# Coupled fixed and coincidence points for monotone operators in partial metric spaces

- Saud M Alsulami
^{1}, - Nawab Hussain
^{1}and - Abdullah Alotaibi
^{1}Email author

**2012**:173

https://doi.org/10.1186/1687-1812-2012-173

© Alsulami et al.; licensee Springer 2012

**Received: **26 August 2012

**Accepted: **25 September 2012

**Published: **9 October 2012

## Abstract

In this paper, we prove some coupled fixed point results for $(\varphi ,\phi )$-weakly contractive mappings in ordered partial metric spaces. As an application, we establish coupled coincidence results without any type of commutativity of the concerned maps. Consequently, the results of Luong and Thuan (Nonlinear Anal. 74:983-992, 2011), Alotaibi and Alsulami (Fixed Point Theory Appl. 2011:44, 2011) and many others are extended to the class of ordered partial metric spaces.

## Keywords

## 1 Introduction

The Banach contraction principle is the most celebrated fixed point theorem. Afterward many authors obtained various important extensions of this principle (see [1]). The concept of partial metric spaces was introduced by Matthews [2] in 1994. A partial metric space is a generalized metric space in which each object does not necessarily have to have a zero distance from itself. A motivation behind introducing the concept of a partial metric was to obtain appropriate mathematical models in the theory of computation and, in particular, to give a modified version of the Banach contraction principle [3, 4]. Subsequently, several authors studied the problem of existence and uniqueness of a fixed point for mappings satisfying different contractive conditions on partial metric spaces (*e.g.*, [5–7]).

Recently, Bhaskar and Lakshmikantham [8] presented coupled fixed point theorems for contractions in partially ordered metric spaces. Luong and Thuan [9] presented nice generalizations of these results. Alotaibi and Alsulami [10] further extended the work of Luong and Thuan to coupled coincidences. For more related work on coupled coincidences we refer the readers to recent work in [11–16]. Our main aim in this paper is to extend Luong and Thuan [9] results to ordered partial metric spaces. We shall also establish coupled coincidence results and show that main results in [10] hold in ordered partial metric spaces without the compatibility of maps.

## 2 Basic concepts

We start by recalling some definitions and properties of partial metric spaces.

**Definition 2.1** A *partial metric* on a nonempty set *X* is a function $p:X\times X\u27f6{\mathbb{R}}^{+}$ such that for all $x,y,z\in X$,

p1. $x=y\iff p(x,x)=p(x,y)=p(y,y)$.

p2. $p(x,x)\le p(x,y)$.

p3. $p(x,y)=p(y,x)$.

p4. $p(x,z)\le p(x,y)+p(y,z)-p(y,y)$.

A partial metric space is a pair $(X,p)$ such that *X* is a nonempty set and *p* is a partial metric on *X*.

From the above definition, if $p(x,y)=0$ , then $x=y$. But if $x=y$, $p(x,y)$ may not be 0 in general. A trivial example of a partial metric space is the pair $({\mathbb{R}}^{+},p)$ , where $p:{\mathbb{R}}^{+}\times {\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is defined as $p(x,y)=max\{x,y\}$. For some more examples of partial metric spaces, we refer to [4, 6].

Each partial metric *p* on *X* generates a ${T}_{0}$ topology ${\tau}_{p}$ on *X* which has as a base the family of open *p*-balls $\{{B}_{p}(x,\u03f5):x\in X,\u03f5>0\}$, where ${B}_{p}(x,\u03f5)=\{y\in X:p(x,y)<p(x,x)+\u03f5\}$ for all $x\in X$ and $\u03f5>0$. A sequence $\{{x}_{n}\}$ in *X* converges to a point $x\in X$, with respect to ${\tau}_{p}$ if and only if $p(x,x)={lim}_{n\to \mathrm{\infty}}p(x,{x}_{n})$. A sequence $\{{x}_{n}\}$ in *X* is called a Cauchy sequence if ${lim}_{n,m\to \mathrm{\infty}}p({x}_{n},{x}_{m})$ exists and is finite.

*p*is a partial metric on

*X*, then the function ${p}^{s}:X\times X\u27f6{\mathbb{R}}^{+}$ given by

is a metric on *X*.

*Let*$(X,p)$

*be a partial metric space*.

*Then*

- (a)
$\{{x}_{n}\}$

*is a Cauchy sequence in*$(X,p)$*if and only if it is a Cauchy sequence in the metric space*$(X,{p}^{s})$. - (b)$(X,p)$
*is complete if and only if the metric space*$(X,{p}^{s})$*is complete*.*Furthermore*, ${lim}_{n\to \mathrm{\infty}}{p}^{s}({x}_{n},x)=0$*if and only if*$p(x,x)=\underset{n\to \mathrm{\infty}}{lim}p({x}_{n},x)=\underset{n,m\to \mathrm{\infty}}{lim}p({x}_{n},{x}_{m}).$

*q*defined as follows:

A mapping $F:X\times X\to X$ is said to be continuous at $(x,y)\in X\times X$ if for each $\u03f5>0$, there exists $\delta >0$ such that $F({B}_{q}((x,y),\delta ))\subseteq {B}_{p}(F(x,y),\u03f5)$.

**Definition 2.2** (Mixed monotone property)

*F*has the mixed monotone property if

*F*is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument. That is, for any $x,y\in X$,

**Definition 2.3** [11]

Let $F:X\times X\to X$. We say that $(x,y)\in X\times X$ is a coupled fixed point of *F* if $F(x,y)=x$ and $F(y,x)=y$.

**Definition 2.4** (Mixed *g*-monotone property [11])

*F*has the mixed

*g*-monotone property if

*F*is monotone

*g*-non-decreasing in its first argument and is monotone

*g*-non-increasing in its second argument. That is, for any $x,y\in X$,

**Definition 2.5** [11]

Let $F:X\times X\to X$ and $g:X\to X$. We say that $(x,y)\in X\times X$ is a coupled coincidence point of *F* and *g* if $F(x,y)=gx$ and $F(y,x)=gy$.

## 3 Coupled fixed point results

Let Φ denote all functions $\varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ which satisfy

(*ϕ* 1) *ϕ* is continuous and non-decreasing,

(*ϕ* 2) $\varphi (t)=0$ if and only if $t=0$,

(*ϕ* 3) $\varphi (t+s)\le \varphi (t)+\varphi (s)$, $\mathrm{\forall}t,s\in [0,\mathrm{\infty})$,

(*ϕ* 4) $\varphi (\alpha x)\le \alpha \varphi (x)$ for $\alpha \in (0,\mathrm{\infty})$,

and let Ψ denote all functions $\psi :[0,\mathrm{\infty})\to (0,\mathrm{\infty})$ which satisfy ${lim}_{t\to r}\psi (t)>0$ for all $r>0$ and ${lim}_{t\to {0}^{+}}\psi (t)=0$.

Now, we state and prove our main result.

**Theorem 3.1**

*Let*$(X,\u2aaf)$

*be a partially ordered set and suppose there is a partial metric*

*p*

*on*

*X*

*such that*$(X,d)$

*is a complete partial metric space*.

*Let*$F:X\times X\to X$

*be a mapping having the mixed monotone property on*

*X*.

*Assume that there exist two elements*${x}_{0},{y}_{0}\in X$

*with*

*Suppose there exist*$\varphi \in \mathrm{\Phi}$

*and*$\psi \in \mathrm{\Psi}$

*such that*

*for all*$x,y,u,v\in X$

*with*$x\u2ab0u$

*and*$y\u2aafv$.

*Suppose either*

- (a)
*F**is continuous or* - (b)
*X**has the following property*: - (i)
*if a non*-*decreasing sequence*$\{{x}_{n}\}\to x$,*then*${x}_{n}\u2aafx$,*for all**n*, - (ii)
*if a non*-*increasing sequence*$\{{y}_{n}\}\to y$,*then*$y\u2aaf{y}_{n}$,*for all**n*.

*Then there exist*$x,y\in X$

*such that*

*that is*, *F* *has a coupled fixed point in* *X*.

*Proof*Let ${x}_{0},{y}_{0}\in X$ be such that ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ and ${y}_{0}\u2ab0F({y}_{0},{x}_{0})$. We construct sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ in

*X*as

For this we shall use mathematical induction.

Let $n=0$. Since ${x}_{0}\u2aafF({x}_{0},{y}_{0})$ and ${y}_{0}\u2ab0F({y}_{0},{x}_{0})$ and as ${x}_{1}=F({x}_{0},{y}_{0})$ and ${y}_{1}=F({y}_{0},{x}_{0})$, we have ${x}_{0}\u2aaf{x}_{1}$ and ${y}_{0}\u2ab0{y}_{1}$. Thus (5) and (6) hold for $n=0$.

*ϕ*3), we have

*ψ*is a non-negative function,

*ϕ*is non-decreasing, we get

*ϕ*is continuous, we have

*ϕ*, we have

*ϕ*and

*ψ*together with the inequalities established above, we have

We now show that $x=F(x,y)$ and $y=F(y,x)$. Suppose that the assumption (a) holds.

*F*is continuous at $(x,y)$, so for any $\u03f5>0$ , there exists $\delta >0$ such that if $(u,v)\in X\times X$ with $\nu ((x,y),(u,v))<\nu ((x,y),(x,y))+\delta =\delta $, meaning that

Thus, we have $F(x,y)=x$. Similarly, we can show that $F(y,x)=y$.

Taking limit as $n\to \mathrm{\infty}$ in the above inequality, using (31) and (29) and the properties of *ϕ* and *ψ*, we get $\varphi (p(x,F(x,y)))=0$, which implies $p(x,F(x,y))=0$. Hence, $x=F(x,y)$. Similarly, we can show that $y=F(y,x)$. Thus *F* has a coupled fixed point. □

**Remark 3.1** Note that the property (*ϕ* 4) is utilized only to get the inequality (25). Thus the conclusion of Theorem 3.1 holds if we drop property (*ϕ* 4) and assume the additivity in (*ϕ* 3), *i.e.*, $\varphi (t+s)=\varphi (t)+\varphi (s)$, $\mathrm{\forall}t,s\in [0,\mathrm{\infty})$.

As an immediate consequence of the above theorem, by taking $\varphi (t)=t$, we have:

**Corollary 3.1**

*Let*$(X,\u2aaf)$

*be a partially ordered set and suppose there is a partial metric*

*p*

*on*

*X*

*such that*$(X,d)$

*is a complete partial metric space*.

*Let*$F:X\times X\to X$

*be a mapping having the mixed monotone property on*

*X*.

*Assume that there exist two elements*${x}_{0},{y}_{0}\in X$

*with*

*Suppose there exist*$\varphi \in \mathrm{\Phi}$

*and*$\psi \in \mathrm{\Psi}$

*such that*

*for all*$x,y,u,v\in X$

*with*$x\u2ab0u$

*and*$y\u2aafv$.

*Suppose either*

- (a)
*F**is continuous or* - (b)
*X**has the following property*: - (i)
*if a non*-*decreasing sequence*$\{{x}_{n}\}\to x$,*then*${x}_{n}\u2aafx$*for all**n*, - (ii)
*if a non*-*increasing sequence*$\{{y}_{n}\}\to y$,*then*$y\u2aaf{y}_{n}$*for all**n*.

*Then there exist*$x,y\in X$

*such that*

*that is*, *F* *has a coupled fixed point in* *X*.

Moreover, if we take $\psi (t)=\frac{1-k}{2}t$ where $k\in [0,1)$ in Corollary 3.1, we get:

**Corollary 3.2**

*Let*$(X,\u2aaf)$

*be a partially ordered set and suppose there is a partial metric*

*p*

*on*

*X*

*such that*$(X,d)$

*is a complete partial metric space*.

*Let*$F:X\times X\to X$

*be a mapping having the mixed monotone property on*

*X*.

*Assume that there exist two elements*${x}_{0},{y}_{0}\in X$

*with*

*Suppose there exist*$\varphi \in \mathrm{\Phi}$

*and*$\psi \in \mathrm{\Psi}$

*such that*

*for all*$x,y,u,v\in X$

*with*$x\u2ab0u$

*and*$y\u2aafv$.

*Suppose either*

- (a)
*F**is continuous or* - (b)
*X**has the following property*: - (i)
*if a non*-*decreasing sequence*$\{{x}_{n}\}\to x$,*then*${x}_{n}\u2aafx$*for all**n*, - (ii)
*if a non*-*increasing sequence*$\{{y}_{n}\}\to y$,*then*$y\u2aaf{y}_{n}$*for all**n*.

*Then there exist*$x,y\in X$

*such that*

*that is*, *F* *has a coupled fixed point in* *X*.

Recently, Alotaibi and Alsulami [10] extended Luong and Thuan’s [9] main result to coupled coincidences using the notion of compatible maps. Here we extend these results to partial metric spaces without the condition of compatible maps. We shall need the following lemma.

*Let* *X* *be a nonempty set and* $g:X\to X$ *be a mapping*. *Then there exists a subset* $E\subseteq X$ *such that* $g(E)=g(X)$ *and* $g:E\to X$ *is one*-*to*-*one*.

**Theorem 3.2**

*Let*$(X,\u2aaf)$

*be a partially ordered set and suppose there is a partial metric*

*p*

*on*

*X*

*such that*$(X,d)$

*is a partial metric space*.

*Let*$g:X\to X$

*and*$F:X\times X\to X$

*be a mapping having the mixed*

*g*-

*monotone property on*

*X*

*such that there exist two elements*${x}_{0},{y}_{0}\in X$

*with*

*Suppose there exist*$\varphi \in \mathrm{\Phi}$

*and*$\psi \in \mathrm{\Psi}$

*such that*

*for all*$x,y,u,v\in X$

*with*$gx\u2ab0gu$

*and*$gy\u2aafgv$.

*Suppose*$F(X\times X)\subseteq g(X)$,

*g*

*is continuous and*$g(X)$

*is complete and also suppose either*

- (a)
*F**is continuous or* - (b)
*X**has the following property*: - (i)
*if a non*-*decreasing sequence*$\{{x}_{n}\}\to x$,*then*${x}_{n}\u2aafx$*for all**n*, - (ii)
*if a non*-*increasing sequence*$\{{y}_{n}\}\to y$,*then*$y\u2aaf{y}_{n}$*for all**n*.

*Then there exist*$x,y\in X$

*such that*

*that is*, *F* *and* *g* *have a coupled coincidence point in* *X*.

*Proof*Using Lemma 3.1, there exists $E\subseteq X$ such that $g(E)=g(X)$ and $g:E\to X$ is one-to-one. We define a mapping $G:g(E)\times g(E)\to X$ by

*g*is one-to-one on $g(E)$, so

*G*is well defined. Thus, it follows from (33) and (34) that

*F*has the mixed

*g*-monotone property, for all $gx,gy\in g(X)$,

*G*has the mixed monotone property. Also, there exist ${x}_{0},{y}_{0}\in X$ such that

Suppose that the assumption (a) holds. Since *F* is continuous, *G* is also continuous. Using Theorem 3.1 with the mapping *G*, it follows that *G* has a coupled fixed point $(u,v)\in g(X)\times g(X)$.

*G*has a coupled fixed point $(u,v)\in g(X)\times g(X)$. Finally, we prove that

*F*and

*g*have a coupled coincidence point. Since $(u,v)$ is a coupled fixed point of

*G*, we get

Thus, $({u}_{0},{v}_{0})$ is a required coupled coincidence point of *F* and *g*. This completes the proof. □

The following coupled coincidence point theorems are obtained respectively from Corollaries 3.1 and 3.2 in a similar way as Theorem 3.2 from Theorem 3.1.

**Theorem 3.3**

*Let*$(X,\u2aaf)$

*be a partially ordered set*,

*and suppose there is a partial metric*

*p*

*on*

*X*

*such that*$(X,p)$

*is a partial metric space*.

*Let*$g:X\to X$

*and*$F:X\times X\to X$

*be a mapping having the mixed*

*g*-

*monotone property on*

*X*

*such that there exist two elements*${x}_{0},{y}_{0}\in X$

*with*

*Suppose there exist*$\varphi \in \mathrm{\Phi}$

*and*$\psi \in \mathrm{\Psi}$

*such that*

*for all*$x,y,u,v\in X$

*with*$gx\u2ab0gu$

*and*$gy\u2aafgv$.

*Suppose*$F(X\times X)\subseteq g(X)$,

*g*

*is continuous and*$g(X)$

*is complete and also suppose either*

- (a)
*F**is continuous or* - (b)
*X**has the following property*: - (i)
*if a non*-*decreasing sequence*$\{{x}_{n}\}\to x$,*then*${x}_{n}\u2aafx$*for all**n*, - (ii)
*if a non*-*increasing sequence*$\{{y}_{n}\}\to y$,*then*$y\u2aaf{y}_{n}$*for all**n*.

*Then there exist*$x,y\in X$

*such that*

*that is*, *F* *and* *g* *have a coupled coincidence point in* *X*.

**Theorem 3.4**

*Let*$(X,\u2aaf)$

*be a partially ordered set and suppose there is a partial metric*

*p*

*on*

*X*

*such that*$(X,d)$

*is a complete partial metric space*.

*Let*$g:X\to X$

*and*$F:X\times X\to X$

*be a mapping having the mixed*

*g*-

*monotone property on*

*X*

*such that there exist two elements*${x}_{0},{y}_{0}\in X$

*with*

*Suppose there exist*$\varphi \in \mathrm{\Phi}$

*and*$\psi \in \mathrm{\Psi}$

*such that*

*for all*$x,y,u,v\in X$

*with*$gx\u2ab0gu$

*and*$gy\u2aafgv$.

*Suppose*$F(X\times X)\subseteq g(X)$,

*g*

*is continuous and*$g(X)$

*is complete and also suppose either*

- (a)
*F**is continuous or* - (b)
*X**has the following property*: - (i)
*if a non*-*decreasing sequence*$\{{x}_{n}\}\to x$,*then*${x}_{n}\u2aafx$*for all**n*, - (ii)
*if a non*-*increasing sequence*$\{{y}_{n}\}\to y$,*then*$y\u2aaf{y}_{n}$*for all**n*.

*Then there exist*$x,y\in X$

*such that*

*that is*, *F* *and* *g* *have a coupled coincidence point in* *X*.

**Remark 3.2** From the proof of Theorem 3.2 we conclude that Theorems 3.3, 4.4 and 5.4 in [6] hold without the compatibility of the maps $(F,g)$.

## Declarations

### Acknowledgements

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. (113-130-D1432). The authors, therefore, acknowledge with thanks DSR technical and financial support.

## Authors’ Affiliations

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