# Some new common fixed point results through generalized altering distances on partial metric spaces

- Abd Ghafur Bin Ahmad
^{1}, - Zaid Mohammed Fadail
^{1}Email author, - Hemant Kumar Nashine
^{2}, - Zoran Kadelburg
^{3}and - Stojan Radenović
^{4}

**2012**:120

https://doi.org/10.1186/1687-1812-2012-120

© Ahmad et al.; licensee Springer 2012

**Received: **21 March 2012

**Accepted: **4 July 2012

**Published: **23 July 2012

## Abstract

We establish common fixed point results for two pairs of weakly compatible mappings on a partial metric space, satisfying a weak contractive condition involving generalized control functions. The presented theorems extend and unify various known fixed point results. Examples are given to show that our results are proper extensions of the known ones.

**MSC:**47H10, 54H25, 54H10.

## Keywords

## 1 Introduction

In [1], Matthews introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks. He showed that the Banach contraction mapping theorem can be generalized to the partial metric context for applications in program verification.

Subsequently, several authors (see, *e.g.*, Altun and Erduran [2], Oltra *et al.* [3], Romaguera and Schellekens [4], Romaguera and Valero [5], Rus [6], Djukić *et al.* [7], Nashine *et al.* [8], Di Bari and Vetro [9], Paesano and Vetro [10], Shatanawi *et al.* [11], Shatanawi and Nashine [12], Aydi *et al.* [13]) derived fixed point theorems in partial metric spaces.

Altering distance functions (also called control functions) were introduced by Khan *et al.* [14]. Subsequently, they were used by many authors to obtain fixed point results, including those in partial metric spaces (*e.g.*, Abdeljawad [15], Abdeljawad *et al.* [16, 17], Altun *et al.* [18], Ćirić *et al.* [19], Karapinar and Yüksel [20]). Generalized altering distance functions with several variables were used on metric spaces by Berinde [21], Choudhury [22] and Rao *et al.* [23].

In this paper, an attempt has been made to derive some common fixed point theorems for two pairs of weakly compatible mappings on partial metric spaces, satisfying a weak contractive condition involving generalized control functions. The presented theorems extend and unify various known fixed point results. Examples are given to show that our results are proper extensions of the known ones.

## 2 Preliminaries

The following definitions and details about partial metrics can be seen, *e.g.*, in [1, 24–28].

**Definition 1** A *partial metric* on a nonempty set *X* is a function $p:X\times X\to {\mathbb{R}}^{+}$ such that for all $x,y,z\in X$:

(p_{1}) $x=y\u27fap(x,x)=p(x,y)=p(y,y)$,

(p_{2}) $p(x,x)\le p(x,y)$,

(p_{3}) $p(x,y)=p(y,x)$,

(p_{4}) $p(x,y)\le p(x,z)+p(z,y)-p(z,z)$.

The pair $(X,p)$ is called a *partial metric space*.

It is clear that, if $p(x,y)=0$, then from (p_{1}) and (p_{2}), it follows that $x=y$. But $p(x,x)$ may not be 0.

*p*on

*X*generates a ${T}_{0}$ topology ${\tau}_{p}$ on

*X*which has as a base the family of open

*p*-balls $\{{B}_{p}(x,\epsilon ):x\in X,\epsilon >0\}$, where ${B}_{p}(x,\epsilon )=\{y\in X:p(x,y)<p(x,x)+\epsilon \}$ for all $x\in X$ and $\epsilon >0$. A sequence $\{{x}_{n}\}$ in $(X,p)$ converges to a point $x\in X$, with respect to ${\tau}_{p}$, if ${lim}_{n\to \mathrm{\infty}}p(x,{x}_{n})=p(x,x)$. This will be denoted as ${x}_{n}\to x$, $n\to \mathrm{\infty}$ or ${lim}_{n\to \mathrm{\infty}}{x}_{n}=x$. If $(X,p)$ is a partial metric space, and $T:X\to X$ is a mapping, continuous at ${x}_{0}\in X$ (in ${\tau}_{p}$) then, for each sequence $\{{x}_{n}\}$ in

*X*, we have

Clearly, a limit of a sequence in a partial metric space need not be unique. Moreover, the function $p(\cdot ,\cdot )$ need not be continuous in the sense that ${x}_{n}\to x$ and ${y}_{n}\to y$ implies $p({x}_{n},{y}_{n})\to p(x,y)$.

**Definition 2**Let $(X,p)$ be a partial metric space. Then:

- 1
A sequence $\{{x}_{n}\}$ in $(X,p)$ is called a

*Cauchy sequence*if ${lim}_{n,m\to \mathrm{\infty}}p({x}_{n},{x}_{m})$ exists (and is finite). - 2
The space $(X,p)$ is said to be

*complete*if every Cauchy sequence $\{{x}_{n}\}$ in*X*converges, with respect to ${\tau}_{p}$, to a point $x\in X$ such that $p(x,x)={lim}_{n,m\to \mathrm{\infty}}p({x}_{n},{x}_{m})$.

It is easy to see that every closed subset of a complete partial metric space is complete.

*p*is a partial metric on

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

*X*. Furthermore, ${lim}_{n\to \mathrm{\infty}}{p}^{s}({x}_{n},x)=0$ if and only if

**Lemma 1**

*Let*$(X,p)$

*be a partial metric space*.

- (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)
*The space*$(X,p)$*is complete if and only if the metric space*$(X,{p}^{s})$*is complete*.

*generalized altering distance function*if:

- 1
*ψ*is continuous; - 2
*ψ*is increasing in each of its variables; - 3
$\psi ({t}_{1},\dots ,{t}_{n})=0$ if and only if ${t}_{1}=\dots ={t}_{n}=0$.

The set of generalized altering distance functions with *n* variables will be denoted by ${\mathcal{F}}_{n}$. If $\psi \in {\mathcal{F}}_{n}$, we will write $\mathrm{\Psi}(t)=\psi (t,t,\dots ,t)$ (obviously, this function belongs to ${\mathcal{F}}_{1}$).

Recall also the following notions. Let *X* be a nonempty set and ${T}_{1},{T}_{2}:X\to X$ be given self-maps on *X*. If $w={T}_{1}x={T}_{2}x$ for some $x\in X$, then *x* is called a coincidence point of ${T}_{1}$ and ${T}_{2}$, and *w* is called a point of coincidence of ${T}_{1}$ and ${T}_{2}$. The pair $\{{T}_{1},{T}_{2}\}$ is said to be weakly compatible if ${T}_{1}{T}_{2}t={T}_{2}{T}_{1}t$, whenever ${T}_{1}t={T}_{2}t$ for some *t* in *X*.

## 3 Results

### 3.1 Some auxiliary results

Assertions similar to the following lemma (see, *e.g.*, [29]) were used (and proved) in the course of proofs of several fixed point results in various papers.

**Lemma 2**

*Let*$(X,d)$

*be a metric space and let*$\{{y}_{n}\}$

*be a sequence in*

*X*

*such that*$\{d({y}_{n+1},{y}_{n})\}$

*is nonincreasing and*

*If*$\{{y}_{2n}\}$

*is not a Cauchy sequence*,

*then there exist*$\epsilon >0$

*and two sequences*$\{m(k)\}$

*and*$\{n(k)\}$

*of positive integers such that*$n(k)>m(k)>k$

*and the following four sequences tend to*

*ε*

*when*$k\to \mathrm{\infty}$:

As a corollary (putting $d={p}^{s}$ for a partial metric *p*), we obtain

**Lemma 3**

*Let*$(X,p)$

*be a partial metric space and let*$\{{y}_{n}\}$

*be a sequence in*

*X*

*such that*$\{p({y}_{n+1},{y}_{n})\}$

*is nonincreasing and*

### 3.2 Main results

**Theorem 1**

*Let*$(X,p)$

*be a complete partial metric space*.

*Let*$T,S,I,J:X\to X$

*be given mappings satisfying for every pair*$(x,y)\in X\times X$:

*where*${\psi}_{1}\in {\mathcal{F}}_{4}$

*and*${\psi}_{2}\in {\mathcal{F}}_{3}$

*are generalized altering distance functions*,

*and*${\mathrm{\Psi}}_{1}(t)={\psi}_{1}(t,t,t,t)$.

*Suppose that*

- (i)
$TX\subseteq IX$

*and*$SX\subseteq JX$; - (ii)
*one of the ranges**IX*,*JX*,*TX**and**SX**is a closed subset of*$(X,p)$.

*Then*

- (a)
*I**and**S**have a coincidence point*, - (b)
*J**and**T**have a coincidence point*.

*Moreover*, *if the pairs* $\{I,S\}$ *and* $\{J,T\}$ *are weakly compatible*, *then* *I*, *J*, *T* *and* *S* *have a unique common fixed point*.

*Proof*Let ${x}_{0}$ be an arbitrary point in

*X*. Since $TX\subseteq IX$ and $SX\subseteq JX$, we can define sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ in

*X*by

*n*. Taking $x={x}_{2n}$ and $y={x}_{2n+1}$, from (3.4) and the considered contraction condition (3.3), we have

which is a contradiction. It follows that $p({y}_{2n+1},{y}_{2n+2})=0$ and hence ${y}_{2n+1}={y}_{2n+2}$. Following similar arguments, we obtain ${y}_{2n+2}={y}_{2n+3}$. Thus $\{{y}_{n}\}$ becomes an eventually constant sequence and ${y}_{2n}$ is a point of coincidence of *I* and *S*, while ${y}_{2n+1}$ is a point of coincidence of *J* and *T*.

which implies that ${\psi}_{2}(r,r,r)=0$, and thus $r=0$. Hence, (3.7) is proved.

*ε*when $i\to \mathrm{\infty}$. Applying condition (3.3) to elements $x={x}_{2m(i)}$ and $y={x}_{2n(i)-1}$, we get that

which implies that ${\psi}_{2}(\epsilon ,0,0)=0$, that is a contradiction since $\epsilon >0$. We deduce that $\{{y}_{n}\}$ is a Cauchy sequence.

Finally, we prove the existence of a common fixed point of the four mappings *I*, *J*, *S* and *T*.

_{2}) and (3.7), we have $p({y}_{n},{y}_{n})\le p({y}_{n},{y}_{n+1})\to 0$, $n\to \mathrm{\infty}$ and hence

*IX*is a closed subset of the partial metric space $(X,p)$. From (3.15), there exists $u\in X$ such that $z=Iu$. We claim that $Su=z$. Suppose, to the contrary, that $p(Su,z)>0$. By (p

_{4}) we get

We get that $Su=Iu=z$, so *u* is a coincidence point of *I* and *S*.

We get that $Jv=Tv=z$, so *v* is a coincidence point of *J* and *S*.

so *z* is a common fixed point of the four mappings *I*, *J*, *S* and *T*.

*S*,

*T*,

*I*and

*J*. Assume on contrary that, $Su=Tu=Iu=Ju=u$ and $Sv=Tv=Iv=Jv=v$ with $p(u,v)>0$. By supposition, we can replace

*x*by

*u*and

*y*by

*v*in (3.3) to obtain

a contradiction. Hence $p(u,v)=0$, that is, $u=v$. We conclude that *S*, *T*, *I* and *J* have only one common fixed point in *X*. The proof is complete. □

It is easy to state the corollary of Theorem 1 involving a contraction of integral type.

**Corollary 1**

*Let*

*T*,

*S*,

*I*

*and*

*J*

*as well as*${\psi}_{1}$, ${\psi}_{2}$

*satisfy the conditions of Theorem*1,

*except that condition*(3.3)

*is replaced by the following*:

*there exists a positive Lebesgue integrable function*

*u*

*on*${\mathbb{R}}^{+}$

*such that*${\int}_{0}^{\epsilon}u(t)\phantom{\rule{0.2em}{0ex}}dt>0$

*for each*$\epsilon >0$

*and that*

*for all* $x,y\in X$. *Then*, *S*, *T*, *I* *and* *J* *have a unique common fixed point*.

If in Theorem 1 $I=J$ is the identity mapping on *X*, then we have the following consequence:

**Theorem 2**

*Let*$(X,p)$

*be a complete partial metric space*.

*Let*$T,S:X\to X$

*be given mappings satisfying for every pair*$(x,y)\in X\times X$

*where* ${\psi}_{1}\in {\mathcal{F}}_{4}$ *and* ${\psi}_{2}\in {\mathcal{F}}_{3}$ *are altering distance functions*, *and* ${\mathrm{\Psi}}_{1}(t)={\psi}_{1}(t,t,t,t)$. *Then* *T* *and* *S* *have a unique common fixed point*.

**Remark 1** Several corollaries of Theorems 1 and 2 could be derived for particular choices of ${\psi}_{1}$ and ${\psi}_{2}$. We state some of them.

Putting ${\psi}_{1}({t}_{1},{t}_{2},{t}_{3},{t}_{4})=\psi (max\{{t}_{1},{t}_{2},{t}_{3},{t}_{4}\})$ and ${\psi}_{2}({t}_{1},{t}_{2},{t}_{3})=\varphi (max\{{t}_{1},{t}_{2},{t}_{3}\})$ for $\psi ,\varphi \in {\mathcal{F}}_{1}$, [15], Theorem 9] is obtained.

where ${\psi}_{2}\in {\mathcal{F}}_{4}$. Hence, Theorem 1 can be considered an extension of [23], Theorem 2.1] to the frame of partial metric spaces (since semi-compatible mappings are weakly compatible).

Putting ${\psi}_{1}({t}_{1},{t}_{2},{t}_{3},{t}_{4})=max\{{t}_{1},{t}_{2},{t}_{3},{t}_{4}\}$ and ${\psi}_{2}=(1-r){\psi}_{1}$, with $0\le r<1$, in Theorem 2 (with condition (3.25)), we obtain [20], Theorem 8]. The same substitution in Theorem 1 (with (3.24)) gives an improvement of [20], Theorem 12] (since only weak compatibility and not commutativity of the respective mappings is assumed).

Putting ${\psi}_{1}({t}_{1},{t}_{2},{t}_{3},{t}_{4})=max\{{t}_{1},{t}_{2},{t}_{3},{t}_{4}\}$ and ${\psi}_{2}=\phi \circ {\psi}_{1}$ for $\phi \in {\mathcal{F}}_{1}$ in Theorem 2 (with condition (3.25)), [16], Theorem 5] is obtained.

Of course, several known results from the frame of standard metric spaces (see, *e.g.*, [30] and [31]) are also special cases of these theorems. For example, the following corollary can be obtained as a consequence of Theorem 2, which is a generalization and extension of [31], Corollary 3.2].

**Corollary 2**

*Let*$(X,p)$

*be a complete partial metric space*.

*Let*$T,S:X\to X$

*be given mappings satisfying for every pair*$(x,y)\in X\times X$

*where* *m* *and* *n* *are positive integers*, ${\psi}_{1}\in {\mathcal{F}}_{4}$ *and* ${\psi}_{2}\in {\mathcal{F}}_{3}$ *are altering distance functions*, *and* ${\mathrm{\Psi}}_{1}(t)={\psi}_{1}(t,t,t,t)$. *Then* *T* *and* *S* *have a unique common fixed point*.

**Remark 2** However, it is not possible to use ${\psi}_{1},{\psi}_{2}\in {\mathcal{F}}_{5}$ in Theorems 1 and 2, as the following example, adapted from [23], Example 2.3], shows.

**Example 1**Let $X=\{1,2,3,4\}$ and $p:X\times X\to X$ be given by $p(x,x)=\frac{1}{2}$ for $x\in X$, $p(1,2)=p(3,4)=2$, $p(1,3)=p(2,4)=1$, $p(1,4)=p(2,3)=\frac{3}{2}$ and $p(y,x)=p(x,y)$ for $x,y\in X$. Then $(X,p)$ is a (complete) partial metric space. Consider the mappings $S,T:X\to X$ defined by

whatever ${\psi}_{2}\in {\mathcal{F}}_{4}$ is chosen.

This example also shows (as in [23], Remark 2.4]) the importance of the second generalized altering distance function ${\psi}_{2}$ in Theorems 1 and 2.

The next example shows that Theorems 1 and 2 are proper extensions of the respective results in standard metric spaces.

**Example 2**Let $X=[0,1]$ be endowed with the partial metric $p(x,y)=max\{x,y\}$. Consider the mappings $S,T:X\to X$ defined by

*X*. Then

Hence, condition (3.25) is satisfied, as well as other conditions of Theorem 2. Mappings $S,T$ have a common fixed point $z=0$.

Thus, condition (3.25) for $p=d$ does not hold and the existence of a common fixed point of these mappings cannot be derived from [23], Theorem 2.1].

## Declarations

### Acknowledgements

The first and second author would like to acknowledge the financial support received from University Kebangsaan Malaysia under the research grant OUP-UKM-FST-2012. The fourth and fifth author are thankful to the Ministry of Science and Technological Development of Serbia.

## Authors’ Affiliations

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