# Retracted Article: Coupled fixed point theorems without continuity and mixed monotone property

- Ramesh Kumar Vats
^{1}, - Vizender Sihag
^{1}and - Yeol Je Cho
^{2}Email author

**2013**:217

https://doi.org/10.1186/1029-242X-2013-217

© Vats et al.; licensee Springer 2013

**Received: **17 October 2012

**Accepted: **16 April 2013

**Published: **30 April 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. 74:7347-7355, 2011). The example demonstrates that our main result is an actual improvement over the results which are generalized.

**MSC:**47H10, 54H25.

### Keywords

partially ordered set*G*-metric space coupled fixed point mixed monotone property

## 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 [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 g-monotone 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 *G*-metric 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 *G*-metric 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 *G-metric* on *X* and the pair $(X,G)$ is called a *G-metric space*.

**Definition 1.2** (see [1])

*G*-metric 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 *G-convergent* to *x* or $\{{x}_{n}\}$ *G-converges* to *x*.

Thus ${x}_{n}\to x$ in a *G*-metric 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*

*G*-

*metric space*.

*Then the following are equivalent*:

- (1)
$\{{x}_{n}\}$

*is**G*-*convergent 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* *G*-*metric space*. *Then* $f:X\to X$ *is* *G*-*continuous at a point* $x\in X$ *if and only if it is* *G*-*sequentially continuous at* *x*, *that is*, *whenever* $\{{x}_{n}\}$ *is* *G*-*convergent to* *x*, $\{f({x}_{n})\}$ *is* *G*-*convergent to* $f(x)$.

**Proposition 1.5** (see [1])

*Let* $(X,G)$ *be a* *G*-*metric 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 G-metric space. A mapping $F:X\times X\to X$ is said to be *continuous* on $X\times X$ if, for any two *G*-convergent sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ converging to *x* and *y*, respectively, $\{F({x}_{n},{y}_{n})\}$ is *G*-convergent to $F(x,y)$.

**Definition 1.7** (see [1])

A *G*-metric space $(X,G)$ is called *G-complete* if every *G*-Cauchy sequence is *G*-convergent in $(X,G)$.

**Definition 1.8** A *G*-metric 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**G*-*metric 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**G*-*metric 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**G*-*metric 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])

*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 *g*-mixed monotone mapping.

**Definition 1.11** (see [26])

*F*is said to have the

*mixed*

*g-monotone property*if $F(x,y)$ is monotone

*g*-nondecreasing in

*x*and is monotone

*g*-nonincreasing 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.

*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])

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*

*G*-

*metric on*

*X*

*such that*$(X,G)$

*is a complete*

*G*-

*metric 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 *G*-metric 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*

*G*-

*metric on*

*X*

*such that*$(X,G)$

*is a complete G*-

*metric 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

*G*-metric on ${X}^{2}$ and, moreover, if $(X,G)$ is complete, then $({X}^{2},G)$ is a complete

*G*-metric space, too. We consider the mapping $T:{X}^{2}\to {X}^{2}$ defined by

*G*-metric space as follows:

*T*and the initial approximation ${Z}_{0}$, that is, the sequence $\{{Z}_{n}\}\subset {X}^{2}$ is defined by

*X*has the condition (1.1), we have

*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

*G*-metric space established by Mustafa and Sims [32]. Taking $Y={Z}_{n}\ge U={Z}_{n-1}=V$ in (2.6), we have

for all $n\ge 1$.

*G*-metric space $({X}^{2},{G}_{3})$ and hence it is convergent. Therefore, there exists $\overline{Z}\in {X}^{2}$ such that

*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,

*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 $G(x,y,z)=(|x-y|+|y-z|+|z-x|)$ for all $x,y\in X$ be a

*G*-metric defined on

*X*. Also, let $F:X\times X\to X$ be a mapping defined by

*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

Thus we have $\frac{6}{5}\le k<1$, which is a contradiction.

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]:

*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.

which is a contradiction since $0\le k<1$.

*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,

*F*, we have

*G*-metric

*G*, we have

This completes the proof. □

## Notes

## Declarations

### 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/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).

## Authors’ Affiliations

## References

- Mustafa Z, Sims B: A new approach to generalized metric spaces.
*J. Nonlinear Convex Anal.*2006, 7: 289–297.MathSciNetMATHGoogle Scholar - Mustafa Z, Sims B: Some remarks concerning
*D*-metric spaces. In*Proceedings of the International Conference on Fixed Point Theory Appl.*. Yokohama Publ., Yokohama; 2004:189–198.Google Scholar - Abbas M, Cho YJ, Nazir T: Common fixed points of Ćirić-type contractive mappings in two ordered generalized metric spaces.
*Fixed Point Theory Appl.*2012., 2012: Article ID 139Google Scholar - Abbas M, Khan AR, Nazir T: Coupled common fixed point results in two generalized metric spaces.
*Appl. Math. Comput.*2011. 10.1016/j.amc.2011.01.006Google Scholar - Abbas M, Rhoades BE: Common fixed point results for non-commuting mappings without continuity in generalized metric spaces.
*Appl. Math. Comput.*2009, 215: 262–269. 10.1016/j.amc.2009.04.085MathSciNetView ArticleMATHGoogle Scholar - Aydi H, Damjanović B, Samet B, Shatanawi W: Coupled fixed point theorems for nonlinear contractions in partially ordered
*G*-metric spaces.*Math. Comput. Model.*2011, 54: 2443–2450. 10.1016/j.mcm.2011.05.059MathSciNetView ArticleMATHGoogle Scholar - Dhage BC: Generalized metric space and mapping with fixed point.
*Bull. Calcutta Math. Soc.*1992, 84: 329–336.MathSciNetMATHGoogle Scholar - Dhage BC: Generalized metric spaces and topological structure I.
*An. ştiinţ. Univ. “Al.I. Cuza” Iaşi, Mat.*2000, 46: 3–24.MathSciNetMATHGoogle Scholar - Dhage BC: On generalized metric spaces and topological structure II.
*Pure Appl. Math. Sci.*1994, 40(1–2):37–41.MathSciNetMATHGoogle Scholar - Dhage BC: On continuity of mappings in
*D*-metric spaces.*Bull. Calcutta Math. Soc.*1994, 86(6):503–508.MathSciNetMATHGoogle Scholar - Mustafa Z, Obiedat H, Awawdeh F: Some fixed point theorem for mapping on complete
*G*-metric spaces.*Fixed Point Theory Appl.*2008., 2008: Article ID 189870Google Scholar - Mustafa Z, Sims B: Fixed point theorems for contractive mappings in complete
*G*-metric spaces.*Fixed Point Theory Appl.*2009., 2009: Article ID 917175Google Scholar - Mustafa Z, Shatanawi W, Bataineh M: Existence of fixed point results in
*G*-metric spaces.*Int. J. Math. Math. Sci.*2009., 2009: Article ID 283028Google Scholar - Shatanawi W: Fixed point theory for contractive mappings satisfying Φ-maps in
*G*-metric spaces.*Fixed Point Theory Appl.*2010., 2010: Article ID 181650Google Scholar - Shatanawi W: Partially ordered cone metric spaces and coupled fixed point results.
*Comput. Math. Appl.*2010, 60: 2508–2515. 10.1016/j.camwa.2010.08.074MathSciNetView ArticleMATHGoogle Scholar - Saadati R, Vaezpour SM, Vetro P, Rhoades BE: Fixed point theorems in generalized partially ordered
*G*-metric spaces.*Math. Comput. Model.*2010, 52: 797–801. 10.1016/j.mcm.2010.05.009MathSciNetView ArticleMATHGoogle Scholar - Abbas M, Sintunavarat W, Kumam P: Coupled fixed points of generalized contractive mappings on partially ordered
*G*-metric spaces.*Fixed Point Theory Appl.*2012., 2012: Article ID 31Google Scholar - Cho YJ, Rhoades BE, Saadati R, Samet B, Shantawi W: Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type.
*Fixed Point Theory Appl.*2012., 2012: Article ID 8Google Scholar - Cho YJ, Shah MH, Hussain N: Coupled fixed points of weakly
*F*-contractive mappings in topological spaces.*Appl. Math. Lett.*2011, 24: 1185–1190. 10.1016/j.aml.2011.02.004MathSciNetView ArticleMATHGoogle Scholar - Eshaghi Gordji M, Cho YJ, Ghods S, Ghods M, Hadian Dehkordi H: Coupled fixed-point theorems for contractions in partial ordered metric spaces and applications.
*Math. Probl. Eng.*2012., 2012: Article ID 150363Google Scholar - Huang NJ, Fang YP, Cho YJ: Fixed point and coupled fixed point theorems for multi-valued increasing operators in ordered metric spaces. 3. In
*Fixed Point Theory and Applications*. Edited by: Cho YJ, Kim JK, Kang SM. Nova Science Publishers, New York; 2002:91–98.Google Scholar - Karapinar E, Kumam P, Sintunavarat W: Coupled fixed point theorems in cone metric spaces with a
*c*-distance and applications.*Fixed Point Theory Appl.*2012., 2012: Article ID 194Google Scholar - Sintunavarat W, Cho YJ, Kumam P: Coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces.
*Fixed Point Theory Appl.*2011., 2011: Article ID 81Google Scholar - Sintunavarat W, Kumam P: Coupled coincidence and coupled common fixed point theorems in partially ordered metric spaces.
*Thai J. Math.*2012, 10: 551–563.MathSciNetMATHGoogle Scholar - Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications.
*Nonlinear Anal.*2006, 65: 1379–1393. 10.1016/j.na.2005.10.017MathSciNetView ArticleMATHGoogle Scholar - Lakshmikantham V, Ćirić L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces.
*Nonlinear Anal.*2009, 70: 4341–4349. 10.1016/j.na.2008.09.020MathSciNetView ArticleMATHGoogle Scholar - Choudhury BS, Kundu A: A coupled coincidence point result in partially ordered metric spaces for compatible mappings.
*Nonlinear Anal.*2010, 73: 2524–2531. 10.1016/j.na.2010.06.025MathSciNetView ArticleMATHGoogle Scholar - Berinde V: Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces.
*Nonlinear Anal.*2011, 74: 7347–7355. 10.1016/j.na.2011.07.053MathSciNetView ArticleMATHGoogle Scholar - Doric, D, Kadelburg, Z, Radenovic, S: Coupled fixed point theorems for mappings without mixed monotone property. Appl. Math. Lett. (in press)Google Scholar
- Agarwal RP, Sintunavarat W, Kumam P: Coupled coincidence point and common coupled fixed point theorems lacking the mixed monotone property.
*Fixed Point Theory Appl.*2013., 2013: Article ID 22Google Scholar - Choudhury BS, Maity P: Coupled fixed point results in generalized metric spaces.
*Math. Comput. Model.*2011. 10.1016/j.mcm.2011.01.036Google Scholar - Mustafa, Z: A new structure for generalized metric spaces with applications to fixed point theory. PhD thesis, The University of Newcastle, Callaghan, Australia (2005)Google Scholar
- Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations.
*Proc. Am. Math. Soc.*2004, 132: 1435–1443. 10.1090/S0002-9939-03-07220-4MathSciNetView ArticleMATHGoogle Scholar - Abbas M, Khan MA, Radenović S: Common coupled fixed point theorem in cone metric space for
*w*-compatible mappings.*Appl. Math. Comput.*2010, 217: 195–202. 10.1016/j.amc.2010.05.042MathSciNetView ArticleMATHGoogle Scholar - Aydi H, Samet B, Vetro C:Coupled fixed point results in cone metric spaces for $\tilde{w}$-compatible mappings.
*Fixed Point Theory Appl.*2011., 2011: Article ID 27 10.1186/1687-1812-2011-27Google Scholar - Chugh R, Kadian T, Rani A, Rhoades BE: Property
*P*in*G*-metric spaces.*Fixed Point Theory Appl.*2010., 2010: Article ID 401684Google Scholar - Ćirić L, Cakić N, Rajović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces.
*Fixed Point Theory Appl.*2008., 2008: Article ID 131294Google Scholar - Ćirić L, Mihet D, Saadati R: Monotone generalized contractions in partially ordered probabilistic metric spaces.
*Topol. Appl.*2009, 156(17):2838–2844. 10.1016/j.topol.2009.08.029MathSciNetView ArticleMATHGoogle Scholar

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