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

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R E S E A R C H

Open Access

Common ﬁxed point results under a new

contractive condition without using

continuity

Feng Gu

1

_{, Yunjuan Shen}

1

_{and Lin Wang}

2*

Dedicated to Professor SS Chang on the occasion of his 80th birthday

*_{Correspondence:}

[email protected]

2_{College of Statistics and}

Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, P.R. China Full list of author information is available at the end of the article

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.

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

1 Introduction and preliminaries

In , Mustafa and Sims [] introduced a new structure of generalized metric space which is called aG-metric. Based on the notion of generalized metric spaces, Mustafaet al.[–], Obiedat and Mustafa [], Aydiet al.[, ], Gajić and Stojaković [], Zhou and Gu [], Shatanawi [] obtained some fixed point results for mappings satisfying differ-ent contractive conditions. Chughet al.[] obtained some fixed point results for maps satisfying propertyPinG-metric spaces. The study of common fixed point problems in

G-metric spaces was initiated by Abbas and Rhoades []. Subsequently, many authors obtained many common fixed point theorems for the mappings satisfying different con-tractive conditions; see [–] for more details. Recently, some authors using (E.A) prop-erty in generalized metric space to prove common fixed point, such as Abbaset al.[], Mustafaet al.[], Longet al.[], Gu and Yin [], Gu and Shatanawi [].

Recently, Jleli and Samet [] and Sametet al.[] observed that some fixed point the-orems in the context of aG-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 onG-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 []).

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Very recently, Karapınar and Agarwal suggest new contraction conditions inG-metric space in a way that the techniques in [, ] are not applicable. In this approach [], contraction conditions cannot be expressed in two variables. So, in some cases, as is no-ticed even in Jleli-Samet’s paper [], 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 com-mon 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 .[] LetXbe a nonempty set, and letG:X×X×X→R+_{be a function}

satisfying the following axioms:

(G) G(x,y,z) = ifx=y=z;

(G)  <G(x,y,z)for allx,y∈Xwithx=y;

(G) G(x,x,y)≤G(x,y,z)for allx,y,z∈Xwithz=y;

(G) G(x,y,z) =G(x,z,y) =G(y,z,x)· · · (symmetry in all three variables); (G) G(x,y,z)≤G(x,a,a) +G(a,y,z)for allx,y,z,a∈X(rectangle inequality).

Then the functionGis called a generalized metric or aG-metric onX, and the pair (X,G) is called aG-metric space.

Deﬁnition .[] Let (X,G) be aG-metric space, and let{xn}a sequence of points inX, a pointxinXis said to be the limit of the sequence{xn},limn→∞G(x,xn,xm) = , and one says that sequence{xn}isG-convergent tox.

Thus, ifxn→xorlimn→∞xn=xin aG-metric space (X,G), then if for each> , there exists a positive integerNsuch thatG(x,xn,xm) <for alln,m≥N.

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

() {xn}isG-convergent tox.

() G(xn,xn,x)→asn→ ∞.

() G(xn,x,x)→asn→ ∞.

() G(xm,xn,x)→asm,n→ ∞.

Deﬁnition .[] Let (X,G) be aG-metric space. A sequence{xn}is calledG-Cauchy if, for each> , there exists a positive integerNsuch thatG(xm,xn,x) <for alln,m,l≥N;

i.e.ifG(xm,xn,xl)→ asm,n,l→ ∞.

Proposition .[] If(X,G)is a G-metric space then the following are equivalent:

() The sequence{xn}isG-Cauchy.

() For each> ,there exists a positive integerNsuch thatG(xm,xn,x) <for all

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Proposition .[] 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 .[] AG-metric space (X,G) is said to beG-complete if everyG-Cauchy sequence in (X,G) isG-convergent inX.

Deﬁnition .[] Letf andgbe self-maps of a setX. Ifw=fx=gxfor somexinX, then

xis called a coincidence point off andg, andwis called a point of coincidence off andg.

Deﬁnition .[] Two self-mappingsf andgonXare said to be weakly compatible if they commute at coincidence points.

Deﬁnition .[] LetXbe aG-metric space. Self-mapsf andgonXare said to satisfy theG-(E.A) property if there exists a sequence {xn}inX such that{fxn}and{gxn} are

G-convergent to somet∈X.

Deﬁnition .[] Let (X,d) be aG-metric space andA,B,S, andT be four self-maps onX. The pairs (A,S) and (B,T) are said to satisfy the common (E.A) property if there exist two sequences{xn}and{yn}inXsuch that

lim

n→∞Axn=nlim→∞Sxn=nlim→∞Byn=nlim→∞Tyn=t

for somet∈X.

Deﬁnition .[] Self-mappingsf andgof aG-metric space (X,G) are said to be com-patible iflim_n_→∞G(fgxn,gfxn,gfxn) =  andlimn→∞G(gfxn,fgxn,fgxn) = , whenever{xn} is a sequence inXsuch that

lim

n→∞fxn=nlim→∞gxn=t

for somet∈X.

2 Main results

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

G(fx,gy,hz)≤kmax

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

G(Rx,Sy,Tz)G(fx,Rx,Rx),

G(gy,Sy,Sy)G(hz,Tz,Tz),

G(fx,Sy,Tz)G(Rx,gy,Tz),

G(Rx,Sy,hz)G(fx,gy,Tz),

G(fx,Sy,hz)G(Rx,gy,hz)

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(.)

for all x,y,z∈X, ≤k< .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.

(i) The subspaceRXis closed inX,fX⊆SX,gX⊆TX,and the two pairs of(f,R)and

(g,S)satisfy the common(E.A)property.

(ii) The subspaceSXis closed inX,gX⊆TX,hX⊆RX,and the two pairs of(g,S)and

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(iii) The subspaceTXis closed inX,fX⊆SX,hX⊆RX,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 thatRXis closed inX,fX⊆SX,gX⊆TX, and the two pairs of (f,R) and (g,S) satisfy the common (E.A) property, then by Deﬁnition . we know that there exist two sequences{xn}and{yn}inXsuch that

lim

n→∞fxn=nlim→∞Rxn=nlim→∞gyn=nlim→∞Syn=t

for somet∈X.

Since gX⊆ TX, there exists a sequence {zn} in X such that gyn =Tzn. So we get limn→∞Tzn=limn→∞gyn=t. By the condition (.) we have

G(fxn,gyn,hzn)≤kmax

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

G(Rxn,Syn,Tzn)G(fxn,Rxn,Rxn),

G(gyn,Syn,Syn)G(hzn,Tzn,Tzn),

G(fxn,Syn,Tzn)G(Rxn,gyn,Tzn),

G(Rxn,Syn,hzn)G(fxn,gyn,Tzn),

G(fxn,Syn,hzn)G(Rxn,gyn,hzn)

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ .

Lettingn→ ∞, we have

G t,t, lim n→∞hzn

≤kG t,t, lim n→∞hzn

;

this givesG₍_t_,_t_,_lim

n→∞hzn) = , since ≤k< . Hencelimn→∞hzn=t.

SinceRXis a closed subspace ofX, andlimn→∞Rxn=t, there exists a pointu∈Xsuch thatRu=t. By the condition (.) we have

G(fu,gyn,hzn)≤kmax

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

G(Ru,Syn,Tzn)G(fu,Ru,Ru),

G(gyn,Syn,Syn)G(hzn,Tzn,Tzn),

G(fu,Syn,Tzn)G(Ru,gyn,Tzn),

G(Ru,Syn,hzn)G(fu,gyn,Tzn),

G(fu,Syn,hzn)G(Ru,gyn,hzn)

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ .

Lettingn→ ∞, we haveG₍_fu_,_t_,_t₎_≤_{, hence}_fu₌_t_{. Thus}_Ru₌_fu₌_t_{, so}_u_{is the}

coinci-dence point of the pair (f,R).

SincefX⊆SXandfu=t, there exists a pointv∈Xsuch thatSv=fu=t. By the condition (.) we have

G(fu,gv,hzn)≤kmax

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

G(Ru,Sv,Tzn)G(fu,Ru,Ru),

G(gv,Sv,Sv)G(hzn,Tzn,Tzn),

G(fu,Sv,Tzn)G(Ru,gv,Tzn),

G(Ru,Sv,hzn)G(fu,gv,Tzn),

G(fu,Sv,hzn)G(Ru,gv,hzn)

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ .

Lettingn→ ∞, we haveG₍_t_,_gv_,_t₎_≤_{, hence}_gv₌_t_{. Thus}_Sv₌_gv₌_t_{, so}_v_{is the}

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SincegX⊆TXandgv=t, there exists a pointw∈Xsuch thatTw=gv=t. By the con-dition (.) we have

G(t,t,hw) =G(fu,gv,hw)

≤kmax

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

G(Ru,Sv,Tw)G(fu,Ru,Ru),

G(gv,Sv,Sv)G(hw,Tw,Tw),

G(fu,Sv,Tw)G(Ru,gv,Tw),

G(Ru,Sv,hw)G(fu,gv,Tw),

G(fu,Sv,hw)G(Ru,gv,hw)

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

=kG₍_t_,_t_,_hw_),

hencehw=t, since ≤k< . ThusTw=hw=t, sowis the coincidence point of the pair (h,T).

In the above proof we getfu=Ru=gv=Sv=hw=Tw=t. Then we getft=Rt,gt=St, andht=Tt, since the pairs (f,R), (g,S), and (h,T) are weakly compatible. By the condition (.), we have

G(ft,t,t) =G(ft,gv,hw)

≤kmax

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

G(Rt,Sv,Tw)G(ft,Rt,Rt),

G(gv,Sv,Sv)G(hw,Tw,Tw),

G(ft,Sv,Tw)G(Rt,gv,Tw),

G(Rt,Sv,hw)G(ft,gv,Tw),

G(ft,Sv,hw)G(Rt,gv,hw)

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

=kG(ft,t,t),

henceG₍_ft_,_t_,_t_{) = , since }_≤_k_{< . Thus}_ft₌_t₌_Rt_{. Similarly, it can be shown that}_gt₌ St=tandht=Tt=t, which means thattis a common ﬁxed point off,g,h,R,S, andT.

Now we prove the uniqueness of the common ﬁxed pointt.

Lettandpbe two common ﬁxed point off,g,h,R,S, andT, then using the condition (.), we have

G(p,t,t) =G(fp,gt,ht)

≤kmax

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

G(Rp,St,Tt)G(fp,Rp,Rp),

G(gt,St,St)G(ht,Tt,Tt),

G(fp,St,Tt)G(Rp,gt,Tt),

G(Rp,St,ht)G(fp,gt,Tt),

G(fp,St,ht)G(Rp,gt,ht)

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

=kG(p,t,t),

henceG₍_p_,_t_,_t_{) = , since }_≤_k_{< . Thus}_p₌_t_{. So common ﬁxed point is unique.}

Example . LetX= [, ] be aG-metric space with

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We deﬁne mappingsf,g,h,R,S, andTonXby

fx=

, x∈[,_],

 , x∈(

 , ],

gx=



, x∈[,  ], 

, x∈(  , ],

hx=



, x∈[,  ], 

, x∈(  , ], Rx= ⎧ ⎪ ⎨ ⎪ ⎩

, x∈[,_],

 , x∈(

 , ), 

, x= ,

Sx= ⎧ ⎪ ⎨ ⎪ ⎩

, x∈[,_]

 , x∈(

 , ),

, x= ,

Tx= ⎧ ⎪ ⎨ ⎪ ⎩

, x∈[,_],

 , x∈(

 , ), 

, x= .

Clearly, from the above functions we know that the subspaceRXis closed inX,fX⊆SX,

gX⊆TX,hX⊆RXand the pairs (f,R), (g,S), (h,T) be weakly compatible. The pairs (f,R) and (g,S) satisfy the common (E.A) property, letxn=_ andyn=_ for eachn∈Nbe the required sequences.

Now we prove that the mappingsf,g,h,R,S, andT are satisfying the condition (.) of Theorem . withk=_∈[, ). Let

M(x,y,z) =max

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ .

Case () Ifx,y,z∈[,_], then we have

G(fx,gy,hz) =G

, ,   =  ,

G(Rx,Sy,Tz)G(fx,Rx,Rx) =G(, , )G(, , ) = .

Thus we have

G(fx,gy,hz) =  <



· =kG(Rx,Sy,Tz)G(fx,Rx,Rx)≤kM(x,y,z).

Case () Ifx,y∈[,_],z∈(_, ], then we have

, ,   =  .

Ifz= , then

G(Rx,Sy,Tz)G(fx,Rx,Rx) =G

, , 

G(, , ) = .

Ifz∈(

, ), then

G(Rx,Sy,Tz)G(fx,Rx,Rx) =G

, , 

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So we knowG(Rx,Sy,Tz)G(fx,Rx,Rx) = . Thus we have

G(fx,gy,hz) =  <



· =kG(Rx,Sy,Tz)G(fx,Rx,Rx)≤kM(x,y,z).

Case () Ifx,z∈[,_],y∈(_, ], then we have

, ,

 

=  .

Ify= , then

G(fx,Sy,Tz)G(Rx,gy,Tz) =G(, , )G

, , 

= .

Ify∈(_, ), then

G(fx,Sy,Tz)G(Rx,gy,Tz) =G

, , 

G

, , 

=  .

So we knowG(Rx,Sy,Tz)G(fx,Rx,Rx) =_. Thus we have

G(fx,gy,hz) =  <

 ·



 =kG(fx,Sy,Tz)G(Rx,gy,Tz)≤kM(x,y,z).

Case () Ify,z∈[,_],x∈(_, ], then we have

 ,

 ,

 

=  ,.

Ifx= , then

 , , 

G

 ,

 , 

=  .

Ifx∈(_, ), then

 , , 

G

 ,

 , 

= .

So we knowG(fx,Sy,Tz)G(Rx,gy,Tz)≥_. Thus we have

G(fx,gy,hz) =  , <

 ·



 ≤kG(fx,Sy,Tz)G(Rx,gy,Tz)≤kM(x,y,z).

Case ()x∈[,_],y,z∈(_, ], then we have

, ,

 

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Ify= , then

G(fx,Sy,hz)G(Rx,gy,hz) =G

, ,  G , ,   = .

Ify∈(

, ), then

, ,   G , ,   = .

So we knowG(fx,Sy,hz)G(Rx,gy,hz)≥_. Thus we have

G(fx,gy,hz) =  =

 ·



≤kG(fx,Sy,hz)G(Rx,gy,hz)≤kM(x,y,z).

Case ()y∈[,_],x,z∈(_, ], then we have

 ,  ,   =  .

Ifx= , then

 , ,   G  ,  ,   =  .

Ifx∈(_, ), then

 , ,   G  ,  ,   =  .

So we knowG(fx,Sy,hz)G(Rx,gy,hz)≥_. Thus we have

G(fx,gy,hz) =  <

 ·



≤kG(fx,Sy,hz)G(Rx,gy,hz)≤kM(x,y,z).

Case ()z∈[,_],x,y∈(_, ], then we have

 ,  ,   =  ,.

Ifx= ,y= , then

 , ,  G  ,  ,  = .

Ifx= ,y∈(_, ), then

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Ify= ,x∈(_, ), then

 , , 

G

 ,

 , 

= .

Ifx,y∈(_, ), then

 ,

 , 

G

 ,

 , 

= .

So we knowG(fx,Sy,Tz)G(Rx,gy,Tz)≥_. Thus we have

G(fx,gy,hz) =  , <

 ·



 ≤kG(fx,Sy,Tz)G(Rx,gy,Tz)≤kM(x,y,z).

Case () Ifx,y,z∈(

, ], then

 ,

 

= ≤

M(x,y,z) =kM(x,y,z).

Then in all the above cases, the mappingsf,g,h,R,T, andSare satisfying the condition (.) of Theorem . withk=_, so that all the conditions of Theorem . are satisﬁed. Moreover,

is the unique common ﬁxed point off,g,h,R,T, andS.

The following example supports the usability of our results for nonsymmetric general-ized metric spaces.

Example . LetX={, , }be a set withG-metric deﬁned by Table . It is easy to see that (X,G) is a nonsymmetric generalized metric space. Let the mapsf,g,h,R,S,T:X→Xbe deﬁned by Table .

Clearly, the subspaceRX,SX, andTXare closed inX,fX⊆SX,gX⊆TX, andhX⊆RX

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,xn=  andyn=  for eachn∈Nare the required sequences.

[image:9.595.172.424.580.648.2]

To check the contractive condition (.) for allx,y∈X, we consider the following cases.

Table 1 The deﬁnition ofG-metric onX

(x,y,z) G(x,y,z)

(0, 0, 0), (1, 1, 1), (2, 2, 2) 0 (0, 0, 1), (0, 1, 0), (1, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0) 1 (1, 2, 2), (2, 1, 2), (2, 2, 1) 2 (0, 0, 2), (0, 2, 0), (2, 0, 0), (0, 2, 2), (2, 0, 2), (2, 2, 0) 3 (1, 1, 2), (1, 2, 1), (2, 1, 1), (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0) 4

Table 2 The deﬁnition of mapsf,g,h,R,SandTonX

x f(x) g(x) h(x) R(x) S(x) T(x)

[image:9.595.219.376.684.731.2]

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Note that for Case ()x=y=z= , ()x=y= ,z= , ()x=z= ,y= , ()x= ,

y= ,z= , ()x= ,y=z= , ()x= ,y= ,z= , ()x=y= ,z= , ()x=y= ,z= , ()x= ,y=z= , ()x=z= ,y= , ()x= ,y= ,z= , and ()x=z= ,y= .

We haveG(fx,gy,hz) =G(, , ) = , and hence (.) is obviously satisﬁed.

Case () Ifx=y= ,z= , thenfx=gy= ,hz= ,Rx=Sy= ,Tz= , hence we have

G(f,g,h) =G(, , ) = 

<  × =



G(, , )G(, , )

= 

G(f,S,T)G(R,g,T)

≤ 

M(, , ).

Case () Ifx= ,y=z= , thenfx=gy= ,hz= ,Tz= , hence we have

G(f,g,h) =G(, , ) = 

<  × =



G(, , )G(, , )

= 

G(f,S,T)G(R,g,T)

≤ 

M(, , ).

Case () Ifx=z= ,y= , thenfx=hz= ,gy= ,Sy= , hence we have

G(f,g,h) =G(, , ) = 

=   × =



G(, , )G(, , )

= 

G(R,S,h)G(f,g,T)

≤ 

M(, , ).

Case () Ifx= ,y= ,z= , thenfx= ,gy=hz= ,Rx= ,Sy=Tz= , hence we have

G(f,g,h) =G(, , ) = 

<  × =



G(, , )G(, , )

= 

G(R,S,h)G(f,g,T)

≤ 

M(, , ).

Case () Ifx= ,y=z= , thenfx=hz= ,gy= ,Sy= , hence we have

G(f,g,h) =G(, , ) = 

=  × =



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= 

G(R,S,h)G(f,g,T)

≤ 

M(, , ).

Case () Ifx=z= ,y= , thenfx=gy= ,hz= ,Tz= , hence we have

G(f,g,h) =G(, , ) = 

<  × =



G(, , )G(, , )

= 

G(f,S,T)G(R,g,T)

≤ 

M(, , ).

Case ()x=y=z= , thenfx=gy= ,hz= ,Tz= , hence we have

G(f,g,h) = G(, , ) = 

<   × =



G(, , )G(, , )

= 

G(f,S,T)G(R,g,T)

≤ 

M(, , ).

Case () Ifx= ,y= ,z= , thenfx=hz= ,gy= ,Sy= , hence we have

G₍_f_,_g_,_h_{) =}_G_{(, , ) = }

<  × =



G(, , )G(, , )

= 

G(R,S,h)G(f,g,T)

≤ 

M(, , ).

Case () Ifx=z= ,y= , thenfx= ,gy=hz= ,Tz= , hence we have

ψG(fx,gy,hz)= G(, , ) = 

<   × =



G(, , )

=

G(hz,Tz,Tz)≤ 

M(x,y,z) =ψM(x,y,z)–φM(x,y,z).

Case () Ifx= ,y=z= , thenfx=hz= ,gy= ,Sy= , hence we have

ψG(fx,gy,hz)= G(, , ) = 

<   × =



(12)

=

G(gy,Sy,Sy)≤ 

Case () Ifx= ,y= ,z= , thenfx=gy= ,hz= ,Tz= , hence we have

ψG(fx,gy,hz)= G(, , ) = 

<   × =



G(, , )

=

Case () Ifx= ,y=z= , thenfx=gy= ,hz= ,Tz= , hence we have

ψG(fx,gy,hz)= G(, , ) = 

<   × =



G(, , )

=

Case ()x=y= ,z= , thenfx=hz= ,gy= ,Sy= , hence we have

ψG(fx,gy,hz)= G(, , ) = 

<   × =



G(, , )

=

Case ()x=y= ,z= , thenfx= ,gy=hz= ,Tz= , hence we have

ψG(fx,gy,hz)= G(, , ) = 

<   × =



G(, , )

=

Case () Ifx=y=z= , thenfx=hz= ,gy= ,Sy= , hence we have

ψG(fx,gy,hz)= G(, , ) = 

<   × =



(13)

=

Hence, all of the conditions of Theorem . are satisﬁed. Moreover,  is the unique com-mon ﬁxed point off,g,h,R,S, andT.

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

G(fx,gy,hz)≤aG(Rx,Sy,Tz)G(fx,Rx,Rx) +aG(gy,Sy,Sy)G(hz,Tz,Tz)

+aG(fx,Sy,Tz)G(Rx,gy,Tz) +aG(Rx,Sy,hz)G(fx,gy,Tz)

+aG(fx,Sy,hz)G(Rx,gy,hz) (.)

for all x,y,z∈X.Here ai≥ (i= , , , , )and≤a+a+a+a+a< .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.

(i) The subspaceRXis closed inX,fX⊆SX,gX⊆TX,and the two pairs of(f,R)and

(g,S)satisfy the common(E.A)property.

(ii) The subspaceSXis closed inX,gX⊆TX,hX⊆RX,and the two pairs of(g,S)and

(iii) The subspaceTXis closed inX,fX⊆SX,hX⊆RX,and the two pairs of(f,R)and

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

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

.

Then

aG(Rx,Sy,Tz)G(fx,Rx,Rx) +aG(gy,Sy,Sy)G(hz,Tz,Tz)

+aG(fx,Sy,Tz)G(Rx,gy,Tz) +aG(Rx,Sy,hz)G(fx,gy,Tz)

+aG(fx,Sy,hz)G(Rx,gy,hz)

≤(a+a+a+a+a)M(x,y,z).

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Competing interests

The authors declare that they have no competing interests. Authors’ contributions

The authors contributed equally to the writing of the present article. They also read and approved the ﬁnal manuscript. Author details

1_{Department of Mathematics, Institute of Applied Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang}

310036, China.2_{College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan}

650221, P.R. China. 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. Received: 2 April 2014 Accepted: 10 November 2014 Published:26 Nov 2014

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Cite this article as:Gu et al.:Common ﬁxed point results under a new contractive condition without using