1.1.

Suppose that all the assumptions of Theorem 1.1 are satisfied. Recall that if $S:X\to X$ is a given map, then there exists a subset E of X such that $SE=SX$ and $S:E\to X$ is one-to-one. For the proof of this result, we refer to . Due to this remark, there exists $E\subseteq X$ such that $gE=gX$ and $g:E\to X$ is one-to-one. Let us define the map $T:gE\to gE$ by
$T\left(gx\right)=fx,\phantom{\rule{1em}{0ex}}x\in E.$
Notice that the mapping T is well defined since g is one-to-one on E. From condition (ii) of Theorem 1.1, the metric space $\left(gE,d\right)$ is complete. From condition (iii) of Theorem 1.1, the mapping T is nondecreasing. Observe also that T is an ordered quasicontraction. Indeed, if $u,v\in gE$ such that $v⪯u$, from condition (iv) of Theorem 1.1 and the definition of gE, there exist $x,y\in E$ with $v=gy⪯gx=u$ such that
$\begin{array}{rcl}d\left(Tu,Tv\right)& =& d\left(fx,fy\right)\\ \le & \lambda M\left(f,g,x,y\right)\\ =& \lambda max\left\{d\left(gx,gy\right),d\left(gx,fx\right),d\left(gy,fy\right),d\left(gx,fy\right),d\left(gy,fx\right)\right\}\\ =& \lambda max\left\{d\left(u,v\right),d\left(u,Tu\right),d\left(v,Tv\right),d\left(u,Tv\right),d\left(v,Tu\right)\right\}.\end{array}$

From condition (v) of Theorem 1.1, there exists ${x}_{0}\in X$ such that $g{x}_{0}⪯f{x}_{0}$. Let ${u}_{0}=g{x}_{0}\in gE$, we have ${u}_{0}⪯T{u}_{0}$. Finally, from condition (iv) of Theorem 1.1, if $\left\{{u}_{n}\right\}\subset gE$ is a nondecreasing sequence that converges to some $u\in gE$, then ${u}_{n}⪯u$ for each $n\in \mathbb{N}$. Thus we proved that T satisfies all the conditions of Theorem 1.2. Then we deduce that T has a fixed point ${u}^{\ast }\in gE$. This means that there exists some ${x}^{\ast }\in X$ such that $f{x}^{\ast }=T\left(g{x}^{\ast }\right)=g{x}^{\ast }$, that is, ${x}^{\ast }\in X$ is a coincidence point of f and g.

Step 2. Theorem 1.1 Theorem 1.3.

Suppose that all the assumptions of Theorem 1.3 are satisfied. Define the function ${d}_{\psi }:X×X\to \left[0,\mathrm{\infty }\right)$ by
In , we proved that ${d}_{\psi }$ is a metric on X. Moreover, $\left(X,d\right)$ is complete if and only if $\left(X,{d}_{\psi }\right)$ is complete. Then from condition (iv) of Theorem 1.3, we deduce that f is an ordered g-quasicontraction with respect to the new metric ${d}_{\psi }$. More precisely, we have
${d}_{\psi }\left(fx,fy\right)\le \lambda max\left\{{d}_{\psi }\left(gx,gy\right),{d}_{\psi }\left(gx,fx\right),{d}_{\psi }\left(gy,fy\right),{d}_{\psi }\left(gx,fy\right),{d}_{\psi }\left(gy,fx\right)\right\}$

for all $x,y\in X$ such that $gy⪯gx$. Now, applying Theorem 1.1 with the metric space $\left(X,{d}_{\psi }\right)$, we obtain the result of Theorem 1.3.

Step 3. Theorem 1.3 Theorem 1.2.

Taking $g=i{d}_{X}$ and $\psi \left(t\right)=t$ in Theorem 1.3, we obtain immediately the result of Theorem 1.2. □

## Declarations

### Acknowledgements

The authors thank the Visiting Professor Programming at King Saud University for funding this work. The authors thank the anonymous referees for their remarkable comments, suggestions and ideas that helped to improve this paper.

## Authors’ Affiliations

(1)
Department of Mathematics, Atilim University, İncek, Ankara, 06836, Turkey
(2)
Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia
(3)
Department of Mathematics, King Saud University, Riyadh, Saudi Arabia

## References 