Abstract
In this work, we show the existence of a coupled coincidence point and a coupled common fixed point for a ϕ-contractive mapping in G-metric spaces without the mixed g-monotone property, using the concept of a -invariant set. We also show the uniqueness of a coupled coincidence point and give some examples, which are not applied to the existence of a coupled coincidence point by using the mixed g-monotone property. Further, we apply our results to the existence and uniqueness of a coupled coincidence point of the given mapping in partially ordered G-metric spaces.
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1 Introduction
In 2004, the existence and uniqueness of a fixed point for contraction type of mappings in partially ordered complete metric spaces has been first considered by Ran and Reurings [1]. Following this initial work, Nieto and Lopez [2, 3] extended the results in [1] for a non-decreasing mapping. Later, Agarwal et al. [4] presented some new results for contractions in partially ordered metric spaces.
One of the interesting concepts, a coupled fixed point theorem, was introduced by Guo and Lakshmikantham [5]. Afterwards, Bhaskar and Lakshmikantham [6] introduced the concept of the mixed monotone property and also proved some coupled fixed point theorems for mappings satisfying the mixed monotone property in partially ordered metric spaces. Lakshimikantham and Ćirić [7] extended the results in [6] by defining the mixed g-monotone property and proved the existence and uniqueness of a coupled coincidence point for such mapping satisfying the mixed g-monotone property in partially ordered metric spaces. As a continuation of this work, several results of a coupled fixed point and a coupled coincidence point have been discussed in the recent literature (see, e.g., [7–25]).
Recently, Sintunavarat et al. [23] proved the existence and uniqueness of a coupled fixed point for nonlinear contractions in partially ordered metric spaces without mixed monotone property and extended some coupled fixed point theorems of Bhaskar and Lakshmikantham [6]. Later, Charoensawan and Klanarong [17] proved the existence and uniqueness of a coupled coincidence point in partially ordered metric space without the mixed g-monotone property which extended some coupled fixed point theorems of Sintunavarat et al. [23].
The concept of a new class of generalized metric spaces, called G-metric space, was introduced by Mustafa and Sims [26]. Choudhury and Maily [27] proved the existence of a coupled fixed point of nonlinear contraction mappings with mixed monotone property in partially ordered G-metric spaces. Later, Abbas et al. [28] extended the results of a coupled fixed point for a mixed monotone mapping obtained in [27].
In the case of the coupled coincidence point theory in partially ordered G-metric space, Aydi et al. [29] established some coupled coincidence and coupled common fixed point theory for a mixed g-monotone mapping satisfying nonlinear contractions in partially ordered G-metric spaces. They extended the results obtained in [27]. Later, Karapınar et al. [30] extended the results of coupled coincidence and coupled common fixed point theorems for a mixed g-monotone mapping obtained in [29]. Some examples dealing with G-metric spaces are discussed in [21, 30–44].
In this work, we generalize the results of Aydi et al. [29] by extending the coupled coincidence point theorem of nonlinear contraction mappings in partially ordered G-metric spaces without the mixed g-monotone property using the concept of a -invariant set in partially ordered G-metric spaces.
2 Preliminaries
In this section, we give some definitions, propositions, examples, and remarks which are used in our main results. Throughout this paper, denotes a partially ordered set with the partial order ⪯. By , we mean . A mapping is said to be non-decreasing (resp., non-increasing) if for all , implies (resp. ).
Definition 2.1 [26]
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 specifically, 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 [26]
Let be a G-metric space, and let be a sequence of point 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 [26]
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 [26]
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 [26]
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 [26]
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 [26]
A G-metric space is called G-complete if every G-Cauchy sequence is G-convergent in .
Definition 2.9 [27]
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 .
Bhaskar and Lakshmikantham in [6] introduced the following condition.
Definition 2.10 [6]
Let be a partially ordered set and . We say F has the mixed monotone property if for any
and
Definition 2.11 [6]
An element is called a coupled fixed point of a mapping if and .
Lakshmikantham and Ćirić [7] introduced the concept of a mixed g-monotone mapping and a coupled coincidence point as follows.
Definition 2.12 [7]
Let be a partially ordered set and and . We say F has the mixed g-monotone property if for any
and
Definition 2.13 [7]
An element is called a coupled coincidence point of a mapping and if and .
Definition 2.14 [7]
Let X be a nonempty set and and . We say F and g are commutative if for all .
The following class of functions was considered by Lakshmikantham and Ćirić in [7].
Let Φ denote the set of functions satisfying
-
1.
,
-
2.
for all ,
-
3.
for all .
Lemma 2.15 [7]
Let . For all , we have .
Aydi et al. [29] proved the following theorem.
Theorem 2.16 [29]
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 exist such that
then there exists such that and .
Definition 2.17 [29]
Let be a partially ordered set and G be a G-metric on X. We say that is regular if the following conditions hold:
-
1.
if a non-decreasing sequence , then for all n,
-
2.
if a non-increasing sequence , then for all n.
Theorem 2.18 [29]
Let be a partially ordered set and G be a G-metric on X such that is regular. Suppose that there exist , , and such that
for all for which and .
Suppose also that is complete, F has the mixed g-monotone property, , and g is continuous and commutes with F. If there exist such that
then there exists such that and .
Batra and Vashistha [45] introduced an -invariant set which is a generalization of the F-invariant set introduced by Samet and Vetro [46].
Definition 2.19 [45]
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)
.
Now, we give the notion of an -invariant set and an -invariant set, which is useful for our main results.
Definition 2.20 Let be a G-metric space and be given mapping. Let M be a nonempty subset of . We say that M is an -invariant subset of if and only if, for all ,
-
1.
;
-
2.
.
Definition 2.21 Let be a G-metric space and and are given mapping. Let M be a nonempty subset of . We say that M is an -invariant subset of if and only if, for all ,
-
1.
;
-
2.
.
Definition 2.22 Let be a G-metric space and M be a subset of . We say that satisfies the transitive property if and only if, for all ,
Remarks
-
1.
The set is trivially -invariant, which satisfies the transitive property.
-
2.
Every -invariant set is -invariant when denote identity map on X.
Example 2.23 Let be a partially ordered set and suppose there is a G-metric d on X such that is a complete G-metric space. Let and be a mapping satisfying the mixed g-monotone property. Define a subset by . Then M is an -invariant subset of , which satisfies the transitive property.
Example 2.24 Let and be defined by . Let be given by . Then it is easy to show that is -invariant subset of but not -invariant subset of as but .
3 Main results
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 there exists and also suppose that and such that
for all .
Suppose also that F is continuous, , and g is continuous and commutes with F. If there exist such that
and M is an -invariant set which satisfies the transitive property, then there exist such that and , that is, F has a coupled coincident point.
Proof Let . Since , we can choose such that
Similarly, we can choose such that
Continuing this process we can construct sequences and in X such that
If there exists such that then and . Thus, is a coupled coincidence point of F. The proof is completed.
Now we assume that for all . Thus, we have either or for all . Since
and M is an -invariant set, we have
By repeating this argument, we get
and
From (1), (2) and (3), we have
From (3) and using the fact that M is an -invariant set and (1), we have
and
Let
Adding (4) with (5) which 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 (7) and using the properties of the map ϕ, we get
A contradiction, thus and hence
Next, we prove that and are Cauchy sequence in the G-metric space . Suppose, to the contrary, that is the least of and is not a Cauchy sequence in . Then there exists an for which we can find subsequences and of , and of with such that
Further, corresponding to , we can choose in such a way that it is the smallest integer with and satisfying (9). Then
Using the rectangle inequality, we get
Letting in the above inequality and using (8), we get
Again, by the rectangle inequality, we have
Using the fact that for any , we obtain
Since , using (3), we have
and
From the fact that M is an -invariant set which satisfies the transitive property, we have
Again from
we get
By this process, we can get
Now, using (1), we have
Since and M is an -invariant set, we have
and
Adding (14) to (15), we get
From (13) and (16), it follows that
Letting in (17) and using (8), (12) and for all , we have
This is a contradiction. This shows that and are Cauchy sequence in the G-metric space . Since is complete, and are G-convergent, there exist such that and . That is, from Proposition 2.4, we have
From (18), (19), continuity of g, and Proposition (2.7), we get
From (2) and commutativity of F and g, we have
We now show that and .
Taking the limit as in (22) and (23), by (20), (21), and continuity of F, we get
and
Thus we prove that and . □
In the next theorem, we omit the continuity hypothesis of F.
Theorem 3.2 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 there exists and also suppose that and such that
for all .
Suppose also that is complete and g is continuous and commutes with F; if we have any two sequences , with
and for all , then for all . If there exists such that
and M is an -invariant set which satisfies the transitive property, then there exist such that and .
Proof Proceeding exactly as in Theorem 3.1, we find that and are Cauchy sequences in the complete G-metric space . Then there exist such that and and
by the assumption, we have
for all .
Since we have the -invariant set property,
for all . By the rectangle inequality, (1), and for all , we get
Taking the limit as in the above inequality, we obtain
This implies that and . Thus we prove that is a coupled coincidence point of F and g. □
The following example is valid for Theorem 3.1.
Example 3.3 Let . Define by and let be defined by
and by . Let and . Then we have , but , and so the mapping F does not satisfy the mixed g-monotone property.
Letting , we have
and we have
Now, let such that , then
Therefore, if we apply Theorem 3.1 with , we know that F has a coupled coincidence point .
Next, we give a sufficient condition for the uniqueness of the coupled coincidence point in Theorem 3.1.
Theorem 3.4 In addition to the hypotheses of Theorem 3.1, suppose that for every there exists such that
Suppose also that ϕ is a non-decreasing function. Then F and g have a unique coupled common fixed point, that is, there exists a unique such that and .
Proof From Theorem 3.1, the set of coupled coincidence points is nonempty. Suppose and are coupled coincidence points of F, that is,
We shall show that
By assumption there is such that
Put , and choose , such that and . Then similarly as in the proof of Theorem 3.1, we can inductively define sequences and such that
Since M is -invariant and , we have
That is, .
From , if we use again the property of -invariance, then it follows that
By repeating this process, we get
Since M is -invariant, we get
Thus from (1), (25), and (26), we have
Thus from(27), we have
Since ϕ is non-decreasing and (28), we get
for each . Letting in (29) and using Lemma 2.15, this implies
Similarly, we obtain
Hence, from (30), (31), and Proposition 2.4, we get and .
Since and , by commutativity of F and g, we have
Denote and . Then from (32)
Therefore, is a coupled coincidence fixed point of F and g. Then from (24) with and , it follows that and , that is,
From (33) and (34), and . Therefore, is a coupled common fixed point of F and g.
To prove the uniqueness, assume that is another coupled common fixed point. Then by (24) we have and . □
Next, we give a simple application of our results to coupled coincidence point theorems in partially ordered metric spaces with the mixed g-monotone property.
Corollary 3.5 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 exist such that
then there exists such that and .
Proof We define the subset by . From Example 2.23, M is an -invariant set which satisfies the transitive property. By (1), we have
for all .
Since such that
We have . Since F is continuous, all the hypotheses of Theorem 3.1 hold, and we have and . □
Corollary 3.6 Let be a partially ordered set and G be a G-metric on X such that is regular. Suppose that there exist , , and such that
for all for which and .
Suppose also that is complete, F has the mixed g-monotone property, , and g is continuous and commutes with F. If there exist such that
then there exists such that and .
Proof As in Corollary 3.5, we get
Since any two sequences , in X such that is non-decreasing sequence with and is non-increasing sequence with , for all .
Since is regular, we have
and
Therefore, we have for all , and so the whole assumption of Theorem 3.2 holds, thus F has a coupled coincidence point. □
Next, we show the uniqueness of a coupled fixed point of F.
Corollary 3.7 In addition to the hypothesis of Corollary 3.5 (Corollary 3.6), suppose that for every there exists a such that is comparable to and . Suppose also that ϕ is a non-decreasing function. Then F and g have a unique coupled common fixed point, that is, there exists a unique such that and .
Proof We define the subset by . From Example 2.23, M is an -invariant set which satisfies the transitive property. Thus, the proof of the existence of a coupled fixed point is straightforward by following the same lines as in the proof of Corollary 3.5 (Corollary 3.6).
Next, we show the uniqueness of a coupled fixed point of F.
Since for all , there exists such that , and , we can conclude that
and
Therefore, since all the hypotheses of Theorem 3.4 hold, and F has a unique coupled fixed point. The proof is completed. □
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Acknowledgements
This research was supported by Chiang Mai University and the author would like to express sincere appreciation to Prof. Suthep Suantai for very helpful suggestions and many kind comments.
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Thangthong, C., Charoensawan, P. Coupled coincidence point theorems for a ϕ-contractive mapping in partially ordered G-metric spaces without mixed g-monotone property. Fixed Point Theory Appl 2014, 128 (2014). https://doi.org/10.1186/1687-1812-2014-128
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DOI: https://doi.org/10.1186/1687-1812-2014-128