Open Access

Remarks on ‘Coupled coincidence point results for a generalized compatible pair with applications’

  • İnci M Erhan1,
  • Erdal Karapınar1, 2,
  • Antonio-Francisco Roldán-López-de-Hierro3 and
  • Naseer Shahzad4Email author
Fixed Point Theory and Applications20142014:207

https://doi.org/10.1186/1687-1812-2014-207

Received: 19 May 2014

Accepted: 6 August 2014

Published: 13 October 2014

Abstract

Very recently, Hussain et al. (Fixed Point Theory Appl. 2014:62, 2014) announced the existence and uniqueness of some coupled coincidence point. In this short note we remark that the announced results can be derived from the coincidence point results in the literature.

MSC: 47H10, 54H25.

Keywords

fixed pointcoupled coincidence pointordered metric space

1 Introduction

Recently, a number of studies related to fixed points, coupled fixed points and coupled coincidence points of maps defined via auxiliary functions have appeared in the literature. In particular, the so-called weak φ-contractions, contractions defined by means of altering distance functions, α ψ -type contractions have been a subject of considerable interest. Studies of this type aim to generalize and improve contractive condition on the maps (see, e.g., [115]).

A great deal of these studies investigate contractions on partially ordered metric spaces because of their applicability to initial value problems defined by differential or integral equations. This is the case of the following result.

Theorem 1.1 (Hussain et al. [16], Theorem 15)

Let ( X , ) be a partially ordered set such that there exists a complete metric d on X. Assume that F , G : X × X X are two generalized compatible mappings such that F is G-increasing with respect to , G is continuous and has the mixed monotone property, and there exist two elements x 0 , y 0 X such that
G ( x 0 , y 0 ) F ( x 0 , y 0 ) and G ( y 0 , x 0 ) F ( y 0 , x 0 ) .
(1)
Suppose that there exist ϕ Φ and ψ Ψ such that
ϕ ( d ( F ( x , y ) , F ( u , v ) ) ) 1 2 ϕ ( d ( G ( x , y ) , G ( u , v ) ) + d ( G ( y , x ) , G ( v , u ) ) ) ψ ( d ( G ( x , y ) , G ( u , v ) ) + d ( G ( y , x ) , G ( v , u ) ) 2 )
(2)
for all x , y , u , v X with G ( x , y ) G ( u , v ) and G ( y , x ) G ( v , u ) . Suppose that for any x , y X , there exist u , v X such that
F ( x , y ) = G ( u , v ) and F ( y , x ) = G ( v , u ) .
(3)
Also suppose that either
  1. (a)

    F is continuous, or

     
  2. (b)

    X has the following property:

     
  3. (i)

    if a -non-decreasing sequence { x n } x , then x n x for all n N ,

     
  4. (ii)

    if a -non-increasing sequence { y n } y , then y y n for all n N .

     

Then F and G have a coupled coincidence point in X.

In this paper we show that the previous result can be easily improved because of the following facts.
  1. (1)

    The mixed monotone property is not necessary since F is G-increasing with respect to .

     
  2. (2)

    It is possible to consider a pair of mappings satisfying a weaker condition than the generalized compatible property (using monotone sequences).

     
  3. (3)

    In fact, Theorem 1.1 is not a true advance because it can be reduced to its corresponding unidimensional coincidence point theorem.

     

To prove our main claims, we will show a unidimensional proof of the mentioned theorem.

2 Preliminaries

Firstly, we recall some basic definitions and elementary results needed throughout the paper. Some of them can be found in [17]. In the sequel, we denote by X a nonempty set. Given a natural number n N , let X n be the nth Cartesian product X × X × × X (n times). We employ mappings T , g : X X and F : X n X . For simplicity, if x X , we denote T ( x ) by Tx.

Definition 2.1 (Khan et al. [18])

An altering distance function is a continuous, non-decreasing function ϕ : [ 0 , ) [ 0 , ) such that ϕ ( t ) = 0 if and only if t = 0 . Let F alt denote the family of all altering distance functions.

A function ϕ : [ 0 , ) [ 0 , ) is said to be subadditive if ϕ ( t + s ) ϕ ( t ) + ϕ ( s ) for all t , s 0 . Following [16], we introduce the following families of control functions. Let Φ denote the family of all subadditive altering distance functions, that is, functions ϕ : [ 0 , ) [ 0 , ) which satisfy the following:

( ϕ 1 ) ϕ is continuous and non-decreasing;

( ϕ 2 ) ϕ ( t ) = 0 if and only if t = 0 ;

( ϕ 3 ) ϕ ( t + s ) ϕ ( t ) + ϕ ( s ) for all t , s [ 0 , ) .

We denote by Ψ the family of all functions ψ : [ 0 , ) [ 0 , ) which satisfy the following:
  1. (1)

    lim t r ψ ( t ) > 0 for all r > 0 ;

     
  2. (2)

    lim t 0 + ψ ( t ) = 0 .

     

Remark 2.1 Let ψ Ψ , c > 0 and define ψ c : [ 0 , ) [ 0 , ) by ψ c ( t ) = c ψ ( t / c ) for all t 0 . Then ψ c Ψ .

Definition 2.2 (see [19, 20])

A coincidence point of two mappings T , g : X X is a point x X such that T x = g x .

Definition 2.3 (Hussain et al. [16], Definition 10)

A coupled coincidence point of two mappings F , G : X 2 X is a point ( x , y ) X such that
F ( x , y ) = G ( x , y ) and F ( y , x ) = G ( y , x ) .

Definition 2.4 An ordered metric space ( X , d , ) is a metric space ( X , d ) provided with a partial order .

Definition 2.5 ([16, 21])

An ordered metric space ( X , d , ) is said to be non-decreasing-regular (respectively, non-increasing-regular) if for every sequence { x m } X such that { x m } x and x m x m + 1 (respectively, x m x m + 1 ) for all m, we have that x m x (respectively, x m x ) for all m. ( X , d , ) is said to be regular if it is both non-decreasing-regular and non-increasing-regular.

Remark 2.2 Notice that condition (b) in Theorem 1.1 means that ( X , d , ) is regular.

Definition 2.6 Let ( X , ) be a partially ordered set, and let T , g : X X be two mappings. We say that T is ( g , ) -non-decreasing if T x T y for all x , y X such that g x g y . If g is the identity mapping on X, we say that T is -non-decreasing.

Remark 2.3 If T is ( g , ) -non-decreasing and g x = g y , then T x = T y . It follows that
g x = g y { g x g y g y g x } { T x T y T y T x } T x = T y .

Definition 2.7 (Hussain et al. [16], Definition 7)

Suppose that F , G : X × X X are two mappings, and let be a partial order on X. The mapping F is said to be G-increasing with respect to if for all x , y , u , v X with G ( x , y ) G ( u , v ) we have F ( x , y ) F ( u , v ) .

Lemma 2.1 (see [22])

Let ( X , d ) be a metric space and define Δ n : X n × X n [ 0 , ) , for all A = ( a 1 , a 2 , , a n ) , B = ( b 1 , b 2 , , b n ) X n , by
Δ n ( A , B ) = i = 1 n d ( a i , b i ) .

Then Δ n is metric on X n and ( X , d ) is complete if and only if ( X , Δ n ) is complete.

Consider on the product space X 2 the following partial order: for ( x , y ) , ( u , v ) X 2 ,
( x , y ) ( u , v ) [ x u  and  y v ] .
(4)

Definition 2.8 ([17, 2325])

Let ( X , d , ) be an ordered metric space. Two mappings T , g : X X are said to be O-compatible if
lim m d ( g T x m , T g x m ) = 0
provided that { x m } is a sequence in X such that { g x m } is -monotone, that is, it is either non-increasing or non-decreasing with respect to , and
lim m T x m = lim m g x m X .

Definition 2.9 (Hussain et al. [16], Definition 12)

Let F , G : X × X X . We say that the pair { F , G } is generalized compatible if for all sequences { x n } , { y n } X such that
lim n F ( x n , y n ) = lim n G ( x n , y n ) = t 1 X and lim n F ( y n , x n ) = lim n G ( y n , x n ) = t 2 X ,
we have that
lim n d ( F ( G ( x n , y n ) , G ( y n , x n ) ) , G ( F ( x n , y n ) , F ( y n , x n ) ) ) = 0 and lim n d ( F ( G ( y n , x n ) , G ( x n , y n ) ) , G ( F ( y n , x n ) , F ( x n , y n ) ) ) = 0 .

3 Main results

To start with, we highlight the weakness of Theorem 1.1 using the following example.

Example 3.1 Let X = [ 0 , ) endowed with the standard metric d ( x , y ) = | x y | for all x , y X . Consider the maps F , G : X × X X defined by
F ( x , y ) = 3 5 x 1 5 y and G ( x , y ) = x y 2 for all  x , y X .
Then, for all x , y , u , v X with y = v , we have
d ( F ( x , y ) , F ( u , v ) ) = 3 5 | x u | and d ( G ( x , y ) , G ( u , v ) ) + d ( G ( y , x ) , G ( v , u ) ) = | x u | .
Thus,
d ( F ( x , y ) , F ( u , v ) ) > 1 2 ( d ( G ( x , y ) , G ( u , v ) ) + d ( G ( y , x ) , G ( v , u ) ) ) .
Regarding the properties of the functions in Φ, we derive that
φ ( d ( F ( x , y ) , F ( u , v ) ) ) > 1 2 φ ( d ( G ( x , y ) , G ( u , v ) ) + d ( G ( y , x ) , G ( v , u ) ) ) .

Since the function in the class Ψ takes values on [ 0 , ) , it is impossible to verify inequality (2). Hence, Theorem 1.1 cannot be applied to get a coupled coincidence point. However, it is easy to see that ( 0 , 0 ) is a coupled coincidence point of F and G.

Next, we show a unidimensional version of Theorem 1.1. Notice that, indeed, the following result is better than Theorem 1.1 because we reorder the hypotheses obtaining that, in some cases, neither the continuity of, at least, one mapping (T or g) nor the O-compatibility of the pair ( T , g ) is necessary. In fact, both hypotheses are omitted in case (c).

Theorem 3.1 Let ( X , d , ) be an ordered metric space, and let T , g : X X be two mappings such that the following properties are fulfilled:
  1. (i)

    T ( X ) g ( X ) ;

     
  2. (ii)

    T is ( g , ) -non-decreasing;

     
  3. (iii)

    there exists x 0 X such that g x 0 T x 0 ;

     
  4. (iv)
    there exist ϕ Φ and ψ Ψ verifying
    ϕ ( d ( T x , T y ) ) ϕ ( d ( g x , g y ) ) ψ ( d ( g x , g y ) ) for all  x , y X  such that  g x g y .
     
Also assume that, at least, one of the following conditions holds.
  1. (a)

    ( X , d ) is complete, T and g are continuous and the pair ( T , g ) is O-compatible;

     
  2. (b)

    ( X , d ) is complete and T and g are continuous and commuting;

     
  3. (c)

    ( g ( X ) , d ) is complete and ( X , d , ) is non-decreasing-regular;

     
  4. (d)

    ( X , d ) is complete, g ( X ) is closed and ( X , d , ) is non-decreasing-regular;

     
  5. (e)

    ( X , d ) is complete, g is continuous and monotone -non-decreasing, the pair ( T , g ) is O-compatible and ( X , d , ) is non-decreasing-regular.

     

Then T and g have, at least, a coincidence point.

We omit the proof of the previous result since its proof is similar to the main theorem in [17] and it can be concluded by following, point by point, all of its arguments.

Next, we show how to deduce an appropriate version of Theorem 1.1 from Theorem 3.1. Given the ordered metric space ( X , d , ) , let us consider the ordered metric space ( X 2 , Δ 2 , ) , where Δ 2 was defined in Lemma 2.1 and was introduced in (4). We define the mappings T F , T G : X 2 X 2 , for all ( x , y ) X 2 , by
T F ( x , y ) = ( F ( x , y ) , F ( y , x ) ) and T G ( x , y ) = ( G ( x , y ) , G ( y , x ) ) .

Under these conditions, the following properties hold.

Lemma 3.1 Let ( X , d , ) be an ordered metric space, and let F , G : X 2 X be two mappings. Then the following properties hold.
  1. (1)

    ( X , d ) is complete if and only if ( X 2 , Δ 2 ) is complete.

     
  2. (2)

    If ( X , d , ) is regular, then ( X 2 , Δ 2 , ) is also regular.

     
  3. (3)

    If F is d-continuous, then T F is Δ 2 -continuous.

     
  4. (4)

    If F is G-increasing with respect to , then T F is ( T G , ) -non-decreasing.

     
  5. (5)

    Condition (1) is equivalent to the existence of a point ( x 0 , y 0 ) X 2 such that T G ( x 0 , y 0 ) T F ( x 0 , y 0 ) .

     
  6. (6)

    Condition (3) is equivalent to T F ( X 2 ) T G ( X 2 ) .

     
  7. (7)
    If there exist ϕ Φ and ψ Ψ such that (2) holds, then
    ϕ ( Δ 2 ( T F ( x , y ) , T F ( u , v ) ) ) ϕ ( Δ 2 ( T G ( x , y ) , T G ( u , v ) ) ) ψ 2 ( Δ 2 ( T G ( x , y ) , T G ( u , v ) ) )
     
for all ( x , y ) , ( u , v ) X 2 such that T G ( x , y ) T G ( u , v ) , where ψ 2 Ψ was defined in Remark  2.1.
  1. (8)

    If the pair { F , G } is generalized compatible, then the mappings T F and T G are O-compatible in ( X 2 , Δ 2 , ) .

     
  2. (9)

    A point ( x , y ) X 2 is a coupled coincidence point of F and G if and only if it is a coincidence point of T F and T G .

     
Proof Item (1) follows from Lemma 2.1 and items (2), (3), (5), (6) and (9) are obvious.
  1. (4)

    Assume that F is G-increasing with respect to , and let ( x , y ) , ( u , v ) X 2 be such that T G ( x , y ) T G ( u , v ) . Then G ( x , y ) G ( u , v ) and G ( y , x ) G ( v , u ) . Since F is G-increasing with respect to , we deduce that F ( x , y ) F ( u , v ) and F ( y , x ) F ( v , u ) . Therefore, T F ( x , y ) T F ( u , v ) and this means that T F is ( T G , ) -non-decreasing.

     
  2. (7)
    Suppose that there exist ϕ Φ and ψ Ψ such that (2) holds, and let ( x , y ) , ( u , v ) X 2 be such that T G ( x , y ) T G ( u , v ) . Therefore G ( x , y ) G ( u , v ) and G ( y , x ) G ( v , u ) . Using (2), we have that
    ϕ ( d ( F ( x , y ) , F ( u , v ) ) ) 1 2 ϕ ( d ( G ( x , y ) , G ( u , v ) ) + d ( G ( y , x ) , G ( v , u ) ) ) ψ ( d ( G ( x , y ) , G ( u , v ) ) + d ( G ( y , x ) , G ( v , u ) ) 2 ) .
    (5)
     
Furthermore, taking into account that G ( v , u ) G ( y , x ) and G ( u , v ) G ( x , y ) , the contractivity condition (2) also guarantees that
ϕ ( d ( F ( v , u ) , F ( y , x ) ) ) 1 2 ϕ ( d ( G ( v , u ) , G ( y , x ) ) + d ( G ( u , v ) , G ( x , y ) ) ) ψ ( d ( G ( v , u ) , G ( y , x ) ) + d ( G ( u , v ) , G ( x , y ) ) 2 ) .
(6)
Since ϕ is subadditive, it follows from (5) and (6) that
ϕ ( Δ 2 ( T F ( x , y ) , T F ( u , v ) ) ) = ϕ ( Δ 2 [ ( F ( x , y ) , F ( y , x ) ) , ( F ( u , v ) , F ( v , u ) ) ] ) = ϕ ( d ( F ( x , y ) , F ( u , v ) ) + d ( F ( y , x ) , F ( v , u ) ) ) ϕ ( d ( F ( x , y ) , F ( u , v ) ) ) + ϕ ( d ( F ( y , x ) , F ( v , u ) ) ) ϕ ( d ( G ( x , y ) , G ( u , v ) ) + d ( G ( y , x ) , G ( v , u ) ) ) 2 ψ ( d ( G ( v , u ) , G ( y , x ) ) + d ( G ( u , v ) , G ( x , y ) ) 2 ) = ϕ ( Δ 2 ( T G ( x , y ) , T G ( u , v ) ) ) ψ 2 ( Δ 2 ( T G ( x , y ) , T G ( u , v ) ) ) .
  1. (8)
    Let { ( x m , y m ) } X 2 be any sequence such that { T F ( x m , y m ) } Δ 2 ( x , y ) and { T G ( x m , y m ) } Δ 2 ( x , y ) (notice that we do not need to suppose that { T G ( x m , y m ) } is -monotone). Therefore,
    { ( F ( x m , y m ) , F ( y m , x m ) ) } Δ 2 ( x , y ) [ { F ( x m , y m ) } d x  and  { F ( y m , x m ) } d y ] ; { ( G ( x m , y m ) , G ( y m , x m ) ) } Δ 2 ( x , y ) [ { G ( x m , y m ) } d x  and  { G ( y m , x m ) } d y ] .
     
Therefore
lim m F ( x m , y m ) = lim m G ( x m , y m ) = x X and lim m F ( y m , x m ) = lim m G ( y m , x m ) = y X .
Since the pair { F , G } is generalized compatible, we deduce that
lim m d ( F ( G ( x m , y m ) , G ( y m , x m ) ) , G ( F ( x m , y m ) , F ( y m , x m ) ) ) = 0 and lim m d ( F ( G ( y m , x m ) , G ( x m , y m ) ) , G ( F ( y m , x m ) , F ( x m , y m ) ) ) = 0 .
In particular,
lim m Δ 2 ( T G T F ( x m , y m ) , T F T G ( x m , y m ) ) = lim m Δ 2 ( T G ( F ( x m , y m ) , F ( y m , x m ) ) , T F ( G ( x m , y m ) , G ( y m , x m ) ) ) = lim m Δ 2 ( ( G ( F ( x m , y m ) , F ( y m , x m ) ) , G ( F ( y m , x m ) , F ( x m , y m ) ) ) , ( F ( G ( x m , y m ) , G ( y m , x m ) ) , F ( G ( y m , x m ) , G ( x m , y m ) ) ) ) = lim m [ d ( ( G ( F ( x m , y m ) , F ( y m , x m ) ) , F ( G ( x m , y m ) , G ( y m , x m ) ) ) ) + d ( ( G ( F ( y m , x m ) , F ( x m , y m ) ) , F ( G ( y m , x m ) , G ( x m , y m ) ) ) ) ] = 0 .

Hence, the mappings T F and T G are O-compatible in ( X 2 , Δ 2 , ) . □

As a consequence, we conclude that Hussain et al.’s result can be deduced from the corresponding unidimensional result. Furthermore, as we have pointed out, it is not necessary for G to have the mixed monotone property because F is G-increasing with respect to .

Corollary 3.1 Theorem  1.1, even avoiding the assumption that G has the mixed monotone property, is a consequence of Theorem  3.1.

Proof It is only necessary to apply Theorem 3.1 to the mappings T = T F and g = T G in the ordered metric space ( X 2 , Δ 2 , ) , taking into account all items of Lemma 3.1. □

The following result is an improved version of Theorem 1.1 in which the contractivity condition is replaced by a more convenient one, which is symmetric on the variables ( x , y ) and ( u , v ) .

Corollary 3.2 Let ( X , ) be a partially ordered set such that there exists a complete metric d on X. Assume that F , G : X × X X are two generalized compatible mappings such that F is G-increasing with respect to , G is continuous and there exist two elements x 0 , y 0 X such that
G ( x 0 , y 0 ) F ( x 0 , y 0 ) and G ( y 0 , x 0 ) F ( y 0 , x 0 ) .
Suppose that there exist ϕ Φ and ψ Ψ such that
ϕ ( d ( F ( x , y ) , F ( u , v ) ) + d ( F ( y , x ) , F ( v , u ) ) 2 ) ϕ ( d ( G ( x , y ) , G ( u , v ) ) + d ( G ( y , x ) + G ( v , u ) ) 2 ) ψ ( d ( G ( x , y ) , G ( u , v ) ) + d ( G ( y , x ) + G ( v , u ) ) 2 )
(7)
for all x , y , u , v X with G ( x , y ) G ( u , v ) and G ( y , x ) G ( v , u ) . Suppose that for any x , y X , there exist u , v X such that
F ( x , y ) = G ( u , v ) and G ( y , x ) F ( y , x ) .
Also assume that either
  1. (a)

    F is continuous, or

     
  2. (b)

    ( X , d , ) is regular.

     

Then F and G have, at least, a coupled coincidence point, that is, there exist x , y X such that G ( x , y ) = F ( x , y ) and G ( y , x ) = F ( y , x ) .

Proof It is only necessary to apply Theorem 3.1 to the mappings T = T F and g = T G in the ordered metric space ( X 2 , Δ 2 , ) , where Δ 2 = Δ 2 / 2 , taking into account all items of Lemma 3.1. □

In the following example we show that Corollary 3.2 is applicable to the mappings of Example 3.1, when Theorem 1.1 is not useful.

Example 3.2 Let X = [ 0 , ) endowed with the Euclidean metric d ( x , y ) = | x y | for all x , y X . Consider the maps F , G : X × X X defined by
F ( x , y ) = 3 5 x 1 5 y and G ( x , y ) = x y 2 for all  x , y X .
Then, for all x , y , u , v X with y = v , we have
d ( F ( x , y ) , F ( u , v ) ) + d ( F ( y , x ) , F ( v , u ) ) = 4 5 ( | x u | + | y v | ) and d ( G ( x , y ) , G ( u , v ) ) + d ( G ( y , x ) , G ( v , u ) ) = | x u | + | y v | .
Thus,
d ( F ( x , y ) , F ( u , v ) ) < d ( G ( x , y ) , G ( u , v ) ) + d ( G ( y , x ) , G ( v , u ) ) .
Regarding the properties of the functions in Φ, we derive that
ϕ ( d ( F ( x , y ) , F ( u , v ) ) + d ( F ( y , x ) , F ( v , u ) ) 2 ) ϕ ( d ( G ( x , y ) , G ( u , v ) ) + d ( G ( y , x ) + G ( v , u ) ) 2 ) .

To provide inequality (7), it is sufficient to choose ψ ( t ) = t 10 . Hence, Theorem 3.2 can be applied in order to guarantee that F and G have a coupled coincidence point. Indeed, it is easy to check that ( 0 , 0 ) is a coupled coincidence point of F and G.

To finish the paper, we want to point out a pair of details.
  1. (1)

    The function in ϕ in Theorem 3.1 is not a true generalization because if ϕ Φ , then the mapping d ϕ : X × X [ 0 , ) , defined by d ϕ ( x , y ) = ϕ ( d ( x , y ) ) for all x , y X , is also a metric on X. For more details, see [26]. Notice also that the assumption of sub-additivity ( ϕ 2 ) is superfluous in most of the published results (see, e.g., [27]).

     
  2. (2)

    Using the same techniques that can be found in [17, 22, 2831], it is possible to deduce, from Theorem 3.1, tripled, quadrupled and, in general, multidimensional coincidence point theorems.

     

Declarations

Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. N Shahzad acknowledges with thanks DSR for financial support. Antonio Roldán has been partially supported by Junta de Andalucía by project FQM-268 of the Andalusian CICYE.

Authors’ Affiliations

(1)
Department of Mathematics, Atilim University, Incek, Ankara, Turkey
(2)
Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia
(3)
Department of Mathematics, University of Jaén, Jaén, Spain
(4)
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

References

  1. Luong NV, Thuan NX: Coupled fixed points in partially ordered metric spaces and application. Nonlinear Anal. 2011, 74: 983–992. 10.1016/j.na.2010.09.055View ArticleMathSciNetGoogle Scholar
  2. Abbas M, Rhoades BE: Common fixed point results for noncommuting mappings without continuity in generalized metric spaces. Appl. Math. Comput. 2009, 215: 262–269. 10.1016/j.amc.2009.04.085View ArticleMathSciNetGoogle Scholar
  3. Haghi RH, Rezapour S, Shahzad N: Some fixed point generalizations are not real generalizations. Nonlinear Anal. 2011, 74: 1799–1803. 10.1016/j.na.2010.10.052View ArticleMathSciNetGoogle Scholar
  4. Berinde V: Coupled fixed point theorems for ϕ -contractive mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal. 2012, 75: 3218–3228. 10.1016/j.na.2011.12.021View ArticleMathSciNetGoogle Scholar
  5. Bilgili N, Karapınar E, Turkoglu D:A note on common fixed points for ( ψ , α , β ) -weakly contractive mappings in generalized metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 287Google Scholar
  6. Harjani J, Sadarangani K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal. 2010, 72: 1188–1197. 10.1016/j.na.2009.08.003View ArticleMathSciNetGoogle Scholar
  7. Turkoglu D, Sangurlu M: Coupled fixed point theorems for mixed g -monotone mappings in partially ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 348Google Scholar
  8. Roldán A, Martínez-Moreno J, Roldán C: Multidimensional fixed point theorems in partially ordered complete metric spaces. J. Math. Anal. Appl. 2012, 396(2):536–545. 10.1016/j.jmaa.2012.06.049View ArticleMathSciNetGoogle Scholar
  9. Roldán A, Martínez-Moreno J, Roldán C, Karapınar E:Multidimensional fixed point theorems in partially ordered complete partial metric spaces under ( ψ , φ ) -contractivity conditions. Abstr. Appl. Anal. 2013., 2013: Article ID 634371Google Scholar
  10. Karapınar E, Roldán A, Martínez-Moreno J, Roldán C: Meir-Keeler type multidimensional fixed point theorems in partially ordered metric spaces. Abstr. Appl. Anal. 2013., 2013: Article ID 406026Google Scholar
  11. Pathak HK, Shahzad N: Fixed point results for set-valued contractions by altering distances in complete metric spaces. Nonlinear Anal. 2009, 70(7):2634–2641. 10.1016/j.na.2008.03.050View ArticleMathSciNetGoogle Scholar
  12. Alshehri S, Aranđelović I, Shahzad N: Symmetric spaces and fixed points of generalized contractions. Abstr. Appl. Anal. 2014., 2014: Article ID 763547Google Scholar
  13. Asl JH, Rezapour S, Shahzad N: On fixed points of α - ψ -contractive multifunctions. Fixed Point Theory Appl. 2012., 2012: Article ID 212 10.1186/1687-1812-2012-212Google Scholar
  14. Mohammadi B, Rezapour S, Shahzad N: Some results on fixed points of α - ψ -Ćirić generalized multifunctions. Fixed Point Theory Appl. 2013., 2013: Article ID 24 10.1186/1687-1812-2013-24Google Scholar
  15. Alikhani H, Rezapour S, Shahzad N: Fixed points of a new type contractive mappings and multifunctions. Filomat 2013, 27(7):1315–1319. 10.2298/FIL1307315AView ArticleMathSciNetGoogle Scholar
  16. Hussain N, Abbas M, Azam A, Ahmad J: Coupled coincidence point results for a generalized compatible pair with applications. Fixed Point Theory Appl. 2014., 2014: Article ID 62Google Scholar
  17. Al-Mezel SA, Alsulami H, Karapınar E, Roldán A: Discussion on ‘Multidimensional coincidence points’ via recent publications. Abstr. Appl. Anal. 2014., 2014: Article ID 287492. http://www.hindawi.com/journals/aaa/aip/287492/Google Scholar
  18. Khan MS, Swaleh M, Sessa S: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 1984, 30: 1–9. 10.1017/S0004972700001659View ArticleMathSciNetGoogle Scholar
  19. Goebel K: A coincidence theorem. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 1968, 16: 733–735.MathSciNetGoogle Scholar
  20. Agarwal RP, Bisht RK, Shahzad N: A comparison of various noncommuting conditions in metric fixed point theory and their applications. Fixed Point Theory Appl. 2014., 2014: Article ID 38Google Scholar
  21. Bhaskar TG, Lakshmikantham V: Fixed point theory in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017View ArticleMathSciNetGoogle Scholar
  22. Roldán A, Martínez-Moreno J, Roldán C, Karapınar E: Some remarks on multidimensional fixed point theorems. Fixed Point Theory Appl. 2013., 2013: Article ID 158Google Scholar
  23. 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.2010.06.025View ArticleMathSciNetGoogle Scholar
  24. Luong NV, Thuan NX: Coupled points in ordered generalized metric spaces and application to integro-differential equations. Comput. Math. Appl. 2011, 62(11):4238–4248. 10.1016/j.camwa.2011.10.011View ArticleMathSciNetGoogle Scholar
  25. Hung NM, Karapınar E, Luong NV: Coupled coincidence point theorem for O -compatible mappings via implicit relation. Abstr. Appl. Anal. 2012., 2012: Article ID 796964Google Scholar
  26. Karapınar E, Samet B: A note on ψ -Geraghty contractions. Fixed Point Theory Appl. 2014., 2014: Article ID 26Google Scholar
  27. Karapınar E: A discussion on α - ψ Geraghty contracton type mappings and some related fixed point results. Filomat 2014, 28(1):37–48. 10.2298/FIL1401037KView ArticleMathSciNetGoogle Scholar
  28. Karapınar E, Roldán A: A note on ‘ n -Tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces’. J. Inequal. Appl. 2013., 2013: Article ID 567Google Scholar
  29. Karapınar E, Roldán A, Roldán C, Martínez-Moreno J: A note on ‘ N -Fixed point theorems for nonlinear contractions in partially ordered metric spaces’. Fixed Point Theory Appl. 2013., 2013: Article ID 310Google Scholar
  30. Karapınar E, Roldán A, Shahzad N, Sintunavarat W: Discussion on coupled and tripled coincidence point theorems for ϕ -contractive mappings without the mixed g -monotone property. Fixed Point Theory Appl. 2014., 2014: Article ID 92Google Scholar
  31. Samet B, Karapınar E, Aydi H, Rajic VC: Discussion on some coupled fixed point theorems. Fixed Point Theory Appl. 2013., 2013: Article ID 50Google Scholar

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© Erhan et al.; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.