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On coupled coincidence point theorems on partially ordered G-metric spaces without mixed g-monotone

Abstract

In this work, we prove the existence of a coupled coincidence point theorem of nonlinear contraction mappings in G-metric spaces without the mixed g-monotone property and give some examples of a nonlinear contraction mapping, which is not applied to the existence of coupled coincidence point by using the mixed monotone property. We also show the uniqueness of a coupled coincidence point of the given mapping. Further, we apply our results to the existence and uniqueness of a coupled coincidence point of the given mapping in partially ordered G-metric spaces.

1 Introduction

The existence of a fixed point for the contraction type of mappings in partially ordered metric spaces has been studied by Ran and Reurings [1] and they established some new results for contractions in partially ordered metric spaces and presented applications to matrix equations. Later, Nieto and Rodriguez-Lopez [2, 3] and Agarwal et al. [4] presented some new results for contractions in partially ordered metric spaces. Examples of extensions and applications of these works see in [59].

The concept of coupled fixed point was introduced by Guo and Lakshmikantham [10]. Later, Bhaskar and Lakshmikantham [11] introduced the concept of mixed monotone property for contractive operators in partially ordered metric spaces. They also gave some applications in the existence and uniqueness of the coupled fixed point theorems for mappings which satisfy the mixed monotone property. Lakshimikantham and Ćirić [12] extended the results in [11] by defining the mixed g-monotone and to study the existence and uniqueness of coupled coincidence point for such mapping which satisfy the mixed monotone property in partially ordered metric space. As a continuation of this work, many authors conducted research on the coupled fixed point theory and coupled coincidence point theory in partially ordered metric spaces and different spaces. For example see [1347].

Recently, Sintunavarat et al. [45, 46] proved some coupled fixed point theorems for nonlinear contractions without mixed monotone property and extended some coupled fixed point theorems of Bhaskar and Lakshmikantham [11] by using the concept of F-invariant set due to Samet and Vetro [48]. Later, in 2013, Batra and Vashistha [19] introduced the concept of (F,g)-invariant set which is a generalization of an F-invariant set introduced by Samet and Vetro [48] and proved theorems on the existence of coupled fixed points for nonlinear contractions under c-distance in cone metric spaces having an (F,g)-invariant subset. Very recently, Charoensawan and Klanarong [25] proved theorems on the existence of coupled coincidence points in partially ordered metric spaces without mixed g-monotone property which extended some coupled fixed point theorems of Sintunavarat et al. [45]. They also proved uniqueness of coupled common fixed point theorems for nonlinear contractions.

In 2006, Mustafa and Sims [49] introduced the notion of a G-metric spaces as a generalization of the concept of a metric space and proved the analog of the Banach contraction mapping principle in the context of G-metric spaces. Following this initial research, many authors discussed research on the fixed point theory in partially ordered G-metric space (see, e.g., [5066]).

Recently, Jleli and Samet [54] showed the weakness of the fixed point theory in G-metric by introducing the concept of a quasi-metric space and showed that the result of Mustafa et al. [57] can be deduced by some well-known results in the literature in the setting of a usual (quasi) metric space. Later, Samet et al. [63] established some propositions to show that many fixed point theorems on (nonsymmetric) G-metric spaces given recently by many authors follow directly from well-known theorems on metric spaces. However, Karapinar and Agarwal [55] noticed that the techniques used in [54, 63] are valid if the contraction condition in the statement of the theorem can be expressed in two variables and they proved some theorems on the existence and uniqueness of a common fixed point for which the techniques of the papers [54, 63] are not applicable.

In recent times, coupled fixed point and coupled coincidence point theory has been developed in partially ordered G-metric space. Some authors have studied coupled fixed point theory. For example, Choudhury and Maity [27] proved the existence of a coupled fixed point theorem of nonlinear contraction mappings with mixed monotone property in partially ordered G-metric space. Later, Abbas et al. [13] extended the results of a coupled fixed point theorem for a mixed monotone mapping obtained by Choudhury and Maity [27].

On the other hand, some authors have studied coupled coincidence point theory in partially ordered G-metric space. In 2011, Aydi et al. [15] established coupled coincidence and coupled common fixed point results for a mixed g-monotone mapping satisfying nonlinear contractions in a partially ordered G-metric space. They generalized the results obtained by Choudhury and Maity [27]. Later, Karapinar et al. [34] extended the results of coupled coincidence and coupled common fixed point theorem for a mixed g-monotone mapping obtained by Aydi et al. [15]. As a continuation of this trend, many authors have studied coupled coincidence point and coupled common fixed point results for a mixed g-monotone mapping satisfying nonlinear contractions in a partially ordered G-metric space (see, for example, [1618, 24, 26, 28, 34, 43, 44, 50, 66]). However, very recently, Agarwal and Karapinar [50] introduced the concept of g-ordered completeness and showed that the weaknesses of some of the coupled fixed point theorems and coupled coincidence point theorems in [13, 15, 26, 27, 43, 67] are in fact immediate consequences of well-known fixed point theorems in the literature.

In this work, we generalize and extend the coupled coincidence point theorem of nonlinear contraction mappings in partially ordered G-metric spaces without the mixed g-monotone property.

2 Preliminaries

In this section, we give some definitions, proposition, examples, and remarks which are useful for the main results in this paper. Throughout this paper, (X,) denotes a partially ordered set with the partial order ≤. By xy, we mean yx. A mapping f:XX is said to be non-decreasing (resp., non-increasing) if, for all x,yX, xy implies f(x)f(y) (resp. f(y)f(x)).

Definition 2.1 [49]

Let X be a nonempty set, and 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 specially, a G-metric on X, and the pair (X,G) is called a G-metric space.

Example 2.2 Let (X,d) be a metric space. The function G:X×X×X[0,+), defined by G(x,y,z)=d(x,y)+d(y,z)+d(z,x), for all x,y,zX, is a G-metric space on X.

Definition 2.3 [49]

Let (X,G) be a G-metric space, and let ( x n ) be a sequence of points of X. We say that ( x n ) is G-convergent to xX if lim n , m G(x, x n , x m )=0, that is, for any ε>0, there exists NN such that G(x, x n , x m )<ε, for all n,mN. We call x the limit of the sequence and write x n x or lim n x n =x.

Proposition 2.4 [49]

Let (X,G) be a G-metric space; 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+.

Definition 2.5 [49]

Let (X,G) be a G-metric space. A sequence ( x n ) is called a G-Cauchy sequence if, for any ε>0, there exists NN such that G( x n , x m , x l )<ε, for all n,m,lN. That is, G( x n , x m , x l )0 as n,m,l+.

Proposition 2.6 [49]

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

  1. (1)

    the sequence ( x n ) is G-Cauchy;

  2. (2)

    for any ε>0, there exists NN such that G( x n , x m , x m )<ε, for all n,mN.

Proposition 2.7 [49]

Let (X,G) be a G-metric space. A mapping f:XX is G-continuous at 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).

Definition 2.8 [49]

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

Definition 2.9 [27]

Let (X,G) be a G-metric space. A mapping F:X×XX is said to be continuous 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).

The concept of a mixed monotone property and a coupled fixed point have been introduced by Bhaskar and Lakshmikantham in [11].

Definition 2.10 [11]

Let (X,) be a partially ordered set and F:X×XX. We say F has the mixed monotone property if for any x,yX

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

and

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

Definition 2.11 [11]

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.

Lakshmikantham and Ćirić in [12] introduced the concept of a mixed g-monotone mapping and a coupled coincidence point.

Definition 2.12 [12]

Let (X,) be a partially ordered set and F:X×XX and g:XX. We say F has the mixed g-monotone property if for any x,yX

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

and

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

Definition 2.13 [12]

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

Definition 2.14 [12]

Let X be a nonempty set and F:X×XX and g:XX. We say F and g are commutative if g(F(x,y))=F(g(x),g(y)) for all x,yX.

Now, we give the notion of an F -invariant set and an ( F ,g)-invariant set which is useful for our main results.

Definition 2.15 Let (X,d) be a metric space and F:X×XX be mapping. Let M be a nonempty subset of X 6 . We say that M is an F -invariant subset of X 6 if and only if, for all x,y,z,u,v,wX,

  1. 1.

    (x,u,y,v,z,w)M(w,z,v,y,u,x)M;

  2. 2.

    (x,u,y,v,z,w)M(F(x,u),F(u,x),F(y,v),F(v,y),F(z,w),F(w,z))M.

Definition 2.16 Let (X,d) be a metric space and F:X×XX and g:XX are given mapping. Let M be a nonempty subset of X 6 . We say that M is an ( F ,g)-invariant subset of X 6 if and only if, for all x,y,z,u,v,wX,

  1. 1.

    (x,u,y,v,z,w)M(w,z,v,y,u,x)M;

  2. 2.

    (g(x),g(u),g(y),g(v),g(z),g(w))M(F(x,u),F(u,x),F(y,v),F(v,y),F(z,w),F(w,z))M.

Definition 2.17 Let (X,d) be a metric space and M be a subset of X 6 . We say that M satisfies the transitive property if and only if, for all x,y,w,z,a,b,c,d,e,fX,

(x,y,w,z,a,b)Mand(a,b,c,d,e,f)M(x,y,w,z,e,f)M.

Remark

  1. 1.

    The set M= X 6 is trivially ( F ,g)-invariant, which satisfies the transitive property.

  2. 2.

    Every F -invariant set is ( F , I X )-invariant when I X denote identity map on X.

Example 2.18 Let (X,) be a partially ordered set and suppose there is a metric d on X such that (X,d) is a complete metric space. Let F:X×XX and g:XX be a mapping satisfying the mixed g-monotone property. Define a subset M X 6 by M={(a,b,c,d,e,f) X 6 ,ace,bdf}. Then M is an ( F ,g)-invariant subset of X 6 , which satisfies the transitive property.

Example 2.19 Let X=R and F:X×XX be defined by F(x,y)=1 x 2 . Let g:XX be given by g(x)=x1. Then it is easy to show that M={(x,0,0,0,0,w) X 6 :x=w} is an ( F ,g)-invariant subset of X 6 but not an F -invariant subset of X 6 as (1,0,0,0,0,1)M but (F(1,0),F(0,1),F(0,0),F(0,0),F(0,1),F(1,0))=(0,1,1,1,1,0)M.

Let Φ denote the set of functions ϕ:[0,)[0,) satisfying

  1. 1.

    ϕ 1 ({0})={0},

  2. 2.

    ϕ(t)<t for all t>0,

  3. 3.

    lim r t + ϕ(r)<t for all t>0.

Lemma 2.20 [12]

Let ϕΦ. For all t>0, we have lim n ϕ n (t)=0.

Karapinar et al. [34] proved the following theorem.

Theorem 2.21 [34]

Let (X,) be a partially ordered set and G be a G-metric on X such that (X,G) is a complete G-metric space. Suppose that there exist ϕΦ, F:X×XX, and g:XX such that

[ G ( F ( x , u ) , F ( y , v ) , F ( z , w ) ) + G ( F ( u , x ) , F ( v , y ) , F ( w , z ) ) ] ϕ ( G ( g ( x ) , g ( y ) , g ( z ) ) + G ( g ( u ) , g ( v ) , g ( w ) ) )

for all x,y,z,u,v,wX for which g(x)g(y)g(z) and g(u)g(v)g(w).

Suppose also that F is continuous and has the mixed g-monotone property, F(X×X)G(X) and g is continuous and commutes with F. If there exist x 0 , y 0 X such that

g( x 0 )F( x 0 , y 0 )andg( y 0 )F( y 0 , x 0 ),

then there exist (x,y)X×X such that g(x)=F(x,y) and g(y)=F(y,x).

Definition 2.22 [34]

Let (X,) be a partially ordered set and G be a G-metric on X. We say that (X,G,) is regular if the following conditions hold:

  1. 1.

    if a non-decreasing sequence ( x n )x, then x n x for all n,

  2. 2.

    if a non-increasing sequence ( y n )y, then y y n for all n.

Theorem 2.23 [34]

Let (X,) be a partially ordered set and G be a G-metric on X such that (X,G,) is regular. Suppose that there exist ϕΦ, F:X×XX, and g:XX such that

[ G ( F ( x , u ) , F ( y , v ) , F ( z , w ) ) + G ( F ( u , x ) , F ( v , y ) , F ( w , z ) ) ] ϕ ( G ( g ( x ) , g ( y ) , g ( z ) ) + G ( g ( u ) , g ( v ) , g ( w ) ) )

for all x,y,z,u,v,wX for which g(x)g(y)g(z) and g(u)g(v)g(w).

Suppose also that (g(X),G) is complete, F has the mixed g-monotone property, F(X×X)G(X), and g is continuous and commutes with F. If there exist x 0 , y 0 X such that

g( x 0 )F( x 0 , y 0 )andg( y 0 )F( y 0 , x 0 ),

then there exist (x,y)X×X such that g(x)=F(x,y) and g(y)=F(y,x).

The purpose of this paper is to present some coupled coincidence point theorems without a mixed g-monotone, using the concept of ( F ,g)-invariant set in complete metric space which are generalizations of the results of Karapinar et al. [34].

3 Main results

Theorem 3.1 Let (X,) be a partially ordered set and G be a G-metric on X such that (X,G) is a complete G-metric space and M be a nonempty subset of X 6 . Assume that there exists ϕΦ and, also, suppose that F:X×XX and g:XX such that

[ G ( F ( x , u ) , F ( y , v ) , F ( z , w ) ) + G ( F ( u , x ) , F ( v , y ) , F ( w , z ) ) ] ϕ ( G ( g ( x ) , g ( y ) , g ( z ) ) + G ( g ( u ) , g ( v ) , g ( w ) ) )
(1)

for all (g(x),g(u),g(y),g(v),g(z),g(w))M.

Suppose also that F is continuous, F(X×X)G(X) and g is continuous and commutes with F. If there exist x 0 , y 0 X×X such that

( F ( x 0 , y 0 ) , F ( y 0 , x 0 ) , F ( x 0 , y 0 ) , F ( y 0 , x 0 ) , g ( x 0 ) , g ( y 0 ) ) M

and M is an ( F ,g)-invariant set which satisfies the transitive property. Then there exist x,yX such that g(x)=F(x,y) and g(y)=F(y,x).

Proof Let ( x 0 , y 0 )X×X. Since F(X×X)g(X), we can choose x 1 , y 1 X such that

g( x 1 )=F( x 0 , y 0 )andg( y 1 )=F( y 0 , x 0 ).

Again from F(X×X)g(X) we can choose x 2 , y 2 X such that

g( x 2 )=F( x 1 , y 1 )andg( y 2 )=F( y 1 , x 1 ).

Continuing this process we can construct sequences {g( x n )} and {g( y n )} in X such that

g( x n )=F( x n 1 , y n 1 )andg( y n )=F( y n 1 , x n 1 )for all n1.
(2)

If there exists kN such that (g( x k + 1 ),g( y k + 1 ))=(g( x k ),g( y k )) then g( x k )=g( x k + 1 )=F( x k , y k ) and g( y k )=g( y k + 1 )=F( y k , x k ). Thus, ( x k , y k ) is a coupled coincidence point of F. The proof is completed.

Now we assume that (g( x k + 1 ),g( y k + 1 ))(g( x k ),g( y k )) for all n0. Thus, we have either g( x n + 1 )=F( x n , y n )g( x n ) or g( y n + 1 )=F( y n , x n )g(y) for all n0. Since

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

and M is an ( F ,g)-invariant set, we have

( F ( x 1 , y 1 ) , F ( y 1 , x 1 ) , F ( x 1 , y 1 ) , F ( y 1 , x 1 ) , F ( x 0 , y 0 ) , F ( y 0 , x 0 ) ) = ( g ( x 2 ) , g ( y 2 ) , g ( x 2 ) , g ( y 2 ) , g ( x 1 ) , g ( y 1 ) ) M .

Again, using the fact that M is an ( F ,g)-invariant set, we have

( F ( x 2 , y 2 ) , F ( y 2 , x 2 ) , F ( x 2 , y 2 ) , F ( y 2 , x 2 ) , F ( x 1 , y 1 ) , F ( y 1 , x 1 ) ) = ( g ( x 3 ) , g ( y 3 ) , g ( x 3 ) , g ( y 3 ) , g ( x 2 ) , g ( y 2 ) ) M .

By repeating this argument, we get

( F ( x n , y n ) , F ( y n , x n ) , F ( x n , y n ) , F ( y n , x n ) , F ( x n 1 , y n 1 ) , F ( y n 1 , x n 1 ) ) = ( g ( x n ) , g ( y n ) , g ( x n ) , g ( y n ) , g ( x n 1 ) , g ( y n 1 ) ) M .
(3)

From (1), (2), and (3), we have

[ G ( g ( x n + 1 ) , g ( x n + 1 ) , g ( x n ) ) + G ( g ( y n + 1 ) , g ( y n + 1 ) , g ( y n ) ) ] = G ( F ( x n , y n ) , F ( x n , y n ) , F ( x n 1 , y n 1 ) ) + G ( F ( y n , x n ) , F ( y n , x n ) , F ( y n 1 , x x 1 ) ) ϕ ( G ( g ( x n ) , g ( x n ) , g ( x n 1 ) ) + G ( g ( y n ) , g ( y n ) , g ( y n 1 ) ) ) .
(4)

Let

t n =G ( g ( x n + 1 ) , g ( x n + 1 ) , g ( x n ) ) +G ( g ( y n + 1 ) , g ( y n + 1 ) , g ( y n ) ) .
(5)

This implies that

t n ϕ( t n 1 ).
(6)

Since ϕ(t)<t for all t>0, it follows that { t n } is decreasing sequence. Therefore, there is some δ0 such that lim n t n =δ.

We shall prove that δ=0. Assume, to the contrary, that δ>0. Then by letting n in (6) and using the properties of the map ϕ, we get

δ= lim n t n lim n ϕ( t n 1 )= lim t n 1 δ + ϕ( t n 1 )<δ.

This is a contradiction. Thus δ=0 and hence

lim n t n = lim n [ G ( g ( x n + 1 ) , g ( x n + 1 ) , g ( x n ) ) + G ( g ( y n + 1 ) , g ( y n + 1 ) , g ( y n ) ) ] =0.
(7)

Next, we prove that {g( x n )} and {g( y n )} are Cauchy sequences in the G-metric space (X,G). Suppose, to the contrary, that the least of {g( x n )} and {g( y n )} is not a Cauchy sequence in (X,G). Then there exists an ε>0 for which we can find subsequences {g( x m ( k ) )} and {g( x n ( k ) )} of {g( x n )}, {g( y m ( k ) )} and {g( y n ( k ) )} of {g( y n )} with m(k)>n(k)K such that

G ( g ( x m ( k ) ) , g ( x m ( k ) ) , g ( x n ( k ) ) ) +G ( g ( y m ( k ) ) , g ( y m ( k ) ) , g ( y n ( k ) ) ) ε.
(8)

Further, corresponding to n(k), we can choose m(k) in such a way that it is the smallest integer with m(k)>n(k)K and satisfying (8). Then

G ( g ( x m ( k ) 1 ) , g ( x m ( k ) 1 ) , g ( x n ( k ) ) ) +G ( g ( y m ( k ) 1 ) , g ( y m ( k ) 1 ) , g ( y n ( k ) ) ) <ε.
(9)

Using the rectangle inequality, we get

ε r k : = G ( g ( x m ( k ) ) , g ( x m ( k ) ) , g ( x n ( k ) ) ) + G ( g ( y m ( k ) ) , g ( y m ( k ) ) , g ( y n ( k ) ) ) G ( g ( x m ( k ) ) , g ( x m ( k ) ) , g ( x m ( k ) 1 ) ) + G ( g ( x m ( k ) 1 ) , g ( x m ( k ) 1 ) , g ( x n ( k ) ) ) + G ( g ( y m ( k ) ) , g ( y m ( k ) ) , g ( y m ( k ) 1 ) ) + G ( g ( y m ( k ) 1 ) , g ( y m ( k ) 1 ) , g ( y n ( k ) ) ) < t m ( k ) 1 + ε .
(10)

Letting k+ and using (7), we get

lim k r k = lim k + G ( g ( x m ( k ) ) , g ( x m ( k ) ) , g ( x n ( k ) ) ) +G ( g ( y m ( k ) ) , g ( y m ( k ) ) , g ( y n ( k ) ) ) =ε.
(11)

Again, by the rectangle inequality, we have

r k : = G ( g ( x m ( k ) ) , g ( x m ( k ) ) , g ( x n ( k ) ) ) + G ( g ( y m ( k ) ) , g ( y m ( k ) ) , g ( y n ( k ) ) ) t n ( k ) + G ( g ( x m ( k ) ) , g ( x m ( k ) ) , g ( x m ( k ) + 1 ) ) + G ( g ( x m ( k ) + 1 ) , g ( x m ( k ) + 1 ) , g ( x n ( k ) + 1 ) ) + G ( g ( y m ( k ) ) , g ( y m ( k ) ) , g ( y m ( k ) + 1 ) ) + G ( g ( y m ( k ) + 1 ) , g ( y m ( k ) + 1 ) , g ( y n ( k ) + 1 ) ) .

Using the fact that G(x,x,y)2G(x,y,y) for any x,yX, we obtain

r k t n ( k ) + 2 t m ( k ) + G ( g ( x m ( k ) + 1 ) , g ( x m ( k ) + 1 ) , g ( x n ( k ) + 1 ) ) + G ( g ( y m ( k ) + 1 ) , g ( y m ( k ) + 1 ) , g ( y n ( k ) + 1 ) ) .

Since m(k)>n(k),

( g ( x m ( k ) ) , g ( y m ( k ) ) , g ( x m ( k ) ) , g ( y m ( k ) ) , g ( x m ( k ) 1 ) , g ( y m ( k ) 1 ) ) M

and

( g ( x m ( k ) 1 ) , g ( y m ( k ) 1 ) , g ( x m ( k ) 1 ) , g ( y m ( k ) 1 ) , g ( x m ( k ) 2 ) , g ( y m ( k ) 2 ) ) M.

From M being an ( F ,g)-invariant set which satisfies the transitive property, we have

( g ( x m ( k ) ) , g ( y m ( k ) ) , g ( x m ( k ) ) , g ( y m ( k ) ) , g ( x m ( k ) 2 ) , g ( y m ( k ) 2 ) ) M.

Again from

( g ( x m ( k ) 2 ) , g ( y m ( k ) 2 ) , g ( x m ( k ) 2 ) , g ( y m ( k ) 2 ) , g ( x m ( k ) 3 ) , g ( y m ( k ) 3 ) ) M

we get

( g ( x m ( k ) ) , g ( y m ( k ) ) , g ( x m ( k ) ) , g ( y m ( k ) ) , g ( x n ( k ) ) , g ( y n ( k ) ) ) M.

Now, using (1), we have

G ( g ( x m ( k ) + 1 ) , g ( x m ( k ) + 1 ) , g ( x n ( k ) + 1 ) ) + G ( g ( y m ( k ) + 1 ) , g ( y m ( k ) + 1 ) , g ( y n ( k ) + 1 ) ) = G ( F ( x m ( k ) , y m ( k ) ) , F ( x m ( k ) , y m ( k ) ) , F ( x n ( k ) , y n ( k ) ) ) + G ( F ( y m ( k ) , x m ( k ) ) , F ( y m ( k ) , x m ( k ) ) , F ( y n ( k ) , x n ( k ) ) ) ϕ ( G ( g ( x m ( k ) ) , g ( x m ( k ) ) , g ( x n ( k ) ) ) + G ( g ( y m ( k ) ) , g ( y m ( k ) ) , g ( y n ( k ) ) ) ) ϕ ( r k ) .
(12)

Letting k+ in (12) and using (7) and (11) and lim r t + ϕ(r)<t for all t>0, we have

ε= lim k r k lim n ϕ( r k )= lim r k ε + ϕ( r k )<ε.

This is a contradiction. This shows that {g( x n )} and {g( y n )} are Cauchy sequences in the G-metric space (X,G). Since (X,G) is complete, {g( x n )} and {g( y n )} are G-convergent; there exist x,yX such that lim n g( x n )=x and lim n g( y n )=y. That is, from Proposition 2.4, we have

lim n G ( g ( x n ) , g ( x n ) , x ) = lim n G ( g ( x n ) , x , x ) =0,
(13)
lim n G ( g ( y n ) , g ( y n ) , y ) = lim n G ( g ( y n ) , y , y ) =0.
(14)

From (13), (14), continuity of g, and Proposition 2.7, we get

lim n G ( g ( g ( x n ) ) , g ( g ( x n ) , g ( x ) ) ) = lim n G ( g ( g ( x n ) ) , g ( x ) , g ( x ) ) =0,
(15)
lim n G ( g ( g ( y n ) ) , g ( g ( y n ) ) , g ( y ) ) = lim n G ( g ( g ( y n ) ) , g ( y ) , g ( y ) ) =0.
(16)

From (2) and commutativity of F and g,

g ( g ( x n + 1 ) ) =g ( F ( x n , y n ) ) =F ( g ( x n ) , g ( y n ) ) ,
(17)
g ( g ( y n + 1 ) ) =g ( F ( y n , x n ) ) =F ( g ( y n ) , g ( x n ) ) .
(18)

We now show that F(x,y)=g(x) and F(y,x)=g(y).

Taking the limit as n+ in (17) and (18), by (15), (16), continuity of F, and commutativity of F and g, we get

g ( x ) = g ( lim n g ( x n + 1 ) ) = lim n g ( g ( x n + 1 ) ) = lim n g ( F ( x n , y n ) ) = lim n F ( g ( x n ) , g ( y n ) ) = F ( x , y )

and

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

Thus we prove that F(x,y)=g(x) and F(y,x)=g(y). □

Theorem 3.2 Let (X,) be a partially ordered set and G be a G-metric on X such that (X,G) is a complete G-metric space and M be a nonempty subset of X 6 . Assume that there exists ϕΦ and, also, suppose that F:X×XX and g:XX such that

[ G ( F ( x , u ) , F ( y , v ) , F ( z , w ) ) + G ( F ( u , x ) , F ( v , y ) , F ( w , z ) ) ] ϕ ( G ( g ( x ) , g ( y ) , g ( z ) ) + G ( g ( u ) , g ( v ) , g ( w ) ) )

for all (g(x),g(u),g(y),g(v),g(z),g(w))M.

Suppose also that (g(X),G) is complete F(X×X)G(X) and g is continuous and commutes with F and if any two sequences { x n }, { y n } with ( x n + 1 , y n + 1 , x n + 1 , y n + 1 , x n , y n )M, { x n }x and { y n }y for all n1, then (x,y, x n , y n , x n , y n )M. If there exist ( x 0 , y 0 )X×X such that (F( x 0 , y 0 ),F( y 0 , x 0 ),F( x 0 , y 0 ),F( y 0 , x 0 ),g( x 0 ),g( y 0 ))M and M is an ( F ,g)-invariant set which satisfies the transitive property. Then there exist x,yX such that g(x)=F(x,y) and g(y)=F(y,x).

Proof Consider a Cauchy sequences {g( x n )}, {g( y n )} as in the proof of Theorem 3.1. Since (g(X),G) is a complete metric space, there exists x,yX such that {g( x n )}g(x) and {g( y n )}g(y) by the assumption, and we have (g(x),g(y),g( x n ),g( y n ),g( x n ),g( y n ))M for all n1; by the rectangle inequality, (1), and ϕ(t)<t for all t>0, we get

G ( F ( x , y ) , g ( x ) , g ( x ) ) + G ( F ( y , x ) , g ( y ) , g ( y ) ) G ( F ( x , y ) , g ( x n + 1 ) , g ( x n + 1 ) ) + G ( g ( x n + 1 ) , g ( x ) , g ( x ) ) + G ( F ( y , x ) , g ( y n + 1 ) , g ( y n + 1 ) ) + G ( g ( y n + 1 ) , g ( y ) , g ( y ) ) = G ( F ( x , y ) , F ( x n , y n ) , F ( x n , y n ) ) + G ( g ( x n + 1 ) , g ( x ) , g ( x ) ) + G ( F ( y , x ) , F ( y n , x n ) , F ( y n , x n ) ) + G ( g ( y n + 1 ) , g ( y ) , g ( y ) ) ϕ ( G ( g ( x ) , g ( x n ) , g ( x n ) ) + G ( g ( y ) , g ( y n ) , g ( y n ) ) ) + G ( g ( x n + 1 ) , g ( x ) , g ( x ) ) + G ( g ( y n + 1 ) , g ( y ) , g ( y ) ) < G ( g ( x ) , g ( x n ) , g ( x n ) ) + G ( g ( y ) , g ( y n ) , g ( y n ) ) + G ( g ( x n + 1 ) , g ( x ) , g ( x ) ) + G ( g ( y n + 1 ) , g ( y ) , g ( y ) ) .

Taking the limit as n in the above inequality, we obtain

G ( F ( x , y ) , g ( x ) , g ( x ) ) +G ( F ( y , x ) , g ( y ) , g ( y ) ) =0.

This implies that g(x)=F(x,y) and g(y)=F(y,x). □

Example 3.3 Let X=R. Define G:X×X×X[0,+) by G(x,y,z)=|xy|+|xz|+|yz| and let F:X×XX be defined by

F(x,y)= x + 2 y 4 ,(x,y) X 2 ,

and g:XX by g(x)= 3 x 2 . Let y 1 =2 and y 2 =4. Then we have g( y 1 )g( y 2 ), but F(x, y 1 )F(x, y 2 ), and so the mapping F does not satisfy the mixed g-monotone property.

Letting x,u,y,v,z,wX, we have

[ G ( F ( x , u ) , F ( y , v ) , F ( z , w ) ) + G ( F ( u , x ) , F ( v , u ) , F ( w , z ) ) ] = | x + 2 u 4 y + 2 v 4 | + | x + 2 u 4 z + 2 w 4 | + | y + 2 v 4 z + 2 w 4 | + | u + 2 x 4 v + 2 y 4 | + | u + 2 x 4 w + 2 z 4 | + | v + 2 y 4 w + 2 z 4 | 3 | x y 4 | + 3 | x z 4 | + 3 | y z 4 | + 3 | u v 4 | + 3 | u w 4 | + 3 | v w 4 | = 3 4 ( | x y | + | x z | + | y z | ) + 3 4 ( | u v | + | u w | + | v w | )

and we have

G ( g ( x ) , g ( y ) , g ( z ) ) + G ( g ( u ) , g ( v ) , g ( w ) ) = G ( 3 x 2 , 3 y 2 , 3 z 2 ) + G ( 3 u 2 , 3 v 2 , 3 w 2 ) = 3 2 ( | x y | + | x z | + | y z | ) + 3 2 ( | u v | + | u w | + | v w | ) .

Put ϕ(t)=t/2, then

[ G ( F ( x , u ) , F ( y , v ) , F ( z , w ) ) + G ( F ( u , x ) , F ( v , u ) , F ( w , z ) ) ] ϕ ( G ( g ( x ) , g ( y ) , g ( z ) ) + G ( g ( u ) , g ( v ) , g ( w ) ) ) .

If we apply Theorem 3.1 with M= X 6 then F satisfy (1). So by our theorem we see that F has a coupled coincidence point (0,0).

Remark Although the mixed monotone property is an essential tool in the partially ordered G-metric spaces to show the existence of coupled coincidence points, the mappings do not have the mixed g-monotone property in the general case as in the above example. Therefore, Theorem 3.1 and Theorem 3.2 are interesting, as a new auxiliary tool, in showing the existence of a coupled coincidence point.

Theorem 3.4 In addition to the hypotheses of Theorem  3.1, suppose that for every (x,y),( x , y )X×X there exist (u,v)X×X such that

( g ( u ) , g ( v ) , g ( x ) , g ( y ) , g ( x ) , g ( y ) ) Mand ( g ( u ) , g ( v ) , g ( x ) , g ( y ) , g ( x ) , g ( y ) ) M.

Suppose also that ϕ is a non-decreasing function. Then F and g have a unique coupled common fixed point, that is, there exist unique (x,y)X×X such that x=g(x)=F(x,y) and y=g(y)=F(y,x).

Proof From Theorem 3.1 the set of coupled coincidence point is nonempty. Suppose (x,y) and ( x , y ) are coupled coincidence point of F, that is,

g(x)=F(x,y),g(y)=F(y,x),g ( x ) =F ( x , y ) andg ( y ) =F ( y , x ) .

We shall show that

g ( x ) =g(x)andg ( y ) =g(y).
(19)

By assumption there is (u,v)X×X such that

( g ( u ) , g ( v ) , g ( x ) , g ( y ) , g ( x ) , g ( y ) ) Mand ( g ( u ) , g ( v ) , g ( x ) , g ( y ) , g ( x ) , g ( y ) ) M.

Put u 0 =u, v 0 =v and choose u 1 , v 1 X such that g( u 1 )=F( u 0 , v 0 ) and g( v 1 )=F( v 0 , u 0 ). Then similarly as in Theorem 3.1, we can inductively define sequences {g( u n )} and {g( v n )} such that

g( u n )=F( u n 1 , v n 1 )andg( v n )=F( v n 1 , u n 1 )for all n1.

Since M is ( F ,g)-invariant and (g( u 0 ),g( v 0 ),g(x),g(y),g(x),g(y))M, we have

( F ( u 0 , v 0 ) , F ( v 0 , u 0 ) , F ( x , y ) , F ( y , x ) , F ( x , y ) , F ( y , x ) ) M.

That is (g( u 1 ),g( v 1 ),g(x),g(y),g(x),g(y))M.

From (g( u 1 ),g( v 1 ),g(x),g(y),g(x),g(y))M, if we use again the property of ( F ,g)-invariance, then it follows that

( F ( u 1 , v 1 ) , F ( v 1 , u 1 ) , F ( x , y ) , F ( y , x ) , F ( x , y ) , F ( y , x ) ) M

and so (g( u 2 ),g( v 2 ),g(x),g(y),g(x),g(y))M.

By repeating this process, we get

( g ( u n ) , g ( v n ) , g ( x ) , g ( y ) , g ( x ) , g ( y ) ) Mfor all n1.
(20)

Thus from (1) and (20), we have

G ( g ( u n + 1 ) , g ( x ) , g ( x ) ) + G ( g ( v n + 1 ) , g ( y ) , g ( y ) ) = G ( F ( u n , v n ) , F ( x , y ) , F ( x , y ) ) + G ( F ( v n , u n ) , F ( y , x ) , F ( y , x ) ) ϕ ( G ( g ( u n ) , g ( x ) , g ( x ) ) + G ( g ( v n ) , g ( y ) , g ( y ) ) ) .
(21)

Since ϕ is non-decreasing from (21), we get

G ( g ( u n + 1 ) , g ( x ) , g ( x ) ) + G ( g ( v n + 1 ) , g ( y ) , g ( y ) ) ϕ n ( G ( g ( u 1 ) , g ( x ) , g ( x ) ) + G ( g ( v 1 ) , g ( y ) , g ( y ) ) ) .
(22)

This holds for each n1. Letting n+ in (22), using Lemma 2.20 implies

lim n G ( g ( u n + 1 ) , g ( x ) , g ( x ) ) = lim n G ( g ( v n + 1 ) , g ( y ) , g ( y ) ) =0.
(23)

Similarly, we obtain

lim n G ( g ( u n + 1 ) , g ( x ) , g ( x ) ) = lim n G ( g ( v n + 1 ) , g ( y ) , g ( y ) ) =0.
(24)

Hence, from (23), (24), and Proposition 2.4, we get g( x )=g(x) and g( y )=g(y).

Since g(x)=F(x,y) and g(y)=F(y,x), by commutativity of F and g, we have

g ( g ( x ) ) =g ( F ( x , y ) ) =F ( g ( x ) , g ( y ) ) andg ( g ( y ) ) =g ( F ( y , x ) ) =F ( g ( y ) , g ( x ) ) .
(25)

Denote g(x)=z and g(y)=w. Then from (25)

g(z)=F(z,w)andg(w)=F(w,z).
(26)

Therefore, (z,w) is a coupled coincidence fixed point of F and g. Then from (19) with x =z and y =w, it follows that g(z)=g(x) and g(w)=g(y), that is,

g(z)=zandg(w)=w.
(27)

From (26) and (27), z=g(z)=F(z,w) and w=g(w)=F(w,z). Therefore, (z,w) is a coupled common fixed point of F and g.

To prove the uniqueness, assume that (p,q) is another coupled common fixed point. Then by (19) we have p=g(p)=g(z)=z and q=g(q)=g(w)=w. □

Next, we give a simple application of our results to coupled coincidence point theorems in partially ordered metric spaces.

Corollary 3.5 Let (X,) be a partially ordered set and G be a G-metric on X such that (X,G) is a complete G-metric space. Suppose that there exist ϕΦ, F:X×XX, and g:XX such that

[ G ( F ( x , u ) , F ( y , v ) , F ( z , w ) ) + G ( F ( u , x ) , F ( v , y ) , F ( w , z ) ) ] ϕ ( G ( g ( x ) , g ( y ) , g ( z ) ) + G ( g ( u ) , g ( v ) , g ( w ) ) )

for all x,y,z,u,v,wX for which g(x)g(y)g(z) and g(u)g(v)g(w).

Suppose also that F is continuous and has the mixed g-monotone property, F(X×X)G(X) and g is continuous and commutes with F. If there exist x 0 , y 0 X such that

g( x 0 )F( x 0 , y 0 )andg( y 0 )F( y 0 , x 0 ),

then there exist (x,y)X×X such that g(x)=F(x,y) and g(y)=F(y,x).

Proof We define the subset M X 6 by

M= { ( x , u , y , v , z , w ) X 6 : x y z , u v w } .

From Example 2.18, M is an ( F ,g)-invariant set which satisfies the transitive property. By (1), we have

[ G ( F ( x , u ) , F ( y , v ) , F ( z , w ) ) + G ( F ( u , x ) , F ( v , y ) , F ( w , z ) ) ] ϕ ( G ( g ( x ) , g ( y ) , g ( z ) ) + G ( g ( u ) , g ( v ) , g ( w ) ) ) .

Since x 0 , y 0 X such that

g( x 0 )F( x 0 , y 0 )andg( y 0 )F( y 0 , x 0 ).

We have (F( x 0 , y 0 ),F( y 0 , x 0 ),F( x 0 , y 0 ),F( y 0 , x 0 ),g( x 0 ),g( y 0 ))M because F is continuous. By Theorem 3.1, we have x=F(x,y) and y=F(y,x). □

Corollary 3.6 Let (X,) be a partially ordered set and G be a G-metric on X such that (X,G,) is regular. Suppose that there exists ϕΦ, F:X×XX and g:XX such that

[ G ( F ( x , u ) , F ( y , v ) , F ( z , w ) ) + G ( F ( u , x ) , F ( v , u ) , F ( w , z ) ) ] ϕ ( G ( g ( x ) , g ( y ) , g ( z ) ) + G ( g ( u ) , g ( v ) , g ( w ) ) )

for all x,y,z,u,v,wX for which g(x)g(y)g(z) and g(u)g(v)g(w).

Suppose also that (g(X),G) is complete, F has the mixed g-monotone property, F(X×X)G(X), and g is continuous and commutes with F. If there exist x 0 , y 0 X such that

g( x 0 )F( x 0 , y 0 )andg( y 0 )F( y 0 , x 0 ),

then there exist (x,y)X×X such that g(x)=F(x,y) and g(y)=F(y,x).

Proof As in Corollary 3.5, we get

( F ( x 0 , y 0 ) , F ( y 0 , x 0 ) , F ( x 0 , y 0 ) , F ( y 0 , x 0 ) , g ( x 0 ) , g ( y 0 ) ) M.

For any two sequences {g( x n )}, {g( y n )} such that {g( x n )} is non-decreasing sequence {g( x n )}g(x) and {g( y n )} is non-increasing sequence {g( y n )}g(y). We have

g( x 1 )g( x 2 )g( x n )g(x)

and

g( y 1 )g( y 2 )g( y n )g(y)for all n1.

Therefore, we have (g(x),g(y),g( x n ),g( y n ),g( x n ),g( y n ))M for all n1. So the assumption of Theorem 3.2 holds and hence F has a coupled coincidence point. □

Corollary 3.7 In addition to the hypothesis of Corollary  3.5, suppose that for every (x,y),( x , y )X×X there exists a (u,v)X×X such that (F(u,v),F(v,u)) is comparable to (F(x,y),F(y,x)) and (F( x , y ),F( y , x )). Suppose also that ϕ is a non-decreasing function. Then F and g have a unique coupled common fixed point, that is, there exists a unique (x,y)X×X such that x=g(x)=F(x,y) and y=g(y)=F(y,x).

Proof We define the subset M X 6 by M={(x,u,y,v,z,w) X 6 :xyz,uvw}. From Example 2.18, M is an ( F ,g)-invariant set which satisfies the transitive property. Thus, the proof of the existence of a coupled fixed point is straightforward by following the same lines as in the proof of Corollary 3.5.

Next, we show the uniqueness of a coupled fixed point of F.

Since for all (x,y),( x , y )X×X, there exist (u,v)X×X such that g(x)g(u), g(y)g(v) and g( x )g(u), g( y )g(v), we can conclude that

( g ( u ) , g ( v ) , g ( x ) , g ( y ) , g ( x ) , g ( y ) ) M

and

( g ( u ) , g ( v ) , g ( x ) , g ( y ) , g ( x ) , g ( y ) ) M.

Therefore, since all the hypotheses of Theorem 3.4 hold F has a unique coupled fixed point. The proof is completed. □

References

  1. 1.

    Ran A, Reurings M: 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

    MATH  MathSciNet  Article  Google Scholar 

  2. 2.

    Nieto JJ, Rodriguez-Lopez R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equation. Order 2005, 22: 223-239. 10.1007/s11083-005-9018-5

    MATH  MathSciNet  Article  Google Scholar 

  3. 3.

    Nieto JJ, Rodriguez-Lopez R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equation. Acta Math. Sin. Engl. Ser. 2007,23(12):2205-2212. 10.1007/s10114-005-0769-0

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 1-8. 10.1080/00036810701714164

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Alghamdi MA, Karapinar E: G - β - ψ Contractive type mappings and related fixed point theorems. J. Inequal. Appl. 2013., 2013: Article ID 70

    Google Scholar 

  6. 6.

    Aydi H, Karapinar E, Kumam P: A note on ‘Modified proof of Caristi’s fixed point theorem on partial metric spaces, Journal of Inequalities and Applications 2013, 2013:210’. J. Inequal. Appl. 2013., 2013: Article ID 355 10.1186/1029-242X-2013-355

    Google Scholar 

  7. 7.

    Chaipunya P, Sintunavarat W, Kumam P:On -contractions in ordered metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 219 10.1186/1687-1812-2012-219

    Google Scholar 

  8. 8.

    Kumam P, Vetro C, Vetro F: Fixed points for weak α - ψ -contractions in partial metric spaces. Abstr. Appl. Anal. 2013., 2013: Article ID 986028

    Google Scholar 

  9. 9.

    Sintunavarat W, Kumam P: Some fixed point results for weakly isotone mappings in ordered Banach spaces. Appl. Math. Comput. 2013,224(1):826-834.

    MathSciNet  Article  Google Scholar 

  10. 10.

    Guo D, Lakshmikantham V: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. 1987, 11: 623-632. 10.1016/0362-546X(87)90077-0

    MATH  MathSciNet  Article  Google Scholar 

  11. 11.

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

    MATH  MathSciNet  Article  Google Scholar 

  12. 12.

    Lakshmikantham V, Ćirić L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. TMA 2009, 70: 4341-4349. 10.1016/j.na.2008.09.020

    MATH  Article  MathSciNet  Google Scholar 

  13. 13.

    Abbas M, Sintunavarat W, Kumam P: Coupled fixed point of generalized contractive mappings on partially ordered G -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 31

    Google Scholar 

  14. 14.

    Amini-Harandi A: Coupled and tripled fixed point theory in partially ordered metric spaces with application to initial value problem. Math. Comput. Model. 2013,57(9-10):2343-2348. 10.1016/j.mcm.2011.12.006

    MATH  Article  Google Scholar 

  15. 15.

    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

    MATH  Article  MathSciNet  Google Scholar 

  16. 16.

    Aydi H, Postolache M, Shatanawi W: Coupled fixed point results for (ψ,φ) -weakly contractive mappings in ordered G -metric spaces. Comput. Math. Appl. 2012,63(1):298-309. 10.1016/j.camwa.2011.11.022

    MATH  MathSciNet  Article  Google Scholar 

  17. 17.

    Aydi H, Karapinar E, Shatanawi W: Tripled fixed point results in generalized metric spaces. J. Appl. Math. 2012., 2012: Article ID 314279

    Google Scholar 

  18. 18.

    Aydi H, Karapinar E, Shatanawi W: Tripled common fixed point results for generalized contractions in ordered generalized metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 101

    Google Scholar 

  19. 19.

    Batra R, Vashistha S:Coupled coincidence point theorems for nonlinear contractions under (F,g)-invariant set in cone metric spaces. J. Nonlinear Sci. Appl. 2013, 6: 86-96.

    MATH  MathSciNet  Google Scholar 

  20. 20.

    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

    MATH  MathSciNet  Article  Google Scholar 

  21. 21.

    Berinde V: Coupled coincidence point theorems for mixed monotone nonlinear operators. Comput. Math. Appl. 2012,64(6):1770-1777. 10.1016/j.camwa.2012.02.012

    MATH  MathSciNet  Article  Google Scholar 

  22. 22.

    Berinde V: Coupled fixed point theorems for φ -contractive mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal., Theory Methods Appl. 2012,75(6):3218-3228. 10.1016/j.na.2011.12.021

    MATH  MathSciNet  Article  Google Scholar 

  23. 23.

    Berzig M, Samet B: An extension of coupled fixed point’s concept in higher dimension and applications. Comput. Math. Appl. 2012, 63: 1319-1334. 10.1016/j.camwa.2012.01.018

    MATH  MathSciNet  Article  Google Scholar 

  24. 24.

    Chandok S, Sintunavarat W, Kumam P: Some coupled common fixed points for a pair of mappings in partially ordered G -metric spaces. Math. Sci. 2013., 7: Article ID 24 10.1186/2251-7456-7-24

    Google Scholar 

  25. 25.

    Charoensawan P, Klanarong C: Coupled coincidence point theorems for φ -contractive under (f,g) -invariant set in complete metric space. Int. J. Math. Anal. 2013,7(33-36):1685-1701.

    MATH  MathSciNet  Google Scholar 

  26. 26.

    Cho YJ, Rhoades BE, Saadati R, Samet B, Shatanawi 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 

  27. 27.

    Choudhury BS, Maity P: Coupled fixed point results in generalized metric spaces. Math. Comput. Model. 2011,54(1-2):73-79. 10.1016/j.mcm.2011.01.036

    MATH  MathSciNet  Article  Google Scholar 

  28. 28.

    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

    MATH  MathSciNet  Article  Google Scholar 

  29. 29.

    Ding H-S, Li L: Coupled fixed point theorems in partially ordered cone metric spaces. Filomat 2011,25(2):137-149. 10.2298/FIL1102137D

    MATH  MathSciNet  Article  Google Scholar 

  30. 30.

    Hussain N, Latif A, Shah MH: Coupled and tripled coincidence point results without compatibility. Fixed Point Theory Appl. 2012., 2012: Article ID 77

    Google Scholar 

  31. 31.

    Karapinar E, Luong NV, Thuan NX, Hai TT: Coupled coincidence points for mixed monotone operators in partially ordered metric spaces. Arab. J. Math. 2012, 1: 329-339. 10.1007/s40065-012-0027-0

    MATH  Article  MathSciNet  Google Scholar 

  32. 32.

    Karapinar E: Coupled fixed point theorems for nonlinear contractions in cone metric spaces. Comput. Math. Appl. 2010, 59: 3656-3668. 10.1016/j.camwa.2010.03.062

    MATH  MathSciNet  Article  Google Scholar 

  33. 33.

    Karapinar E: Couple fixed point on cone metric spaces. Gazi Univ. J. Sci. 2011, 24: 51-58.

    Google Scholar 

  34. 34.

    Karapinar E, Kaymakcalan B, Tas K: On coupled fixed point theorems on partially ordered G -metric spaces. J. Inequal. Appl. 2012., 2012: Article ID 200

    Google Scholar 

  35. 35.

    Karapinar E, Kumam P, Erhan I: Coupled fixed points on partially ordered G -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 174 10.1186/1687-1812-2012-174

    Google Scholar 

  36. 36.

    Karapinar E, Roldan A, Martinez-Moreno J, Roldan C: Meir-Keeler type multidimensional fixed point theorems in partially ordered metric spaces. Abstr. Appl. Anal. 2013., 2013: Article ID 406026

    Google Scholar 

  37. 37.

    Kaushik P, Kumar S, Kumam P: Coupled coincidence point theorems for α - ψ -contractive type mappings in partially ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 325 10.1186/1687-1812-2013-325

    Google Scholar 

  38. 38.

    Luong NV, Thuan NX: Coupled fixed points in partially ordered metric spaces and application. Nonlinear Anal. 2011, 74: 983-992. 10.1016/j.na.2010.09.055

    MATH  MathSciNet  Article  Google Scholar 

  39. 39.

    Mursaleen M, Mohiuddine SA, Agarwal RP: Coupled fixed point theorems for α - ψ -contractive type mappings in partially ordered metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 228 (Corrigendum to Coupled fixed point theorems for α-ψ-contractive type mappings in partially ordered metric spaces. Fixed Point Theory Appl. 2013, 127 (2013))

    Google Scholar 

  40. 40.

    Roldan A, Martinez-Moreno J, Roldan C: Multidimensional fixed point theorems in partially ordered complete metric spaces. J. Math. Anal. Appl. 2012,396(2):536-545. 10.1016/j.jmaa.2012.06.049

    MATH  MathSciNet  Article  Google Scholar 

  41. 41.

    Roldan A, Martinez-Moreno J, Roldan C, Karapinar E:Multidimensional fixed point theorems in partially ordered complete partial metric spaces under (ψ,φ)-contractivity conditions. Abstr. Appl. Anal. 2013., 2013: Article ID 634371

    Google Scholar 

  42. 42.

    Roldan A, Karapinar E:Some multidimensional fixed point theorems on partially preordered G -metric spaces under (ψ,φ)-contractivity conditions. Fixed Point Theory Appl. 2013., 2013: Article ID 158

    Google Scholar 

  43. 43.

    Shatanawi W: Coupled fixed point theorems in generalized metric spaces. Hacet. J. Math. Stat. 2011,40(3):441-447.

    MATH  MathSciNet  Google Scholar 

  44. 44.

    Shatanawi W, Abbas M, Nazir T: Common coupled coincidence and coupled fixed point results in two generalized metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 80

    Google Scholar 

  45. 45.

    Sintunavarat W, Kumam P, Cho YJ: Coupled fixed point theorems for nonlinear contractions without mixed monotone property. Fixed Point Theory Appl. 2012., 2012: Article ID 170

    Google Scholar 

  46. 46.

    Sintunavarat W, Radenović S, Golubović Z, Kuman P: Coupled fixed point theorems for F -invariant set. Appl. Math. Inf. Sci. 2013,7(1):247-255. 10.12785/amis/070131

    MathSciNet  Article  Google Scholar 

  47. 47.

    Sintunavarat W, Kumam P: Coupled coincidence and coupled common fixed point theorems in partially ordered metric spaces. Thai J. Math. 2012,10(3):551-563.

    MATH  MathSciNet  Google Scholar 

  48. 48.

    Samet B, Vetro C: Coupled fixed point F -invariant set and fixed point of N -order. Ann. Funct. Anal. 2010, 1: 46-56.

    MATH  MathSciNet  Article  Google Scholar 

  49. 49.

    Mustafa Z, Sims B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 2006,7(2):289-297.

    MATH  MathSciNet  Google Scholar 

  50. 50.

    Agarwal RP, Karapinar E: Remarks on some coupled fixed point theorems in G -metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 2

    Google Scholar 

  51. 51.

    Alghamdi MA, Karapinar E: G - β - ψ Contractive type mappings in G -metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 123

    Google Scholar 

  52. 52.

    Bilgili N, Karapinar E: Cyclic contractions via auxiliary functions on G -metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 49

    Google Scholar 

  53. 53.

    Ding H-S, Karapinar E: Meir-Keeler type contractions in partially ordered G -metric space. Fixed Point Theory Appl. 2013., 2013: Article ID 35

    Google Scholar 

  54. 54.

    Jleli M, Samet B: Remarks on G -metric spaces and fixed point theorems. Fixed Point Theory Appl. 2012., 2012: Article ID 210

    Google Scholar 

  55. 55.

    Karapinar E, Agarwal RP: Further fixed point results on G -metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 154

    Google Scholar 

  56. 56.

    Mustafa Z, Aydi H, Karapinar E: On common fixed points in image-metric spaces using (E.A) property. Comput. Math. Appl. 2012. 10.1016/j.camwa.2012.03.051

    Google Scholar 

  57. 57.

    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 

  58. 58.

    Mustafa Z, Khandaqji M, Shatanawi W: Fixed point results on complete G -metric spaces. Studia Sci. Math. Hung. 2011, 48: 304-319.

    MATH  MathSciNet  Google Scholar 

  59. 59.

    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 

  60. 60.

    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 

  61. 61.

    Mustafa Z, Aydi H, Karapinar E: Generalized Meir-Keeler type contractions on G -metric spaces. Appl. Math. Comput. 2013,219(21):10441-10447. 10.1016/j.amc.2013.04.032

    MATH  MathSciNet  Article  Google Scholar 

  62. 62.

    Roldan A, Karapinar E, Kumam P: G -Metric spaces in any number of arguments and related fixed point theorems. Fixed Point Theory Appl. 2014., 2014: Article ID 13 10.1186/1687-1812-2014-13

    Google Scholar 

  63. 63.

    Samet B, Vetro C, Vetro F: Remarks on G -metric spaces. Int. J. Anal. 2013., 2013: Article ID 917158

    Google Scholar 

  64. 64.

    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 

  65. 65.

    Shatanawi W: Some fixed point theorems in ordered G -metric spaces and applications. Abstr. Appl. Anal. 2011., 2011: Article ID 126205

    Google Scholar 

  66. 66.

    Tahat N, Aydi H, Karapınar E, Shatanawi W: Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in G -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 48

    Google Scholar 

  67. 67.

    Nashine HK: Coupled common fixed point results in ordered G -metric spaces. J. Nonlinear Sci. Appl. 2012, 1: 1-13.

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

This research was supported by Chiang Mai University and the authors would like to express sincere gratitude to Prof. Suthep Suantai and the referees for being kind enough to give very helpful suggestions and make many comments.

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Correspondence to Phakdi Charoensawan.

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Charoensawan, P., Thangthong, C. On coupled coincidence point theorems on partially ordered G-metric spaces without mixed g-monotone. J Inequal Appl 2014, 150 (2014). https://doi.org/10.1186/1029-242X-2014-150

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Keywords

  • coupled fixed point
  • coupled coincidence point
  • invariant set
  • mixed g-monotone
  • partially ordered set
  • G-metric