Abstract
In this work, we present a notion of an -closed set and prove the existence of a coupled coincidence point theorem for a pair of mappings with φ-contraction mappings in partially ordered metric spaces without H-increasing property of F and mixed monotone property of H. We give some examples of a nonlinear contraction mapping, which is not applied to the existence of coupled coincidence point by H using the mixed monotone property and H-increasing property of F. We also show the uniqueness of a coupled coincidence point of the given mappings. Further, we apply our results to the existence and uniqueness of a coupled coincidence point of the given mappings in partially ordered G-metric spaces with H-increasing property of F and mixed monotone property of H. These results generalize some recent results in the literature.
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1 Introduction
The existence of a fixed point for the contraction type of mappings in partially ordered metric spaces has been first studied by Ran and Reurings [1]. Moreover, they established some new results and presented some applications to matrix equations. In 1987, Guo and Lakshmikantham [2] introduced the concept of a coupled fixed point. Later, Bhaskar and Lakshmikantham [3] introduced the concept of the mixed monotone property for contractive operators. They also showed some applications on the existence and uniqueness of the coupled fixed point theorems for mappings which satisfy the mixed monotone property in partially ordered metric spaces. Lakshimikantham and Ćirić [4] extended the results in [3] by defining the mixed g-monotonicity and studied the existence and uniqueness of coupled coincidence point for such a mappings which satisfy the mixed monotone property in partially ordered metric spaces. As a continuation of this work, many authors conducted research on the coupled fixed point theory and coupled coincidence point theory in partially ordered metric spaces and different spaces. We refer the reader for example to [4–31].
In 2006, Mustafa and Sims [32] introduced the notion of a G-metric spaces as a generalization of the concept of a metric spaces and proved the analog of the Banach contraction mapping principle in the context of G-metric spaces. For examples of extensions and applications of these works see [33–46]. In 2011, Choudhury and Maity [47] proved the existence of a coupled fixed point theorem of nonlinear contraction mappings with mixed monotone property in partially ordered G-metric spaces. Aydi et al. [48] established coupled coincidence and coupled common fixed point results for a mixed g-monotone mapping satisfying nonlinear contractions in partially ordered G-metric spaces. They generalized the results obtained by Choudhury and Maily [47]. Later, Karapınar et al. [49] extended the results of coupled coincidence and coupled common fixed point theorem for a mixed g-monotone mapping obtained by Aydi et al. [48]. Many authors have studied coupled coincidence point and coupled common fixed point results for a mixed g-monotone mapping satisfying nonlinear contractions in partially ordered G-metric spaces (see, for example, [47–64]).
One of the interesting ways to developed coupled fixed point theory is to consider the mapping without the mixed monotone property. Recently, Sintunavarat et al. [29, 30] proved some coupled fixed point theorems for nonlinear contractions without mixed monotone property which extended the results of Bhaskar and Lakshmikantham [3] by using the concept of an F-invariant set due to Samet and Vetro [28]. Later, Batra and Vashistha [6] introduced an -invariant set which is a generalization of an F-invariant set. Recently, Kutbi et al. [22] introduced the concept of an F-closed set which is weaker than the concept of an F-invariant set and proved some coupled fixed point theorems without the condition of F-invariant set and mixed monotone property. Very recently, Charoensawan and Thangthong [55] generalized and extended the coupled coincidence point theorem of nonlinear contraction mappings in partially ordered G-metric spaces without the mixed g-monotone property by using the concept of -invariant set in partially ordered G-metric spaces which are generalizations of the results of Aydi et al. [48]. In 2014, Hussain et al. [16] presented the new concept of generalized compatibility of a pair of mappings and proved some coupled coincidence point results of such a mapping without the mixed G-monotone property of F in partially ordered metric spaces which generalized some recent comparable results in the literature.
In this work, we introduce the concept of -closed set and the notion of generalized compatibility of a pair of mapping in the setting of G-metric spaces. We also obtain a coupled coincidence point theorem for a pair with φ-contraction mappings in partially ordered metric spaces without H-increasing property of F and mixed monotone property of H. Our theorem generalizes and extends the very recent results obtained by Hussain et al. [16] and Karapınar et al. [49].
2 Preliminaries
In this section, we give some definitions, propositions, examples and remarks which are useful for main results in our paper. Throughout this paper, denotes a partially ordered set with the partial order ⪯. By , we mean . Let be a partially ordered set, the partial order ⪯2 for the product set defined in the following way, for all :
where is one-one.
We say that is comparable to if either or .
Definition 2.1 [32]
Let X be a nonempty set and be a function satisfying the following properties:
-
(G1) if .
-
(G2) for all with .
-
(G3) for all with .
-
(G4) (symmetry in all three variables).
-
(G5) for all (rectangle inequality).
Then the function G is called a generalized metric, or more specially, a G-metric on X, and the pair is called a G-metric space.
Example 2.2 Let be a metric space. The function , defined by , for all , is a G-metric on X.
Definition 2.3 [32]
Let be a G-metric space, and let be a sequence of points of X. We say that is G-convergent to if , that is, for any , there exists such that , for all . We call x the limit of the sequence and write or .
Proposition 2.4 [32]
Let be a G-metric space, the following are equivalent:
-
(1)
is G-convergent to x.
-
(2)
as .
-
(3)
as .
-
(4)
as .
Definition 2.5 [32]
Let be a G-metric space. A sequence is called a G-Cauchy sequence if, for any , there exists such that , for all . That is, as .
Proposition 2.6 [32]
Let be a G-metric space, the following are equivalent:
-
(1)
the sequence is G-Cauchy;
-
(2)
for any , there exists such that , for all .
Proposition 2.7 [32]
Let be a G-metric space. A mapping is G-continuous at if and only if it is G-sequentially continuous at x, that is, whenever is G-convergent to x, is G-convergent to .
Definition 2.8 [32]
A G-metric space is called G-complete if every G-Cauchy sequence is G-convergent in .
Definition 2.9 [47]
Let be a G-metric space. A mapping is said to be continuous if for any two G-convergent sequences and converging to x and y, respectively, is G-convergent to .
In 2009, Lakshmikantham and Ćirić [4] introduced the concept of a mixed g-monotone mapping and a coupled coincidence point as follows.
Definition 2.10 [4]
Let be a partially ordered set and and . We say F has the mixed g-monotone property if for any
and
Definition 2.11 [4]
An element is called a coupled coincidence point of mappings , and if and .
Definition 2.12 [4]
Let X be a nonempty set and and . We say F and g are commutative if for all .
Let Φ denote the set of functions satisfying
-
1.
for all ,
-
2.
for all .
In 2012, Karapınar et al. [49] proved the following theorems.
Theorem 2.13 [49]
Let be a partially ordered set and G be a G-metric on X such that is a complete G-metric space. Suppose that there exist , , and such that
for all for which and . Suppose also that F is continuous and has the mixed g-monotone property, , and g is continuous and commutes with F. If there exists such that
then there exists such that and , that is, F and g have a coupled coincidence point.
Hussain et al. [16] introduced the concept of H-increasing and generalized compatible as follows.
Definition 2.14 [16]
Suppose that are two mappings. F is said to be H-increasing with respect to ⪯ if for all , with , we have .
Definition 2.15 [16]
An element is called a coupled coincidence point of mappings if and .
Definition 2.16 [16]
Let be a metric space and . We say that the pair is generalized compatible if
whenever and are sequences in X such that
Definition 2.17 [16]
Let be two maps. We say that the pair is commuting if
It is easy to see that a commuting pair is generalized compatible but the converse is not true in general.
Let ϒ denote the set of all functions such that:
-
(i)
ϕ is continuous and increasing,
-
(ii)
if and only if ,
-
(iii)
, for all .
Let Ψ be the set of all functions such that for all and .
Recently, Hussain et al. [16] proved the coupled coincidence point for such mappings involving -contractive condition as follows.
Theorem 2.18 [16]
Let be a partially ordered set and M be a nonempty subset of and let there exists d, a metric on X such that is a complete metric space. Assume that are two generalized compatible mappings such that F is H-increasing with respect to ⪯, H is continuous and has the mixed monotone property. Suppose that for any , there exist such that and . Suppose that there exist and such that the following holds:
for all with and .
Also suppose that either
-
(a)
F is continuous or
-
(b)
X has the following properties: for any two sequences and with
-
(i)
if a non-decreasing sequence , then for all n,
-
(ii)
if a non-increasing sequence , then for all n.
-
(i)
If there exists with
then there exists such that and , that is, F and H have a coupled coincidence point.
In order to remove the mixed monotone property, Batra and Vashistha [6] introduced the following property.
Definition 2.19 [6]
Let be a metric space and , be given mappings. Let M be a nonempty subset of . We say that M is an -invariant subset of if and only if, for all ,
-
(i)
.
-
(ii)
.
Kutbi et al. [22] introduced the notion of F-closed set which extended the notion of F-invariant set as follows.
Definition 2.20 [22]
Let be a mapping, and let M be a subset of . We say that M is an F-closed subset of if, for all ,
Inspired by above definitions, we give the notion of a -closed set which is useful for our main results.
Definition 2.21 Let be two mappings and let M be a subset of . We say that M is an -closed subset of if, for all ,
Definition 2.22 Let be a mapping and M be a subset of . We say that M satisfies the transitive property if and only if for all ,
Definition 2.23 Let be two mappings. We say that the pair is generalized compatible if and are sequences in X such that for some
imply
Remark The set is trivially -closed set, which satisfies the transitive property.
Example 2.24 Let be a G-metric space endowed with a partial order ⪯. Let are two generalized compatible mappings such that F is H-increasing with respect to ⪯, H is continuous and has the mixed monotone property. Define a subset by
Let . It is easy to see that, since F is H-increasing with respect to ⪯, we have and , this implies that
Then M is -closed subset of , which satisfies the transitive property.
3 Main results
Now, we state our first result which successively guarantees a coupled coincidence point.
Theorem 3.1 Let be a partially ordered set and G be a G-metric on X such that is a complete G-metric space and M be a nonempty subset of . Assume that are two generalized compatible mappings such that H is continuous and for any , there exists such that and . Suppose that there exists such that the following holds:
for all with .
Suppose also that either
-
(a)
F is continuous;
-
(b)
for any two sequences and with for all
If there exists such that
and M is an -closed, then there exists such that and , that is, F and H have a coupled coincidence point.
Proof Let be such that
From the assumption, there exists such that
Again from the assumption, we can choose such that
By repeating this argument, we can construct two sequences and in X such that
Since
and M is an -closed, we get
Again, using the fact that M is a -closed, we have
Continuing this process, for all we obtain
Let
We can suppose that for all . If not, will be a coupled coincidence point and the proof is finished. From (1), (2), and (3), we have
This implies that
Since for all , it follows that is decreasing sequence. Therefore, there is some such that .
We shall prove that . Assume, to the contrary, that . Then by letting in (6) and using the properties of the map φ, we get
A contradiction, thus , and hence
Next, we prove that and are Cauchy sequences in the G-metric space . Suppose, to the contrary, that at least of and is not Cauchy sequence in . Then there exists an for which we can find subsequences , of and , of , respectively, with such that
Further, corresponding to , we can choose in such a way that it is the smallest integer with and satisfying (8). Then
Using the rectangle inequality and (9), we have
Letting and using (7), we obtain
Again, by the rectangle inequality, we have
Using the fact that for any , we obtain
Since and using (3), we have
and
From the fact that M is an -closed set which satisfies the transitive property, we have
By this process, we can get
Now, using (1), we have
From (12) and (13), it follows that
Letting in (14) and using (7) and (11) and for all , we have
which is a contradiction. This shows that and are Cauchy sequences in the G-metric space . Since is complete and from (2), and are G-convergent, there exist such that
Since the pair satisfies the generalized compatibility, from (15), we have
Suppose that assumption (a) holds. For all , from (16), we have
and
We have
Therefore, is a coupled coincidence point of F and H.
Suppose now assumption (b) holds. Since converges to x, converges to y, the pair satisfies the generalized compatibility, H is continuous and by (15), we have
and
From (3), (17), (18), and assumption (b), for all , we have
Then, by (1), (2), (19), and the triangle inequality, we have
Letting now in the above inequality and using property of φ such that , we have
which implies that and . □
Next, we give an example to validate Theorem 3.1.
Example 3.2 Let , , and be defined by
and
Clearly, H does not satisfy the mixed monotone property and if , , consider
Then F is not H-increasing.
Now, we prove that for any , there exist such that and . It is easy to see that we have the following cases.
Case 1: If , then we have .
Case 2: If , then and we have
and
Case 3: If , then and we have
and
Now, we prove that the pair satisfies the generalized compatibility hypothesis. Let and be two sequences in X such that
Then we must have and it is easy to prove that
Now, for all with and let be a function defined by , we have
Therefore, condition (1) is satisfied. Thus, all the requirements of Theorem 3.1 are satisfied and is a coupled coincidence point of F and G.
Next, we show the uniqueness of the coupled coincidence point and coupled fixed point of F and G.
Theorem 3.3 In addition to the hypotheses of Theorem 3.1, suppose that for every , there exists such that
Then F and H have a unique coupled coincidence point. Moreover, if the pair is commuting, then F and H have a unique coupled fixed point, that is, there exists a unique such that
Proof From Theorem 3.1, we know that F and H have a coupled coincidence point. Suppose that , are coupled coincidence points of F and H, that is,
Now, we show that and . By the hypothesis there exists such that
We put and and define two sequences and as follows:
Since M is -closed and
we have
From , if we use again the property of -closedness, then
By repeating this process, we get
Using (1), (20), and (21), for all , we have
Using property that and repeating this process, for all , we get
From and , it follows that for each . Therefore, from (23), we have
This implies that
Similarly, we show that
From (25) and (26), we have
Now let the pair be commuting, we shall prove that F and H have a unique coupled fixed point. Since
and F and H commutes, we have
Denote and . Then, by (28) and (29), one gets
Therefore, is a coupled coincidence point of F and H. Then, by (27) with and , it follows that
Thus, is a coupled fixed point of H, by (28), is also a coupled fixed point of F. To prove the uniqueness, assume is another coupled fixed point of F and H. Then, by (27) and (31), we have
□
Next, we give some applications of our results to coupled coincidence point theorems.
Corollary 3.4 Let be a partially ordered set and M be a nonempty subset of and let there exists G be a G-metric on X such that is a complete G-metric space. Assume that are two generalized compatible mappings such that F is H-increasing with respect to ⪯, H is continuous and has the mixed monotone property. Suppose that for any , there exist such that and . Suppose that there exists such that the following holds:
for all with and .
Also suppose that either
-
(a)
F is continuous or
-
(b)
X has the following properties: for any two sequences and with
-
(i)
if a non-decreasing sequence , then for all n,
-
(ii)
if a non-increasing sequence , then for all n.
-
(i)
If there exists with
Then there exists such that and , that is, F and H have a coupled coincidence point.
Proof We define the subset by
From Example 2.24, M is a -closed set which satisfies the transitive property. For all with and , we have . By (1), we get
Since with
We have
Assumption (a) holds, and F is continuous. By assumption (a) of Theorem 3.1, we have and .
Next, assumption (b) holds; since F is H-increasing with respect to ⪯, using (32) and (2), we have
Therefore
From H is continuous and by (15), we have
For any two sequences and such that is a non-decreasing sequence in X with and is a non-increasing sequence in X with . Using assumption (b), we have
Since H has the mixed monotone property, we have
Therefore, we have
and so assumption (b) of Theorem 3.1 holds. Now, since all the hypotheses of Theorem 3.1 hold, then F and H have a coupled coincidence point. The proof is completed. □
Corollary 3.5 In addition to the hypotheses of Corollary 3.4, suppose that for every , there exists which is comparable to and . Then F and H have a unique coupled coincidence point.
Proof We define the subset by
From Example 2.24, M is an -closed set which satisfies the transitive property. Thus, the proof of the existence of a coupled coincidence point is straightforward by following the same lines as in the proof of Corollary 3.4.
Next, we show the uniqueness of a coupled coincidence point of F and H.
Since for all , there exists such that
and
we can conclude that
Therefore, since all the hypotheses of Theorem 3.3 hold, F and H have a unique coupled coincidence point. The proof is completed. □
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Acknowledgements
This research was supported by Chiang Mai University and the authors would like to express sincere appreciation to Prof. Suthep Suantai for very helpful suggestions and many kind comments.
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Na Nan, N., Charoensawan, P. -Closed set and coupled coincidence point theorems for a generalized compatible in partially G-metric spaces. J Inequal Appl 2014, 342 (2014). https://doi.org/10.1186/1029-242X-2014-342
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DOI: https://doi.org/10.1186/1029-242X-2014-342