Abstract
In this work, we give the notions of coupled g-coincidence point and -increasing property of F for mappings and and prove the existence and uniqueness of a coupled g-coincidence point theorem for mappings and with φ-contraction mappings in complete metric spaces without the -increasing property of F and the mixed monotone property of G. Further, we apply our results to the existence and uniqueness of a coupled g-coincidence point of the given mappings with the -increasing property of F and the mixed monotone property of H in partially ordered metric spaces.
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1 Introduction
In 2004, the study of a fixed point in partially ordered metric spaces was initiated by Ran and Reurings [1], and continued by Nieto and Lopez [2, 3]. Agarwal et al.[4] and many other authors presented some new results for contractions in partially ordered metric spaces.
In 1987, Guo and Lakshmikantham [5] introduced the concept of coupled fixed point. Afterwards, Bhaskar and Lakshmikantham [6] introduced the concept of mixed monotone property for contractive operators in partially ordered metric spaces and proved coupled fixed point theorems for mappings which satisfy the mixed monotone property. They also gave some applications on the existence and uniqueness of the coupled fixed point theorems for such mappings. As a continuation of this trend, Lakshimikantham and Ćirić [7] extended the results in [6] by defining the mixed g-monotonicity and studied the existence and uniqueness of a coupled coincidence point for such mappings which satisfy the mixed monotone property in partially ordered metric spaces. For more work on the coupled fixed point theory and coupled coincidence point theory in partially ordered metric spaces and different spaces, we refer to the reviews (see, e.g., [8–48]).
One of the interesting ways to develop coupled fixed point theory in partially ordered metric spaces is to consider the mapping without the mixed monotone property. In 2012, Sintunavarat et al. [44, 45] proved some coupled fixed point theorems for nonlinear contractions without the mixed monotone property and extended some coupled fixed point theorems of Bhaskar and Lakshmikantham [6] by using the concept of F-invariant set due to Samet and Vetro [49]. Later, Charoensawan and Klanarong [22] proved the existence and uniqueness of a coupled coincidence point in partially ordered metric spaces without the mixed g-monotone property of and . Recently, Kutbi et al. [48] introduced the concept of F-closed set which is weaker than the concept of F-invariant set and proved some coupled fixed point theorems without the condition of mixed monotone property. Following this trend, many authors have studied fixed point theorems for nonlinear contractions without the monotone property in several spaces (see, e.g., [14, 50–56]).
Very recently, Hussain et al. [47] presented the new concept of generalized compatibility of a pair of mappings and proved some coupled coincidence point results of such mappings without the mixed G-monotone property of F, which generalized some recent comparable results in the literature. They also gave some examples and an application to integral equations to support the result.
In this work, we give the notion of coupled g-coincidence point and the -increasing property of F for mappings and . We apply our results to the existence and uniqueness of a coupled g-coincidence point of the given mapping with the -increasing property of F and the mixed monotone property of H in partially ordered metric spaces which generalize and extend the coupled coincidence point theorem in [47].
2 Preliminaries
In this section, we give some definitions, propositions, examples, and remarks which are useful for the main results in this 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 .
Guo and Lakshmikantham [5] introduced the concept of coupled fixed point as follows.
Definition 2.1 [5]
An element is called a coupled fixed point of a mapping if and .
The concept of mixed monotone property was introduced by Bhaskar and Lakshmikantham in [6].
Definition 2.2 [6]
Let be a partially ordered set and . We say that F has the mixed monotone property if for any ,
and
Lakshmikantham and Ćirić in [7] introduced the concepts of mixed g-monotone mapping and coupled coincidence point.
Definition 2.3 [7]
Let be a partially ordered set and and . We say that F has the mixed g-monotone property if for any ,
and
Definition 2.4 [7]
An element is called a coupled coincidence point of mappings and if and .
Definition 2.5 [7]
Let X be a non-empty set and and . We say that F and g are commutative if for all .
Hussain et al. [47] introduced the concept of G-increasing and generalized compatibility as follows.
Definition 2.6 [47]
Suppose that are two mappings. F is said to be G-increasing with respect to ⪯ if for all , with , we have .
Definition 2.7 [47]
An element is called a coupled coincidence point of mappings if and .
Definition 2.8 [47]
Let . We say that the pair is generalized compatible if
whenever and are sequences in X such that
Definition 2.9 [47]
Let be two maps. We say that the pair is commuting if
Let Θ 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 .
Hussain et al. [47] proved the coupled coincidence point for such mappings involving the -contractive condition as follows.
Theorem 2.10 [47]
Let be a partially ordered set and M be a non-empty subset of , and let d be a metric on X such that is a complete metric space. Assume that are two generalized compatible mappings such that F is G-increasing with respect to ⪯, G 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
-
(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 G have a coupled coincidence point.
Kutbi et al. [48] introduced the notion of F-closed set which extended the notion of F-invariant set as follows.
Definition 2.11 [48]
Let be a mapping, and let M be a subset of . We say that M is an F-closed subset of if, for all ,
Now, we give the notion of a coupled g-coincidence point and a -closed set which is useful for our main results.
Definition 2.12 Suppose that are two mappings and . F is said to be -increasing with respect to ⪯ if for all , with , we have .
Example 2.13 Let endowed with the natural ordering of real number ≤. Define the mappings and by , and for all . Note that F is -increasing with respect to ≤.
Example 2.14 Let endowed with the natural ordering of real number ≤. Define the mappings and by , and for all . Note that F is -increasing with respect to ≤ but not H-increasing. If , then , but if , then .
Definition 2.15 An element is called a coupled g-coincidence point of mappings and if and .
Example 2.16 Let endowed with the natural ordering of real number ≤. Define the mappings and by , and for all . Note that is a coupled g-coincidence point.
Definition 2.17 Let be a metric space and be two mappings and . Let M be a subset of . We say that M is an -closed subset of if for all ,
Definition 2.18 Let be a metric space and be a given mapping. Let M be a subset of . We say that M satisfies the transitive property if and only if, for all ,
Remark The set is a trivially -closed set, which satisfies the transitive property.
Example 2.19 Let endowed with the usual metric and be defined by
and
Let be defined by
It is easy to see that is an -closed set but not an F-closed set.
Definition 2.20 Let and . We say that the pair is g-generalized compatible if
whenever ,,, and are sequences in X such that
Example 2.21 Let be a metric space endowed with a partial order ⪯. Let and be two mappings such that F is -increasing with respect to ⪯. Define a subset by
Let . Since F is -increasing with respect to ⪯, we have and , and this implies that . Then M is an -closed subset of which satisfies the transitive property.
3 Main results
Let Φ denote the set of functions satisfying
-
1.
for all ,
-
2.
for all .
Now, we state our first main theorem which guarantees a coupled g-coincidence point.
Theorem 3.1 Let be a complete metric space and M be a non-empty subset of . Assume that is continuous and are two generalized compatible mappings such that H is continuous, and for any , there exist such that and . Suppose that there exists such that the following holds:
for all with .
Also suppose that F is continuous. If there exists such that
and M is -closed, then there exists such that and , that is, F and H have a coupled g-coincidence point.
Proof Let be such that
From the assumption, there exist such that
Again from assumption, we can choose such that
By repeating this argument, we can construct sequences such that for all ,
Since and M is -closed, we get
Again, using the fact that M is -closed, we have
Continuing this process, for all , we get
For all , denote
We can suppose that for all . If not, will be a coupled g-coincidence point and the proof is finished. From (1), (2) and (3), we have
Therefore, the sequence satisfies
Using the property of φ, it follows that the sequence is decreasing. Therefore, there exists some such that
We shall prove that . Assume, to the contrary, that . Then by letting in (6) and using the property of φ, we have
a contradiction. Thus and hence
We now prove that and are Cauchy sequences in . Suppose, to the contrary, that at least one of the sequences or is not a Cauchy sequence. Then there exists 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 satisfying (9). Then
Using (9), (10), and the triangle inequality, we have
Letting in (11) and using (8), we get
Again, for all , we have
From (3) and , we have
and
Using M has the transitive property, we get
Continuing this process, we have
From (1),(2) and (14), we have
which, by (13), yields
Letting in the above inequality and using (8) and (12), we get
which is a contradiction. Hence, and are Cauchy sequences in . Since is complete and (2), there exist such that
Since the pair satisfies the generalized compatibility, from (17) we have
Since F is continuous, for all , by the triangle inequality, we have
and
Taking the limit as in (19) and (20), using (17),(18), and the fact that H, F and g are continuous, we have
Therefore, is a coupled g-coincidence point of F and H. □
In our next theorem, we drop the continuity of F.
Theorem 3.2 Let be a complete metric space and M be a non-empty subset of . Assume that is continuous and are two generalized compatible mappings such that H is continuous and for any , there exist such that and . Suppose that there exists such that the following holds:
for all with .
Also suppose that is a complete metric space, and any two sequences and with and , for all implies
for all . If there exist such that
and M is -closed, then there exists such that and , that is, F and H have a coupled g-coincidence point.
Proof As in the proof of Theorem 3.1. Since is complete, there exist such that , and we have
Since the pair satisfies the generalized compatibility, H is continuous and by (3), we have
and
From (3), (23), (24), by assumption, for all , we have
Then, by (1), (2), (25), and the triangle inequality, we have
Letting now in the above inequality and using the property of φ that , we have
which implies that and . □
Next, we give an example to validate Theorem 3.1.
Example 3.3 Let , and be defined by
Let be defined by . Clearly, H does not satisfy the mixed monotone property. Now, we prove that for any , there exist such that and . It is easy to see that there exist , such that
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 , then we have
Therefore, condition (1) is satisfied. Thus, all the requirements of Theorem 3.1 are satisfied and is a coupled g-coincidence point of F and H.
Example 3.4 Let , and be defined by
and
Let by . Clearly, H does not satisfy the mixed monotone property and if , , consider
Then F is not -increasing.
Now, we prove that for any , there exist such that and . It is easy to see 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 , then we have
Therefore, condition (1) is satisfied. Thus, all the requirements of Theorem 3.1 are satisfied and is a coupled g-coincidence point of F and H.
Next, we show the uniqueness of the coupled coincidence point and the coupled fixed point of F and H.
Theorem 3.5 In addition to the hypotheses of Theorem 3.1, suppose that for every , there exists such that
Then F and G have a unique coupled g-coincidence point.
Proof From Theorem 3.1, we know that F and H have a coupled g-coincidence point. Suppose that are coupled g-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 -closed, then
By repeating this process, we get
Using (1), (26) and (27), we have
Using the property that and repeating this process, we get
From and , it follows that for each . Therefore, from (29) we have
This implies that
Similarly, we show that
From (31) and (32), we have
□
Next, we give some application of our results to coupled coincidence point theorems in partially metric spaces with F is -increasing with respect to ⪯ and H has the mixed monotone property.
Corollary 3.6 Let be a partially ordered set and M be a non-empty subset of , and let d be a metric on X such that is a complete metric space. Assume that is continuous and are two generalized compatible mappings such that F is -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 F is continuous. If there exist such that
and M is -closed, then there exists such that and , that is, F and H have a coupled g-coincidence point.
Proof We define the subset by
From Example 2.21, M is an -closed set which satisfies the transitive property. For all with ( and ), we have . By (1),we get
Since with
we have
For the assumption holds, F is continuous. By the assumption of Theorem 3.1, we have and . □
Corollary 3.7 Let be a partially ordered set and M be a non-empty subset of , and let d be a metric on X such that is a complete metric space. Assume that is continuous and are two generalized compatible mappings such that F is -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 is a complete metric space and X has the following properties: for any two sequences and ,
-
(i)
if a non-decreasing sequence , then for all n,
-
(ii)
if a non-increasing sequence , then for all n.
If there exist such that
and M is -closed, then there exists such that and , that is, F and H have a coupled g-coincidence point.
Proof We define the subset by
From Example 2.21, M is an -closed set which satisfies the transitive property. For all with and , we have . Let with
Using (2) and F is -increasing with respect to ⪯, we have
Therefore, .
From is complete, as in Theorem 3.1, we have two Cauchy sequences and such that is a non-decreasing sequence in X with and is a non-increasing sequence in X with . Using assumption, we have
Since H has the mixed monotone property, we have
Therefore, we have
for all . Now, since all the hypotheses of Theorem 3.2 hold, then F and H have a coupled g-coincidence point. The proof is completed. □
Corollary 3.8 In addition to the hypotheses of Corollary 3.6, suppose that for every , there exists such that is comparable to and . Then F and H have a unique coupled g-coincidence point.
Proof We define the subset by
From Example 2.21, 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 3.6.
Next, we show the uniqueness of a coupled g-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 3.5 hold, F and H have a unique g-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. Coupled g-coincidence point theorems for a generalized compatible pair in complete metric spaces. Fixed Point Theory Appl 2014, 201 (2014). https://doi.org/10.1186/1687-1812-2014-201
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DOI: https://doi.org/10.1186/1687-1812-2014-201