In this paper, we extend the p-metric space to an M-metric space, and we shall show that the definition we give is a real generalization of the p-metric 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.
Hinweise
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors have read and approved the final manuscript.
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 fixed-point 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 fixed-point generalizations to partial metric spaces can be obtained from the corresponding results in metric spaces.
In this paper, we extend the p-metric space to an M-metric space, and we shall show that our definition is a real generalization of the p-metric 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.
A partial metric on a nonempty set X is a function such that for all :
(p1) ,
(p2) ,
(p3) ,
(p4) .
A partial metric space is a pair 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.
,
2.
.
Anzeige
Now we want to extend Definition 1.1 as follows.
Definition 1.2 Let X be a nonempty set. A function is called an m-metric if the following conditions are satisfied:
(m1) ,
(m2) ,
(m3) ,
(m4) .
Then the pair is called an M-metric space.
According to the above definition the condition (p1) in the definition of [1] changes to (m1), and (p2) is expressed for where may become . Thus we improve that condition by replacing it by , 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 m-metric but not for the p-metric.
Remark 1.1 For every
1.
,
2.
,
3.
.
The next examples show that and are ordinary metrics.
Example 1.1 Let . Then on X is an m-metric.
Example 1.2 Let m be an m-metric. Put
1.
,
2.
when and if .
Then and are ordinary metrics.
Proof If , then
(1)
But from equation (1) and we get , so by equation (1) we obtain and therefore . For the triangle inequality it is enough that we consider Remark 1.1 and (m4). □
Remark 1.2 For every
1.
,
2.
.
In other words
In the following example we present an example of an m-metric which is not a p-metric.
Example 1.3 Let ; define
So m is an m-metric, but it is not p-metric.
Example 1.4 Let be a metric space. Let be a one to one and nondecreasing or strictly increasing mapping, with defined such that
Then is an m-metric.
Proof (m1), (m2), and (m3) are clear. For (m4) we have
□
Example 1.5 Let be a metric space. Then where is an m-metric, because we can put .
Remark 1.3 According to Example 1.5, by the Banach contraction
we have
which does not imply the ordinary Banach contraction
for all self-maps T on X. Thus, this states that even if the m-metric m and the ordinary metric d have the same topology, the Banach contraction of the m-metric does not imply the Banach contraction of the ordinary metric d.
Lemma 1.1Everyp-metric is anm-metric.
Proof Let m be a p-metric. It is enough that we consider the following cases:
1.
,
2.
,
3.
,
4.
,
5.
,
6.
.
For example, to prove (2), we have
□
2 Topology for M-metric space
It is clear that each m-metric p on X generates a topology on X. The set
where
for all and , forms a base of .
Definition 2.1 Let be a m-metric space. Then:
1.
A sequence in a M-metric space converges to a point if and only if
(2)
2.
A sequence in a M-metric space is called an m-Cauchy sequence if
(3)
exist (and are finite).
3.
An M-metric space is said to be complete if every m-Cauchy sequence in X converges, with respect to , to a point such that
Lemma 2.1Letbe am-metric space. Then:
1.
is anm-Cauchy sequence inif and only if it is a Cauchy sequence in the metric space .
2.
AnM-metric spaceis complete if and only if the metric spaceis complete. Furthermore,
Likewise the above definition holds also for .
Lemma 2.2Assume thatandasin anM-metric space . Then
Proof We have
□
From Lemma 2.2 we deduce the following lemma.
Lemma 2.3Assume thatasin anM-metric space . Then
for all .
Lemma 2.4Assume thatandasin anM-metric space . Then . Furthermore, if , then .
Proof By Lemma 2.2 we have
□
Lemma 2.5Letbe a sequence in anm-metric space , such that
without loss of generality we assume that for each we have . We want to show that the intersection of the following neighborhoods is not empty:
To prove , we have
and for
so we can find such that for all nonempty neighborhoods of x and of y we have . □
3 Fixed point results on M-metric space
Theorem 3.1Letbe a completeM-metric space and letbe a mapping satisfying the following condition:
(5)
ThenThas a unique fixed point.
Proof Let and , so we have
(6)
and so (A), (B), (C), and (D) of Lemma 2.5 hold. By completeness of X we get for some . Thus by equation (5) . Hence by (m2) so by equation (2) .
Contraction (5) implies that and . Since , by Lemma 2.2, we get .
On the other hand, by Lemma 2.2 and ,
thus . Since now by (m1) . Uniqueness by the contraction (5) is clear. □
Theorem 3.2Letbe a completeM-metric space and letbe a mapping satisfying the following condition:
(7)
ThenThas an unique fixed point.
Proof Let and , so we have
So
where .
By Lemma 2.5 and completeness of X, for some . So
and since , we have and . Therefore by Remark 1.1, ;
hence by
On the other hand
implies that
because and . So . Now by contraction (7) we have , so , thus by (m1). □
The next theorem is still open.
Theorem 3.3Letbe a completeM-metric space and letbe a mapping satisfying the following condition:
(8)
ThenThas a unique fixed point.
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.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors have read and approved the final manuscript.