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Some coupled fixed-point theorems in two quasi-partial metric spaces
Fixed Point Theory and Applications volume 2014, Article number: 19 (2014)
Abstract
The purpose of this paper is to prove some new coupled common fixed-point theorems for mappings defined on a set equipped with two quasi-partial metrics. We also provide illustrative examples in support of our new results.
MSC:47H10, 54H25.
1 Introduction and preliminaries
In 1994, Matthews [1] introduced the notion of partial metric spaces as follows.
Definition 1.1 [1]
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.
In [1], Matthews extended the Banach contraction principle from metric spaces to partial metric spaces. Based on the notion of partial metric spaces, several authors (for example, [2–32]) obtained some fixed-point results for mappings satisfying different contractive conditions. Very recently, Haghi et al. [33] showed in their interesting paper that some fixed-point theorems in partial metric spaces can be obtained from metric spaces.
Karapınar et al. [34] introduced the concept of quasi-partial metric spaces and studied some fixed-point problems on quasi-partial metric spaces. The notion of a quasi-partial metric space is defined as follows.
Definition 1.2 [34]
A quasi-partial metric on nonempty set X is a function which satisfies:
(QPM1) If , then ,
(QPM2) ,
(QPM3) , and
(QPM4)
for all .
A quasi-partial metric space is a pair such that X is a nonempty set and q is a quasi-partial metric on X.
Let q be a quasi-partial metric on set X. Then
is a metric on X.
Definition 1.3 [34]
Let be a quasi-partial metric space. Then
-
(i)
A sequence converges to a point if and only if
-
(ii)
A sequence is called a Cauchy sequence if and exist (and are finite).
-
(iii)
The quasi-partial metric space is said to be complete if every Cauchy sequence in X converges, with respect to , to a point such that
Bhaskar and Lakshmikantham [35] introduced the concept of a coupled fixed point and studied some nice coupled fixed-point theorems. Later, Lakshmikantham and Ćirić [36] introduced the notion of a coupled coincidence point of mappings. For some works on a coupled fixed point, we refer the reader to [37–62].
Definition 1.4 [35]
Let X be a nonempty set. We call an element a coupled fixed point of the mapping if and .
Definition 1.5 [36]
An element is called
-
(i)
a coupled coincidence point of the mapping and if and ; in this case is called coupled point of coincidence of mappings F and g;
-
(ii)
a common coupled fixed point of mappings and if and ;
-
(iii)
a common coupled fixed point of mappings and if and .
Abbas et al. [37] introduced the concept of w-compatible mappings as follows.
Definition 1.6 [37]
Let X be a nonempty set. We say that the mappings and are w-compatible if whenever and .
Very recently, Shatanawi and Pitea [38] obtained some common coupled fixed-point results for a pair of mappings in quasi-partial metric space.
Theorem 1.1 (see [[38], Theorem 2.1])
Let be a quasi-partial metric space, and be two mappings. Suppose that there exist , , and in with such that the condition
holds for all . Also, suppose we have the following hypotheses:
-
(i)
.
-
(ii)
is a complete subspace of X with respect to the quasi-partial metric q.
Then the mappings F and g have a coincidence point satisfying and .
Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point of the form .
The aim of this article is to prove some new coupled common fixed-point theorems for mappings defined on a set equipped with two quasi-partial metrics.
The following lemma is crucial in our work.
Lemma 1.1 [38]
Let be a quasi-partial metric space. Then the following statements hold true:
-
(i)
If , then .
-
(ii)
If , then and .
In this manuscript, we generalize, improve, enrich, and extend the above coupled common fixed-point results. We also state some examples to illustrate our results. This paper can be considered as a continuation of the remarkable works of Aydi [12], Karapınar et al. [34], and Shatanawi and Pitea [38].
2 Main results
Now we shall prove our main results.
Theorem 2.1 Let and be two quasi-metrics on X such that , for all , and let , be two mappings. Suppose that there exist , , , , and in with
such that the condition
holds for all . Also, suppose we have the following hypotheses:
-
(i)
.
-
(ii)
is a complete subspace of X with respect to the quasi-partial metric .
Then the mappings F and g have a coincidence point satisfying .
Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point of the form .
Proof Let . Since , we can choose such that and . Similarly, we can choose such that and . Continuing in this way we construct two sequences and in X such that
It follows from (2.2) and (QPM4) that
which implies that
Put . Obviously, . By repetition of the above inequality (2.4) n times, we get
Next, we shall prove that and are Cauchy sequences in .
In fact, for each , , from (QPM4) and (2.5) we have
This implies that
and so
By similar arguments as above, we can show that
Hence and are Cauchy sequences in . Since is complete, there exist such that and converge to gx and gy with respect to , that is,
and
Combining (2.7)-(2.10), we have
and
By (QPM4) we obtain
Letting in the above inequalities and using (2.11), we have
That is,
Similarly, using (2.12) we have
Now we prove that and . In fact, it follows from (2.2) and (2.3) that
Letting in the above inequality, using (2.11)-(2.14), we obtain
By (2.1) we have . Hence, it follows from (2.15) that . By Lemma 1.1, we get and . Hence, is a coupled point of coincidence of mappings F and g.
Next, we will show that the coupled point of coincidence is unique. Suppose that with and . Using (2.2), (2.11), (2.12), and (QPM3), we obtain
This implies that
Similarly, we have
Substituting (2.17) into (2.16), we obtain
Since , from (2.18), we must have . By Lemma 1.1, we get and , which implies the uniqueness of the coupled point of coincidence of F and g, that is, .
Next, we will show that . In fact, from (2.2), (2.11), and (2.12) we have
Since , we have . By Lemma 1.1, we get .
Finally, assume that g and F are w-compatible. Let , then we have , so that
Consequently, is a coupled coincidence point of F and g, and therefore is a coupled point of coincidence of F and g, and by its uniqueness, we get . Thus, we obtain . Therefore, is the unique common coupled fixed point of F and g. This completes the proof of Theorem 2.1. □
In Theorem 2.1, if we take for all , then we get the following.
Corollary 2.1 Let be a quasi-partial metric space, and be two mappings. Suppose that there exist , , , and in with such that the condition
holds for all . Also, suppose we have the following hypotheses:
-
(i)
.
-
(ii)
is a complete subspace of X with respect to the quasi-partial metric q.
Then the mappings F and g have a coincidence point satisfying .
Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point of the form .
Remark 2.1 Corollary 2.1 improve and extend Theorem 2.1 of Shatanawi and Pitea [38]; the contractive condition defined by (1.1) is replaced by the new contractive condition defined by (2.23).
Corollary 2.2 Let and be two quasi-metrics on X such that , for all , and , be two mappings. Suppose that there exist () with
such that the condition
holds for all . Also, suppose we have the following hypotheses:
-
(i)
.
-
(ii)
is a complete subspace of X with respect to the quasi-partial metric .
Then the mappings F and g have a coincidence point satisfying .
Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point of the form .
Proof Given . It follows from (2.23) that
and
Adding inequality (2.24) to inequality (2.25), we get
Therefore, the result follows from Theorem 2.1. □
Remark 2.2 If we take for all and , then Corollary 2.2 is reduced to Corollary 2.1 of Shatanawi and Pitea [38].
Corollary 2.3 Let and be two quasi-metrics on X such that , for all , and , be two mappings. Suppose that there exists such that the condition
holds for all . Also, suppose we have the following hypotheses:
-
(i)
.
-
(ii)
is a complete subspace of X with respect to the quasi-partial metric .
Then the mappings F and g have a coincidence point satisfying .
Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point of the form .
Remark 2.3 If we take for all , then Corollary 2.3 is reduced to Corollary 2.2 of Shatanawi and Pitea [38].
Corollary 2.4 Let and be two quasi-metrics on X such that , for all , and , be two mappings. Suppose that there exists such that the condition
holds for all . Also, suppose we have the following hypotheses:
-
(i)
.
-
(ii)
is a complete subspace of X with respect to the quasi-partial metric .
Then the mappings F and g have a coincidence point satisfying .
Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point of the form .
Remark 2.4 If we take for all , then Corollary 2.4 is reduced to Corollary 2.3 of Shatanawi and Pitea [38].
Corollary 2.5 Let and be two quasi-metrics on X such that , for all , and , be two mappings. Suppose that there exists such that the condition
holds for all . Also, suppose we have the following hypotheses:
-
(i)
.
-
(ii)
is a complete subspace of X with respect to the quasi-partial metric .
Then the mappings F and g have a coincidence point satisfying .
Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point of the form .
Remark 2.5 If we take for all , then Corollary 2.5 is reduced to Corollary 2.4 of Shatanawi and Pitea [38].
Corollary 2.6 Let and be two quasi-metrics on X such that , for all , and , be two mappings. Suppose that there exists such that the condition
holds for all . Also, suppose we have the following hypotheses:
-
(i)
.
-
(ii)
is a complete subspace of X with respect to the quasi-partial metric .
Then the mappings F and g have a coincidence point satisfying .
Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point of the form .
Corollary 2.7 Let and be two quasi-metrics on X such that , for all , and , be two mappings. Suppose that there exists such that the condition
holds for all . Also, suppose we have the following hypotheses:
-
(i)
.
-
(ii)
is a complete subspace of X with respect to the quasi-partial metric .
Then the mappings F and g have a coincidence point satisfying .
Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point of the form .
Let (the identity mapping) in Theorem 2.1 and Corollaries 2.1-2.7. Then we have the following results.
Corollary 2.8 Let and be two quasi-metrics on X such that , for all , and be a mapping. Suppose that there exist , , , , and in with such that the condition
holds for all . If is a complete quasi-partial metric space, then the mapping F has a unique coupled fixed point of the form .
Corollary 2.9 Let be a complete quasi-partial metric space, be a mapping. Suppose that there exist , , , , and in with such that the condition
holds for all . Then F has a unique coupled fixed point of the form .
Remark 2.6 Corollary 2.9 improve and extend Corollary 2.5 of Shatanawi and Pitea [38], the contractive condition is replaced by the new contractive condition defined by (2.35).
Corollary 2.10 Let and be two quasi-metrics on X such that , for all , and be a mapping. Suppose that there exist () with
such that the condition
holds for all . If is a complete quasi-partial metric space. Then the mapping F has a unique coupled fixed point of the form .
Remark 2.7
-
(1)
If we take for all and , then Corollary 2.10 is reduced to Corollary 2.6 of Shatanawi and Pitea [38].
-
(2)
If we take for all and (), then Corollary 2.10 extends Theorem 2.1 of Aydi [12] on the class of quasi-partial metric spaces.
-
(3)
If we take for all , and (), then Corollary 2.10 extends the Corollary 2.2 of Aydi [12] on the class of quasi-partial metric spaces.
-
(4)
If we take for all and (), then Corollary 2.10 extends Theorem 2.4 of Aydi [12] on the class of quasi-partial metric spaces.
-
(5)
If we take for all and (), then Corollary 2.10 extends Theorem 2.5 of Aydi [12] on the class of quasi-partial metric spaces.
-
(6)
If we take for all , and (), then Corollary 2.10 extends Corollary 2.6 of Aydi [12] on the class of quasi-partial metric spaces.
-
(7)
If we take for all , and (), then Corollary 2.10 extends Corollary 2.7 of Aydi [12] on the class of quasi-partial metric spaces.
Corollary 2.11 Let and be two quasi-metrics on X such that , for all , and be a mapping. Suppose that there exists such that the condition
holds for all . If is a complete quasi-partial metric space. Then the mapping F has a unique coupled fixed point of the form .
Remark 2.8 If we take for all , then Corollary 2.11 is reduced to Corollary 2.7 of Shatanawi and Pitea [38].
Corollary 2.12 Let and be two quasi-metrics on X such that , for all , and be a mapping. Suppose that there exists such that the condition
holds for all . If is a complete quasi-partial metric space, then the mapping F has a unique coupled fixed point of the form .
Remark 2.9 If we take for all , then Corollary 2.12 is reduced to Corollary 2.8 of Shatanawi and Pitea [38].
Corollary 2.13 Let and be two quasi-metrics on X such that , for all , and be a mapping. Suppose that there exists such that the condition
holds for all . If is a complete quasi-partial metric space, then the mapping F has a unique coupled fixed point of the form .
Remark 2.10 If we take for all , then Corollary 2.13 is reduced to Corollary 2.9 of Shatanawi and Pitea [38].
Corollary 2.14 Let and be two quasi-metrics on X such that , for all , and be a mapping. Suppose that there exists such that the condition
holds for all . If is a complete quasi-partial metric space, then the mapping F has a unique coupled fixed point of the form .
Corollary 2.15 Let and be two quasi-metrics on X such that , for all , and be a mapping. Suppose that there exists such that the condition
holds for all . If is a complete quasi-partial metric space, then the mapping F has a unique coupled fixed point of the form .
Now, we introduce an example to support our results.
Example 2.1 Let , and two quasi-partial metrics , on X be given as
for all . Also, define and as
for all . Then
-
(1)
is a complete quasi-partial metric space.
-
(2)
.
-
(3)
F and g is w-compatible.
-
(4)
For any , we have
Proof The proofs of (1), (2), and (3) are clear. Next we show that (4). In fact, for , we have
Thus, F and g satisfy all the hypotheses of Corollary 2.3. So, F and g have a unique common coupled fixed point. Here is the unique common coupled fixed point of F and g. □
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (11271105, 11361070), the Natural Science Foundation of Zhejiang Province (Y6110287, LY12A01030), and the Natural Science Foundation of Shandong Province (ZR2013AL015).
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Gu, F., Wang, L. Some coupled fixed-point theorems in two quasi-partial metric spaces. Fixed Point Theory Appl 2014, 19 (2014). https://doi.org/10.1186/1687-1812-2014-19
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DOI: https://doi.org/10.1186/1687-1812-2014-19