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Retracted Article: Coupled fixed point theorems without continuity and mixed monotone property

This article was retracted on 23 January 2014

Abstract

In this paper, we generalize some coupled fixed point theorems for the mixed monotone operators F:X×XX 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.

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 [115]. 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 ([1724] 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×X×X 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,yX with xy,

(G3) G(x,x,y)G(x,y,z) for all x,y,zX with yz,

(G4) G(x,y,z)=G(x,z,y)=G(y,z,x)= (symmetry in all three variables),

(G5) G(x,y,z)G(x,a,a)+G(a,y,z) for all x,y,z,aX (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])

Let (X,G) be a G-metric space and let { x n } be a sequence in X. A point xX is said to be the limit of the sequence { x n } if

lim n , m + G(x, x n , x m )=0.

We say that the sequence { x n } is G-convergent to x or { x n } G-converges to x.

Thus x n x in a G-metric space (X,G) if, for any ε>0, there exists kN such that G(x, x n , x m )<ε for all m,nk.

Proposition 1.3 (see [1])

Let (X,G) be a G-metric space. Then the following are equivalent:

  1. (1)

    { x n } is G-convergent to x.

  2. (2)

    G( x n , x n ,x)0 as n+.

  3. (3)

    G( x n ,x,x)0 as n+.

  4. (4)

    G( x n , x m ,x)0 as n,m+.

Proposition 1.4 (see [1])

Let (X,G) be a G-metric space. Then f:XX is G-continuous at a point xX 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×XX is said to be continuous on X×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,yX.

Proposition 1.9 (see [1])

  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,yX.

  2. (2)

    If a G-metric space (X,G) is symmetric, then d G (x,y)=2G(x,y,y) for all x,yX.

  3. (3)

    However, if (X,G) is not symmetric, then it follows from G-metric properties that

    3 2 G(x,y,y) d G (x,y)3G(x,y,y)

for all x,yX.

The concept of a mixed monotone property has been introduced by Bhaskar and Lakshmikantham in [25].

Definition 1.10 (see [25])

Let (X,) be a partially ordered set. A mapping F:X×XX 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,yX,

x 1 , x 2 X , x 1 x 2 F ( x 1 , y ) F ( x 2 , y ) , y 1 , y 2 X , y 1 y 2 F ( x , y 2 ) F ( x , y 1 ) .

Lakshmikantham and Ćirić in [26] introduced the concept of a g-mixed monotone mapping.

Definition 1.11 (see [26])

Let (X,) be a partially ordered set. Let us consider the mappings F:X×XX and g:XX. The mapping 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,yX,

x 1 , x 2 X , g x 1 g x 2 F ( x 1 , y ) F ( x 2 , y ) , y 1 , y 2 X , g y 1 g y 2 F ( x , y 2 ) F ( x , y 1 ) .

Definition 1.12 (see [25])

An element (x,y)X×X is called a coupled fixed point of a mapping F:X×XX if F(x,y)=x and F(y,x)=y.

Definition 1.13 (see [26])

An element (x,y)X×X is called a coupled coincidence point of the mappings F:X×XX and g:XX 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,) are comparable (that is, xy or yx), then we write xy. Let F:X×XX be a mapping. Then consider the following condition:

if x,y,vX are such that xF(x,y), then F(x,y)F ( F ( x , y ) , v ) .
(1.1)

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

X = { a , b , c , d } , = { ( a , a ) , ( b , b ) , ( c , c ) , ( d , d ) , ( a , b ) , ( c , d ) } , F : ( ( a , y ) ( b , y ) ( c , y ) ( d , y ) a b c d )

for all yX. Then F does not have the mixed monotone property since ab and F(a,y)=ba=F(b,y), while cd and F(c,y)=cd=F(d,y). But it has the condition (1.1) since aF(a,y)=b and F(a,y)=ba=F(b,v)=F(F(a,y),v) and ba=F(b,y) and F(b,y)=ab=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,) 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×XX be a continuous mapping having the mixed monotone property on X. Assume that there exists k[0,1) such that, for all x,y,u,v,w,zX,

G ( F ( x , y ) , F ( u , v ) , F ( w , z ) ) k 2 [ G ( x , u , w ) + G ( y , v , z ) ]
(1.2)

for all x,y,u,v,w,zX with xuw and yvz, where either uw or vz. If there exist x 0 and y 0 X such that x 0 F( x 0 , y 0 ) and y 0 F( y 0 , x 0 ), then F has a coupled fixed point in X, that is, there exist x,yX 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,) 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×XX be a mapping satisfying the property (1.1). Assume that there exists k[0,1) such that for x,y,u,v,w,zX, then the following holds:

G ( F ( x , y ) , F ( u , v ) , F ( w , z ) ) + G ( F ( y , x ) , F ( v , u ) , F ( z , w ) ) k [ G ( x , u , w ) + G ( y , v , z ) ]
(2.1)

for all wux and yvz, where either uw or vz. If there exist x 0 , y 0 X such that

x 0 F( x 0 , y 0 )andF( y 0 , x 0 ) y 0 ,
(2.2)

then there exists ( x ¯ , y ¯ )X×X such that x ¯ =F( x ¯ , y ¯ ) and y ¯ =F( y ¯ , x ¯ ).

Proof Consider the functional G 3 : X 2 × X 2 × X 2 R + defined by

G 3 (Y,U,V)= 1 2 [ G ( x , u , w ) + G ( y , v , z ) ]
(2.3)

for all Y=(x,y),U=(u,v),V=(w,z) X 2 . It is simple to check that G 3 is a 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 X 2 defined by

T(Y)= ( F ( x , y ) , F ( y , x ) )
(2.4)

for all Y=(x,y) X 2 . Clearly, for all Y=(x,y),U=(u,v),V=(w,z) X 2 , in view of the definition of G 3 , we have

G 3 ( T ( Y ) , T ( U ) , T ( V ) ) = G ( F ( x , y ) , F ( u , v ) , F ( w , z ) ) + G ( F ( y , x ) , F ( v , u ) , F ( z , w ) ) 2

and

G 3 (Y,U,V)= G ( x , u , w ) + G ( y , v , z ) 2 .

Hence, by the contractive condition (2.1), we obtain the Banach-type contractive condition in a G-metric space as follows:

G 3 ( T ( Y ) , T ( U ) , T ( V ) ) k G 3 (Y,U,V)
(2.5)

for all Y,U,V X 2 with YU and UV. Assume that (2.2) holds. Then there exist x 0 and y 0 in X such that

x 0 F( x 0 , y 0 )andF( y 0 , x 0 ) y 0 .

Denote Z 0 =( x 0 , y 0 ) X 2 and consider the Picard iteration associated to T and the initial approximation Z 0 , that is, the sequence { Z n } X 2 is defined by

Z n + 1 =T( Z n )
(2.6)

for all n0, where Z n =( x n , y n ) X 2 for all n0. Since X has the condition (1.1), we have

Z 0 =( x 0 , y 0 ) ( F ( x 0 , y 0 ) , F ( y 0 , x 0 ) ) =( x 1 , y 1 )= Z 1 ,

and so, by induction,

Z n =( x n , y n ) ( F ( x n , y n ) , F ( y n , x n ) ) =( x n + 1 , y n + 1 )= Z n + 1 ,

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

G 3 ( T ( Z n ) , T ( Z n 1 ) , T ( Z n 1 ) ) k G 3 ( Z n , Z n 1 , Z n 1 )
(2.7)

for all n1, which implies that

G 3 ( Z n + 1 , Z n , Z n )k G 3 ( Z n , Z n 1 , Z n 1 )
(2.8)

for all n1. Thus, by induction, we have

G 3 ( Z n + 1 , Z n , Z n ) k n G 3 ( Z 1 , Z 0 , Z 0 )
(2.9)

for all n1.

Now, we claim that { Z n } is a Cauchy sequence in ( X 2 , G 3 ). Let m<n. Then, by (2.9), we have

G 3 ( Z n , Z m , Z m ) i = m + 1 n G 3 ( Z i , Z i 1 , Z i 1 ) ( k m + k m + 1 + + k n m 1 ) G 3 ( Z 1 , Z 0 , Z 0 ) k n 1 k n m 1 1 k G 3 ( Z 1 , Z 0 , z 0 ) .

So, { Z n } is indeed a Cauchy sequence in a complete G-metric space ( X 2 , G 3 ) and hence it is convergent. Therefore, there exists Z ¯ X 2 such that

lim n Z n = Z ¯ .

Since T is continuous in ( X 2 , G 3 ), by virtue of the Lipschitzian type conditions (2.1) and (2.7), it follows that Z ¯ is a fixed point of T, that is,

T( Z ¯ )= Z ¯ .

Let Z ¯ =( x ¯ , y ¯ ). Then, by the definition of T, we obtain

x ¯ =F( x ¯ , y ¯ )and y ¯ =F( y ¯ , x ¯ ),

that is, ( x ¯ , 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,yX be a G-metric defined on X. Also, let F:X×XX be a mapping defined by

F(x,y)= x + 3 y 5

for all (x,y) 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, 0k<1, such that (1.2) holds. This means

G ( F ( x , y ) , F ( u , v ) , F ( w , z ) ) = G ( x + 3 y 5 , u + 3 v 5 , w + 3 z 5 ) = | x + 3 y 5 u + 3 v 5 | + | u + 3 v 5 w + 3 z 5 | + | w + 3 z 5 x + 3 y 5 | k 2 [ | x u | + | y v | + | u w | + | v z | + | w x | + | z y | ]

for all x,y,u,v,w,zX with xuw and yvz. From this, in particular, for x=u=w and y=vz, we get

6 5 |vz|k|vz|.

Thus we have 6 5 k<1, which is a contradiction.

Now, we show that (2.1) holds. Indeed, since we have, for x=u and y=v,

2 5 |u+3vw3z| 2 5 |uw|+ 6 5 |vz|

and

2 5 |v+3uz3w| 2 5 |vz|+ 6 5 |uw|,

by adding up the above two inequalities, we get exactly (2.1) with k= 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), Y ¯ =( x ¯ , y ¯ ) X 2 ,

there exists Z=( z 1 , z 2 ) X 2  that is comparable to Y and  Y ¯ .
(2.10)

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 =( x , y ) X 2 is a coupled fixed point of F different from Z ¯ =( x ¯ , y ¯ ). This means, by (G2), that G 3 ( Z , Z ¯ , Z ¯ )>0.

Now, we discuss two cases.

Case 1. Z is comparable to Z ¯ . Since Z is comparable to Z ¯ with respect to the ordering in X 2 , by taking Y= Z and V=U= Z ¯ (or U=V= Z and Y= Z ¯ ) in (2.6), we obtain

G 3 ( T ( Z ) , T ( Z ¯ ) , T ( Z ¯ ) ) = G 3 ( Z , Z ¯ , Z ¯ ) k G 3 ( Z , Z ¯ , Z ¯ ) ,

which is a contradiction since 0k<1.

Case 2. Z and Z ¯ are not comparable. In this case, there exists an upper bound or a lower bound Z=( z 1 , z 2 ) X 2 of Z and Z ¯ . Then, in view of the monotonicity of T, T n (Z) is comparable to T n ( Z )= Z and T n ( Z ¯ )= Z ¯ . Now, again, by the contractive condition (2.6), we have

G 3 ( Z , Z ¯ , Z ¯ ) = G 3 ( T n ( Z ) , T n ( Z ¯ ) , T n ( Z ¯ ) ) G 3 ( T n ( Z ) , T n ( Z ) , T n ( Z ) ) + G 3 ( T n ( Z ) , T n ( Z ¯ ) , T n ( Z ¯ ) ) k n [ G 3 ( Z , Z , Z ) ) + G 3 ( Z , Z ¯ , Z ¯ ) 0

as n, 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 X are comparable. Then, for a coupled fixed point ( x ¯ , y ¯ ), we have x ¯ = y ¯ , that is, F has a fixed point such that F( x ¯ , x ¯ )= x ¯ .

Proof Consider the condition (2.2), that is,

x 0 F( x 0 , y 0 )and y 0 F( y 0 , x 0 ).

Since x 0 and y 0 are comparable, we have x 0 y 0 . Then, by the condition (1.1) of F, we have

x 1 =F( x 0 , y 0 )F( y 0 , x 0 )= y 1

and hence, by induction,

x n y n
(2.11)

for all n0. Now, since

x ¯ = lim n F( x n , y n ); y ¯ = lim n F( y n , x n ),

by the continuity of the G-metric G, we have

G ( x ¯ , y ¯ , y ¯ ) = G ( lim n F ( x n , y n ) , lim n F ( y n , x n ) , lim n F ( y n , x n ) ) = lim n G ( F ( x n , y n ) , F ( y n , x n ) , F ( y n , x n ) ) = lim n G ( F ( x n + 1 , y n + 1 ) , y n + 1 ) .

On the other hand, by taking Y=( x n , y n ) and U=V=( y n , x n ) in (2.4), we have

G ( F ( x n , y n ) , F ( y n , x n ) , F ( y n , x n ) ) kG( x n , y n , y n )

for all n0, which actually means that

G( x n + 1 , y n + 1 , y n + 1 )kG( x n , y n , y n )

for all n0. Therefore, we have

G( x ¯ , y ¯ , y ¯ )= lim n G( x n + 1 , y n + 1 , y n + 1 ) lim n k n G( x 1 , y 1 , y 1 )=0.

This completes the proof. □

References

  1. Mustafa Z, Sims B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 2006, 7: 289–297.

    MathSciNet  MATH  Google Scholar 

  2. 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 

  3. 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 139

    Google Scholar 

  4. Abbas M, Khan AR, Nazir T: Coupled common fixed point results in two generalized metric spaces. Appl. Math. Comput. 2011. 10.1016/j.amc.2011.01.006

    Google Scholar 

  5. 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.2009.04.085

    MathSciNet  Article  MATH  Google Scholar 

  6. 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.2011.05.059

    MathSciNet  Article  MATH  Google Scholar 

  7. Dhage BC: Generalized metric space and mapping with fixed point. Bull. Calcutta Math. Soc. 1992, 84: 329–336.

    MathSciNet  MATH  Google Scholar 

  8. Dhage BC: Generalized metric spaces and topological structure I. An. ştiinţ. Univ. “Al.I. Cuza” Iaşi, Mat. 2000, 46: 3–24.

    MathSciNet  MATH  Google Scholar 

  9. Dhage BC: On generalized metric spaces and topological structure II. Pure Appl. Math. Sci. 1994, 40(1–2):37–41.

    MathSciNet  MATH  Google Scholar 

  10. Dhage BC: On continuity of mappings in D -metric spaces. Bull. Calcutta Math. Soc. 1994, 86(6):503–508.

    MathSciNet  MATH  Google Scholar 

  11. 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 189870

    Google Scholar 

  12. Mustafa Z, Sims B: Fixed point theorems for contractive mappings in complete G -metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 917175

    Google Scholar 

  13. Mustafa Z, Shatanawi W, Bataineh M: Existence of fixed point results in G -metric spaces. Int. J. Math. Math. Sci. 2009., 2009: Article ID 283028

    Google Scholar 

  14. Shatanawi W: Fixed point theory for contractive mappings satisfying Φ-maps in G -metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 181650

    Google Scholar 

  15. Shatanawi W: Partially ordered cone metric spaces and coupled fixed point results. Comput. Math. Appl. 2010, 60: 2508–2515. 10.1016/j.camwa.2010.08.074

    MathSciNet  Article  MATH  Google Scholar 

  16. 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.2010.05.009

    MathSciNet  Article  MATH  Google Scholar 

  17. 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 31

    Google Scholar 

  18. 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 8

    Google Scholar 

  19. 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.2011.02.004

    MathSciNet  Article  MATH  Google Scholar 

  20. 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 150363

    Google Scholar 

  21. 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 

  22. 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 194

    Google Scholar 

  23. 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 81

    Google Scholar 

  24. Sintunavarat W, Kumam P: Coupled coincidence and coupled common fixed point theorems in partially ordered metric spaces. Thai J. Math. 2012, 10: 551–563.

    MathSciNet  MATH  Google Scholar 

  25. Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017

    MathSciNet  Article  MATH  Google Scholar 

  26. 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.2008.09.020

    MathSciNet  Article  MATH  Google Scholar 

  27. 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.2010.06.025

    MathSciNet  Article  MATH  Google Scholar 

  28. 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.2011.07.053

    MathSciNet  Article  MATH  Google Scholar 

  29. Doric, D, Kadelburg, Z, Radenovic, S: Coupled fixed point theorems for mappings without mixed monotone property. Appl. Math. Lett. (in press)

  30. 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 22

    Google Scholar 

  31. Choudhury BS, Maity P: Coupled fixed point results in generalized metric spaces. Math. Comput. Model. 2011. 10.1016/j.mcm.2011.01.036

    Google Scholar 

  32. Mustafa, Z: A new structure for generalized metric spaces with applications to fixed point theory. PhD thesis, The University of Newcastle, Callaghan, Australia (2005)

  33. 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/S0002-9939-03-07220-4

    MathSciNet  Article  MATH  Google Scholar 

  34. 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.2010.05.042

    MathSciNet  Article  MATH  Google Scholar 

  35. Aydi H, Samet B, Vetro C:Coupled fixed point results in cone metric spaces for w ˜ -compatible mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 27 10.1186/1687-1812-2011-27

    Google Scholar 

  36. Chugh R, Kadian T, Rani A, Rhoades BE: Property P in G -metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 401684

    Google Scholar 

  37. 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 131294

    Google Scholar 

  38. 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.2009.08.029

    MathSciNet  Article  MATH  Google Scholar 

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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/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).

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Correspondence to Yeol Je Cho.

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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

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Keywords

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