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Coupled common fixed point theorems for generalized nonlinear contraction mappings with the mixed monotone property in partially ordered metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 307 (2013)
Abstract
The purpose of this paper is to establish some coupled coincidence point theorems for generalized nonlinear contraction mappings with the mixed g-monotone property in the framework of metric spaces endowed with partial order. The theorems presented in this paper are generalizations and improvements of the several well-known results in the literature.
MSC:46N40, 47H10, 54H25, 46T99.
1 Introduction and preliminaries
The Banach contraction principle is one of very popular tools in solving the existence in many problems of mathematical analysis. Due to its simplicity and usefulness, there are a lot of generalizations of this principle in the literature. Ran and Reurings [1] extended the Banach contraction principle in partially ordered sets with some applications to linear and nonlinear matrix equations. While Nieto and López [2] extended the result of Ran and Reurings and applied their main theorems to obtain a unique solution for a first-order ordinary differential equation with periodic boundary conditions. Bhaskar and Lakshmikantham [3] introduced the concept of mixed monotone mappings and obtained some coupled fixed point results. Also, they applied their results to a first-order differential equation with periodic boundary conditions. Recently, many researchers have obtained fixed point, common fixed point, coupled fixed point and coupled common fixed point results in cone metric spaces, partially ordered metric spaces and others (see [1–27]).
Definition 1 Let be a metric space and and , F and g are said to commute if for all .
Definition 2 Let be a partially ordered set and . The mapping F is said to be non-decreasing if for , implies and non-increasing if for , implies .
Definition 3 Let be a partially ordered set and and . The mapping F is said to have the mixed g-monotone property if is monotone g-non-decreasing in x and monotone g-non-increasing in y, that is, for any ,
and
If mapping in Definition 3, then the mapping F is said to have the mixed monotone property.
Definition 4 An element is called a coupled coincidence point of the mappings and if and .
If g is the identity mapping in Definition 4, then is called a coupled fixed point.
Geraghty [16] introduced an extension of the Banach contraction principle in which the contraction constant was replaced by a function having some specified properties.
Definition 5 Let Θ be the class of functions with
-
(i)
;
-
(ii)
implies .
The method applied by Geraghty [16] was utilized to obtain further new fixed point result works like [6, 7, 15].
The purpose of this paper is to establish some coupled coincidence point results for a pair of mappings with the mixed g-monotone property satisfying a generalized contractive condition by using the ideas of Geraghty [16] in partially ordered metric spaces. Also we give some examples to illustrate the main results.
2 Main results
Theorem 6 Let be a partially ordered set and suppose that there exists a metric d on X such that is a complete metric space. Suppose that and are self-mappings on X such that F has the mixed g-monotone property on X such that there exist two elements with and . Suppose that there exists such that
for all with and . Further suppose that , g is continuous and commutes with F, and also suppose that either
-
(a)
F is continuous, or
-
(b)
X has the following properties:
-
(i)
if is a non-decreasing sequence with in , then for every n;
-
(ii)
if is a non-increasing sequence with in , then for every n.
Then there exist two elements such that and , that is, F and g have a coupled coincidence point .
Proof Let be such that and . Since , we can construct sequences and in X such that
We claim that for all ,
and
We use the mathematical induction. Let . Since and , in view of and , we have and , that is, (2.3) and (2.4) hold for . Suppose that (2.3) and (2.4) hold for some . As F has the mixed g-monotone property and and , from (2.2), we get
and
Now from (2.5) and (2.6), we obtain that and . Thus, by the mathematical induction, we conclude that (2.3) and (2.4) hold for all . Therefore
and
Assume that there is some such that , that is, and . Then and , and hence we get the result.
For simplicity, let . Now, we assume that
for all n. Since and , from (2.1) and (2.2), we have
which implies that . It follows that is a monotone decreasing sequence of non-negative real numbers. Therefore, there is some such that .
Now, we show that . Assume to the contrary that , then from (2.9) we have
which yields that . This implies that and . Therefore , that is,
Next, we prove that and are Cauchy sequences. On the contrary, assume that at least one of or is not a Cauchy sequence. Then there exists an for which we can find subsequences and of , and and of with such that for every k,
Further, corresponding to , we can choose in such a way that it is the smallest integer with and satisfies (2.11). Then
Using (2.11) and (2.12), we have
Letting and using (2.10), we have
Also, by the triangle inequality, we have
Since , and , from (2.1) and (2.2), we have
Therefore, we have
This implies that
Taking , we get
Since , we get
which is a contradiction. This implies that and are Cauchy sequences in X. Since X is a complete metric space, there is such that and . Since g is continuous, and .
First, suppose that F is continuous. Then and . As F commutes with g, we have and . By the uniqueness of the limit, we get and .
Second, suppose that (b) holds. Since is a non-decreasing sequence such that and is a non-increasing sequence such that , and g is a non-increasing function, we get and hold for all . Hence, by (2.1), we have
Taking , we get , and hence and . Thus F and g have a coupled coincidence point. □
If , where , then we have the following result.
Corollary 7 Let be a partially ordered set and suppose that there exists a metric d on X such that is a complete metric space. Suppose that and are self-mappings on X such that F has the mixed g-monotone property on X such that there exist two elements with and . Suppose that there exists such that
satisfies for all with and . Further suppose that , g is continuous non-decreasing and commutes with F and also suppose that either
-
(a)
F is continuous, or
-
(b)
X has the following properties:
-
(i)
if is a non-decreasing sequence with in , then for every n;
-
(ii)
if is a non-increasing sequence with in , then for every n.
Then there exist two elements such that and , that is, F and g have a coupled coincidence point .
If g is an identity mapping, we have the following result of Bhaskar and Lakshmikantham [3].
Corollary 8 Let be a partially ordered set and suppose that there exists a metric d on X such that is a metric space. Suppose that is a mapping on X and has the mixed monotone property on X such that there exist two elements with and . Suppose that there exists such that
for all with and . Further suppose that either
-
(a)
F is continuous, or
-
(b)
X has the following properties:
-
(i)
if is a non-decreasing sequence with in X, then for each ;
-
(ii)
if is a non-increasing sequence with in X, then for each .
Then there exist two elements such that and , that is, F has a coupled fixed point .
Example 9 Let . Then is a partially ordered set with the natural ordering of real numbers. Let for all . Define a mapping by and a mapping by
Then it is easy to prove that is a complete metric space, is complete, , X satisfies conditions (1) and (2) of Theorem 6 and F has the g-monotone property. Let be defined by
Now, we verify the inequality (2.1) of Theorem 6 for all with and .
Now, we consider the following cases.
Case 1. , or , , we have
Hence inequality (2.1) holds.
Case 2. , , we have
and
Hence inequality (2.1) holds.
Case 3. , , we have
and
Hence inequality (2.1) holds.
Case 4. , , we have
and
Hence inequality (2.1) holds.
Thus, in all the cases, inequality (2.1) of Theorem 6 is satisfied. Hence, by Theorem 6, is a coupled coincidence point of F and g.
Theorem 10 Let be a partially ordered set and suppose that there exists a metric d on X such that is a complete metric space. Suppose that and are self-mappings on X such that F has the mixed g-monotone property on X such that there exist two elements with and . Suppose that there exists such that
where
for all with and . Further suppose that , g is continuous non-decreasing and commutes with F and also suppose that either
-
(a)
F is continuous, or
-
(b)
X has the following properties:
-
(i)
if is a non-decreasing sequence with in , then for every n;
-
(ii)
if is a non-increasing sequence with in , then for every n.
Then there exist two elements such that and , that is, F and g have a coupled coincidence point .
Proof Following the proof of Theorem 6, we have an increasing sequence and a decreasing sequence in X. Now, we assume that
for all n.
Since and , from (2.16) and (2.2), we have
It follows that is a monotone decreasing sequence of non-negative real numbers. Therefore, there is some such that .
Next, we show that . Assume to the contrary that , then from (2.17) we have
which yields that . This implies that and . Therefore , that is,
Now, we prove that and are Cauchy sequences. On the contrary, assume that at least one of or is not a Cauchy sequence. Then there exists an for which we can find subsequences and of and and of with such that for every k,
Further, corresponding to , we can choose in such a way that it is the smallest integer with and satisfies (2.19). Then
Using (2.19) and (2.20), we have
Letting and using (2.18), we have
Also, by the triangle inequality, we have
Since , and , from (2.16) and (2.2), we have
and similarly,
Therefore, we have
Taking , we have , and
Hence, we get , which is a contradiction. This implies that and are Cauchy sequences in .
Since X is a complete metric space, there is such that and . Since g is continuous, and .
First, suppose that F is continuous. Then and . As F commutes with g, we have
and
By the uniqueness of the limit, we get and .
Second, suppose that (b) holds. Since is a non-decreasing sequence such that and is a non-increasing sequence such that , and g is a non-decreasing function, we get that and hold for all . Hence, by (2.16), we have
and
Taking , we get , and hence and . Thus F and g have a coupled coincidence point. □
Example 11 Let . Then is a partially ordered set with the natural ordering of real numbers. Let for all . Define a mapping by and a mapping by
Then it is easy to prove that is a complete metric space, is complete, , X satisfies conditions (1) and (2) of Theorem 10 and F has the g-monotone property. Let be defined as
Now, we verify inequality (2.16) of Theorem 10 for all with and .
Now, we consider the following cases.
Case 1. , or , , we have
Hence, inequality (2.16) holds.
Case 2. , , we have
and
Hence, inequality (2.16) holds.
Case 3. , , we have
and
Hence, inequality (2.16) holds.
Case 4. , , we have
and
Hence, inequality (2.16) holds.
Thus, in all the cases, inequality (2.16) of Theorem 10 is satisfied. Hence, by Theorem 10, is a coupled coincidence point of F and g.
Next, we prove the existence of a coupled coincidence point theorem, where we do not require that F and g are commuting.
The following lemma proved by Haghi et al. [17] is useful for our results.
Lemma 12 [17]
Let X be a nonempty set, and let be a mapping. Then there exists a subset such that and is one-to-one.
Theorem 13 Let be a partially ordered set and suppose that there exists a metric d on X such that is a metric space. Suppose that and are self-mappings on X such that F has the mixed g-monotone property on X such that there exist two elements with and . Suppose that there exists such that
for all with and . Further suppose that and is a complete subspace of X. Also assume that either
-
(a)
F is continuous, or
-
(b)
X has the following properties:
-
(i)
if is a non-decreasing sequence with in , then for every n;
-
(ii)
if is a non-increasing sequence with in , then for every n.
Then there exist two elements such that and , that is, F and g have a coupled coincidence point .
Proof Using Lemma 12, there exists such that and is one-to-one. We define a mapping by
As g is one-to-one on , so A is well defined.
Since F has the mixed g-monotone property, for all , we have
and
Thus, it follows from (2.23), (2.24) and (2.25) that for all ,
and
which implies that A has the mixed monotone property.
Suppose that the assumption (a) holds. Since F is continuous, A is also continuous. Using Theorem 2.1 of [15] with the mapping A, it follows that A has a coupled fixed point .
Suppose that the assumption (b) holds. We can conclude similarly to the proof of Theorem 2.1 of [15] that the mapping A has a coupled fixed point .
Finally, we prove that F and g have a coupled coincidence point in X. Since is a coupled fixed point of A, we get
Since , there exists a point such that
Thus, it follows from (2.28) and (2.29) that
Also, from (2.23) and (2.30), we get
Therefore, is a coupled coincidence point of F and g. This completes the proof. □
Theorem 14 Let be a partially ordered set and suppose that there exists a metric d on X such that is a metric space. Suppose that and are self-mappings on X such that F has the mixed g-monotone property on X such that there exist two elements with and . Suppose that there exists such that
where
for all with and . Further suppose that and is a complete subspace of X. Also assume that either
-
(a)
F is continuous, or
-
(b)
X has the following properties:
-
(i)
if is a non-decreasing sequence with in , then for every n;
-
(ii)
if is a non-increasing sequence with in , then for every n.
Then there exist two elements such that and , that is, F and g have a coupled coincidence point .
Proof Following similar arguments to those in Theorem 13 and using Theorem 2.2 of [15], we get the result. □
Remark 15 Although Theorem 6 and Theorem 10 are an essential tool in the partially ordered metric spaces to claim the existence of coupled coincidence points of two mappings, some mappings do not have the commutative property. For example, see the following.
Example 16 Let . Then is a partially ordered set with the natural ordering of real numbers. Let for all . Define mappings and by for all and for each . Since for all , the mappings F and g do not satisfy the commutative condition. Hence, the above two theorems cannot be applied to this example. But, by a simple calculation, we see that , g and F are continuous and F has the mixed g-monotone property. Moreover, there exist and with and .
Therefore, it is very interesting to use Theorems 13 and 14 as another auxiliary tool to claim the existence of a coupled coincidence point.
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Acknowledgements
This work was supported by the Kyungnam University Research Fund 2013. The authors are thankful to the referees for valuable suggestions.
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The main idea of this paper was proposed by JKK. JKK and SC prepared the manuscript initially and performed all the steps of the proof in this research. All authors read and approved the final manuscript.
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Kim, J.K., Chandok, S. Coupled common fixed point theorems for generalized nonlinear contraction mappings with the mixed monotone property in partially ordered metric spaces. Fixed Point Theory Appl 2013, 307 (2013). https://doi.org/10.1186/1687-1812-2013-307
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DOI: https://doi.org/10.1186/1687-1812-2013-307