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Retracted Article: Coupled fixed point theorems without continuity and mixed monotone property
Journal of Inequalities and Applications volume 2013, Article number: 217 (2013)
In this paper, we generalize some coupled fixed point theorems for the mixed monotone operators obtained in (Choudhury and Maity in Math. Comput. Model., 2011, doi:10.1016/j.mcm.2011.01.036) by significantly weakening the contractive condition involved and by replacing the mixed monotone property with another property which is automatically satisfied in the case of a totally ordered space. The proof follows a different and more natural new technique recently introduced by Berinde (Nonlinear Anal. 74:7347-7355, 2011). The example demonstrates that our main result is an actual improvement over the results which are generalized.
1 Introduction and preliminaries
Banach’s contraction principle is the most celebrated fixed point theorem. Since this principle, many authors have improved, extended and generalized this principle in many ways. Recently, Mustafa and Sims [1, 2] introduced an improved version of the generalized metric space structure, which they called a G-metric space, and established Banach’s contraction principle in this work. For more details on G-metric spaces, one can refer to the papers [1–15]. Since then, some fixed point theorems in partially ordered G-metric spaces have been considered in  and others.
Studies on coupled fixed point problems in partially ordered metric spaces and ordered cone metric spaces have received considerable attention in recent years ([17–24] and others). One of the reasons for this interest is their potential applicability. Specifically, Bhaskar and Lakshmikanthan  established coupled fixed point theorems for a mixed monotone operator in partially ordered metric spaces. Afterward, Lakshmikanthan and Ciric  extended the results of  by furnishing coupled coincidence and a coupled fixed point theorem for two commuting mappings having the mixed g-monotone property. In a subsequent series, Choudhary and Kundu  introduced the concept of compatibility and proved the result of  under a different set of some conditions. Very recently, Berinde  extended the results of  by weakening the contractive condition using a different and more natural technique, and Doric et al.  and Agarwal et al.  established coupled fixed points results without the mixed monotone property.
Recently, Choudhary and Maity  published coupled fixed point results in partially ordered G-metric spaces. Following the new technique of Berinde , we extend the result of Choudhary and Maity  by weakening the contractive condition involving and relaxing the mixed monotone property and continuity requirement. An illustrative example is discussed which shows that the above mentioned improvements are actual.
In what follows, we collect some related definitions and results for our further use. In 2004, Mustafa and Sims  introduced the concept of G-metric spaces as follows.
Definition 1.1 (see )
Let X be a nonempty set and let 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.
Definition 1.2 (see )
Let be a G-metric space and let be a sequence in X. A point is said to be the limit of the sequence if
We say that the sequence is G-convergent to x or G-converges to x.
Thus in a G-metric space if, for any , there exists such that for all .
Proposition 1.3 (see )
Let be a G-metric space. Then the following are equivalent:
is G-convergent to x.
Proposition 1.4 (see )
Let be a G-metric space. Then is G-continuous at a point if and only if it is G-sequentially continuous at x, that is, whenever is G-convergent to x, is G-convergent to .
Proposition 1.5 (see )
Let be a G-metric space. Then the function is jointly continuous in all three of its variables.
Definition 1.6 (see )
Let be a G-metric space. A mapping is said to be continuous on if, for any two G-convergent sequences and converging to x and y, respectively, is G-convergent to .
Definition 1.7 (see )
A G-metric space is called G-complete if every G-Cauchy sequence is G-convergent in .
Definition 1.8 A G-metric space is called symmetric if for all .
Proposition 1.9 (see )
Every G-metric space defines a metric space by for all .
If a G-metric space is symmetric, then for all .
However, if is not symmetric, then it follows from G-metric properties that
for all .
The concept of a mixed monotone property has been introduced by Bhaskar and Lakshmikantham in .
Definition 1.10 (see )
Let be a partially ordered set. A mapping is said to have the mixed monotone property if is monotone nondecreasing in x and is monotone nonincreasing in y, that is, for any ,
Lakshmikantham and Ćirić in  introduced the concept of a g-mixed monotone mapping.
Definition 1.11 (see )
Let be a partially ordered set. Let us consider the mappings and . The mapping F is said to have the mixed g-monotone property if is monotone g-nondecreasing in x and is monotone g-nonincreasing in y, that is, for any ,
Definition 1.12 (see )
An element is called a coupled fixed point of a mapping if and .
Definition 1.13 (see )
An element is called a coupled coincidence point of the mappings and if and .
To relax the mixed monotone property, Doric et al.  introduced the following condition.
If the elements x, y of a partially ordered set are comparable (that is, or ), then we write . Let be a mapping. Then consider the following condition:
The following example shows that this condition may be satisfied when F does not have the mixed monotone property.
Example 1.14 (see )
for all . Then F does not have the mixed monotone property since and , while and . But it has the condition (1.1) since and and and (the other two cases are trivial).
Using the concepts of continuity, mixed monotone property and coupled fixed point, Choudhary and Maity  introduced the following theorem.
Theorem 1.15 Let be a partially ordered set and let G be a G-metric on X such that is a complete G-metric space. Let be a continuous mapping having the mixed monotone property on X. Assume that there exists such that, for all ,
for all with and , where either or . If there exist and such that and , then F has a coupled fixed point in X, that is, there exist such that and .
In , Choudhary and Maity established some coupled fixed point theorems in the setting of G-metric spaces. Starting from the results in , our main aim of this paper is to obtain more general coupled fixed point theorems for the mappings having no mixed monotone property and satisfying a contractive condition which is more general than (1.2). Following the same approach as in , we weaken the contractive condition satisfied by F. Also, we relax the continuity requirement of F. The techniques of the proofs are simpler and different from those of the results in [29, 31] and others.
2 Main results
Theorem 2.1 Let be a partially ordered set and let G be a G-metric on X such that is a complete G-metric space. Let be a mapping satisfying the property (1.1). Assume that there exists such that for , then the following holds:
for all and , where either or . If there exist such that
then there exists such that and .
Proof Consider the functional defined by
for all . It is simple to check that is a G-metric on and, moreover, if is complete, then is a complete G-metric space, too. We consider the mapping defined by
for all . Clearly, for all , in view of the definition of , we have
Hence, by the contractive condition (2.1), we obtain the Banach-type contractive condition in a G-metric space as follows:
for all with and . Assume that (2.2) holds. Then there exist and in X such that
Denote and consider the Picard iteration associated to T and the initial approximation , that is, the sequence is defined by
for all , where for all . Since X has the condition (1.1), we have
and so, by induction,
which shows that T is monotone and the sequence is nondecreasing. We now follow the steps as in the proof of Banach’s contraction principle in a G-metric space established by Mustafa and Sims . Taking in (2.6), we have
for all , which implies that
for all . Thus, by induction, we have
for all .
Now, we claim that is a Cauchy sequence in . Let . Then, by (2.9), we have
So, is indeed a Cauchy sequence in a complete G-metric space and hence it is convergent. Therefore, there exists such that
Since T is continuous in , by virtue of the Lipschitzian type conditions (2.1) and (2.7), it follows that is a fixed point of T, that is,
Let . Then, by the definition of T, we obtain
that is, is a coupled fixed point of F. This completes the proof. □
Remark 2.2 Theorem 2.1 is more general than Theorem 1.15 which was established by Choudhary and Maity  since the contractive condition (2.1) is more general than the contractive condition (1.2) of Theorem 1.15. This fact is clearly illustrated by the following example.
Example 2.3 Let and let for all be a G-metric defined on X. Also, let be a mapping defined by
for all . Then F satisfies the conditions (2.1) and (1.1), but not (1.2) of Theorem 1.15 of . Indeed, assume that there exists k, , such that (1.2) holds. This means
for all with and . From this, in particular, for and , we get
Thus we have , which is a contradiction.
Now, we show that (2.1) holds. Indeed, since we have, for and ,
by adding up the above two inequalities, we get exactly (2.1) with . Also, by Theorem 2.1, we obtain that F has a unique coupled fixed point, that is, , but Theorem 1.16 cannot be applied to this example.
Every pair of elements in has either a lower bound or an upper bound, i.e., for all , ,
Theorem 2.4 Adding the condition (2.9) to the hypothesis of Theorem (2.1), we obtain the uniqueness of a coupled fixed point of F.
Proof Assume that is a coupled fixed point of F different from . This means, by (G2), that .
Now, we discuss two cases.
Case 1. is comparable to . Since is comparable to with respect to the ordering in , by taking and (or and ) in (2.6), we obtain
which is a contradiction since .
Case 2. and are not comparable. In this case, there exists an upper bound or a lower bound of and . Then, in view of the monotonicity of T, is comparable to and . Now, again, by the contractive condition (2.6), we have
as , which leads to a contradiction. This completes the proof. □
Next, as in , we show that even the components of coupled fixed points are equal.
Theorem 2.5 In addition to the hypothesis of Theorem 2.1, suppose that are comparable. Then, for a coupled fixed point , we have , that is, F has a fixed point such that .
Proof Consider the condition (2.2), that is,
Since and are comparable, we have . Then, by the condition (1.1) of F, we have
and hence, by induction,
for all . Now, since
by the continuity of the G-metric G, we have
On the other hand, by taking and in (2.4), we have
for all , which actually means that
for all . Therefore, we have
This completes the proof. □
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The first author gratefully acknowledges financial assistance of the Council of Scientific and Industrial Research, Government of India, under research project No.25 (0197)/11/EMR-II. The second author was also supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170).
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
An erratum to this article can be found at http://dx.doi.org/10.1186/1029-242X-2014-25.
One of the authors (Yeol Je Cho) found mistakes during the proofreading. He was advised to retract his paper and resubmit the corrected version later for consideration.
A retraction note to this article can be found online at http://dx.doi.org/10.1186/1029-242X-2014-25.
An erratum to this article is available at http://dx.doi.org/10.1186/1029-242X-2014-25.
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Vats, R.K., Sihag, V. & Cho, Y.J. Retracted Article: Coupled fixed point theorems without continuity and mixed monotone property. J Inequal Appl 2013, 217 (2013). https://doi.org/10.1186/1029-242X-2013-217
- partially ordered set
- G-metric space
- coupled fixed point
- mixed monotone property