Tripled fixed point of W-compatible mappings in abstract metric spaces

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R E S E A R C H

Open Access

Tripled ﬁxed point of

W

-compatible

mappings in abstract metric spaces

Hassen Aydi

1

_{, Mujahid Abbas}

2

_{, Wutiphol Sintunavarat}

3*

_{and Poom Kumam}

3*

*_{Correspondence:} [email protected]; [email protected] 3_{Department of Mathematics,} Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, 10140, Thailand Full list of author information is available at the end of the article

Abstract

In this paper, we introduce the concepts ofW-compatible mappings for mappings

F:X×X×X→Xandg:X→X, where (X,d) is an abstract metric space. We establish tripled coincidence and common tripled ﬁxed point theorems in such spaces. The presented theorems generalize and extend several well-known comparable results in literature, in particular the results of Abbas, Ali and Radenovi´c (Appl. Math. Comput. 217:195-202, 2010). We also provide an example to illustrate our obtained results.

MSC: 54H25; 47H10

Keywords: W-compatible mappings; tripled coincidence point; common tripled

ﬁxed point; cone metric space

1 Introduction

Fixed point theory has fascinated many researchers since  with the celebrated Banach fixed point theorem. This theorem provides a technique for solving a variety of applied problems in mathematical sciences and engineering. There exists vast literature on this topic and this is a very active area of research at present. Banach contraction principle has been generalized in different directions in different spaces by mathematicians over the years, for more details on this and related topics, we refer to [, , , , , , , , , –] and references therein.

Fixed point theory inK-metric andK-normed spaces was developed by Perovet al.[], Mukhamadijev and Stetsenko [], Vandergraft [] and others. For more details on ﬁxed point theory inK-metric andK-normed spaces, we refer the reader to a ﬁne survey paper by Zabreˇiko []. The main idea is to use an ordered Banach space instead of the set of real numbers, as the codomain for a metric.

In  Huang and Zhang [] reintroduced such spaces under the name of cone metric spaces and reintroduced the definition of convergent and Cauchy sequences in terms of interior points of the underlying cone. They also proved some fixed point theorems in such spaces in the same work. Afterwards, many papers about fixed point theory in cone metric spaces appeared (see, for example, [–, , , , , , ]). In , Abbaset al.

[] introduced the concept ofw-compatible mappings and obtained a coupled coincidence point and a coupled point of coincidence for mappings satisfying a contractive condition in cone metric spaces. Very recently, Aydiet al.[] introduced the concepts ofw-compatible mappings and generalized the results in [].

The aim of this paper is to introduce the concepts ofW-compatible mappings. Based on this notion, a tripled coincidence point and a common tripled ﬁxed point for mappings

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F:X×X×X→Xandg:X→Xare obtained, where (X,d) is a cone metric space. It is worth mentioning that our results do not rely on the assumption of normality condition of the cone. The presented theorems generalize and extend several well-known comparable results in literature. An example is also given in support of our results.

The following deﬁnitions and results will be needed in the sequel.

Deﬁnition ([, ]) LetEbe a real Banach space. A subsetPofEis called a cone if and only if:

(a) Pis closed, non-empty andP={E},

(b) a,b∈R,a,b≥,x,y∈Pimply thatax+by∈P, (c) P∩(–P) ={E},

where Eis the zero vector ofE.

Given a cone, deﬁne a partial orderingwith respect toPbyxyif and only ify–x∈P. We shall writexyfory–x∈IntP, whereIntPstands for interior ofP. Also we will use

x≺yto indicate thatxyandx=y.

The conePin the normed space (E, · ) is called normal whenever there is a number

k≥ such that for allx,y∈E, Exyimpliesx ≤ky. The least positive number satisfying this norm inequality is called a normal constant ofP.

Deﬁnition  LetXbe a non-empty set. Suppose thatd:X×X→Esatisﬁes: (d) Ed(x,y)for allx,y∈Xandd(x,y) = Eif and only ifx=y,

(d) d(x,y) =d(y,x)for allx,y∈X,

(d) d(x,y)d(x,z) +d(z,y)for allx,y,z∈X.

Thendis called a cone metric [] orK-metric [] onXand (X,d) is called a cone metric space [] orK-metric space [].

The concept of aK-metric space is more general than that of a metric space, because each metric space is aK-metric space whereX=RandP= [, +∞). For other examples ofK-metric spaces, we refer to [], pp. and .

Deﬁnition ([]) LetXbe aK-metric space,{xn}a sequence inXandx∈X. For every

c∈EwithcE, we say that{xn}is

(C) a Cauchy sequence if there is somek∈Nsuch that, for alln,m≥k,d(xn,xm)c, (C) a convergent sequence if there is somek∈Nsuch that, for alln≥k,d(xn,x)c.

This limit is denoted bylim_n→+∞xn=xorxn→xasn→+∞.

Note that every convergent sequence in aK-metric spaceXis a Cauchy sequence. AK -metric spaceXis said to be complete if every Cauchy sequence inXis convergent inX.

2 Main results

For simplicity, we denote from now onX ×X· · ·X×X

kterms

byXk _where_k_∈_N_and_X_{is a} non-empty set. We start by recalling some deﬁnitions.

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Deﬁnition (Lakshmikantham and Ćirić []) An element (x,y)∈X_{is called} (i) a coupled coincidence point of mappingsF:X_→_X_and_g_:_X_→_X_if_gx₌_F₍_x_,_y₎

andgy=F(y,x), and(gx,gy)is called a coupled point of coincidence; (ii) a common coupled ﬁxed point of mappingsF:X→Xandg:X→Xif

x=gx=F(x,y)andy=gy=F(y,x).

Note that ifgis the identity mapping, then Deﬁnition  reduces to Deﬁnition .

Deﬁnition (Abbas, Khan and Radenović []) The mappingsF:X_→_X_and_g_:_X_→_X are calledw-compatible ifg(F(x,y)) =F(gx,gy) whenevergx=F(x,y) andgy=F(y,x).

In , Samet and Vetro [] introduced a ﬁxed point of orderN≥. In particular, for

N= , we have the following deﬁnition.

Deﬁnition (Samet and Vetro []) An element (x,y,z)∈X_{is called a tripled ﬁxed point} of a given mappingF:X_→_X_if_x₌_F₍_x_,_y_,_z_),_y₌_F₍_y_,_z_,_x_{) and}_z₌_F₍_z_,_x_,_y_).

Note that, Berinde and Borcut [] defined differently the notion of a tripled fixed point in the case of ordered sets in order to keep true the mixed monotone property. For more details, see [].

Now, we introduce the following deﬁnitions.

Deﬁnition  An element (x,y,z)∈X_{is called}

(i) a tripled coincidence point of mappingsF:X_→_X_and_g_:_X_→_X_if_gx₌_F₍_x_,_y_,_z₎_,

gy=F(y,x,z)andgz=F(z,x,y). In this case(gx,gy,gz)is called a tripled point of coincidence;

(ii) a common tripled ﬁxed point of mappingsF:X_→_X_and_g_:_X_→_X_if

x=gx=F(x,y,z),y=gy=F(y,z,x)andz=gz=F(z,x,y).

Example  LetX=R. We deﬁneF:X_→_X_and_g_:_X_→_X_{as follows:}

F(x,y,z) =sin(x)sin(y)sin(z) and gx=  +x–π 

for allx,y,z∈X. Then (π

,

π

,

π

) is a tripled coincidence point ofFandg, and (, , ) is a tripled point of coincidence.

Deﬁnition  MappingsF:X_→_X_and_g_:_X_→_X_{are called}_W_{-compatible if}

F(gx,gy,gz) =gF(x,y,z)

wheneverF(x,y,z) =gx,F(y,z,x) =gyandF(z,y,x) =gz.

Example  LetX=R. DeﬁneF:X_→_X_and_g_:_X_→_X_{as follows:}

F(x,y,z) =x–y+z

 and gx=

x



for allx,y,z∈X. One can show that (x,y,z) is a tripled coincidence point ofFandgif and only ifx=y=z= . Here (, , ) is a common tripled ﬁxed point ofFandg. Note thatF

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Now we prove our ﬁrst result.

Theorem  Let(X,d)be a K -metric space with a cone P having non-empty interior and F :X _→_{X and g}_:_X_→_{X be mappings such that F}₍_X₎_⊆_g₍_X₎_{. Suppose that for any}

x,y,z,u,v,w∈X, the following condition

dF(x,y,z),F(u,v,w)ad

F(x,y,z),gx+ad

F(y,z,x),gy+ad

F(z,x,y),gz

+ad

F(u,v,w),gu+ad

F(v,w,u),gv

+ad

F(w,u,v),gw+ad

F(u,v,w),gx

+ad

F(v,w,u),gy+ad

F(w,u,v),gz

+ad

F(x,y,z),gu+ad

F(y,z,x),gv

+ad

F(z,x,y),gw+ad(gx,gu) +ad(gy,gv) +ad(gz,gw)

holds, where ai, i= , . . . , are nonnegative real numbers such that i=ai< . Then F and

g have a tripled coincidence point provided that g(X)is a complete subspace of X. Proof Letx,yandzbe three arbitrary points inX. By given assumption, there exists (x,y,z) such that

F(x,y,z) =gx, F(y,z,x) =gy and F(z,x,y) =gz.

Continuing this process, we can construct three sequences{xn},{yn}and{zn}inXsuch that

F(xn,yn,zn) =gxn+, F(yn,zn,xn) =gyn+ and F(zn,xn,yn) =gzn+ ∀n∈N. ()

Now, taking (x,y,z) = (xn,yn,zn) and (u,v,w) = (xn+,yn+,zn+) in the considered contrac-tive condition and using (), we have

d(gxn+,gxn+) =dF(xn,yn,zn),F(xn+,yn+,zn+)

ad

F(xn,yn,zn),gxn

+ad

F(yn,zn,xn),gyn

+ad

F(zn,xn,yn),gzn

+ad

F(xn+,yn+,zn+),gxn+

+ad

F(yn+,zn+,xn+),gyn+

+ad

F(zn+,xn+,yn+),gzn+

+ad

F(xn+,yn+,zn+),gxn

+ad

F(yn+,zn+,xn+),gyn

+ad

F(zn+,xn+,yn+),gzn

+ad

F(xn,yn,zn),gxn+

+ad

F(yn,zn,xn),gyn+

+ad

F(zn,xn,yn),gzn+

+ad(gxn,gxn+) +ad(gyn,gyn+) +ad(gzn,gzn+)

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+ad(gxn+,gxn) +ad(gyn+,gyn) +ad(gzn+,gzn) +ad(gxn,gxn+) +ad(gyn,gyn+) +ad(gzn,gzn+). Then, using the triangular inequality one can write for anyn∈N

( –a–a)d(gxn+,gxn+)(a+a+a)d(gxn+,gxn) + (a+a+a)d(gyn,gyn+) + (a+a+a)d(gzn,gzn+) + (a+a)d(gyn+,gyn+) + (a+a)d(gzn+,gzn+). () Similarly, following similar arguments to those given above, we obtain

( –a–a)d(gyn+,gyn+)(a+a+a)d(gyn+,gyn) + (a+a+a)d(gzn,gzn+) + (a+a+a)d(gxn,gxn+) + (a+a)d(gzn+,gzn+) + (a+a)d(gxn+,gxn+) () and

( –a–a)d(gzn+,gzn+)(a+a+a)d(gzn+,gzn) + (a+a+a)d(gxn,gxn+) + (a+a+a)d(gyn,gyn+) + (a+a)d(gxn+,gxn+) + (a+a)d(gyn+,gyn+). () Denote

δn=d(gxn+,gxn) +d(gyn+,gyn) +d(gzn+,gzn). Adding () to (), we have

( –a–a–a–a–a–a)δn+

(a+a+a+a+a+a+a+a+a)δn. ()

On the other hand, we have

d(gxn+,gxn+) =dF(xn+,yn+,zn+),F(xn,yn,zn)

ad

F(xn+,yn+,zn+),gxn+

+ad

F(yn+,zn+,xn+),gyn+

+ad

F(zn+,xn+,yn+),gzn+

+ad

F(xn,yn,zn),gxn

+ad

F(yn,zn,xn),gyn

+ad

F(zn,xn,yn),gzn

+ad

F(xn,yn,zn),gxn+

+ad

F(yn,zn,xn),gyn+

+ad

F(zn,xn,yn),gzn+

+ad

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+ad

F(yn+,zn+,xn+),gyn

+ad

F(zn+,xn+,yn+),gzn

+ad(gxn+,gxn) +ad(gyn+,gyn) +ad(gzn+,gzn) =ad(gxn+,gxn+) +ad(gyn+,gyn+)

+ad(gzn+,gzn+) +ad(gxn+,gxn) +ad(gyn+,gyn) +ad(gzn+,gzn) +ad(gxn+,gxn) +ad(gyn+,gyn) +ad(gzn+,gzn) +ad(gxn,gxn+) +ad(gyn,gyn+) +ad(gzn,gzn+). Thus

( –a–a)d(gxn+,gxn+)

(a+a+a)d(gxn+,gxn) + (a+a+a)d(gyn,gyn+) + (a+a+a)d(gzn,gzn+) + (a+a)d(gyn+,gyn+)

+ (a+a)d(gzn+,gzn+). ()

Similarly,

( –a–a)d(gyn+,gyn+)

(a+a+a)d(gyn+,gyn) + (a+a+a)d(gzn,gzn+) + (a+a+a)d(gxn,gxn+) + (a+a)d(gzn+,gzn+)

+ (a+a)d(gxn+,gxn+) ()

and

( –a–a)d(gzn+,gzn+)

(a+a+a)d(gzn+,gzn) + (a+a+a)d(gxn,gxn+) + (a+a+a)d(gyn,gyn+) + (a+a)d(gxn+,gxn+)

+ (a+a)d(gyn+,gyn+). ()

Adding () to (), we obtain that

( –a–a–a–a–a–a)δn+

(a+a+a+a+a+a+a+a+a)δn. ()

Finally, from () and (), for anyn∈N, we have

 – 

i=

ai

δn+ _

i=

ai

+a+a+a

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that is

δn+κδn ∀n∈N, ()

where

κ=( 

i=ai) +a+a+a  – _i=ai

.

Consequently, we have

Eδnκδn– · · · κnδ. ()

Ifδ= E, we getd(gx,gx) +d(gy,gy) =d(gz,gz) = E, that is,gx=gx,gy=gyand

gz=gz. Therefore, from () we have

F(x,y,z) =gx=gx, F(y,z,x) =gy=gy and F(z,x,y) =gz=gz,

that is, (x,y,z) is a tripled coincidence point ofFandg. Now, assume thatδE. If

m>n, we have

d(gxm,gxn)d(gxm,gxm–) +d(gxm–,gxm–) +· · ·+d(gxn+,gxn),

d(gym,gyn)d(gym,gym–) +d(gym–,gym–) +· · ·+d(gyn+,gyn),

and

d(gzm,gzn)d(gzm,gzm–) +d(gzm–,gzm–) +· · ·+d(gzn+,gzn).

Adding above inequalities and using (), we obtain

d(gxm,gxn) +d(gym,gyn) +d(gzm,gzn)δm–+δm–+· · ·+δn

κm–+κm–+· · ·+κnδ

κn

 –κδ.

As ≤ _i=ai< , we have ≤κ< . Hence for anyc∈EwithcE, there existsN∈N such that for anyn≥Nwe have _–κn_κδc. Furthermore, for anym>n≥N, we get

d(gxm,gxn) +d(gym,gyn) +d(gzm,gzn)c.

This implies that{gxn},{gyn}and{gzn}are Cauchy sequences ing(X). By completeness of

g(X), there existx,y,z∈Xsuch that

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Now, we prove thatF(x,y,z) =gx,F(y,z,x) =gyandF(z,x,y) =gz. Note that

dF(x,y,z),gxdF(x,y,z),F(xn,yn,zn)

+dF(xn,yn,zn),gx

=dF(x,y,z),F(xn,yn,zn)

+d(gxn+,gx). () On the other hand, applying given contractive condition, we obtain

dF(x,y,z),F(xn,yn,zn)

ad

F(x,y,z),gx+ad

F(y,z,x),gy+ad

F(z,x,y),gz

+ad

F(xn,yn,zn),gxn

+ad

F(yn,zn,xn),gyn

+ad

F(zn,xn,yn),gzn

+ad

F(xn,yn,zn),gx

+ad

F(yn,zn,xn),gy

+ad

F(zn,xn,yn),gz

+ad

F(x,y,z),gxn

+ad

F(y,z,x),gyn

+ad

F(z,x,y),gzn

+ad(gx,gxn) +ad(gy,gyn) +ad(gz,gzn) =ad

F(x,y,z),gx+ad

F(y,z,x),gy+ad

F(z,x,y),gz

+ad(gxn+,gxn) +ad(gyn+,gyn)

+ad(gzn+,gzn) +ad(gxn+,gx) +ad(gyn+,gy) +ad(gzn+,gz) +ad

F(x,y,z),gxn

+ad

F(y,z,x),gyn

+ad

F(z,x,y),gzn

+ad(gx,gxn) +ad(gy,gyn) +ad(gz,gzn).

Combining above inequality with (), and using triangular inequality, we have

dF(x,y,z),gxad

F(x,y,z),gx+ad

F(y,z,x),gy

+ad

F(z,x,y),gz+ad(gxn+,gxn)

+ad(gyn+,gyn) +ad(gzn+,gzn) +ad(gxn+,gx) +ad(gyn+,gy) +ad(gzn+,gz)

+ad

F(x,y,z),gx+ad(gx,gxn) +ad

F(y,z,x),gy

+ad(gy,gyn) +ad

F(z,x,y),gz

+ad(gz,gzn) +ad(gx,gxn) +ad(gy,gyn) +ad(gz,gzn) +d(gxn+,gx).

Therefore,

( –a–a)d

F(x,y,z),gx– (a+a)d

F(y,z,x),gy– (a+a)d

F(z,x,y),gz

ad(gxn+,gxn) +ad(gyn+,gyn) +ad(gzn+,gzn) + ( +a)d(gxn+,gx) +ad(gyn+,gy) +ad(gzn+,gz) + (a+a)d(gx,gxn)

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Similarly, we obtain

( –a–a)d

F(y,z,x),gy– (a+a)d

F(z,x,y),gz– (a+a)d

F(x,y,z),gx

ad(gyn+,gyn) +ad(gzn+,gzn) +ad(gxn+,gxn) + ( +a)d(gyn+,gy) +ad(gzn+,gz) +ad(gxn+,gx) + (a+a)d(gy,gyn)

+ (a+a)d(gz,gzn) + (a+a)d(gx,gxn), ()

and

( –a–a)d

F(z,x,y),gz– (a+a)d

F(x,y,z),gx– (a+a)d

F(y,z,x),gy

ad(gzn+,gzn) +ad(gxn+,gxn) +ad(gyn+,gyn) + ( +a)d(gzn+,gz) +ad(gxn+,gx) +ad(gyn+,gy) + (a+a)d(gz,gzn)

+ (a+a)d(gx,gxn) + (a+a)d(gy,gyn). ()

Adding () and (), we get

( –a–a–a–a–a–a)

dF(x,y,z),gx+dF(y,z,x),gy+dF(z,x,y),gz

(a+a+a)δn+ ( +a+a+a)

d(gxn+,gx) +d(gyn+,gy) +d(gzn+,gz)

+ (a+a+a+a+a+a)

d(gx,gxn) +d(gy,gyn) +d(gz,gzn)

δn+ 

d(gxn+,gx) +d(gyn+,gy) +d(gzn+,gz) +d(gx,gxn) +d(gy,gyn) +d(gz,gzn).

Therefore, we have

αδn+βd(gxn+,gx) +γd(gyn+,gy) +σd(gzn+,gz) +θd(gx,gxn) +ξd(gy,gyn) +ρd(gz,gzn),

where

α=θ=ξ=ρ= 

 –a–a–a–a–a–a ,

β=σ=γ= 

 –a–a–a–a–a–a .

From () and (), for anycEthere existsN∈Nsuch that

δn

c

α, d(gxn,gx)

c

max{β,θ},

d(gyn,gy) c

max{γ,ξ}, d(gzn,gz)

c

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for alln≥N. Thus, for alln≥N, we have

dF(x,y,z),gx+dF(y,z,x),gy+dF(z,x,y),gz c

+

c

+

c

+

c

+

c

+

c

+

c

=c. It follows thatd(F(x,y,z),gx) =d(F(y,z,x),gy) =d(F(z,x,y),gz) = E, that isF(x,y,z) =gx,

F(y,z,x) =gyandF(z,x,y) =gz. As consequences of Theorem , we give the following corollaries.

Corollary  Let(X,d)be a K -metric space with a cone P having non-empty interior. Let F:X_→_{X and g}_:_X_→_{X be mappings such that F}₍_X₎_⊆_g₍_X₎_{and for any x}_,_y_,_z_,_u_,_v_,_w_∈

X, the following condition

dF(x,y,z),F(u,v,w)α

+αdF(u,v,w),gu+dF(v,w,u),gv+dF(w,u,v),gw

+α

dF(u,v,w),gx+dF(v,w,u),gy+dF(w,u,v),gz

+α

dF(x,y,z),gu+dF(y,z,x),gv+dF(z,x,y),gw

+α

d(gx,gu) +d(gy,gv) +d(gz,gw)

holds, whereαi, i= , . . . , are nonnegative real numbers such that i=αi< /. Then F

and g have a tripled coincidence point provided that g(X)is a complete subspace of X. Proof It suﬃces to takea=a=a=α,a=a=a=α,a=a=a=α,a=a=

a=αanda=a=a=αin Theorem  with i=αi< /.

Corollary (Abbas, Khan and Radenović []) Let(X,d)be a K -metric space with a cone P having non-empty interior. LetF˜:X_→_{X and g}_:_X_→_{X be mappings satisfying}_F˜₍_X₎_⊆

g(X),(g(X),d)is a complete subspace of X and for any x,y,u,v∈X,

dF˜(x,y),F˜(u,v)ad _˜

F(x,y),gx+ad _˜

F(u,v),gu+ad _˜

F(u,v),gx

+ad _˜

F(x,y),gu+ad(gx,gu) +ad(gy,gv), ()

where ai, i= , . . . , are nonnegative real numbers such that i=ai< . ThenF and g have˜

a coupled coincidence point(x,y)∈X, that is,F˜(x,y) =gx andF˜(y,x) =gy.

Proof Consider the mappingsF:X_→_X_{deﬁned by}_F₍_x_,_y_,_z_{) =}_F˜₍_x_,_y_{) for all}_x_,_y_,_z_∈_X_. Then, the contractive condition () implies that, for allx,y,z,u,v,w∈X

dF(x,y,z),F(u,v,w)ad

F(x,y,z),gx+ad

F(u,v,w),gu+ad

F(x,y,z),gu

+ad

F(u,v,w),gx+ad(gx,gu) +ad(gy,gv).

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F(x,y,z) =gx, F(y,z,x) =gyandF(z,x,y) =gz, that is,F˜(x,y) =gx andF˜(y,x) =gy. This

makes end to the proof.

Now, we are ready to state and prove the result of a common tripled coupled ﬁxed point.

Theorem  Let F:X_→_{X and g}_:_X_→_{X be two mappings which satisfy all the conditions}

of Theorem . If F and g are W -compatible, then F and g have a unique common tripled ﬁxed point. Moreover, a common tripled ﬁxed point of F and g is of the form(u,u,u)for some u∈X.

Proof First, we will show that the tripled point of coincidence is unique. Suppose that (x,y,z) and (x∗,y∗,z∗)∈X_with

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

gx=F(x,y,z),

gy=F(y,z,x),

gz=F(z,x,y), and

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

gx∗=F(x∗,y∗,z∗),

gy∗=F(y∗,z∗,x∗),

gz∗=F(z∗,x∗,y∗). Using the contractive condition in Theorem , we obtain

dgx,gx∗=dF(x,y,z),Fx∗,y∗,z∗

ad

F(x,y,z),gx+ad

F(y,z,x),gy+ad

F(z,x,y),gz

+ad

Fx∗,y∗,z∗,gx∗+ad

Fy∗,z∗,x∗,gy∗+ad

Fz∗,x∗,y∗,gz∗

+ad

Fx∗,y∗,z∗,gx+ad

Fy∗,z∗,x∗,gy

+ad

Fz∗,x∗,y∗,gz+ad

F(x,y,z),gx∗

+ad

F(y,z,x),gy∗+ad

F(z,x,y),gz∗

+ad

gx,gx∗+ad

gy,gy∗+ad

gz,gz∗

= (a+a+a)d

gx∗,gx+ (a+a+a)d

gy∗,gy

+ (a+a+a)d

gz∗,gz.

Similarly, we have

dgy,gy∗=dF(y,z,x),Fy∗,z∗,x∗

(a+a+a)d

gy∗,gy+ (a+a+a)d

gz∗,gz

+ (a+a+a)d

gx∗,gx

and

dgz,gz∗=dF(z,x,y),Fz∗,x∗,y∗

(a+a+a)d

gz∗,gz+ (a+a+a)d

gx∗,gx

+ (a+a+a)d

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Adding above three inequalities, we get

dgx,gx∗+dgy,gy∗+dgz,gz∗

_

i=

ai

dgx,gx∗+dgy,gy∗+dgz,gz∗.

Since _i=ai< , we obtain

dgx,gx∗+dgy,gy∗+dgz,gz∗= E,

which implies that

gx=gx∗, gy=gy∗ and gz=gz∗, ()

which implies the uniqueness of the tripled point of coincidence of F and g, that is, (gx,gy,gz). Note that

dgx,gy∗=dF(x,y,z),Fy∗,z∗,x∗

ad

F(x,y,z),gx+ad

F(y,z,x),gy+ad

F(z,x,y),gz

+ad

Fy∗,z∗,x∗,gy∗+ad

Fz∗,x∗,y∗,gz∗+ad

Fx∗,y∗,z∗,gx∗

+ad

Fy∗,z∗,x∗,gx+ad

Fz∗,x∗,y∗,gy

+ad

Fx∗,y∗,z∗,gz+ad

F(x,y,z),gy∗

+ad

F(y,z,x),gz∗+ad

F(z,x,y),gx∗

+ad

gx,gy∗+ad

gy,gz∗+ad

gz,gx∗

= (a+a+a)d

gy∗,gx+ (a+a+a)d

gz∗,gy

+ (a+a+a)d

gx∗,gz.

Similarly,

dgy,gz∗(a+a+a)d

gz∗,gy+ (a+a+a)d

gx∗,gz

+ (a+a+a)d

gy∗,gx,

and

dgz,gx∗(a+a+a)d

gx∗,gz+ (a+a+a)d

gy∗,gx

+ (a+a+a)d

gz∗,gy.

Adding above inequalities, we obtain

dgx,gy∗+dgy,gz∗+dgz,gx∗

_

i=

ai

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The fact that _i=ai<  yields that

gx=gy∗, gy=gz∗ and gz=gx∗. ()

In view of () and (), one can assert that

gx=gy=gz. ()

That is, the unique tripled point of coincidence ofFandgis (gx,gy,gz).

Now, letu=gx, then we haveu=gx=F(x,y,z) =gy=F(y,z,x) =gz=F(z,x,y). SinceF

andgareW-compatible, we have

F(gx,gy,gz) =gF(x,y,z),

which due to () gives that

F(u,u,u) =gu.

Consequently, (u,u,u) is a tripled coincidence point ofF andg, and so (gu,gu,gu) is a tripled point of coincidence ofFandg, and by its uniqueness, we getgu=gx. Thus, we obtain

u=gx=gu=F(u,u,u).

Hence, (u,u,u) is the unique common tripled ﬁxed point ofFandg. This completes the

proof.

Corollary (Abbas, Khan and Radenović []) Let(X,d)be a cone metric space with a cone P having non-empty interior. LetF˜:X_→_{X and g}_:_X_→_{X be mappings satisfying}

˜

F(X₎_⊆_g₍_X₎_,₍_g₍_X_),_d₎_{is a complete subspace of X and for any x}_,_y_,_u_,_v_∈_X,

dF˜(x,y),F˜(u,v)ad _˜

F(x,y),gx+ad _˜

F(u,v),gu+ad _˜

F(u,v),gx

+ad _˜

F(x,y),gu+ad(gx,gu) +ad(gy,gv),

where ai, i= , . . . , are nonnegative real numbers such that i=ai< . IfF and g are˜

w-compatible, then F and g have a unique common coupled ﬁxed point. Moreover, the˜ common ﬁxed point ofF and g is of the form˜ (u,u)for some u∈X.

Proof Consider the mappingsF:X_→_X_{deﬁned by}_F₍_x_,_y_,_z_{) =}_F˜₍_x_,_y_{) for all}_x_,_y_,_z_∈_X_. From the proof of Corollary  and the result given by Theorem , we have only to show thatFandgareW-compatible. Let (x,y,z)∈Xsuch thatF(x,y,z) =gx,F(y,z,x) =gyand

F(z,x,y) =gz. From the deﬁnition ofF, we getF˜(x,y) =gxandF˜(y,x) =gy. SinceF˜ andg

arew-compatible, this implies that

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Using (), we have

F(gx,gy,gz) =F˜(gx,gy) =gF˜(x,y)=gF(x,y,z).

Thus, we proved thatFandgareW-compatible mappings, and the desired result follows

immediately from Theorem .

Remark 

• Corollary  extends Theorem . of Samet and Vetro [] toK-metric spaces (corresponding to the caseN= ).

• Theorem  extends Theorem . of Samet and Vetro [] toK-metric spaces (case

N= ).

• Theorem  extends Theorem . of Samet and Vetro [] toK-metric spaces (case

N= ).

Similar to Corollaries  and , by consideringF(x,y,z) =fxfor allx,y,z∈Xwheref :

X→X, we may state the following consequence of Theorem .

Corollary (Olaleru []) Let(X,d)be a K -metric space and f,g:X→X be mappings such that

d(fx,fu)ad(fx,gx) +ad(fu,gu) +ad(fu,gx)

+ad(fx,gu) +ad(gu,gx) ()

for all x,u∈X, where ai∈[, ), i= , . . . , and i=ai< . Suppose that f and g are weakly

compatible, f(X)⊆g(X)and g(X)is a complete subspace of X. Then the mappings f and g have a unique common ﬁxed point.

Now, we give an example to illustrate our obtained results.

Example  LetX= [,∞). TakeE=C_R[, ] endowed with order induced byP={φ∈E: φ(t)≥ fort∈[, ]}. The mappingd:X×X→Eis deﬁned byd(x,y)(t) =|x–y|et_{. In this} case (X,d) is a complete abstract metric space with a non-normal cone having non-empty interior. Deﬁne the mappingsF:X→Xandg:X→Xby

gx=x

 and F(x,y,z) =

x–y+z

 .

We will check that all the hypotheses of Theorem  are satisﬁed. Since, for allx,y,z,u,v,w∈ X, we have

dF(x,y,z),F(u,v,w) =F(x,y,z) –F(u,v,w)et

≤ 

(x–y+z) – (u–v+w)e t

≤ 



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Then, the contractive condition is satisﬁed withai=  for alli= , . . . ,  anda=a=

a= /. All conditions of Theorem  are satisﬁed. Consequently,Fandghave a tripled coincidence point. In this case, (x,y,z) is a tripled coincidence point if and only ifx=y=

z= . This implies thatFandg areW-compatible. Applying our Theorem , we obtain the existence and uniqueness of a common tripled ﬁxed point ofFandg. In this example, (, , ) is the unique common tripled ﬁxed point.

Example  LetX= [,∞) andd:X×X→Ebe deﬁned asd(x,y)(t) =|x–y|t. Deﬁne the mappingsF:X_→_X_and_g_:_X_→_X_by

F(x,y,z) = ⎧ ⎨ ⎩ x + y + z

 ifx∈[, ] andy,z∈[,∞), x

+ y +

z

 ifx∈(,∞) andy,z∈[,∞) and

g(x) = ⎧ ⎨ ⎩

x ifx∈[, ], x ifx∈(,∞).

Leta=a=a=· · ·=a=  anda=a=_,a=a=a=_. Now we shall prove that the contractive condition in Theorem  holds for allx,y,z,u,v,w∈X. By its symmetry and without loss of generality, it suﬃces to prove it forx≤y≤zandu≤v≤z. Deﬁne

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(I):x,y,z∈[, ],

(II):x,y∈[, ],z∈(,∞), (III):x∈[, ],y,z∈(,∞), (IV):x,y,z∈(,∞)

and ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(i):u,v,w∈[, ],

(ii):u,v∈[, ],w∈(,∞), (iii):u∈[, ],v,w∈(,∞), (iv):u,v,w∈(,∞).

There are  possibilities which are (I, i), (I, ii), (I, iii), (I, iv), (II, i), (II, ii), (II, iii), (II, iv), (III, i), (III, ii), (III, iii), (III, iv), (IV, i), (IV, ii), (IV, iii) and (IV, iv).

Case . If (I, i) holds, we have

dF(x,y,z),F(u,v,w)≤

|x–u|

 +

|y–v|

 +

|z–w|



t

≤ad(gx,gu) +ad(gy,gv) +ad(gz,gw).

Case . If (I, ii) holds, we have

dF(x,y,z),F(u,v,w)

≤x

–

u

 +y

–

v

 +z

–

w

 t

≤ad(gx,gu) +ad(gy,gv) +



|z– w|+  w

t

≤ad(gx,gu) +ad(gy,gv) +ad(gz,gw) +   w  – u – v  t

≤ad(gx,gu) +ad(gy,gv) +ad(gz,gw) +ad

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Case . If (I, iii) holds, we have

dF(x,y,z),F(u,v,w)

≤x

–

u

 +y

–

v

 +z

–

w

 t

≤ad(gx,gu) +



|y– v|+  w+



|z– w|+  w

t

≤ad(gx,gu) +ad(gy,gv) +ad(gz,gw) +   w  – u – v  t

=ad(gx,gu) +ad(gy,gv) +ad(gz,gw) +ad

F(w,u,v),gw. Case . If (I, iv) holds, we get

dF(x,y,z),F(u,v,w)

≤x

–

u

 +y

–

v

 +z

– w  t ≤ 

|x– u|+  w+



|y– v|+  w+



|z– w|+  w

t

≤ad(gx,gu) +ad(gy,gv) +ad(gz,gw) +   w  – u – v  t

F(w,u,v),gw. Case . If (II, i) holds, we have

dF(x,y,z),F(u,v,w)

≤

|x–u|

 +

|y–v|

 +

|z–w|



t



|z– w|+  z

t

≤ad(gx,gu) +ad(gy,gv) +ad(gz,gw) +   z  – x – y  t

≤ad(gx,gu) +ad(gy,gv) +ad(gz,gw) +ad

F(z,x,y),gz. Case . If (II, ii) holds, we have

dF(x,y,z),F(u,v,w)

≤x

–

u

 +y

–

v

 +z

–

w

 t



|z– w|+  w+

 w

t

≤ad(gx,gu) +ad(gy,gv) +ad(gz,gw) +   w  – u – v 

t+   z  – x – y  t

F(w,u,v),gw+ad

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Case . If (II, iii) holds, we have

dF(x,y,z),F(u,v,w)

≤x

–

u

 +y

–

v

 +z

–

w

 t

≤ad(gx,gu) +



|y– v|+  w+



|z– w|+  z+

 w

t

≤ad(gx,gu) +ad(gy,gv) +ad(gz,gw) +   w  – u – v 

t+   z  – x – y  t

F(w,u,v),gw+ad

F(z,x,y),gz.

Case . If (II, iv) holds, we get

dF(x,y,z),F(u,v,w)

≤x

–

u

 +y

–

v

 +z

– w  t ≤ 

|x– u|+  w+



|y– v|+  w+



|z– w|+  z+

 w

t

≤ad(gx,gu) +ad(gy,gv) +ad(gz,gw) +   w  – u – v 

t+   z  – x – y  t

F(w,u,v),gw+ad

F(z,x,y),gz.

Case  corresponding to (III, i) is as Case . Case  corresponding to (III, ii) is as Case . Case . If (III, iii) holds, we have

dF(x,y,z),F(u,v,w)

≤x

–

u

 +y

–

v

 +z

– w  t ≤ 

|x– u|+ 

|y– v|+ 

|z– w|+  z+

 w

t

≤ad(gx,gu) +ad(gy,gv) +ad(gz,gw) +   w  – u – v 

t+   z  – x – y  t

F(w,u,v),gw+ad

F(z,x,y),gz.

Case . If (III, iv) holds, we get

dF(x,y,z),F(u,v,w)

≤x

–

u

 +y

–

v

 +z

–

w

(18)

≤ 

|x– u|+ 

|y– v|+ 

|z– w|+  z+

 w 

t

t+   z  – x – y  t

F(w,u,v),gw+ad

F(z,x,y),gz. Case  corresponding to (IV, i) is as Case .

Case  corresponding to (IV, ii) is as Case . Case . If (IV, iii) holds, we have

dF(x,y,z),F(u,v,w)

≤x

–

u

 +y

–

v

 +z

– w  t ≤ 

|x– u|+ 

|y– v|+ 

|z– w|+  z+

 w

t

t+   z  – x – y  t

F(w,u,v),gw+ad

F(z,x,y),gz. Case . If (IV, iv) holds, we have

dF(x,y,z),F(u,v,w)

≤x

–

u

 +y

–

v

 +z

– w  t ≤ 

|x– u|+ 

|y– v|+ 

|z– w|+  z+

 w

t

t+   z  – x – y  t

F(w,u,v),gw+ad

F(z,x,y),gz. All the conditions of Theorem  are fulﬁlled. Moreover, (, , ) is a common tripled coincidence point ofFandg.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Author details

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Acknowledgements

The third author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and the fourth author would like to thank the Commission on Higher Education and the Thailand Research Fund (Grant No. MRG550085) for ﬁnancial support during the preparation of this manuscript.

Received: 14 February 2012 Accepted: 6 August 2012 Published: 23 August 2012

References

1. Abbas, M, Ali Khan, M, Radenovi´c, S: Common coupled ﬁxed point theorems in cone metric spaces forw-compatible mappings. Appl. Math. Comput.217, 195-202 (2010)

2. Aliprantis, CD, Tourky, R: Cones and duality. In: Graduate Studies in Mathematics, vol. 84. American Mathematical Society, Providence (2007)

3. Altun, I, Damjanović, B, Djorić, D: Fixed point and common fixed point theorems on ordered cone metric spaces. Appl. Math. Lett.23, 310-316 (2010)

4. Aydi, H, Nashine, HK, Samet, B, Yazidi, H: Coincidence and common ﬁxed point results in partially ordered cone metric spaces and applications to integral equations. Nonlinear Anal.74(17), 6814-6825 (2011)

5. Aydi, H, Samet, B, Vetro, C: Coupled ﬁxed point results in cone metric spaces forw˜-compatible mappings. Fixed Point Theory Appl.2011, 27 (2011)

6. Aydi, H, Vetro, C, Sintunavarat, W, Kumam, P: Coincidence and ﬁxed points for contractions and cyclical contractions in partial metric spaces. Fixed Point Theory Appl.2012, 124 (2012)

7. Beg, I, Azam, A, Arshad, M: Common ﬁxed points for maps on topological vector space valued cone metric spaces. Int. J. Math. Math. Sci.2009, Article ID 560264 (2009)

8. Berinde, V, Borcut, M: Tripled ﬁxed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal.74(15), 4889-4897 (2011)

9. Bhashkar, TG, Lakshmikantham, V: Fixed point theorems in partially ordered cone metric spaces and applications. Nonlinear Anal.65, 825-832 (2006)

10. Cho, YJ: Fixed points for compatible mappings of type (A). Math. Jpn.18, 497-508 (1993)

11. Cho, YJ, Pathak, HK, Kang, SM, Jung, JS: Common ﬁxed points of compatible maps of type (β) on fuzzy metric spaces. Fuzzy Sets Syst.93, 99-111 (1998)

12. Huang, LG, Zhang, X: Cone metric spaces and ﬁxed point theorems of contractive mappings. J. Math. Anal. Appl.332, 1467-1475 (2007)

13. Jungck, G, Cho, YJ, Murthy, PP: Compatible mappings of type (A) and common ﬁxed points. Math. Jpn.38, 381-390 (1993)

14. Kaewkhao, A, Sintunavarat, W, Kumam, P: Common ﬁxed point theorems ofc-distance on cone metric spaces. J. Nonlinear Anal. Appl.2012, Article ID jnaa-00137 (2012). doi:10.5899/2012/jnaa-00137

15. Lakshmikantham, V, Ćirić, L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal.70, 4341-4349 (2009)

16. Mukhamadiev, EM, Stetsenko, VJ: Fixed point principle in generalized metric space. Izv. Akad. Nauk Tadzh. SSR, Otd. Fiz.-Mat. Khim. Geol. Nauk10(4), 8-19 (1969) (in Russian)

17. Olaleru, JO: Some generalizations of ﬁxed point theorems in cone metric spaces. Fixed Point Theory Appl.2009, Article ID 657914 (2009)

18. Perov, AI: The Cauchy problem for systems of ordinary diﬀerential equations. In: Approximate Methods of Solving Diﬀerential Equations, pp. 115-134. Naukova Dumka, Kiev (1964) (in Russian)

19. Rezapour, S, Hamlbarani, R: Some notes on the paper ‘Cone metric spaces and ﬁxed point theorems of contractive mappings’. J. Math. Anal. Appl.345, 719-724 (2008)

20. Samet, B, Vetro, C: Coupled fixed point,f-invariant set and fixed point ofN-order. Ann. Funct. Anal.1(2), 46-56 (2010) 21. Sessa, S, Cho, YJ: Compatible mappings and a common fixed point theorem of Chang type. Publ. Math. (Debr.)43,

289-296 (1993)

22. Sintunavarat, W, Kumam, P: Coincidence and common ﬁxed points for hybrid strict contractions without the weakly commuting condition. Appl. Math. Lett.22, 1877-1881 (2009)

23. Sintunavarat, W, Kumam, P: Common ﬁxed points off-weak contractions in cone metric spaces. Bull. Iran. Math. Soc. 38(2), 293-303 (2012)

24. Sintunavarat, W, Kumam, P: Weak condition for generalized multi-valued (f,α,β)-weak contraction mappings. Appl. Math. Lett.24, 460-465 (2011)

25. Sintunavarat, W, Kumam, P: Gregus type ﬁxed points for a tangential multi-valued mappings satisfying contractive conditions of integral type. J. Inequal. Appl.2011, 3 (2011)

26. Sintunavarat, W, Cho, YJ, Kumam, P: Common ﬁxed point theorems forc-distance in ordered cone metric spaces. Comput. Math. Appl.62, 1969-1978 (2011)

27. Sintunavarat, W, Kumam, P: Common ﬁxed point theorems for hybrid generalized multi-valued contraction mappings. Appl. Math. Lett.25, 52-57 (2012)

28. Sintunavarat, W, Kumam, P: Common ﬁxed point theorems for generalizedJ H-operator classes and invariant approximations. J. Inequal. Appl.2011, 67 (2011)

29. Sintunavarat, W, Kumam, P: Common ﬁxed point theorem for cyclic generalized multi-valued contraction mappings. Appl. Math. Lett.25, 1849-1855 (2012)

30. Sintunavarat, W, Kumam, P: Generalized common ﬁxed point theorems in complex valued metric spaces and applications. J. Inequal. Appl.2012, 84 (2012)

31. Vetro, P: Common ﬁxed points in cone metric spaces. Rend. Circ. Mat. Palermo56, 464-468 (2007) 32. Zabreˇiko, PP:K-metric andK-normed linear spaces: survey. Collect. Math.48, 825-859 (1997)

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doi:10.1186/1687-1812-2012-134