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

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.

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, αψ-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×XX 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 )andG( y 0 , x 0 )F( y 0 , x 0 ).
(1)

Suppose that there exist ϕΦ and ψΨ 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,vX with G(x,y)G(u,v) and G(y,x)G(v,u). Suppose that for any x,yX, there exist u,vX such that

F(x,y)=G(u,v)andF(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 nN,

  4. (ii)

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

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 nN, let X n be the nth Cartesian product X×X××X (n times). We employ mappings T,g:XX and F: X n X. For simplicity, if xX, 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,s0. 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 ψΨ, c>0 and define ψ c :[0,)[0,) by ψ c (t)=cψ(t/c) for all t0. Then ψ c Ψ.

Definition 2.2 (see [19, 20])

A coincidence point of two mappings T,g:XX is a point xX such that Tx=gx.

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)andF(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:XX be two mappings. We say that T is (g,) -non-decreasing if TxTy for all x,yX such that gxgy. 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 gx=gy, then Tx=Ty. It follows that

gx=gy { g x g y g y g x } { T x T y T y T x } Tx=Ty.

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

Suppose that F,G:X×XX 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,vX 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)[xu and yv].
(4)

Definition 2.8 ([17, 2325])

Let (X,d,) be an ordered metric space. Two mappings T,g:XX are said to be O-compatible if

lim m d(gT x m ,Tg 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×XX. 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 Xand 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)=|xy| for all x,yX. Consider the maps F,G:X×XX defined by

F(x,y)= 3 5 x 1 5 yandG(x,y)= x y 2 for all x,yX.

Then, for all x,y,u,vX with y=v, we have

d ( F ( x , y ) , F ( u , v ) ) = 3 5 |xu|andd ( G ( x , y ) , G ( u , v ) ) +d ( G ( y , x ) , G ( v , u ) ) =|xu|.

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:XX 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 ϕΦ and ψΨ verifying

    ϕ ( d ( T x , T y ) ) ϕ ( d ( g x , g y ) ) ψ ( d ( g x , g y ) ) for all x,yX such that gxgy.

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 ϕΦ and ψΨ 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 ϕΦ and ψΨ 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×XX 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 )andG( y 0 , x 0 )F( y 0 , x 0 ).

Suppose that there exist ϕΦ and ψΨ 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,vX with G(x,y)G(u,v) and G(y,x)G(v,u). Suppose that for any x,yX, there exist u,vX such that

F(x,y)=G(u,v)andG(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,yX 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)=|xy| for all x,yX. Consider the maps F,G:X×XX defined by

F(x,y)= 3 5 x 1 5 yandG(x,y)= x y 2 for all x,yX.

Then, for all x,y,u,vX 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 ϕΦ, then the mapping d ϕ :X×X[0,), defined by d ϕ (x,y)=ϕ(d(x,y)) for all x,yX, 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.

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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.

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Correspondence to Naseer Shahzad.

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Erhan, İ.M., Karapınar, E., Roldán-López-de-Hierro, AF. et al. Remarks on ‘Coupled coincidence point results for a generalized compatible pair with applications’. Fixed Point Theory Appl 2014, 207 (2014). https://doi.org/10.1186/1687-1812-2014-207

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

Keywords

  • fixed point
  • coupled coincidence point
  • ordered metric space