Abstract
Matthews (1994) introduced a new distance on a nonempty set
, which is called partial metric. If
is a partial metric space, then
may not be zero for
. In the present paper, we give some fixed point results on these interesting spaces.
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1. Introduction
There are a lot of fixed and common fixed point results in different types of spaces. For example, metric spaces, fuzzy metric spaces, and uniform spaces. One of the most interesting is partial metric space, which is defined by Matthews [1]. In partial metric spaces, the distance of a point in the self may not be zero. After the definition of partial metric space, Matthews proved the partial metric version of Banach fixed point theorem. Then, Valero [2], Oltra and Valero [3], and Altun et al. [4] gave some generalizations of the result of Matthews. Again, Romaguera [5] proved the Caristi type fixed point theorem on this space.
First, we recall some definitions of partial metric spaces and some properties of theirs. See [1–3, 5–7] for details.
A partial metric on a nonempty set is a function
such that for all
(p1),
(p2),
(p3),
(p4).
A partial metric space is a pair such that
is a nonempty set and
is a partial metric on
. It is clear that if
, then from (p1) and (p2)
. But if
,
may not be 0. A basic example of a partial metric space is the pair
, where
for all
. Other examples of partial metric spaces, which are interesting from a computational point of view, may be found in [1, 8].
Each partial metric on
generates a
topology
on
, which has as a base the family open
-balls
, where
for all
and
.
If is a partial metric on
, then the function
given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ1_HTML.gif)
is a metric on .
Let be a partial metric space, then we have the following.
(i)A sequence in a partial metric space
converges to a point
if and only if
.
(ii)A sequence in a partial metric space
is called a Cauchy sequence if there exists (and is finite)
.
-
(iii)
A partial metric space
is said to be complete if every Cauchy sequence
in
converges, with respect to
, to a point
such that
.
-
(iv)
A mapping
is said to be continuous at
, if for every
, there exists
such that
.
Let be a partial metric space.
(a) is a Cauchy sequence in
if and only if it is a Cauchy sequence in the metric space
.
(b)A partial metric space is complete if and only if the metric space
is complete. Furthermore,
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ2_HTML.gif)
On the other hand, existence of fixed points in partially ordered sets has been considered recently in [9], and some generalizations of the result of [9] are given in [10–15] in a partial ordered metric spaces. Also, in [9], some applications to matrix equations are presented; in [14, 15], some applications to ordinary differential equations are given. Also, we can find some results on partial ordered fuzzy metric spaces and partial ordered uniform spaces in [16–18], respectively.
The aim of this paper is to combine the above ideas, that is, to give some fixed point theorems on ordered partial metric spaces.
2. Main Result
Theorem 2.1.
Let be partially ordered set, and suppose that there is a partial metric
on
such that
is a complete partial metric space. Suppose
is a continuous and nondecreasing mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ3_HTML.gif)
for all with
, where
is a continuous, nondecreasing function such that
is convergent for each
. If there exists an
with
, then there exists
such that
. Moreover,
.
Proof.
From the conditions on , it is clear that
for
and
. If
, then the proof is finished, so suppose
. Now, let
for
. If
for some
, then it is clear that
is a fixed point of
. Thus, assume
for all
. Notice that since
and
is nondecreasing, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ4_HTML.gif)
Now, since , we can use the inequality (2.1) for these points, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ5_HTML.gif)
since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ6_HTML.gif)
and is nondecreasing. Now, if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ7_HTML.gif)
for some , then from (2.3) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ8_HTML.gif)
which is a contradiction since . Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ9_HTML.gif)
for all . Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ10_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ11_HTML.gif)
On the other hand, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ12_HTML.gif)
then from (2.9) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ13_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ14_HTML.gif)
This shows that . Now, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ15_HTML.gif)
Since is convergent for each
, then
is a Cauchy sequence in the metric space
. Since
is complete, then, from Lemma 1.1, the sequence
converges in the metric space
, say
. Again, from Lemma 1.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ16_HTML.gif)
Moreover, since is a Cauchy sequence in the metric space
, we have
, and, from (2.11), we have
, thus, from definition
, we have
. Therefore, from (2.14), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ17_HTML.gif)
Now, we claim that . Suppose
. Since
is continuous, then, given
, there exists
such that
. Since
, then there exists
such that
for all
. Therefore, we have
for all
. Thus,
, and so
for all
. This shows that
. Now, we use the inequality (2.1) for
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ18_HTML.gif)
Therefore, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ19_HTML.gif)
and letting , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ20_HTML.gif)
which is a contradiction since . Thus,
, and so
.
In the following theorem, we remove the continuity of . Also, The contractive condition (2.1) does not have to be satisfied for
, but we add a condition on
.
Theorem 2.2.
Let be a partially ordered set, and suppose that there is a partial metric
on
such that
is a complete partial metric space. Suppose
is a nondecreasing mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ21_HTML.gif)
for all with
(i.e.,
and
), where
is a continuous, nondecreasing function such that
is convergent for each
. Also, the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ22_HTML.gif)
holds. If there exists an with
, then there exists
such that
. Moreover,
.
Proof.
As in the proof of Theorem 2.1, we can construct a sequence in
by
for
. Also, we can assume that the consecutive terms of
are different. Otherwise we are finished. Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ23_HTML.gif)
Again, as in the proof of Theorem 2.1, we can show that is a Cauchy sequence in the metric space
, and, therefore, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ24_HTML.gif)
Now, we claim that . Suppose
. Since the condition (2.20) is satisfied, then we can use (2.19) for
. Therefore, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ25_HTML.gif)
using the continuity of and letting
, we have
. Therefore, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ26_HTML.gif)
which is a contradiction. Thus, , and so
.
Example 2.3.
Let and
, then it is clear that
is a complete partial metric space. We can define a partial order on
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ27_HTML.gif)
Let ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ28_HTML.gif)
and ,
. Therefore,
is continuous and nondecreasing. Again we can show by induction that
, and so we have
that is convergent. Also,
is nondecreasing with respect to
, and for
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ29_HTML.gif)
that is, the condition (2.19) of Theorem 2.2 is satisfied. Also, it is clear that the condition (2.20) is satisfied, and for , we have
. Therefore, all conditions of Theorem 2.2 are satisfied, and so
has a fixed point in
. Note that if
and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ30_HTML.gif)
This shows that the contractive condition of Theorem 1 of [4] is not satisfied.
Theorem 2.4.
If one uses the following condition instead of (2.1) in Theorem 2.1, one has the same result.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ31_HTML.gif)
for all with
.
In what follows, we give a sufficient condition for the uniqueness of the fixed point in Theorem 2.4, this condition is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ32_HTML.gif)
In [15], it was proved that condition (2.30) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ33_HTML.gif)
Theorem 2.5.
Adding condition (2.31) to the hypotheses of Theorem 2.4, one obtains uniqueness of the fixed point of .
Proof.
Suppose that there exists are different fixed points of
, then
. Now, we consider the following two cases.
-
(i)
If
and
are comparable, then
and
are comparable for
. Therefore, we can use the condition (2.1), then we have
(2.32)
which is a contradiction.
-
(ii)
If
and
are not comparable, then there exists
comparable to
and
. Since
is nondecreasing, then
is comparable to
and
for
. Moreover,
(2.33)
Now, if for some
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ36_HTML.gif)
which is a contradiction. Thus, for all
, and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ37_HTML.gif)
This shows that is a nonnegative and nondecreasing sequence and so has a limit, say
. From the last inequality, we can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ38_HTML.gif)
hence . Similarly, it can be proven that,
. Finally,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F508730/MediaObjects/13663_2010_Article_1407_Equ39_HTML.gif)
and taking limit , we have
. This contradicts
.
Consequently, has no two fixed points.
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Altun, I., Erduran, A. Fixed Point Theorems for Monotone Mappings on Partial Metric Spaces. Fixed Point Theory Appl 2011, 508730 (2011). https://doi.org/10.1155/2011/508730
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DOI: https://doi.org/10.1155/2011/508730