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Common fixed point results under a new contractive condition without using continuity

Abstract

In this paper, using the concept of the common (E.A) property, we prove a common fixed point theorem for a class of twice power type weakly compatible mappings in generalized metric space. Our results do not rely on any commuting or continuity condition of the mappings. We also state some examples to illustrate our new results in symmetric and nonsymmetric generalized metric spaces. It should be pointed out that this is the first time to use common (E.A) properties to discuss common fixed point problems of contractive mappings for twice power type in generalized metric spaces.

1 Introduction and preliminaries

In 2006, Mustafa and Sims [1] introduced a new structure of generalized metric space which is called a G-metric. Based on the notion of generalized metric spaces, Mustafa et al. [25], Obiedat and Mustafa [6], Aydi et al. [7, 8], Gajić and Stojaković [9], Zhou and Gu [10], Shatanawi [11] obtained some fixed point results for mappings satisfying different contractive conditions. Chugh et al. [12] obtained some fixed point results for maps satisfying property P in G-metric spaces. The study of common fixed point problems in G-metric spaces was initiated by Abbas and Rhoades [13]. Subsequently, many authors obtained many common fixed point theorems for the mappings satisfying different contractive conditions; see [1431] for more details. Recently, some authors using (E.A) property in generalized metric space to prove common fixed point, such as Abbas et al. [32], Mustafa et al. [33], Long et al. [34], Gu and Yin [35], Gu and Shatanawi [36].

Recently, Jleli and Samet [37] and Samet et al. [38] observed that some fixed point theorems in the context of a G-metric space can be proved (by simple transformation) using related existing results in the setting of a (quasi-) metric space. Namely, if the contraction condition of the fixed point theorem on G-metric space can be reduced to two variables, then one can construct an equivalent fixed point theorem in setting of usual metric space. This idea is not completely new, but it was not successfully used before (see [39]).

Very recently, Karapınar and Agarwal suggest new contraction conditions in G-metric space in a way that the techniques in [37, 38] are not applicable. In this approach [40], contraction conditions cannot be expressed in two variables. So, in some cases, as is noticed even in Jleli-Samet’s paper [37], when the contraction condition is of nonlinear type, this strategy cannot be always successfully used. This is exactly the case in our paper.

The purpose of this paper is to use the concept of the common (E.A) property and weakly compatible mappings to discuss common fixed point problem for a class of twice power type contractive mappings in the framework of a generalized metric space. Our results do not rely on any commuting or continuity condition of the mappings. We also state some examples to illustrate our new results in the framework of symmetric and nonsymmetric generalized metric spaces.

As far as we know, this is the first time to use common (E.A) properties to discuss common fixed point problems of contractive mappings for twice power type in generalized metric spaces.

Now we give preliminaries and basic definitions which are used throughout the paper.

Definition 1.1 [1]

Let X be a nonempty set, and let G:X×X×X R + be a function satisfying the following axioms:

  • (G1) G(x,y,z)=0 if x=y=z;

  • (G2) 0<G(x,y,z) for all x,yX with xy;

  • (G3) G(x,x,y)G(x,y,z) for all x,y,zX with zy;

  • (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 a G-metric on X, and the pair (X,G) is called a G-metric space.

Definition 1.2 [1]

Let (X,G) be a G-metric space, and let { x n } a sequence of points in X, a point x in X is said to be the limit of the sequence { x n }, lim n G(x, x n , x m )=0, and one says that sequence { x n } is G-convergent to x.

Thus, if x n x or lim n x n =x in a G-metric space (X,G), then if for each ϵ>0, there exists a positive integer N such that G(x, x n , x m )<ϵ for all n,mN.

Proposition 1.1 [1]

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

  1. (1)

    { x n } is G-convergent to x.

  2. (2)

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

  3. (3)

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

  4. (4)

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

Definition 1.3 [1]

Let (X,G) be a G-metric space. A sequence { x n } is called G-Cauchy if, for each ϵ>0, there exists a positive integer N such that G( x m , x n ,x)<ϵ for all n,m,lN; i.e. if G( x m , x n , x l )0 as m,n,l.

Proposition 1.2 [1]

If (X,G) is a G-metric space then the following are equivalent:

  1. (1)

    The sequence { x n } is G-Cauchy.

  2. (2)

    For each ϵ>0, there exists a positive integer N such that G( x m , x n ,x)<ϵ for all n,m,lN.

Proposition 1.3 [1]

Let (X,G) be a G-metric space. Then the function G(x,y,z) is jointly continuous in all three of its variables.

Definition 1.4 [1]

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

Definition 1.5 [41]

Let f and g be self-maps of a set X. If w=fx=gx for some x in X, then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g.

Definition 1.6 [41]

Two self-mappings f and g on X are said to be weakly compatible if they commute at coincidence points.

Definition 1.7 [32]

Let X be a G-metric space. Self-maps f and g on X are said to satisfy the G-(E.A) property if there exists a sequence { x n } in X such that {f x n } and {g x n } are G-convergent to some tX.

Definition 1.8 [32]

Let (X,d) be a G-metric space and A, B, S, and T be four self-maps on X. The pairs (A,S) and (B,T) are said to satisfy the common (E.A) property if there exist two sequences { x n } and { y n } in X such that

lim n A x n = lim n S x n = lim n B y n = lim n T y n =t

for some tX.

Definition 1.9 [17]

Self-mappings f and g of a G-metric space (X,G) are said to be compatible if lim n G(fg x n ,gf x n ,gf x n )=0 and lim n G(gf x n ,fg x n ,fg x n )=0, whenever { x n } is a sequence in X such that

lim n f x n = lim n g x n =t

for some tX.

2 Main results

Theorem 2.1 Let (X,G) be a G-metric space. Suppose mappings f,g,h,R,S,T:XX satisfying the following conditions:

G 2 (fx,gy,hz)kmax { G ( R x , S y , T z ) G ( f x , R x , R x ) , G ( g y , S y , S y ) G ( h z , T z , T z ) , G ( f x , S y , T z ) G ( R x , g y , T z ) , G ( R x , S y , h z ) G ( f x , g y , T z ) , G ( f x , S y , h z ) G ( R x , g y , h z ) }
(2.1)

for all x,y,zX, 0k<1. If one of the following conditions is satisfied, then the pairs (f,R), (g,S), and (h,T) have a common point of coincidence in X.

  1. (i)

    The subspace RX is closed in X, fXSX, gXTX, and the two pairs of (f,R) and (g,S) satisfy the common (E.A) property.

  2. (ii)

    The subspace SX is closed in X, gXTX, hXRX, and the two pairs of (g,S) and (h,T) satisfy the common (E.A) property.

  3. (iii)

    The subspace TX is closed in X, fXSX, hXRX, and the two pairs of (f,R) and (h,T) satisfy the common (E.A) property.

Further, if the pairs (f,R), (g,S), and (h,T) are weakly compatible, then f, g, h, R, S, and T have a unique common fixed point in X.

Proof First we suppose that RX is closed in X, fXSX, gXTX, and the two pairs of (f,R) and (g,S) satisfy the common (E.A) property, then by Definition 1.8 we know that there exist two sequences { x n } and { y n } in X such that

lim n f x n = lim n R x n = lim n g y n = lim n S y n =t

for some tX.

Since gXTX, there exists a sequence { z n } in X such that g y n =T z n . So we get lim n T z n = lim n g y n =t. By the condition (2.1) we have

G 2 (f x n ,g y n ,h z n )kmax { G ( R x n , S y n , T z n ) G ( f x n , R x n , R x n ) , G ( g y n , S y n , S y n ) G ( h z n , T z n , T z n ) , G ( f x n , S y n , T z n ) G ( R x n , g y n , T z n ) , G ( R x n , S y n , h z n ) G ( f x n , g y n , T z n ) , G ( f x n , S y n , h z n ) G ( R x n , g y n , h z n ) } .

Letting n, we have

G 2 ( t , t , lim n h z n ) k G 2 ( t , t , lim n h z n ) ;

this gives G 2 (t,t, lim n h z n )=0, since 0k<1. Hence lim n h z n =t.

Since RX is a closed subspace of X, and lim n R x n =t, there exists a point uX such that Ru=t. By the condition (2.1) we have

G 2 (fu,g y n ,h z n )kmax { G ( R u , S y n , T z n ) G ( f u , R u , R u ) , G ( g y n , S y n , S y n ) G ( h z n , T z n , T z n ) , G ( f u , S y n , T z n ) G ( R u , g y n , T z n ) , G ( R u , S y n , h z n ) G ( f u , g y n , T z n ) , G ( f u , S y n , h z n ) G ( R u , g y n , h z n ) } .

Letting n, we have G 2 (fu,t,t)0, hence fu=t. Thus Ru=fu=t, so u is the coincidence point of the pair (f,R).

Since fXSX and fu=t, there exists a point vX such that Sv=fu=t. By the condition (2.1) we have

G 2 (fu,gv,h z n )kmax { G ( R u , S v , T z n ) G ( f u , R u , R u ) , G ( g v , S v , S v ) G ( h z n , T z n , T z n ) , G ( f u , S v , T z n ) G ( R u , g v , T z n ) , G ( R u , S v , h z n ) G ( f u , g v , T z n ) , G ( f u , S v , h z n ) G ( R u , g v , h z n ) } .

Letting n, we have G 2 (t,gv,t)0, hence gv=t. Thus Sv=gv=t, so v is the coincidence point of the pair (g,S).

Since gXTX and gv=t, there exists a point wX such that Tw=gv=t. By the condition (2.1) we have

G 2 ( t , t , h w ) = G 2 ( f u , g v , h w ) k max { G ( R u , S v , T w ) G ( f u , R u , R u ) , G ( g v , S v , S v ) G ( h w , T w , T w ) , G ( f u , S v , T w ) G ( R u , g v , T w ) , G ( R u , S v , h w ) G ( f u , g v , T w ) , G ( f u , S v , h w ) G ( R u , g v , h w ) } = k G 2 ( t , t , h w ) ,

hence hw=t, since 0k<1. Thus Tw=hw=t, so w is the coincidence point of the pair (h,T).

In the above proof we get fu=Ru=gv=Sv=hw=Tw=t. Then we get ft=Rt, gt=St, and ht=Tt, since the pairs (f,R), (g,S), and (h,T) are weakly compatible. By the condition (2.1), we have

G 2 ( f t , t , t ) = G 2 ( f t , g v , h w ) k max { G ( R t , S v , T w ) G ( f t , R t , R t ) , G ( g v , S v , S v ) G ( h w , T w , T w ) , G ( f t , S v , T w ) G ( R t , g v , T w ) , G ( R t , S v , h w ) G ( f t , g v , T w ) , G ( f t , S v , h w ) G ( R t , g v , h w ) } = k G 2 ( f t , t , t ) ,

hence G 2 (ft,t,t)=0, since 0k<1. Thus ft=t=Rt. Similarly, it can be shown that gt=St=t and ht=Tt=t, which means that t is a common fixed point of f, g, h, R, S, and T.

Now we prove the uniqueness of the common fixed point t.

Let t and p be two common fixed point of f, g, h, R, S, and T, then using the condition (2.1), we have

G 2 ( p , t , t ) = G 2 ( f p , g t , h t ) k max { G ( R p , S t , T t ) G ( f p , R p , R p ) , G ( g t , S t , S t ) G ( h t , T t , T t ) , G ( f p , S t , T t ) G ( R p , g t , T t ) , G ( R p , S t , h t ) G ( f p , g t , T t ) , G ( f p , S t , h t ) G ( R p , g t , h t ) } = k G 2 ( p , t , t ) ,

hence G 2 (p,t,t)=0, since 0k<1. Thus p=t. So common fixed point is unique. □

Example 2.1 Let X=[0,1] be a G-metric space with

G(x,y,z)=|xy|+|yz|+|zx|.

We define mappings f, g, h, R, S, and T on X by

f x = { 1 , x [ 0 , 1 2 ] , 4 5 , x ( 1 2 , 1 ] , g x = { 5 6 , x [ 0 , 1 2 ] , 4 5 , x ( 1 2 , 1 ] , h x = { 6 7 , x [ 0 , 1 2 ] , 4 5 , x ( 1 2 , 1 ] , R x = { 0 , x [ 0 , 1 2 ] , 4 5 , x ( 1 2 , 1 ) , 6 7 , x = 1 , S x = { 1 , x [ 0 , 1 2 ] 4 5 , x ( 1 2 , 1 ) , 0 , x = 1 , T x = { 0 , x [ 0 , 1 2 ] , 4 5 , x ( 1 2 , 1 ) , 5 6 , x = 1 .

Clearly, from the above functions we know that the subspace RX is closed in X, fXSX, gXTX, hXRX and the pairs (f,R), (g,S), (h,T) be weakly compatible. The pairs (f,R) and (g,S) satisfy the common (E.A) property, let x n = 6 7 and y n = 5 6 for each nN be the required sequences.

Now we prove that the mappings f, g, h, R, S, and T are satisfying the condition (2.1) of Theorem 2.1 with k= 1 4 [0,1). Let

M(x,y,z)=max { G ( R x , S y , T z ) G ( f x , R x , R x ) , G ( g y , S y , S y ) G ( h z , T z , T z ) , G ( f x , S y , T z ) G ( R x , g y , T z ) , G ( R x , S y , h z ) G ( f x , g y , T z ) , G ( f x , S y , h z ) G ( R x , g y , h z ) } .

Case (1) If x,y,z[0, 1 2 ], then we have

G 2 ( f x , g y , h z ) = G 2 ( 1 , 5 6 , 6 7 ) = 1 9 , G ( R x , S y , T z ) G ( f x , R x , R x ) = G ( 0 , 1 , 0 ) G ( 1 , 0 , 0 ) = 4 .

Thus we have

G 2 (fx,gy,hz)= 1 9 < 1 4 4=kG(Rx,Sy,Tz)G(fx,Rx,Rx)kM(x,y,z).

Case (2) If x,y[0, 1 2 ], z( 1 2 ,1], then we have

G 2 (fx,gy,hz)= G 2 ( 1 , 5 6 , 4 5 ) = 4 25 .

If z=1, then

G(Rx,Sy,Tz)G(fx,Rx,Rx)=G ( 0 , 1 , 5 6 ) G(1,0,0)=4.

If z( 1 2 ,1), then

G(Rx,Sy,Tz)G(fx,Rx,Rx)=G ( 0 , 1 , 4 5 ) G(1,0,0)=4.

So we know G(Rx,Sy,Tz)G(fx,Rx,Rx)=4. Thus we have

G 2 (fx,gy,hz)= 4 25 < 1 4 4=kG(Rx,Sy,Tz)G(fx,Rx,Rx)kM(x,y,z).

Case (3) If x,z[0, 1 2 ], y( 1 2 ,1], then we have

G 2 (fx,gy,hz)= G 2 ( 1 , 4 5 , 6 7 ) = 4 25 .

If y=1, then

G(fx,Sy,Tz)G(Rx,gy,Tz)=G(1,0,0)G ( 0 , 4 5 , 0 ) = 16 5 .

If y( 1 2 ,1), then

G(fx,Sy,Tz)G(Rx,gy,Tz)=G ( 1 , 4 5 , 0 ) G ( 0 , 4 5 , 0 ) = 16 5 .

So we know G(Rx,Sy,Tz)G(fx,Rx,Rx)= 16 5 . Thus we have

G 2 (fx,gy,hz)= 4 25 < 1 4 16 5 =kG(fx,Sy,Tz)G(Rx,gy,Tz)kM(x,y,z).

Case (4) If y,z[0, 1 2 ], x( 1 2 ,1], then we have

G 2 (fx,gy,hz)= G 2 ( 4 5 , 5 6 , 6 7 ) = 16 1 , 225 .

If x=1, then

G(fx,Sy,Tz)G(Rx,gy,Tz)=G ( 4 5 , 1 , 0 ) G ( 6 7 , 5 6 , 0 ) = 24 7 .

If x( 1 2 ,1), then

G(fx,Sy,Tz)G(Rx,gy,Tz)=G ( 4 5 , 1 , 0 ) G ( 4 5 , 5 6 , 0 ) = 10 3 .

So we know G(fx,Sy,Tz)G(Rx,gy,Tz) 10 3 . Thus we have

G 2 (fx,gy,hz)= 16 1 , 225 < 1 4 10 3 kG(fx,Sy,Tz)G(Rx,gy,Tz)kM(x,y,z).

Case (5) x[0, 1 2 ], y,z( 1 2 ,1], then we have

G 2 (fx,gy,hz)= G 2 ( 1 , 4 5 , 4 5 ) = 4 25 .

If y=1, then

G(fx,Sy,hz)G(Rx,gy,hz)=G ( 1 , 0 , 4 5 ) G ( 0 , 4 5 , 4 5 ) = 16 5 .

If y( 1 2 ,1), then

G(fx,Sy,hz)G(Rx,gy,hz)=G ( 1 , 4 5 , 4 5 ) G ( 0 , 4 5 , 4 5 ) = 16 25 .

So we know G(fx,Sy,hz)G(Rx,gy,hz) 16 25 . Thus we have

G 2 (fx,gy,hz)= 4 25 = 1 4 16 25 kG(fx,Sy,hz)G(Rx,gy,hz)kM(x,y,z).

Case (6) y[0, 1 2 ], x,z( 1 2 ,1], then we have

G 2 (fx,gy,hz)= G 2 ( 4 5 , 5 6 , 4 5 ) = 1 225 .

If x=1, then

G(fx,Sy,hz)G(Rx,gy,hz)=G ( 4 5 , 1 , 4 5 ) G ( 6 7 , 5 6 , 4 5 ) = 8 175 .

If x( 1 2 ,1), then

G(fx,Sy,hz)G(Rx,gy,hz)=G ( 4 5 , 1 , 4 5 ) G ( 4 5 , 5 6 , 4 5 ) = 2 75 .

So we know G(fx,Sy,hz)G(Rx,gy,hz) 2 75 . Thus we have

G 2 (fx,gy,hz)= 1 225 < 1 4 2 75 kG(fx,Sy,hz)G(Rx,gy,hz)kM(x,y,z).

Case (7) z[0, 1 2 ], x,y( 1 2 ,1], then we have

G 2 (fx,gy,hz)= G 2 ( 4 5 , 4 5 , 6 7 ) = 16 1 , 225 .

If x=1, y=1, then

G(fx,Sy,Tz)G(Rx,gy,Tz)=G ( 4 5 , 0 , 0 ) G ( 6 7 , 4 5 , 0 ) = 96 35 .

If x=1, y( 1 2 ,1), then

G(fx,Sy,Tz)G(Rx,gy,Tz)=G ( 4 5 , 4 5 , 0 ) G ( 6 7 , 4 5 , 0 ) = 96 35 .

If y=1, x( 1 2 ,1), then

G(fx,Sy,Tz)G(Rx,gy,Tz)=G ( 4 5 , 0 , 0 ) G ( 4 5 , 4 5 , 0 ) = 64 25 .

If x,y( 1 2 ,1), then

G(fx,Sy,Tz)G(Rx,gy,Tz)=G ( 4 5 , 4 5 , 0 ) G ( 4 5 , 4 5 , 0 ) = 64 25 .

So we know G(fx,Sy,Tz)G(Rx,gy,Tz) 64 25 . Thus we have

G 2 (fx,gy,hz)= 16 1 , 225 < 1 4 64 25 kG(fx,Sy,Tz)G(Rx,gy,Tz)kM(x,y,z).

Case (8) If x,y,z( 1 2 ,1], then

G 2 (fx,gy,hz)= G 2 ( 4 5 , 4 5 , 4 5 ) =0 1 4 M(x,y,z)=kM(x,y,z).

Then in all the above cases, the mappings f, g, h, R, T, and S are satisfying the condition (2.1) of Theorem 2.1 with k= 1 4 , so that all the conditions of Theorem 2.1 are satisfied. Moreover, 4 5 is the unique common fixed point of f, g, h, R, T, and S.

The following example supports the usability of our results for nonsymmetric generalized metric spaces.

Example 2.2 Let X={0,1,2} be a set with G-metric defined by Table 1. It is easy to see that (X,G) is a nonsymmetric generalized metric space. Let the maps f,g,h,R,S,T:XX be defined by Table 2.

Table 1 The definition of G -metric on X
Table 2 The definition of maps f , g , h , R , S and T on X

Clearly, the subspace RX, SX, and TX are closed in X, fXSX, gXTX, and hXRX with the pairs (f,R), (g,S), and (h,T) being weakly compatible. Also two pairs (f,R) and (g,S) satisfy the common (E.A) property, indeed, x n =0 and y n =1 for each nN are the required sequences.

To check the contractive condition (2.1) for all x,yX, we consider the following cases.

Note that for Case (1) x=y=z=0, (2) x=y=0, z=2, (3) x=z=0, y=1, (4) x=0, y=1, z=2, (5) x=1, y=z=0, (6) x=1, y=0, z=2, (7) x=y=1, z=0, (8) x=y=1, z=2, (9) x=2, y=z=0, (10) x=z=2, y=0, (11) x=2, y=1, z=0, and (12) x=z=2, y=1.

We have G(fx,gy,hz)=G(0,0,0)=0, and hence (2.1) is obviously satisfied.

Case (13) If x=y=0, z=1, then fx=gy=0, hz=1, Rx=Sy=0, Tz=2, hence we have

G 2 ( f 0 , g 0 , h 1 ) = G 2 ( 0 , 0 , 1 ) = 1 < 1 3 × 9 = 1 3 G ( 0 , 0 , 2 ) G ( 0 , 0 , 2 ) = 1 3 G ( f 0 , S 0 , T 1 ) G ( R 0 , g 0 , T 1 ) 1 3 M ( 0 , 0 , 1 ) .

Case (14) If x=0, y=z=1, then fx=gy=0, hz=1, Tz=2, hence we have

G 2 ( f 0 , g 1 , h 1 ) = G 2 ( 0 , 0 , 1 ) = 1 < 1 3 × 9 = 1 3 G ( 0 , 0 , 2 ) G ( 0 , 0 , 2 ) = 1 3 G ( f 0 , S 1 , T 1 ) G ( R 0 , g 1 , T 1 ) 1 3 M ( 0 , 1 , 1 ) .

Case (15) If x=z=0, y=2, then fx=hz=0, gy=1, Sy=2, hence we have

G 2 ( f 0 , g 2 , h 0 ) = G 2 ( 0 , 1 , 0 ) = 1 = 1 3 × 3 = 1 3 G ( 0 , 2 , 0 ) G ( 0 , 1 , 0 ) = 1 3 G ( R 0 , S 2 , h 0 ) G ( f 0 , g 2 , T 0 ) 1 3 M ( 0 , 2 , 0 ) .

Case (16) If x=0, y=2, z=1, then fx=0, gy=hz=1, Rx=0, Sy=Tz=2, hence we have

G 2 ( f 0 , g 2 , h 1 ) = G 2 ( 0 , 1 , 1 ) = 1 < 1 3 × 16 = 1 3 G ( 0 , 2 , 1 ) G ( 0 , 1 , 2 ) = 1 3 G ( R 0 , S 2 , h 1 ) G ( f 0 , g 2 , T 1 ) 1 3 M ( 0 , 2 , 1 ) .

Case (17) If x=0, y=z=2, then fx=hz=0, gy=1, Sy=2, hence we have

G 2 ( f 0 , g 2 , h 2 ) = G 2 ( 0 , 1 , 0 ) = 1 = 1 3 × 3 = 1 3 G ( 0 , 2 , 0 ) G ( 0 , 1 , 1 ) = 1 3 G ( R 0 , S 2 , h 2 ) G ( f 0 , g 2 , T 2 ) 1 3 M ( 0 , 2 , 2 ) .

Case (18) If x=z=1, y=0, then fx=gy=0, hz=1, Tz=2, hence we have

G 2 ( f 1 , g 0 , h 1 ) = G 2 ( 0 , 0 , 1 ) = 1 < 1 3 × 12 = 1 3 G ( 0 , 0 , 2 ) G ( 1 , 0 , 2 ) = 1 3 G ( f 1 , S 0 , T 1 ) G ( R 1 , g 0 , T 1 ) 1 3 M ( 1 , 0 , 1 ) .

Case (19) x=y=z=1, then fx=gy=0, hz=1, Tz=2, hence we have

G 2 ( f 1 , g 1 , h 1 ) = G 2 ( 0 , 0 , 1 ) = 1 < 1 3 × 12 = 1 3 G ( 0 , 0 , 2 ) G ( 1 , 0 , 2 ) = 1 3 G ( f 1 , S 1 , T 1 ) G ( R 1 , g 1 , T 1 ) 1 3 M ( 1 , 1 , 1 ) .

Case (20) If x=1, y=2, z=0, then fx=hz=0, gy=1, Sy=2, hence we have

G 2 ( f 1 , g 2 , h 0 ) = G 2 ( 0 , 1 , 0 ) = 1 < 1 3 × 4 = 1 3 G ( 1 , 2 , 0 ) G ( 0 , 1 , 0 ) = 1 3 G ( R 1 , S 2 , h 0 ) G ( f 1 , g 2 , T 0 ) 1 3 M ( 1 , 2 , 0 ) .

Case (21) If x=z=1, y=2, then fx=0, gy=hz=1, Tz=2, hence we have

ψ ( G ( f x , g y , h z ) ) = 3 G ( 0 , 1 , 1 ) = 3 < 11 4 × 2 = 11 4 G ( 1 , 2 , 2 ) = 11 4 G ( h z , T z , T z ) 11 4 M ( x , y , z ) = ψ ( M ( x , y , z ) ) ϕ ( M ( x , y , z ) ) .

Case (22) If x=1, y=z=2, then fx=hz=0, gy=1, Sy=2, hence we have

ψ ( G ( f x , g y , h z ) ) = 3 G ( 0 , 1 , 0 ) = 3 < 11 4 × 2 = 11 4 G ( 1 , 2 , 2 ) = 11 4 G ( g y , S y , S y ) 11 4 M ( x , y , z ) = ψ ( M ( x , y , z ) ) ϕ ( M ( x , y , z ) ) .

Case (23) If x=2, y=0, z=1, then fx=gy=0, hz=1, Tz=2, hence we have

ψ ( G ( f x , g y , h z ) ) = 3 G ( 0 , 0 , 1 ) = 3 < 11 4 × 2 = 11 4 G ( 1 , 2 , 2 ) = 11 4 G ( h z , T z , T z ) 11 4 M ( x , y , z ) = ψ ( M ( x , y , z ) ) ϕ ( M ( x , y , z ) ) .

Case (24) If x=2, y=z=1, then fx=gy=0, hz=1, Tz=2, hence we have

ψ ( G ( f x , g y , h z ) ) = 3 G ( 0 , 0 , 1 ) = 3 < 11 4 × 2 = 11 4 G ( 1 , 2 , 2 ) = 11 4 G ( h z , T z , T z ) 11 4 M ( x , y , z ) = ψ ( M ( x , y , z ) ) ϕ ( M ( x , y , z ) ) .

Case (25) x=y=2, z=0, then fx=hz=0, gy=1, Sy=2, hence we have

ψ ( G ( f x , g y , h z ) ) = 3 G ( 0 , 1 , 0 ) = 3 < 11 4 × 2 = 11 4 G ( 1 , 2 , 2 ) = 11 4 G ( g y , S y , S y ) 11 4 M ( x , y , z ) = ψ ( M ( x , y , z ) ) ϕ ( M ( x , y , z ) ) .

Case (26) x=y=2, z=1, then fx=0, gy=hz=1, Tz=2, hence we have

ψ ( G ( f x , g y , h z ) ) = 3 G ( 0 , 1 , 1 ) = 3 < 11 4 × 2 = 11 4 G ( 1 , 2 , 2 ) = 11 4 G ( h z , T z , T z ) 11 4 M ( x , y , z ) = ψ ( M ( x , y , z ) ) ϕ ( M ( x , y , z ) ) .

Case (27) If x=y=z=2, then fx=hz=0, gy=1, Sy=2, hence we have

ψ ( G ( f x , g y , h z ) ) = 3 G ( 0 , 1 , 0 ) = 3 < 11 4 × 2 = 11 4 G ( 1 , 2 , 2 ) = 11 4 G ( g y , S y , S y ) 11 4 M ( x , y , z ) = ψ ( M ( x , y , z ) ) ϕ ( M ( x , y , z ) ) .

Hence, all of the conditions of Theorem 2.1 are satisfied. Moreover, 0 is the unique common fixed point of f, g, h, R, S, and T.

Corollary 2.1 Let (X,G) be a G-metric space. Suppose mappings f,g,h,R,S,T:XX satisfying the following conditions:

G 2 ( f x , g y , h z ) a 1 G ( R x , S y , T z ) G ( f x , R x , R x ) + a 2 G ( g y , S y , S y ) G ( h z , T z , T z ) + a 3 G ( f x , S y , T z ) G ( R x , g y , T z ) + a 4 G ( R x , S y , h z ) G ( f x , g y , T z ) + a 5 G ( f x , S y , h z ) G ( R x , g y , h z )
(2.2)

for all x,y,zX. Here a i 0 (i=1,2,3,4,5) and 0 a 1 + a 2 + a 3 + a 4 + a 5 <1. If one of the following conditions is satisfied, then the pairs (f,R), (g,S), and (h,T) have a common point of coincidence in X.

  1. (i)

    The subspace RX is closed in X, fXSX, gXTX, and the two pairs of (f,R) and (g,S) satisfy the common (E.A) property.

  2. (ii)

    The subspace SX is closed in X, gXTX, hXRX, and the two pairs of (g,S) and (h,T) satisfy the common (E.A) property.

  3. (iii)

    The subspace TX is closed in X, fXSX, hXRX, and the two pairs of (f,R) and (h,T) satisfy the common (E.A) property.

Further, if the pairs (f,R), (g,S), and (h,T) are weakly compatible, then f, g, h, R, S, and T have a unique common fixed point in X.

Proof Suppose that

M(x,y,z)=max { G ( R x , S y , T z ) G ( f x , R x , R x ) , G ( g y , S y , S y ) G ( h z , T z , T z ) , G ( f x , S y , T z ) G ( R x , g y , T z ) , G ( R x , S y , h z ) G ( f x , g y , T z ) , G ( f x , S y , h z ) G ( R x , g y , h z ) } .

Then

a 1 G ( R x , S y , T z ) G ( f x , R x , R x ) + a 2 G ( g y , S y , S y ) G ( h z , T z , T z ) + a 3 G ( f x , S y , T z ) G ( R x , g y , T z ) + a 4 G ( R x , S y , h z ) G ( f x , g y , T z ) + a 5 G ( f x , S y , h z ) G ( R x , g y , h z ) ( a 1 + a 2 + a 3 + a 4 + a 5 ) M ( x , y , z ) .

So, if the condition (2.2) holds, then G 2 (fx,gy,hz)( a 1 + a 2 + a 3 + a 4 + a 5 )M(x,y,z). Taking k= a 1 + a 2 + a 3 + a 4 + a 5 in Theorem 2.1, the conclusion of Corollary 2.1 can be obtained from Theorem 2.1, since 0 a 1 + a 2 + a 3 + a 4 + a 5 <1. □

References

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

    MathSciNet  Google Scholar 

  2. Mustafa Z, Obiedat H, Awawdeh F: Some fixed point theorems for mappings on complete G -metric space. Fixed Point Theory Appl. 2008. Article ID 189870, 2008: 10.1155/2008/189870

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  5. Mustafa Z, Khandagji M, Shatanawi W: Fixed point results on complete G -metric spaces. Studia Sci. Math. Hung. 2011,48(3):304-319.

    MathSciNet  Google Scholar 

  6. Obiedat H, Mustafa Z: Fixed point results on a nonsymmetric G -metric spaces. Jordan J. Math. Stat. 2010,3(2):65-79.

    Google Scholar 

  7. Aydi H, Shatanawi W, Vetro C: On generalized weakly G -contraction mapping in G -metric spaces. Comput. Math. Appl. 2011,62(11):4222-4229. 10.1016/j.camwa.2011.10.007

    MathSciNet  Article  Google Scholar 

  8. Aydi H: A fixed point result involving a generalized weakly contractive condition in G -metric spaces. Bull. Math. Anal. Appl. 2011,3(4):180-188.

    MathSciNet  Google Scholar 

  9. Gajić L, Stojaković M: On Ćirić generalization of mappings with a contractive iterate at a point in G -metric spaces. Appl. Math. Comput. 2012,219(1):435-441. 10.1016/j.amc.2012.06.041

    MathSciNet  Article  Google Scholar 

  10. Zhou SH, Gu F: Some new fixed points in G -metric spaces. J. Hangzhou Norm. Univ., Nat. Sci. Ed. 2012,11(1):47-50.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  13. Abbas M, Rhoades BE: Common fixed point results for noncommuting mappings without continuity in generalized metric spaces. Appl. Math. Comput. 2009,215(1):262-269. 10.1016/j.amc.2009.04.085

    MathSciNet  Article  Google Scholar 

  14. Abbas M, Nazir T, Saadati R: Common fixed point results for three maps in generalized metric space. Adv. Differ. Equ. 2011. Article ID 49, 2011:

    Google Scholar 

  15. Abbas M, Nazir T, Radenović S: Some periodic point results in generalized metric spaces. Appl. Math. Comput. 2010,217(8):4094-4099. 10.1016/j.amc.2010.10.026

    MathSciNet  Article  Google Scholar 

  16. Abbas M, Khan SH, Nazir T: Common fixed points of R -weakly commuting maps in generalized metric spaces. Fixed Point Theory Appl. 2011. Article ID 784595, 2011:

    Google Scholar 

  17. Vats RK, Kumar S, Sihag V: Some common fixed point theorems for compatible mappings of type (A) in complete G -metric space. Adv. Fuzzy Math. 2011,6(1):27-38.

    Google Scholar 

  18. Abbas M, Nazir T, Vetro P: Common fixed point results for three maps in G -metric spaces. Filomat 2011,25(4):1-17. 10.2298/FIL1104001A

    MathSciNet  Article  Google Scholar 

  19. Gu F: Common fixed point theorems for six mappings in generalized metric spaces. Abstr. Appl. Anal. 2012. Article ID 379212, 2012: 10.1155/2012/379212

    Google Scholar 

  20. Gu F: Some new common coupled fixed point results in two generalized metric spaces. Fixed Point Theory Appl. 2013. Article ID 181, 2013: 10.1186/1687-1812-2013-181

    Google Scholar 

  21. Gu F, Yang Z: Some new common fixed point results for three pairs of mappings in generalized metric spaces. Fixed Point Theory Appl. 2013. Article ID 174, 2013: 10.1186/1687-1812-2013-174

    Google Scholar 

  22. Gu F, Ye H: Common fixed point theorems of Altman integral type mappings in G -metric spaces. Abstr. Appl. Anal. 2012. Article ID 630457, 2012: 630457 10.1155/2012/630457

    Google Scholar 

  23. Ye H, Gu F: Common fixed point theorems for a class of twice power type contraction maps in G -metric spaces. Abstr. Appl. Anal. 2012. Article ID 736214, 2012: 736214

    Google Scholar 

  24. Yin Y, Gu F: Common fixed point theorem about four mappings in G -metric spaces. J. Hangzhou Norm. Univ., Nat. Sci. Ed. 2012,11(6):511-515.

    Google Scholar 

  25. Ye HQ, Lu J, Gu F: A new common fixed point theorem for noncompatible mappings of type ( A f ) in G -metric space. J. Hangzhou Norm. Univ., Nat. Sci. Ed. 2013,12(1):50-56.

    Google Scholar 

  26. Shen YJ, Lu J, Zheng HH: Common fixed point theorem for converse commuting mappings in generalized metric spaces. J. Hangzhou Norm. Univ., Nat. Sci. Ed. 2014,13(5):542-547.

    Google Scholar 

  27. Hussain N, Parvaneh V, Hoseini Ghoncheh SJ: Generalized contractive mappings and weakly α -admissible pairs in G -metric spaces. Sci. World J. 2014. Article ID 941086, 2014:

    Google Scholar 

  28. Hussain N, Parvaneh V, Roshan JR: Fixed point results for G - α -contractive maps with application to boundary value problems. Sci. World J. 2014. Article ID 585964, 2014:

    Google Scholar 

  29. Hussain N, Roshan JR, Parvaneh V, Latif A: A unification of G -metric, partial metric and b -metric spaces. Abstr. Appl. Anal. 2014. Article ID 180698, 2014:

    Google Scholar 

  30. 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. Article ID 48, 2012: 10.1186/1687-1812-2012-48

    Google Scholar 

  31. Mustafa Z: Common fixed points of weakly compatible mappings in G -metric spaces. Appl. Math. Sci. 2012,6(92):4589-4600.

    MathSciNet  Google Scholar 

  32. Abbas M, Nazir T, Dorić D:Common fixed point of mappings satisfying (E.A) property in generalized metric spaces. Appl. Math. Comput. 2012,218(14):7665-7670. 10.1016/j.amc.2011.11.113

    MathSciNet  Article  Google Scholar 

  33. Mustafa Z, Aydi H, Karapınar E: On common fixed points in G -metric spaces using (E.A) property. Comput. Math. Appl. 2012,64(6):1944-1956. 10.1016/j.camwa.2012.03.051

    MathSciNet  Article  Google Scholar 

  34. Long W, Abbas M, Nazir T, Radenović S:Common fixed point for two pairs of mappings satisfying (E.A) property in generalized metric spaces. Abstr. Appl. Anal. 2012. Article ID 394830, 2012: 10.1155/2012/394830

    Google Scholar 

  35. Gu F, Yin Y:Common fixed point for three pairs of self-maps satisfying common (E.A) property in generalized metric spaces. Abstr. Appl. Anal. 2013. Article ID 808092, 2013: 10.1155/2013/808092

    Google Scholar 

  36. Gu F, Shatanawi W: Common fixed point for generalized weakly G -contraction mappings satisfying common (E.A) property in G -metric spaces. Fixed Point Theory Appl. 2013. Article ID 48, 2013: 10.1186/1687-1812-2013-309

    Google Scholar 

  37. Jleli M, Samet B: Remarks on G -metric spaces and fixed point theorems. Fixed Point Theory Appl. 2012. Article ID 210, 2012: 10.1186/1687-1812-2012-210

    Google Scholar 

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

    Google Scholar 

  39. Mustafa Z, Obiedat H, Awawdeh H: Some fixed point theorem for mappings on complete G -metric spaces. Fixed Point Theory Appl. 2008. Article ID 189870, 2008:

    Google Scholar 

  40. Karapınar E, Agarval R: Further remarks on G -metric spaces. Fixed Point Theory Appl. 2013. Article ID 154, 2013: 10.1186/1687-1812-2013-154

    Google Scholar 

  41. Jungck G, Rhoades BE: Fixed point for set valued functions without continuity. Indian J. Pure Appl. Math. 1998, 29: 227-238.

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The present studies are supported by the National Natural Science Foundation of China (11071169, 11361070), the Natural Science Foundation of Zhejiang Province (Y6110287, LY12A01030), the Natural Science Foundation of Shandong Province (ZR2013AL015) and the Innovation Foundation of Graduate Student of Hangzhou Normal University.

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Gu, F., Shen, Y. & Wang, L. Common fixed point results under a new contractive condition without using continuity. J Inequal Appl 2014, 464 (2014). https://doi.org/10.1186/1029-242X-2014-464

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Keywords

  • generalized metric space
  • weakly compatible mappings
  • contraction mapping for twice power type
  • common (E.A) property