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Retracted Article: Coupled fixed point theorems without continuity and mixed monotone property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)
Abstract
In this paper, we generalize some coupled fixed point theorems for the mixed monotone operators F:X\times X\to X 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. 74:73477355, 2011). The example demonstrates that our main result is an actual improvement over the results which are generalized.
MSC:47H10, 54H25.
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 Gmetric space, and established Banach’s contraction principle in this work. For more details on Gmetric spaces, one can refer to the papers [1–15]. Since then, some fixed point theorems in partially ordered Gmetric spaces have been considered in [16] 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 [25] established coupled fixed point theorems for a mixed monotone operator in partially ordered metric spaces. Afterward, Lakshmikanthan and Ciric [26] extended the results of [25] by furnishing coupled coincidence and a coupled fixed point theorem for two commuting mappings having the mixed gmonotone property. In a subsequent series, Choudhary and Kundu [27] introduced the concept of compatibility and proved the result of [26] under a different set of some conditions. Very recently, Berinde [28] extended the results of [25] by weakening the contractive condition using a different and more natural technique, and Doric et al. [29] and Agarwal et al. [30] established coupled fixed points results without the mixed monotone property.
Recently, Choudhary and Maity [31] published coupled fixed point results in partially ordered Gmetric spaces. Following the new technique of Berinde [28], we extend the result of Choudhary and Maity [31] 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 [4] introduced the concept of Gmetric spaces as follows.
Definition 1.1 (see [1])
Let X be a nonempty set and let G:X\times X\times X\u27f6{R}_{+} be a function satisfying the following properties:
(G1) G(x,y,z)=0 if x=y=z,
(G2) 0<G(x,x,y) for all x,y\in X with x\ne y,
(G3) G(x,x,y)\le G(x,y,z) for all x,y,z\in X with y\ne z,
(G4) G(x,y,z)=G(x,z,y)=G(y,z,x)=\cdots (symmetry in all three variables),
(G5) G(x,y,z)\le G(x,a,a)+G(a,y,z) for all x,y,z,a\in X (rectangle inequality).
Then the function G is called a generalized metric or, more specifically, a Gmetric on X and the pair (X,G) is called a Gmetric space.
Definition 1.2 (see [1])
Let (X,G) be a Gmetric space and let \{{x}_{n}\} be a sequence in X. A point x\in X is said to be the limit of the sequence \{{x}_{n}\} if
We say that the sequence \{{x}_{n}\} is Gconvergent to x or \{{x}_{n}\} Gconverges to x.
Thus {x}_{n}\to x in a Gmetric space (X,G) if, for any \epsilon >0, there exists k\in \mathbb{N} such that G(x,{x}_{n},{x}_{m})<\epsilon for all m,n\ge k.
Proposition 1.3 (see [1])
Let (X,G) be a Gmetric space. Then the following are equivalent:

(1)
\{{x}_{n}\} is Gconvergent to x.

(2)
G({x}_{n},{x}_{n},x)\to 0 as n\to +\mathrm{\infty}.

(3)
G({x}_{n},x,x)\to 0 as n\to +\mathrm{\infty}.

(4)
G({x}_{n},{x}_{m},x)\to 0 as n,m\to +\mathrm{\infty}.
Proposition 1.4 (see [1])
Let (X,G) be a Gmetric space. Then f:X\to X is Gcontinuous at a point x\in X if and only if it is Gsequentially continuous at x, that is, whenever \{{x}_{n}\} is Gconvergent to x, \{f({x}_{n})\} is Gconvergent to f(x).
Proposition 1.5 (see [1])
Let (X,G) be a Gmetric space. Then the function G(x,y,z) is jointly continuous in all three of its variables.
Definition 1.6 (see [31])
Let (X,G) be a Gmetric space. A mapping F:X\times X\to X is said to be continuous on X\times X if, for any two Gconvergent sequences \{{x}_{n}\} and \{{y}_{n}\} converging to x and y, respectively, \{F({x}_{n},{y}_{n})\} is Gconvergent to F(x,y).
Definition 1.7 (see [1])
A Gmetric space (X,G) is called Gcomplete if every GCauchy sequence is Gconvergent in (X,G).
Definition 1.8 A Gmetric space (X,G) is called symmetric if G(x,y,y)=G(y,x,x) for all x,y\in X.
Proposition 1.9 (see [1])

(1)
Every Gmetric space (X,G) defines a metric space (X,{d}_{G}) by {d}_{G}(x,y)=G(x,y,y)+G(y,x,x) for all x,y\in X.

(2)
If a Gmetric space (X,G) is symmetric, then {d}_{G}(x,y)=2G(x,y,y) for all x,y\in X.

(3)
However, if (X,G) is not symmetric, then it follows from Gmetric properties that
\frac{3}{2}G(x,y,y)\le {d}_{G}(x,y)\le 3G(x,y,y)
for all x,y\in X.
The concept of a mixed monotone property has been introduced by Bhaskar and Lakshmikantham in [25].
Definition 1.10 (see [25])
Let (X,\u2aaf) be a partially ordered set. A mapping F:X\times X\to X is said to have the mixed monotone property if F(x,y) is monotone nondecreasing in x and is monotone nonincreasing in y, that is, for any x,y\in X,
Lakshmikantham and Ćirić in [26] introduced the concept of a gmixed monotone mapping.
Definition 1.11 (see [26])
Let (X,\u2aaf) be a partially ordered set. Let us consider the mappings F:X\times X\to X and g:X\to X. The mapping F is said to have the mixed gmonotone property if F(x,y) is monotone gnondecreasing in x and is monotone gnonincreasing in y, that is, for any x,y\in X,
Definition 1.12 (see [25])
An element (x,y)\in X\times X is called a coupled fixed point of a mapping F:X\times X\to X if F(x,y)=x and F(y,x)=y.
Definition 1.13 (see [26])
An element (x,y)\in X\times X is called a coupled coincidence point of the mappings F:X\times X\to X and g:X\to X if F(x,y)=gx and F(y,x)=gy.
To relax the mixed monotone property, Doric et al. [29] introduced the following condition.
If the elements x, y of a partially ordered set (X,\u2aaf) are comparable (that is, x\u2aafy or y\u2aafx), then we write x\asymp y. Let F:X\times X\to X 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 [29])
Let
for all y\in X. Then F does not have the mixed monotone property since a\u2aafb and F(a,y)=b\u2ab0a=F(b,y), while c\u2aafd and F(c,y)=c\u2aafd=F(d,y). But it has the condition (1.1) since a\asymp F(a,y)=b and F(a,y)=b\asymp a=F(b,v)=F(F(a,y),v) and b\asymp a=F(b,y) and F(b,y)=a\asymp b=F(a,v)=F(F(b,y),v) (the other two cases are trivial).
Using the concepts of continuity, mixed monotone property and coupled fixed point, Choudhary and Maity [31] introduced the following theorem.
Theorem 1.15 Let (X,\u2aaf) be a partially ordered set and let G be a Gmetric on X such that (X,G) is a complete Gmetric space. Let F:X\times X\to X be a continuous mapping having the mixed monotone property on X. Assume that there exists k\in [0,1) such that, for all x,y,u,v,w,z\in X,
for all x,y,u,v,w,z\in X with x\u2ab0u\u2ab0w and y\u2aafv\u2aafz, where either u\ne w or v\ne z. If there exist {x}_{0} and {y}_{0}\in X such that {x}_{0}\u2aafF({x}_{0},{y}_{0}) and {y}_{0}\u2ab0F({y}_{0},{x}_{0}), then F has a coupled fixed point in X, that is, there exist x,y\in X such that x=F(x,y) and y=F(y,x).
In [31], Choudhary and Maity established some coupled fixed point theorems in the setting of Gmetric spaces. Starting from the results in [31], 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 [28], 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 (X,\u2aaf) be a partially ordered set and let G be a Gmetric on X such that (X,G) is a complete Gmetric space. Let F:X\times X\to X be a mapping satisfying the property (1.1). Assume that there exists k\in [0,1) such that for x,y,u,v,w,z\in X, then the following holds:
for all w\asymp u\asymp x and y\asymp v\asymp z, where either u\ne w or v\ne z. If there exist {x}_{0},{y}_{0}\in X such that
then there exists (\overline{x},\overline{y})\in X\times X such that \overline{x}=F(\overline{x},\overline{y}) and \overline{y}=F(\overline{y},\overline{x}).
Proof Consider the functional {G}_{3}:{X}^{2}\times {X}^{2}\times {X}^{2}\to {R}_{+} defined by
for all Y=(x,y),U=(u,v),V=(w,z)\in {X}^{2}. It is simple to check that {G}_{3} is a Gmetric on {X}^{2} and, moreover, if (X,G) is complete, then ({X}^{2},G) is a complete Gmetric space, too. We consider the mapping T:{X}^{2}\to {X}^{2} defined by
for all Y=(x,y)\in {X}^{2}. Clearly, for all Y=(x,y),U=(u,v),V=(w,z)\in {X}^{2}, in view of the definition of {G}_{3}, we have
and
Hence, by the contractive condition (2.1), we obtain the Banachtype contractive condition in a Gmetric space as follows:
for all Y,U,V\in {X}^{2} with Y\ge U and U\le V. Assume that (2.2) holds. Then there exist {x}_{0} and {y}_{0} in X such that
Denote {Z}_{0}=({x}_{0},{y}_{0})\in {X}^{2} and consider the Picard iteration associated to T and the initial approximation {Z}_{0}, that is, the sequence \{{Z}_{n}\}\subset {X}^{2} is defined by
for all n\ge 0, where {Z}_{n}=({x}_{n},{y}_{n})\in {X}^{2} for all n\ge 0. Since X has the condition (1.1), we have
and so, by induction,
which shows that T is monotone and the sequence \{{Z}_{n}\} is nondecreasing. We now follow the steps as in the proof of Banach’s contraction principle in a Gmetric space established by Mustafa and Sims [32]. Taking Y={Z}_{n}\ge U={Z}_{n1}=V in (2.6), we have
for all n\ge 1, which implies that
for all n\ge 1. Thus, by induction, we have
for all n\ge 1.
Now, we claim that \{{Z}_{n}\} is a Cauchy sequence in ({X}^{2},{G}_{3}). Let m<n. Then, by (2.9), we have
So, \{{Z}_{n}\} is indeed a Cauchy sequence in a complete Gmetric space ({X}^{2},{G}_{3}) and hence it is convergent. Therefore, there exists \overline{Z}\in {X}^{2} such that
Since T is continuous in ({X}^{2},{G}_{3}), by virtue of the Lipschitzian type conditions (2.1) and (2.7), it follows that \overline{Z} is a fixed point of T, that is,
Let \overline{Z}=(\overline{x},\overline{y}). Then, by the definition of T, we obtain
that is, (\overline{x},\overline{y}) 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 [31] 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 X=R and let G(x,y,z)=(xy+yz+zx) for all x,y\in X be a Gmetric defined on X. Also, let F:X\times X\to X be a mapping defined by
for all (x,y)\in {X}^{2}. Then F satisfies the conditions (2.1) and (1.1), but not (1.2) of Theorem 1.15 of [31]. Indeed, assume that there exists k, 0\le k<1, such that (1.2) holds. This means
for all x,y,u,v,w,z\in X with x\ge u\ge w and y\le v\le z. From this, in particular, for x=u=w and y=v\ne z, we get
Thus we have \frac{6}{5}\le k<1, which is a contradiction.
Now, we show that (2.1) holds. Indeed, since we have, for x=u and y=v,
and
by adding up the above two inequalities, we get exactly (2.1) with k=\frac{8}{11}<1. Also, by Theorem 2.1, we obtain that F has a unique coupled fixed point, that is, (0.0), but Theorem 1.16 cannot be applied to this example.
Now, to ensure the uniqueness of a coupled fixed point, we impose an additional condition used by Bhaskar and Lakshmikantham [25] and Ran and Reurings [33]:
Every pair of elements in {X}^{2} has either a lower bound or an upper bound, i.e., for all Y=(x,y), \overline{Y}=(\overline{x},\overline{y})\in {X}^{2},
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 {Z}^{\ast}=({x}^{\ast},{y}^{\ast})\in {X}^{2} is a coupled fixed point of F different from \overline{Z}=(\overline{x},\overline{y}). This means, by (G2), that {G}_{3}({Z}^{\ast},\overline{Z},\overline{Z})>0.
Now, we discuss two cases.
Case 1. {Z}^{\ast} is comparable to \overline{Z}. Since {Z}^{\ast} is comparable to \overline{Z} with respect to the ordering in {X}^{2}, by taking Y={Z}^{\ast} and V=U=\overline{Z} (or U=V={Z}^{\ast} and Y=\overline{Z}) in (2.6), we obtain
which is a contradiction since 0\le k<1.
Case 2. {Z}^{\ast} and \overline{Z} are not comparable. In this case, there exists an upper bound or a lower bound Z=({z}_{1},{z}_{2})\in {X}^{2} of {Z}^{\ast} and \overline{Z}. Then, in view of the monotonicity of T, {T}^{n}(Z) is comparable to {T}^{n}({Z}^{\ast})={Z}^{\ast} and {T}^{n}(\overline{Z})=\overline{Z}. Now, again, by the contractive condition (2.6), we have
as n\to \mathrm{\infty}, which leads to a contradiction. This completes the proof. □
Next, as in [28], 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 {x}_{0},{y}_{0}\in X are comparable. Then, for a coupled fixed point (\overline{x},\overline{y}), we have \overline{x}=\overline{y}, that is, F has a fixed point such that F(\overline{x},\overline{x})=\overline{x}.
Proof Consider the condition (2.2), that is,
Since {x}_{0} and {y}_{0} are comparable, we have {x}_{0}\asymp {y}_{0}. Then, by the condition (1.1) of F, we have
and hence, by induction,
for all n\ge 0. Now, since
by the continuity of the Gmetric G, we have
On the other hand, by taking Y=({x}_{n},{y}_{n}) and U=V=({y}_{n},{x}_{n}) in (2.4), we have
for all n\ge 0, which actually means that
for all n\ge 0. Therefore, we have
This completes the proof. □
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Acknowledgements
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/EMRII. 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: 20120008170).
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An erratum to this article can be found at http://dx.doi.org/10.1186/1029242X201425.
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/10.1186/1029242X201425.
An erratum to this article is available at http://dx.doi.org/10.1186/1029242X201425.
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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/10.1186/1029242X2013217
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DOI: https://doi.org/10.1186/1029242X2013217
Keywords
 partially ordered set
 Gmetric space
 coupled fixed point
 mixed monotone property