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New extension of pmetric spaces with some fixedpoint results on Mmetric spaces
Journal of Inequalities and Applications volume 2014, Article number: 18 (2014)
Abstract
In this paper, we extend the pmetric space to an Mmetric space, and we shall show that the definition we give is a real generalization of the pmetric by presenting some examples. In the sequel we prove some of the main theorems by generalized contractions for getting fixed points and common fixed points for mappings.
1 Introduction and preliminaries
In 1994, in [1] Matthews introduced the notion of a partial metric space and proved the contraction principle of Banach in this new framework. Next, many fixedpoint theorems in partial metric spaces have been given by several mathematicians. Recently Haghi et al. published [2] a paper which stated that we should ‘be careful on partial metric fixed point results’ along with giving some results. They showed that fixedpoint generalizations to partial metric spaces can be obtained from the corresponding results in metric spaces.
In this paper, we extend the pmetric space to an Mmetric space, and we shall show that our definition is a real generalization of the pmetric by presenting some examples. In the sequel we prove some of the main theorems by generalized contractions for getting fixed points and common fixed points for mappings.
Definition 1.1 ([1], [[3], Definition 1.1])
A partial metric on a nonempty set X is a function $p:X\times X\to {\mathbb{R}}^{+}$ such that for all $x,y,z\in X$:

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

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

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

(p4) $p(x,y)\le p(x,z)+p(z,y)p(z,z)$.
A partial metric space is a pair $(X,p)$ such that X is a nonempty set and p is a partial metric on X.
Notation The following notation is useful in the sequel.

1.
${m}_{xy}:=min\{m(x,x),m(y,y)\}$,

2.
${M}_{xy}:=max\{m(x,x),m(y,y)\}$.
Now we want to extend Definition 1.1 as follows.
Definition 1.2 Let X be a nonempty set. A function $m:X\times X\to {\mathbb{R}}^{+}$ is called an mmetric if the following conditions are satisfied:

(m1) $m(x,x)=m(y,y)=m(x,y)\iff x=y$,

(m2) ${m}_{xy}\le m(x,y)$,

(m3) $m(x,y)=m(y,x)$,

(m4) $(m(x,y){m}_{xy})\le (m(x,z){m}_{xz})+(m(z,y){m}_{zy})$.
Then the pair $(X,m)$ is called an Mmetric space.
According to the above definition the condition (p1) in the definition of [1] changes to (m1), and (p2) is expressed for $p(x,x)$ where $p(y,y)=0$ may become $p(y,y)\ne 0$. Thus we improve that condition by replacing it by $min\{p(x,x),p(y,y)\}\le p(x,y)$, and also we improve the condition (p4) extending it to the form of (m4). In the sequel we present an example that holds for the mmetric but not for the pmetric.
Remark 1.1 For every $x,y\in X$

1.
$0\le {M}_{xy}+{m}_{xy}=m(x,x)+m(y,y)$,

2.
$0\le {M}_{xy}{m}_{xy}=m(x,x)m(y,y)$,

3.
${M}_{xy}{m}_{xy}\le ({M}_{xz}{m}_{xz})+({M}_{zy}{m}_{zy})$.
The next examples show that ${m}^{s}$ and ${m}^{w}$ are ordinary metrics.
Example 1.1 Let $X:=[0,\mathrm{\infty})$. Then $m(x,y)=\frac{x+y}{2}$ on X is an mmetric.
Example 1.2 Let m be an mmetric. Put

1.
${m}^{w}(x,y)=m(x,y)2{m}_{xy}+{M}_{xy}$,

2.
${m}^{s}(x,y)=m(x,y){m}_{xy}$ when $x\ne y$ and ${m}^{s}(x,y)=0$ if $x=y$.
Then ${m}^{w}$ and ${m}^{s}$ are ordinary metrics.
Proof If ${m}^{w}(x,y)=0$, then
But from equation (1) and ${m}_{xy}\le m(x,y)$ we get ${m}_{xy}={M}_{xy}=m(x,x)=m(y,y)$, so by equation (1) we obtain $m(x,y)=m(x,x)=m(y,y)$ and therefore $x=y$. For the triangle inequality it is enough that we consider Remark 1.1 and (m4). □
Remark 1.2 For every $x,y\in X$

1.
$m(x,y){M}_{xy}\le {m}^{w}(x,y)\le m(x,y)+{M}_{xy}$,

2.
$(m(x,y){M}_{xy})\le {m}^{s}(x,y)\le m(x,y)$.
In other words
In the following example we present an example of an mmetric which is not a pmetric.
Example 1.3 Let $X=\{1,2,3\}$; define
So m is an mmetric, but it is not pmetric.
Example 1.4 Let $(X,d)$ be a metric space. Let $\varphi :[0,\mathrm{\infty})\to [\varphi (0),\mathrm{\infty})$ be a one to one and nondecreasing or strictly increasing mapping, with $\varphi (0)$ defined such that
Then $m(x,y)=\varphi (d(x,y))$ is an mmetric.
Proof (m1), (m2), and (m3) are clear. For (m4) we have
□
Example 1.5 Let $(X,d)$ be a metric space. Then $m(x,y)=ad(x,y)+b$ where $a,b>0$ is an mmetric, because we can put $\varphi (t)=at+b$.
Remark 1.3 According to Example 1.5, by the Banach contraction
we have
which does not imply the ordinary Banach contraction
for all selfmaps T on X. Thus, this states that even if the mmetric m and the ordinary metric d have the same topology, the Banach contraction of the mmetric does not imply the Banach contraction of the ordinary metric d.
Lemma 1.1 Every pmetric is an mmetric.
Proof Let m be a pmetric. It is enough that we consider the following cases:

1.
$m(x,x)=m(y,y)=m(z,z)$,

2.
$m(x,x)<m(y,y)<m(z,z)$,

3.
$m(x,x)=m(y,y)<m(z,z)$,

4.
$m(x,x)=m(y,y)>m(z,z)$,

5.
$m(x,x)<m(y,y)=m(z,z)$,

6.
$m(x,x)>m(y,y)=m(z,z)$.
For example, to prove (2), we have
□
2 Topology for Mmetric space
It is clear that each mmetric p on X generates a ${T}_{0}$ topology ${\tau}_{m}$ on X. The set
where
for all $x\in X$ and $\epsilon >0$, forms a base of ${\tau}_{m}$.
Definition 2.1 Let $(X,m)$ be a mmetric space. Then:

1.
A sequence $\{{x}_{n}\}$ in a Mmetric space $(X,m)$ converges to a point $x\in X$ if and only if
$$\underset{n\to \mathrm{\infty}}{lim}(m({x}_{n},x){m}_{{x}_{n},x})=0.$$(2) 
2.
A sequence $\{{x}_{n}\}$ in a Mmetric space $(X,m)$ is called an mCauchy sequence if
$$\underset{n,m\to \mathrm{\infty}}{lim}(m({x}_{n},{x}_{m}){m}_{{x}_{n},{x}_{m}}),\phantom{\rule{2em}{0ex}}\underset{n,m\to \mathrm{\infty}}{lim}({M}_{{x}_{n},{x}_{m}}{m}_{{x}_{n},{x}_{m}})$$(3)exist (and are finite).

3.
An Mmetric space $(X,m)$ is said to be complete if every mCauchy sequence $\{{x}_{n}\}$ in X converges, with respect to ${\tau}_{m}$, to a point $x\in X$ such that
$$(\underset{n\to \mathrm{\infty}}{lim}(m({x}_{n},x){m}_{{x}_{n},x})=0\phantom{\rule{0.25em}{0ex}}\mathrm{\&}\phantom{\rule{0.25em}{0ex}}\underset{n\to \mathrm{\infty}}{lim}({M}_{{x}_{n},x}{m}_{{x}_{n},x})=0).$$
Lemma 2.1 Let $(X,m)$ be a mmetric space. Then:

1.
$\{{x}_{n}\}$ is an mCauchy sequence in $(X,m)$ if and only if it is a Cauchy sequence in the metric space $(X,{m}^{w})$.

2.
An Mmetric space $(X,m)$ is complete if and only if the metric space $(X,{m}^{w})$ is complete. Furthermore,
$$\underset{n\to \mathrm{\infty}}{lim}{m}^{w}({x}_{n},x)=0\phantom{\rule{1em}{0ex}}\iff \phantom{\rule{1em}{0ex}}(\underset{n\to \mathrm{\infty}}{lim}(m({x}_{n},x){m}_{{x}_{n},x})=0,\phantom{\rule{0.25em}{0ex}}\underset{n\to \mathrm{\infty}}{lim}({M}_{{x}_{n},x}{m}_{{x}_{n},x})=0).$$
Likewise the above definition holds also for ${m}^{s}$.
Lemma 2.2 Assume that ${x}_{n}\to x$ and ${y}_{n}\to y$ as $n\to \mathrm{\infty}$ in an Mmetric space $(X,m)$. Then
Proof We have
□
From Lemma 2.2 we deduce the following lemma.
Lemma 2.3 Assume that ${x}_{n}\to x$ as $n\to \mathrm{\infty}$ in an Mmetric space $(X,m)$. Then
for all $y\in X$.
Lemma 2.4 Assume that ${x}_{n}\to x$ and ${x}_{n}\to y$ as $n\to \mathrm{\infty}$ in an Mmetric space $(X,m)$. Then $m(x,y)={m}_{xy}$. Furthermore, if $m(x,x)=m(y,y)$, then $x=y$.
Proof By Lemma 2.2 we have
□
Lemma 2.5 Let $\{{x}_{n}\}$ be a sequence in an mmetric space $(X,m)$, such that
Then

(A)
${lim}_{n\to \mathrm{\infty}}m({x}_{n},{x}_{n1})=0$,

(B)
${lim}_{n\to \mathrm{\infty}}m({x}_{n},{x}_{n})=0$,

(C)
${lim}_{m,n\to \mathrm{\infty}}{m}_{{x}_{m}{x}_{n}}=0$,

(D)
$\{{x}_{n}\}$ is an mCauchy sequence.
Proof From equation (4) we have
thus,
which implies that (A) holds.
From (m2) and (A) we have
That is, (B) holds.
Clearly, (C) holds, since ${lim}_{n\to \mathrm{\infty}}m({x}_{n},{x}_{n})=0$. □
Theorem 2.1 The topology ${\tau}_{m}$ is not Hausdorff.
Proof Let $x,y,z\in X$ be such that
with
and
without loss of generality we assume that for each $\epsilon >0$ we have $\epsilon <r$. We want to show that the intersection of the following neighborhoods is not empty:
To prove $z\in {U}_{x}$, we have
and for $z\in {V}_{y}$
so we can find $x,y\in X$ such that for all nonempty neighborhoods ${U}_{x}$ of x and ${V}_{y}$ of y we have ${U}_{x}\cap {V}_{y}\ne \mathrm{\varnothing}$. □
3 Fixed point results on Mmetric space
Theorem 3.1 Let $(X,m)$ be a complete Mmetric space and let $T:X\to X$ be a mapping satisfying the following condition:
Then T has a unique fixed point.
Proof Let ${x}_{0}\in X$ and ${x}_{n}:=T{x}_{n1}$, so we have
and so (A), (B), (C), and (D) of Lemma 2.5 hold. By completeness of X we get ${x}_{n}\to x$ for some $x\in X$. Thus by equation (5) $m(T{x}_{n},Tx)\le km({x}_{n},x)\to 0$. Hence by (m2) ${m}_{T{x}_{n},Tx}\le m(T{x}_{n},Tx)\to 0$ so by equation (2) $T{x}_{n}\to Tx$.
Contraction (5) implies that $m({x}_{n},T{x}_{n})\to 0$ and $m(Tx,Tx)<m(x,x)$. Since ${m}_{{x}_{n},T{x}_{n}}\to 0$, by Lemma 2.2, we get $m(x,Tx)={m}_{x,Tx}=m(Tx,Tx)$.
On the other hand, by Lemma 2.2 and ${x}_{n}=T{x}_{n1}\to x$,
thus $m(x,x)=m(x,Tx)$. Since $m(x,Tx)={m}_{x,Tx}=m(Tx,Tx)$ now by (m1) $x=Tx$. Uniqueness by the contraction (5) is clear. □
Theorem 3.2 Let $(X,m)$ be a complete Mmetric space and let $T:X\to X$ be a mapping satisfying the following condition:
Then T has an unique fixed point.
Proof Let ${x}_{0}\in X$ and ${x}_{n}:=T{x}_{n1}$, so we have
So
where $0\le r=\frac{k}{1k}<1$.
By Lemma 2.5 and completeness of X, ${x}_{n}\to x$ for some $x\in X$. So
and since ${m}_{{x}_{n},x}\to 0$, we have $m({x}_{n},x)\to 0$ and ${M}_{{x}_{n},x}\to 0$. Therefore by Remark 1.1, $m(x,x)=0={m}_{x,Tx}$;
hence by $m({x}_{n},{x}_{n+1})\to 0$
On the other hand
implies that
because ${m}_{x,Tx}=0$ and $m({x}_{n},x)\to 0$. So $m(x,Tx)=0$. Now by contraction (7) we have $m(Tx,Tx)\le 2km(x,Tx)=0$, so $m(Tx,Tx)=0=m(x,x)=m(x,Tx)$, thus $x=Tx$ by (m1). □
The next theorem is still open.
Theorem 3.3 Let $(X,m)$ be a complete Mmetric space and let $T:X\to X$ be a mapping satisfying the following condition:
Then T has a unique fixed point.
References
 1.
Matthews S: Partial metric topology. Ann. N.Y. Acad. Sci. 1994, 728: 183197. 10.1111/j.17496632.1994.tb44144.x
 2.
Haghi RH, Rezapour Sh, Shahzad N: Be careful on partial metric fixed point results. Topol. Appl. 2013,160(3):450454. 10.1016/j.topol.2012.11.004
 3.
Shatanawi W, Postolache M: Coincidence and fixed point results for generalized weak contractions in the sense of Berinde on partial metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 54
Acknowledgements
The authors express their deep gratitude to the referee for his/her valuable comments and suggestions. This paper has been supported by the I.A.U., Zanjan Branch, Zanjan, Iran. The first author would like to thank for this support. The authors would like to thank Professors William A (Art) Kirk and Billy E. Rhoades for helpful advise which led them to present this paper.
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Asadi, M., Karapınar, E. & Salimi, P. New extension of pmetric spaces with some fixedpoint results on Mmetric spaces. J Inequal Appl 2014, 18 (2014). https://doi.org/10.1186/1029242X201418
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Keywords
 fixed point
 partial metric space