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Some new common coupled fixed point results in two generalized metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 181 (2013)
Abstract
The purpose of this paper is to extend some recent common coupled fixed point theorems in two G-metric spaces in an essentially different and more natural way. We also provide illustrative examples in support of our new results.
MSC:47H10, 54H25.
1 Introduction and preliminaries
In 2006, Mustafa and Sims [1] introduced a new structure of generalized metric spaces, which are called G-metric spaces, as follows.
Definition 1.1 [1]
Let X be a nonempty set, and let be a function satisfying the following axioms:
-
(G1)
=0 if ;
-
(G2)
for all with ;
-
(G3)
for all with z≠ y;
-
(G4)
(symmetry in all three variables);
-
(G5)
for all (rectangle inequality).
Then the function G is called a generalized metric or a G-metric on X and the pair is called a G-metric space.
It is known that the function on a G-metric space X is jointly continuous in all three of its variables, and if and only if (see [1]).
Based on the notion of generalized metric spaces, Mustafa et al. [1–6] obtained some fixed point results for mappings satisfying different contractive conditions. Chugh et al. [7] obtained some fixed point results for maps satisfying property P in G-metric spaces. Shatanawi [8] obtained some fixed point results for contractive mappings satisfying Φ- maps in G-metric spaces.
In 2009, Abbas and Rhoades [9] initiated the study of common fixed point theory in G-metric spaces. Since then, many common fixed point theorems for certain contractive conditions have been established in G-metric spaces (see [10–19]).
Bhaskar and Lakshmikantham [20] introduced the notion of coupled fixed point and proved some interesting coupled fixed point theorems for mappings satisfying the mixed monotone property. Later, Lakshmikantham and Ćirić [21] introduced the concept of mixed g-monotone mapping and proved coupled coincidence and coupled common fixed point theorems that extend theorems due to Bhaskar and Lakshmikantham [20].
In [22, 23], authors established coupled fixed point theorems in cone metric spaces. In 2011, Shatanawi [24] obtained some coupled fixed point results in G-metric spaces. Recently, in [25, 26] authors established some coupled fixed point and common coupled fixed point results in two G-metric spaces. Recently, coupled fixed point and common coupled fixed point problems have also been considered in partially ordered G-metric spaces (see [27–38]).
The aim of this article is to prove some new common coupled fixed point theorems for mappings defined on a set equipped with two generalized metrics.
First, we present some known definitions and propositions.
Definition 1.2 [1]
Let be a G-metric space, be a sequence. Then the sequence is called:
-
(i)
a G-Cauchy sequence if, for any , there is an (the set of natural numbers) such that for all , ;
-
(ii)
a G-convergent sequence if, for any , there are an and an such that for all , .
A G-metric space is said to be G-complete if every G-Cauchy sequence in is G-convergent in X. It is known that is G-convergent to if and only if as .
Proposition 1.3 [1]
Let be a G-metric space. Then the following are equivalent:
-
(1)
is G-convergent to x.
-
(2)
as .
-
(3)
as .
-
(4)
as .
Proposition 1.4 [1]
Let be a G-metric space. Then, for any , we have .
Definition 1.5 [20]
An element is called:
-
(C1) a coupled fixed point of the mapping if and ;
-
(C2) a coupled coincidence point of mappings and if and , and in this case, is called a coupled point of coincidence;
-
(C3) a common coupled fixed point of mappings and if and .
Definition 1.6 [25]
Mappings and are called:
-
(W1) w-compatible if whenever and ;
-
(W2) -compatible if whenever .
Recently, Abbas, Khan and Nazir [25] extended some recent results of Abbas et al. [22] and Sabetghadam et al. [23] to the setting of two generalized metric spaces.
Theorem 1.7 (see [[25], Theorem 2.1])
Let and be two G-metrics on X such that for all , and let , be two mappings satisfying
for all , where , for and . If and is a -complete subspace of X, and F and g are -compatible, then F and g have a unique common coupled fixed point.
Theorem 1.8 (see [[25], Theorem 2.6])
Let and be two G-metrics on X such that for all , and let , be two mappings satisfying
for all , where . If and is a -complete subspace of X, and F and g are -compatible, then F and g have a unique common coupled fixed point.
In this manuscript, we generalize, improve, enrich and extend the above coupled fixed point results. It is worth mentioning that our results do not rely on the continuity of mappings involved therein. We also state some examples to illustrate our results. This paper can be considered as a continuation of the remarkable works of Abbas et al. [22, 23] and Sabetghadam et al. [25].
2 Common coupled fixed points
We begin with an example to illustrate the weakness of Theorem 1.8 above.
Example 2.1 Let . Define by
for all . Then and are two G-metric spaces. Define a map by and for all . For and , we have
and
Then it is easy to see that there is no such that
for all . Thus, Theorem 1.8 cannot be applied to this example. However, it is easy to see that is the unique common coincidence point of F and g. In fact, .
Now we shall prove our main results.
Theorem 2.2 Let and be two G-metrics on X such that for all , and let , be two mappings satisfying
for all , where , for and
If and is a -complete subspace of X, and F and g are -compatible, then F and g have a unique common coupled fixed point.
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 (2.1), (2.3), (G5) and Proposition 1.4 that
which implies that
Similarly, we can prove that
By combining (2.4) and (2.5), we obtain
where . Obviously, .
Repeating the above inequality (2.6) n times, we get
Next, we shall prove that and are G-Cauchy sequences in .
In fact, for each , , from (G5) and (2.7), we have
which implies that
and so
Hence and are -Cauchy sequences in . By -completeness of , there exist such that and converge to gx and gy, respectively.
Now we prove that and . In fact, it follows from (G5) and (2.1) that
Letting in the above inequality, we obtain
By (2.2) we have that . Hence, it follows from (2.8) that , and so . In the same way, we can show that . Hence, is a coupled point of coincidence of mappings F and g.
Next we prove that . In fact, from (2.1) we have
which implies that
In a similar way, we can show that
Since , from (2.9) and (2.10), we must have so that . Thus, is a coupled point of coincidence of mappings F and g.
Now, we claim that a coupled point of coincidence is unique. Suppose that there is another such is a coupled point of coincidence of mappings F and g, then by (2.1) we have
which implies that
In the same way, we can show that
Since , from (2.11) and (2.12), we must have so that . Hence, is a unique coupled point of coincidence of mappings F and g.
Now we show that F and g have a unique common coupled fixed point. For this, let . Then we have . By -compatibility of F and g, we have
Thus, is a coupled point of coincidence of F and g. By the uniqueness of a coupled point of coincidence, we have . Therefore, .
To prove the uniqueness, let with such that
By using (2.1), following the same arguments as in the proof of (2.11) and (2.12), we obtain
and
Since , from (2.13) and (2.14), we must have so that . Thus, F and g have a unique common coupled fixed point. This completes the proof of Theorem 2.1. □
Remark 2.3 Theorem 2.2 improves and extends Theorem 2.1 of Abbas et al. [25], the contractive condition defined by (1.1) is replaced by the new contractive condition defined by (2.1). Theorem 2.1 also improves and extends Theorem 2.4, Corollaries 2.5-2.8 and Theorem 2.11 of Abbas et al. [22]
Now, we introduce an example to support Theorem 2.2.
Example 2.4 Let , and let two G-metrics , on X be given as
for all . Define and as
for all .
Now, for , we have
for all . Thus, (2.1) is satisfied with , , and , where
It is obvious that F is -compatible with g. Hence, all the conditions of Theorem 2.2 are satisfied. Moreover, is the unique common coupled fixed point of F and g.
In Theorem 2.2, take , , , , , and , to obtain Theorem 2.1 of Abbas et al. [25] as the following corollary.
Corollary 2.5 Let and be two G-metrics on X such that for all , and let , be two mappings satisfying
for all , where , for and . If and is a -complete subspace of X, and F and g are -compatible, then F and g have a unique common coupled fixed point.
In Theorem 2.2, take and to obtain the following corollary, which extends and generalizes the corresponding results of [22, 23, 25].
Corollary 2.6 Let and be two G-metrics on X such that for all , and let , be two mappings satisfying
for all , where , for and
If and is a -complete subspace of X, and F and g are -compatible, then F and g have a unique common coupled fixed point.
If we take , , and in Theorem 2.2, then the following corollary, which extends and generalizes the comparable results of [22, 23], is obtained.
Corollary 2.7 Let and be two G-metrics on X such that for all , and let , be two mappings satisfying
for all , where and . If and is a -complete subspace of X, and F and g are -compatible, then F and g have a unique common coupled fixed point.
If we take , , , and in Theorem 2.2, then the following corollary is obtained.
Corollary 2.8 Let and be two G-metrics on X such that for all , and let , be two mappings satisfying
for all , where and . If and is a -complete subspace of X, and F and g are -compatible, then F and g have a unique common coupled fixed point.
If we take , , , , , , and in Theorem 2.2, then the following corollary is obtained.
Corollary 2.9 Let and be two G-metrics on X such that for all , and let , be two mappings satisfying
for all , where , for and
If and is a -complete subspace of X, and F and g are -compatible, then F and g have a unique common coupled fixed point.
If we take , , and in Theorem 2.2, then the following corollary is obtained.
Corollary 2.10 Let and be two G-metrics on X such that for all , and let , be two mappings satisfying
for all , where and . If and is a -complete subspace of X, and F and g are -compatible, then F and g have a unique common coupled fixed point.
Theorem 2.11 Let and be two G-metrics on X such that for all , and let , be two mappings satisfying
for all , where . If and is a -complete subspace of X, and F and g are -compatible, then F and g have a unique common coupled fixed point.
Proof Let . We choose such that and , this can be done in view of . Similarly, we can choose such that and . Continuing this process, we construct two sequences and in X such that and .
By using (2.21) and Proposition 1.4, we obtain
If
then inequality (2.22) becomes
which is a contradiction. So that
This implies that
In a similar way, we obtain
Repeating inequalities (2.23) and (2.24), we obtain
and
By virtue of inequalities (2.25) and (2.26), for each , , repeated use (G5) of a G-metric gives
and
which implies that
Hence and are -Cauchy sequences in . By -completeness of , there exist such that and converge to gx and gy, respectively.
Now, we prove that and . For this, using (G5) and (2.21), we have
On taking the limit as , we obtain that
which implies that , and so . In a similar way, we can show that . Hence, is a coupled point of coincidence of the mappings F and g.
Now, we shall show that . In fact, from (2.21) we have
In the same way, we can show that
If
then by (2.29) we have . This implies that , so that . If
then from (2.30) we obtain , which implies that , so that .
Therefore, is a coupled point of coincidence of mappings F and g.
If there is another such that is a coupled point of coincidence of mappings F and g, then by (2.21) we get
In the same way, we can show that
If
then by (2.31) we have . This implies that , so that . If
then from (2.32) we obtain , which implies that , so that .
Thus, is a unique coupled point of coincidence of mappings F and g.
Now we show that F and g have a unique common coupled fixed point. For this, let . Then we have . By -compatibility of F and g, we have
Thus, is a coupled point of coincidence of F and g. By the uniqueness of a coupled point of coincidence, we have . Therefore, , that is, is the common coupled fixed point of F and g.
To prove the uniqueness, let with such that
By using (2.21), following the same arguments as in the proof of (2.31) and (2.32), we obtain
and
If , then by (2.33) we have , which implies that , so that . If , then from (2.34) we obtain , which implies that , so that .
Thus, is a unique common coupled fixed point of mappings F and g. This completes the proof of Theorem 2.11. □
Remark 2.12 Theorem 2.11 improves and extends Theorem 2.6 of Abbas et al. [25] in the following aspects:
-
(1)
The contractive condition defined by (1.2) is replaced by the new contractive condition defined by (2.21).
-
(2)
The condition is replaced by the new condition .
Corollary 2.13 Let and be two G-metrics on X such that for all , and let , be two mappings satisfying
for all , where . If and is a -complete subspace of X, and F and g are -compatible, then F and g have a unique common coupled fixed point.
Remark 2.14 Corollary 2.13 improves and extends Theorem 2.6 of Abbas et al. [25], the condition is replaced by the new condition .
Next, we introduce two examples to support Corollary 2.13.
Example 2.15 Let us reconsider Example 2.1. For all , we have
Then the statement (2.35) of Corollary 2.13 is satisfied for . Other assumptions of Corollary 2.13 are easy to verify. So, by Corollary 2.13, there exists a unique such that . In fact, it is easy to see that is the unique common coupled fixed point of F and g.
Example 2.16 Let . Define by
for all . Then and are two G-metric spaces. Define a map by and for all . We have
Then the statement (2.35) of Corollary 2.13 is satisfied for . Other assumptions of Corollary 2.13 are easy to verify. So, by Corollary 2.13, there exists a unique such that . In fact, .
Remark 2.17 Theorem 1.8 cannot be applied to Example 2.16 since (1.2) does not hold. In fact, if (1.2) holds for some , then
which is a contradiction.
Corollary 2.18 Let and be two G-metrics on X such that for all , and let , be two mappings satisfying
for all , where . If and is a -complete subspace of X, and F and g are -compatible, then F and g have a unique common coupled fixed point.
Corollary 2.19 Let and be two G-metrics on X such that for all , and let , be two mappings satisfying
for all , where . If and is a -complete subspace of X, and F and g are -compatible, then F and g have a unique common coupled fixed point.
Remark 2.20 Theorem 2.2 and Corollaries 2.5-2.10 improve and extend Theorems 2.2, 2.5, 2.6, Corollary 2.3, 2.7 and 2.8 of Sabetghadam et al. [23].
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Acknowledgements
The author is grateful to the editor and the reviewer for suggestions which improved the contents of the article. This work is supported by the National Natural Science Foundation of China (11271105) and the Natural Science Foundation of Zhejiang Province (Y6110287, LY12A01030).
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Gu, F. Some new common coupled fixed point results in two generalized metric spaces. Fixed Point Theory Appl 2013, 181 (2013). https://doi.org/10.1186/1687-1812-2013-181
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DOI: https://doi.org/10.1186/1687-1812-2013-181