Common fixed point theorem for two pairs of non-self-mappings satisfying generalized Ćirić type contraction condition in cone metric spaces

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R E S E A R C H

Open Access

Common ﬁxed point theorem for two pairs of

non-self-mappings satisfying generalized

´Ciri´c type contraction condition in cone

metric spaces

Xianjiu Huang

1

_{, Jing Luo}

1

_{, Chuanxi Zhu}

1*

_{and Xi Wen}

2

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Nanchang University, Nanchang, Jiangxi 330031, P.R. China Full list of author information is available at the end of the article

Abstract

In this paper, we prove a common ﬁxed point theorem for two pairs of

non-self-mappings satisfying the generalized contraction condition of ´Ciri´c type in cone metric spaces. Our result generalizes and extends some recent results related to non-self-mappings in the setting of cone metric spaces.

MSC: 47H10; 54H25

Keywords: cone metric spaces; common fixed point; non-self-mappings; contraction condition of Ćirić type

1 Introduction and preliminaries

Recently, Huang and Zhang [] generalized the concept of a metric space, replacing the set of real numbers by ordered Banach space and obtained some fixed point theorems for mappings satisfying different contractive conditions. Subsequently, the study of fixed point theorems in such spaces is followed by some other mathematicians; see [–]. The study of fixed point theorems for non-self-mappings in metrically convex metric spaces was initiated by Assad and Kirk []. Utilizing the induction method of Assad and Kirk [], many authors like Assad [], Ćirić [–], Ćirićet al.[–], Kumamet al.[], Hadžić [], Hadžić and Gajić [], Imdad and Kumar [], Rhoades [, ] have ob-tained common fixed point in metrically convex spaces. Recently, Ćirić and Ume [] defined a wide class of multi-valued non-self-mappings which satisfy a generalized con-traction condition and proved a fixed point theorem which generalizes the results of Itoh [] and Khan [].

Very recently, Radenović and Rhoades [] extended the ﬁxed point theorem of Imdad and Kumar [] for a pair of non-self-mappings to non-normal cone metric spaces. Huang

et al.[] proved a fixed point theorem for four non-self-mappings in cone metric spaces which generalizes the result of Radenović and Rhoades []. Jankovićet al.[] proved new common fixed point results for a pair of non-self-mappings defined on a closed sub-set of metrically convex cone metric space which is not necessarily normal by adapting Assad-Kirk’s method. Sumitraet al.[] generalized the fixed point theorems of Ćirić and Ume [] for a pair of non-self-mappings to non-normal cone metric spaces. In the same

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time, Sumitraet al.’s [] results extended the results of Radenović and Rhoades [] and Jankovićet al.[]. The aim of this paper is to prove a common ﬁxed point theorem for two pairs of non-self-mappings on cone metric spaces in which the cone need not be normal and the condition is weaker. This result generalizes the result of Sumitraet al.[] and Huanget al.[].

Consistent with Huang and Zhang [], the following deﬁnitions and results will be needed in the sequel.

LetEbe a real Banach space. A subsetPofEis called a cone if and only if:

(a) Pis closed, nonempty andP={θ};

(b) a,b∈R,a,b≥,x,y∈Pimpliesax+by∈P; (c) P∩(–P) ={θ}.

Given a coneP⊂E, we deﬁne a partial orderingwith respect toPbyxyif and only ify–x∈P. A conePis called normal if there is a numberK>  such that for allx,y∈E,

θxy implies x ≤Ky.

The least positive number satisfying the above inequality is called the normal constant of

P, whilexystands fory–x∈intP(interior ofP).

Deﬁnition .[] LetXbe a nonempty set. Suppose that the mappingd:X×X→E

satisﬁes:

(d) θd(x,y)for allx,y∈Xandd(x,y) =θ if 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 onXand (X,d) is called a cone metric space. The concept of a cone metric space is more general than that of a metric space.

Example .[] LetE=R_,_P₌_{₍_x_,_y₎_∈_E_|_x_,_y_≥__}_,_X₌_R_{, and}_d_:_X_×_X_→_E_{be such}

thatd(x,y) = (|x–y|,α|x–y|), whereα≥ is a constant. Then (X,d) is a cone metric space.

Deﬁnition .[] Let (X,d) be a cone metric space. We say that{xn}is:

(e) a Cauchy sequence if for everyc∈Ewithθc, there is anNsuch that for all

n,m>N,d(xn,xm)c;

(f ) a convergent sequence if for everyc∈Ewithθc, there is anNsuch that for all

n>N,d(xn,x)cfor some ﬁxedx∈X.

A cone metric spaceXis said to be complete if for every Cauchy sequence inXis con-vergent inX. It is well known that{xn}converges tox∈Xif and only ifd(xn,x)→θ as

n→ ∞. It is a Cauchy sequence if and only ifd(xn,xm)→θ (n,m→ ∞).

Remark .[] LetEbe an ordered Banach (normed) space. Thencis an interior point ofP, if and only if [–c,c] is a neighborhood ofθ.

Corollary .[] ()If ab and bc,then ac.

Indeed,c–a= (c–b) + (b–a)c–b implies[–(c–a),c–a]⊇[–(c–b),c–b]. ()If ab and bc,then ac.

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Remark .[, ] Ifc∈intP,θ an, andan→θ, then there exists annsuch that for

alln>nwe haveanc.

Remark .[, ] IfEis a real Banach space with conePand ifakawherea∈Pand  <k< , thena=θ.

Deﬁnition .[] Letf andgbe self-maps on a setX(i.e.,f,g:X→X). Ifw=fx=gx

for somexinX, thenxis called a coincidence point off andg, andwis called a point of coincidence offandg. Self-mapsfandgare said to be weakly compatible if they commute at their coincidence point;i.e., iffx=gxfor somex∈X, thenfgx=gfx.

2 Main results

The following theorem is Sumitraet al.’s [] generalization of Ćirić and Ume’s [] result in cone metric spaces.

Theorem . Let(X,d)be a complete cone metric space,C a nonempty closed subset of X such that for each x∈C and y∈/C there exists a point z∈∂C(the boundary of C)such that

d(x,z) +d(z,y) =d(x,y).

Suppose that f,g:C→X are two non-self-mappings satisfying,for all x,y∈C with x=y,

d(gx,gy)αd(fx,fy) +βu+γv, (.)

where u∈ {d(fx,gx),d(fy,gy)},v∈ {d(fx,gx) +d(fy,gy),d(fx,gy) +d(fy,gx)},andα,β,γ are nonnegative real numbers such that

α+ β+ γ +αγ < . (.)

Also assume that

(i) ∂C⊆fC,gC∩C⊆fC, (ii) fx∈∂Cimplies thatgx∈C, (iii) fCis closed inX.

Then the pair(f,g)has a coincidence point in C.Moreover,if the pair(f,g)is weakly compatible,then f and g have a unique common ﬁxed point in C.

Remark . From the proof of this theorem, it is easy to see that condition (.) can be weakened toα+β+ γ < .

The purpose of this paper is to extend the above theorem for two pairs of non-self-mappings in cone metric spaces with a weaker condition.

We state and prove our main result as follows.

Theorem . Let(X,d)be a complete cone metric space,C a nonempty closed subset of X such that for each x∈C and y∈/C there exists a point z∈∂C such that

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Suppose that F,G,S,T:C→X are two pairs of non-self-mappings satisfying,for all x,y∈C with x=y,

d(Fx,Gy)αd(Tx,Sy) +βu+γv, (.)

where u∈ {d(Tx,Fx),d(Sy,Gy)},v∈ {d(Tx,Fx) +d(Sy,Gy),d(Tx,Gy) +d(Sy,Fx)},andα,β, γ are nonnegative real numbers such that

α+β+ γ< . (.)

Also assume that

(I) ∂C⊆SC∩TC,FC∩C⊆SC,GC∩C⊆TC,

(II) Tx∈∂Cimplies thatFx∈C,Sx∈∂Cimplies thatGx∈C, (III) SCandTC(orFCandGC)are closed inX.

Then

(IV) (F,T)has a point of coincidence, (V) (G,S)has a point of coincidence.

Moreover,if(F,T)and(G,S)are weakly compatible pairs,then F,G,S,and T have a unique common ﬁxed point.

Proof Firstly, we proceed to construct two sequences{xn}and{yn}in the following way. Letx∈∂Cbe arbitrary. Then (due to∂C⊆TC) there exists a pointx∈Csuch that x=Tx. SinceTx⊆∂Cimplies thatFx∈C, one concludes thatFx∈FC∩C⊆SC. Thus,

there existsx∈Csuch thaty=Sx=Fx∈C. Sincey=Fxthere exists a pointy=Gx

such that

d(y,y) =d(Fx,Gx).

Supposey∈C. Theny∈GC∩C⊆TC, which implies that there exists a pointx∈C

such thaty=Tx. otherwise, ify∈/C, then there exists a pointp∈∂Csuch that

d(Sx,p) +d(p,y) =d(Sx,y).

Sincep∈∂C⊆TCthere exists a pointx∈Cwithp=Txso that

d(Sx,Tx) +d(Tx,y) =d(Sx,y).

Lety=Fx be such that d(y,y) =d(Gx,Fx). Thus, repeating the foregoing

argu-ments, one obtains two sequences{xn}and{yn}such that

(a) yn=Gxn–,yn+=Fxn,

(b) yn∈Cimplies thatyn=Txnoryn∈/Cimplies thatTxn∈∂Cand

d(Sxn–,Txn) +d(Txn,yn) =d(Sxn–,yn),

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We denote

P=

Txi∈ {Txn}:Txi=yi

,

P=

Txi∈ {Txn}:Txi=yi

,

Q=

Sxi+∈ {Sxn+}:Sxi+=yi+

,

Q=

Sxi+∈ {Sxn+}:Sxi+=yi+

.

Note that (Txn,Sxn+) /∈P×Q, as ifTxn∈P, thenyn=Txnand one infers that

Txn∈∂C, which implies thatyn+=Fxn∈C. Henceyn+=Sxn+∈Q. Similarly, one

can argue that (Sxn–,Txn) /∈Q×P.

Now, we distinguish the following three cases. Case . If (Txn,Sxn+)∈P×Q, then from (.)

d(Txn,Sxn+) =d(Fxn,Gxn–)αd(Txn,Sxn–) +βun+γvn,

where

un∈

d(Txn,Fxn),d(Sxn–,Gxn–)

=d(Txn,yn+),d(Sxn–,yn)

,

vn∈

d(Txn,Fxn) +d(Sxn–,Gxn–),d(Txn,Gxn–) +d(Sxn–,Fxn)

=d(Txn,yn+) +d(Sxn–,yn),d(Sxn–,yn+)

.

Clearly, there are inﬁnitely many nsuch that at least one of the following four cases holds:

() Ifun=d(Txn,yn+) andvn=d(Txn,yn+) +d(Sxn–,yn), then

d(Txn,Sxn+)αd(Txn,Sxn–) +βd(Txn,yn+)

+γd(Txn,yn+) +d(Sxn–,yn)

=αd(Txn,Sxn–) +βd(Txn,Sxn+)

+γd(Txn,Sxn+) +γd(Sxn–,Txn).

It implies thatd(Txn,Sxn+)_–αβ+γ–γd(Sxn–,Txn).

() Ifun=d(Txn,yn+) andvn=d(Sxn–,yn+), then

d(Txn,Sxn+)αd(Txn,Sxn–) +βd(Txn,yn+) +γd(Sxn–,yn+)

αd(Txn,Sxn–) +βd(Txn,yn+)

+γd(Sxn–,yn) +d(yn,yn+)

=αd(Txn,Sxn–) +βd(Txn,Sxn+)

+γd(Sxn–,Txn) +γd(Txn,Sxn+).

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() Ifun=d(Sxn–,yn) andvn=d(Txn,yn+) +d(Sxn–,yn), then

d(Txn,Sxn+)αd(Txn,Sxn–) +βd(Sxn–,yn)

+γd(Txn,yn+) +d(Sxn–,yn)

=αd(Txn,Sxn–) +βd(Sxn–,Txn)

+γd(Txn,Sxn+) +γd(Sxn–,Txn).

It implies thatd(Txn,Sxn+)α+_–βγ+γd(Sxn–,Txn).

() Ifun=d(Sxn–,yn) andvn=d(Sxn–,yn+), then

d(Txn,Sxn+)αd(Txn,Sxn–) +βd(Sxn–,yn) +γd(Sxn–,yn+)

αd(Txn,Sxn–) +βd(Sxn–,yn)

+γd(Sxn–,yn) +d(yn,yn+)

=αd(Txn,Sxn–) +βd(Sxn–,Txn)

+γd(Sxn–,Txn) +γd(Txn,Sxn+).

It implies thatd(Txn,Sxn+)α+_–βγ+γd(Sxn–,Txn).

From (), (), (), () it follows that

d(Txn,Sxn+)λd(Sxn–,Txn), (.)

whereλ=max{_–α_β+_–γ_γ,α+_–β_γ+γ}<  by (.). Similarly, if (Sxn+,Txn+)∈Q×P, we have

d(Sxn+,Txn+) =d(Fxn,Gxn+)λd(Txn,Sxn+). (.)

If (Sxn–,Txn)∈Q×P, we have

d(Sxn–,Txn) =d(Fxn–,Gxn–)λd(Txn–,Sxn–). (.)

Case . If (Txn,Sxn+)∈P×Q, thenSxn+∈Qand

d(Txn,Sxn+) +d(Sxn+,yn+) =d(Txn,yn+), (.)

which in turns yields

d(Txn,Sxn+)d(Txn,yn+) =d(yn,yn+) (.)

and hence

d(Txn,Sxn+)d(yn,yn+) =d(Fxn,Gxn–). (.)

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If (Sxn+,Txn+)∈Q×P, thenTxn∈P. We show that

d(Sxn+,Txn+)λd(Txn,Sxn–). (.)

Using (.), we get

d(Sxn+,Txn+)d(Sxn+,yn+) +d(yn+,Txn+)

=d(Txn,yn+) –d(Txn,Sxn+) +d(yn+,Txn+). (.)

By noting thatTxn+,Txn∈P, one can conclude that

d(yn+,Txn+) =d(yn+,yn+) =d(Fxn,Gxn+)λd(Txn,Sxn+) (.)

and

d(Txn,yn+) =d(yn,yn+) =d(Fxn,Gxn–)λd(Sxn–,Txn), (.)

in view of Case . Thus,

d(Sxn+,Txn+)λd(Sxn–,Txn) – ( –λ)d(Txn,Sxn+)λd(Sxn–,Txn),

and we proved (.).

Case . If (Txn,Sxn+)∈P×Q, thenSxn–∈Q. We show that

d(Txn,Sxn+)λd(Sxn–,Txn–). (.)

SinceTxn∈P, then

d(Sxn–,Txn) +d(Txn,yn) =d(Sxn–,yn). (.)

From this, we get

d(Txn,Sxn+)d(Txn,yn) +d(yn,Sxn+)

=d(Sxn–,yn) –d(Sxn–,Txn) +d(yn,Sxn+). (.)

By noting thatSxn+,Sxn–∈Q, one can conclude that

d(yn,Sxn+) =d(yn,yn+) =d(Fxn,Gxn–)λd(Sxn–,Txn) (.)

and

d(Sxn–,yn) =d(yn–,yn) =d(Fxn–,Gxn–)λd(Sxn–,Txn–), (.)

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Thus,

d(Txn,Sxn+)λd(Sxn–,Txn–) – ( –λ)d(Sxn–,Txn)λd(Sxn–,Txn–),

and we proved (.).

Similarly, if (Sxn+,Txn+)∈Q×P, thenTxn+∈P, and

d(Sxn+,Txn+) +d(Txn+,yn+) =d(Sxn+,yn+).

From this, we have

d(Sxn+,Txn+)d(Sxn+,yn+) +d(yn+,Txn+)

d(Sxn+,yn+) +d(Sxn+,yn+) –d(Sxn+,Txn+)

= d(Sxn+,yn+) –d(Sxn+,Txn+)

⇒ d(Sxn+,Txn+)d(Sxn+,yn+).

By noting thatSxn+∈Q, one can conclude that

d(Sxn+,Txn+)d(Sxn+,yn+) =d(Fxn,Gxn+)λd(Txn,Sxn+), (.)

in view of Case .

Thus, in all cases ()-(), there existswn∈ {d(Sxn–,Txn),d(Txn–,Sxn–)}such that

d(Txn,Sxn+)λwn

and there existswn+∈ {d(Sxn–,Txn),d(Txn,Sxn+)}such that

d(Sxn+,Txn+)λwn+.

Following the procedure of Assad and Kirk [], it can easily be shown by induction that, forn≥, there existsw∈ {d(Tx,Sx),d(Sx,Tx)}such that

d(Txn,Sxn+)λn–



_w_ _and _d₍_Sx__n_+_,_Tx__n_+₎_λn_w__. _(.)

From (.) and by the triangle inequality, forn>mwe have

d(Txn,Sxm+)d(Txn,Sxn–) +d(Sxn–,Txn–) +· · ·+d(Txm+,Sxm+)

λm+λm+ ₊· · ·₊_λn–_w_ λ m

 –√λw→θ, asm→ ∞.

From Remark . and Corollary .(), it follows thatd(Txn,Sxm+)c.

Thus, the sequence{Tx,Sx,Tx,Sx, . . . ,Sxn–,Txn,Sxn–, . . .}is a Cauchy sequence.

Then, as noted in [], there exists at least one subsequence,{Txnk}or{Sxnk+}, which

is contained in Por Q, respectively, and one ﬁnds its limitz∈C. Furthermore,

sub-sequences{Txnk}and{Sxnk+}both converge toz∈CasCis a closed subset of

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for eachk∈N, thenTxnk =ynk =Gxnk–∈C∩GC⊆TC SinceTC as well asSCare

closed inX and{Txnk}is Cauchy inTC, it converges to a pointz∈TC. Letw∈T –_z_,

thenTw=z. Similarly,{Txnk}and{Sxnk+}being a subsequence of a Cauchy sequence,

{Tx,Sx,Tx,Sx, . . . ,Sxn–,Txn,Sxn–, . . .}also converges tozasSCis closed. Letθc,

thend(z,Sxnk–)

c

_–α_β+_–γ_γ , whereα,β,γare nonnegative real numbers withα+β+γ < .

Using (.), one can write

d(Fw,z)d(Fw,Gxnk–) +d(Gxnk–,z)

αd(Tw,Sxnk–) +βuw+γvw+d(Gxnk–,z)

=αd(z,Sxnk–) +βuw+γvw+d(Gxnk–,z),

where

uw∈

d(Tw,Fw),d(Sxnk–,Gxnk–)

=d(z,Fw),d(Sxnk–,Gxnk–)

,

vw∈

d(Tw,Fw) +d(Sxnk–,Gxnk–),d(Tw,Gxnk–) +d(Fw,Sxnk–)

=d(z,Fw) +d(Sxnk–,Gxnk–),d(z,Gxnk–) +d(Fw,Sxnk–)

.

Letθc. Clearly at least one of the following four cases holds for inﬁnitely manyn: () Ifuw=d(z,Fw) andvw=d(z,Fw) +d(Sxnk–,Gxnk–), then

d(Fw,z)αd(z,Sxnk–) +βd(z,Fw)

+γd(z,Fw) +d(Sxnk–,Gxnk–)

+d(Gxnk–,z)

αd(z,Sxnk–) +βd(z,Fw) +γd(z,Fw)

+γd(Sxnk–,z) +d(z,Gxnk–)

+d(Gxnk–,z)

= (α+γ)d(z,Sxnk–) + (β+γ)d(z,Fw) + (γ + )d(Gxnk–,z)

⇒ d(Fw,z) α+γ

 –β–γd(z,Sxnk–) +

γ + 

 –β–γd(Gxnk–,z)

α+γ

 –β–γ

c

_–α_β+γ_–_γ + γ +   –β–γ

c

_–γ_β+_–_γ =c.

() Ifuw=d(z,Fw) andvw=d(z,Gxnk–) +d(Fw,Sxnk–), then

d(Fw,z)αd(z,Sxnk–) +βd(z,Fw)

+γd(z,Gxnk–) +d(Fw,Sxnk–)

+d(Gxnk–,z)

αd(z,Sxnk–) +βd(z,Fw) +γd(z,Gxnk–)

+γd(Fw,z) +d(z,Sxnk–)

+d(Gxnk–,z)

= (α+γ)d(z,Sxnk–) + (β+γ)d(z,Fw) + (γ + )d(Gxnk–,z)

⇒ d(Fw,z) α+γ

 –β–γd(z,Sxnk–) +

γ + 

 –β–γd(Gxnk–,z)

α+γ

 –β–γ

c

_–α_β+γ_–_γ + γ +   –β–γ

c

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() Ifuw=d(Sxnk–,Gxnk–) andvw=d(z,Fw) +d(Sxnk–,Gxnk–), then

d(Fw,z)αd(z,Sxnk–) +βd(Sxnk–,Gxnk–)

+γd(z,Fw) +d(Sxnk–,Gxnk–)

+d(Gxnk–,z)

αd(z,Sxnk–) +β

d(Sxnk–,z) +d(z,Gxnk–)

+γd(z,Fw)

+γd(Sxnk–,z) +d(z,Gxnk–)

+d(Gxnk–,z)

= (α+β+γ)d(z,Sxnk–) +γd(z,Fw) + (β+γ + )d(Gxnk–,z)

⇒ d(Fw,z)α+β+γ

 –γ d(z,Sxnk–) +

β+γ+ 

 –γ d(Gxnk–,z)

α+β+γ

 –γ

c

α+_–β_γ+γ +

β+γ +   –γ

c

β_–+γ_γ+ =c.

() Ifuw=d(Sxnk–,Gxnk–) andvw=d(z,Gxnk–) +d(Fw,Sxnk–), then

d(Fw,z)αd(z,Sxnk–) +βd(Sxnk–,Gxnk–)

+γd(z,Gxnk–) +d(Fw,Sxnk–)

+d(Gxnk–,z)

αd(z,Sxnk–) +β

d(Sxnk–,z) +d(z,Gxnk–)

+γd(z,Gxnk–)

+γd(Fw,z) +d(z,Sxnk–)

+d(Gxnk–,z)

= (α+β+γ)d(z,Sxnk–) +γd(z,Fw) + (β+γ + )d(Gxnk–,z)

⇒ d(Fw,z)α+β+γ

 –γ d(z,Sxnk–) +

β+γ+ 

 –γ d(Gxnk–,z)

α+β+γ

 –γ

c

α+_–β_γ+γ +

β+γ +   –γ

c

β_–+γ_γ+ =c.

In all cases we obtaind(Fw,z)cfor eachc∈intP. Using Corollary .() it follows that

d(Fw,z) =θ orFw=z. Thus,Fw=z=Tw, that is,zis a coincidence point ofF,T. Further, since Cauchy sequence{Tx,Sx,Tx,Sx, . . . ,Sxn–,Txn,Sxn–, . . .}converges

toz∈Candz=Fw,z∈FC∩C⊆SC, there existsb∈Csuch thatSb=z. Again using (.), we get

d(Sb,Gb) =d(z,Gb) =d(Fw,Gb)αd(Tw,Sb) +βuw+γvw=βuw+γvw,

where

uw∈

d(Tw,Fw),d(Sb,Gb)=θ,d(Sb,Gb),

vw∈

d(Tw,Fw) +d(Sb,Gb),d(Tw,Gb) +d(Sb,Fw)

=d(Sb,Gb),d(z,Gb)=d(Sb,Gb).

Hence, we get the following cases:

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Since ≤γ ≤β+γ <  –α–γ ≤, using Remark . and Corollary .(), it follows thatSb=Gb, therefore,Sb=z=Gb, that is,zis a coincidence point of (G,S).

In caseFCandGCare closed inX, thenz∈FC∩C⊆SC orz∈GC∩C⊆TC. The

analogous arguments establish (IV) and (V). If we assume that there exists a subsequence {Sxnk+} ⊆QwithTCas wellSCare closed inX, then noting that{Sxnk+}is a Cauchy

sequence inSC, the foregoing arguments establish (IV) and (V). Suppose now that (F,T) and (G,S) are weakly compatible pairs, then

z=Fw=Tw ⇒ Fz=FTw=TFw=Tz and

z=Gb=Sb ⇒ Gz=GSb=SGb=Sz.

Then, from (.),

d(Fz,z) =d(Fz,Gb)αd(Tz,Sb) +βu+γv=αd(Fz,z) +βu+γv,

where

u∈d(Tz,Fz),d(Sb,Gb)=d(Fz,Fz),d(z,z)={θ},

v∈d(Tz,Fz) +d(Sb,Gb),d(Tz,Gb) +d(Sb,Fz)

=θ,d(Fz,z) +d(z,Fz)=θ, d(Fz,z).

Hence, we get the following cases:

d(Fz,z)αd(Fz,z) and d(Fz,z)αd(Fz,z) + γd(Fz,z) = (α+ γ)d(z,Fz).

Since ≤α≤α+ γ <  –β≤, using Remark . and Corollary .(), it follows that

Fz=z. Thus,Fz=z=Tz.

Similarly, we can proveGz=z=Sz. Thereforez=Fz=Gz=Sz=Tz, that is,zis a com-mon ﬁxed point ofF,G,S, andT.

Uniqueness of the common ﬁxed point follows easily from (.).

The following example shows that in generalF,G,S, andTsatisfying the hypotheses of Theorem . need not have a common coincidence justifying the two separate conclusions (IV) and (V).

Example . LetE=C_{([, ],}_R_),_P₌_{_ϕ_∈_E_:_ϕ₍_t₎_≥_,_t_∈_{[, ]}_}_,_X_{= [, +}_∞_),_C_{= [, ],}

andd:X×X→Edefined byd(x,y) =|x–y|ϕ, whereϕ∈Pis a fixed function,e.g.,ϕ(t) =et_. Then (X,d) is a complete cone metric space with a non-normal cone having a nonempty interior. DefineF,G,S, andT:C→Xas

Fx=x+

, Gx=x

₊

, Tx= x and Sx= x

_, _x_∈_C_.

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FC∩C= [_,_]∩[, ] = [_, ]⊂SC,GC∩C= [_,_]∩[, ] = [_, ]⊂TC, and,SC,TC,

FC, andGCare closed inX. Also,

T = ∈∂C ⇒ F =

 ∈C,

S = ∈∂C ⇒ G =

∈C,

T

 

= ∈∂C ⇒ F

 

=

 ∈C,

S

 

= ∈∂C ⇒ G

 

=

 ∈C.

Moreover, for eachx,y∈C,

d(Fx,Gy) = x–y ϕ=

d(Tx,Sy),

that is, (.) is satisﬁed withα=_,β=γ= . Obviously,  =T(_) =F(_)=_ and  =S(√

) =G( 

√

)= 

√

. Notice that two separate

coincidence points are not common ﬁxed points asFT(_)=TF(_) andSG(√

)=GS( 

√

),

which shows the necessity of the weakly compatibility property in Theorem ..

Next, we furnish an illustrate example in support of our result. In doing so, we are es-sentially inspired by Imdad and Kumar [].

Example . LetE=C_{([, ],}_R_),_P₌_{_ϕ_∈_E_:_ϕ₍_t₎_≥_,_t_∈_{[, ]}_}_,_X_{= [, +}_∞_),_C_{= [, ],}

andd:X×X→Edefined byd(x,y) =|x–y|ϕ, whereϕ∈Pis a fixed function,e.g.,ϕ(t) =et_. Then (X,d) is a complete cone metric space with a non-normal cone having the nonempty interior. DefineF,G,S, andT:C→Xas

Fx=

x _{if }_≤_x_≤_,

 if  <x≤, Tx=

x_{– } _{if }_≤_x_≤_,

 if  <x≤,

Gx=

x if ≤x≤,

 if  <x≤ and Sx=

x–  if ≤x≤,

 if  <x≤.

Since∂C={, }. Clearly, for eachx∈Candy∈/ Cthere exists a pointz= ∈∂Csuch thatd(x,z) +d(z,y) =d(x,y). Further,SC∩TC= [, ]∩[, ] = [, ]⊃ {, }=∂C,

FC∩C= [, ]∩[, ] = [, ]⊂SC, andGC∩C= [, ]∩[, ] = [, ]⊂TC. Also,

T = ∈∂C ⇒ F = ∈C,

S = ∈∂C ⇒ G = ∈C,

T   

= ∈∂C ⇒ F

   =  ∈C,

S   

= ∈∂C ⇒ G

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Moreover, ifx∈[, ] andy∈[, ], then

d(Fx,Gy) = x–  ϕ=|x

_{– }_|

|x_{+ }|ϕ=

|x_{– }_|

|x_{+ }|ϕ=



(x_{+ )}d(Tx,Sy).

Next, ifx,y∈(, ], then

d(Fx,Gy) =  =α·d(Tx,Sy).

Finally, ifx,y∈[, ], then

d(Fx,Gy) = x–y ϕ=|x

_–_y_|

|x₊_y|ϕ=

|x_–_y_|

|x₊_y|ϕ=



(x₊_y₎d(Tx,Sy).

Therefore, condition (.) is satisﬁed if we chooseα=max{ 

(x_+),_(_x₊_y₎} ∈(,_),β= γ = . Moreover,  is a point of coincidence asT =F as well asS =G, whereas both pairs (F,T) and (G,S) are weakly compatible asTF =  =FT andSG =  =GS. Also,

SC,TC,FC, andGCare closed inX. Thus, all the conditions of Theorem . are satisﬁed and  is the unique common ﬁxed point ofF,G,S, andT. One may note that  is also a point of coincidence for both pairs (F,T) and (G,S).

Remark . . SettingG=F=gandT=S=f in Theorem ., one deduces Theorem . due to Sumitraet al.[] with weaker condition.

. SettingG=F=gandT=S=IXin Theorem ., we obtain the following result.

Corollary . Let(X,d)be a complete cone metric space,and C a nonempty closed subset of X such that for each x∈C and y∈/C there exists a point z∈∂C such that

d(x,z) +d(z,y) =d(x,y).

Suppose that g:C→X satisﬁes,for all x,y∈C with x=y,

d(gx,gy)αd(x,y) +βu+γ,

where

u∈d(x,gx),d(y,gy), v∈d(x,gx) +d(y,gy),d(x,gy) +d(y,gx),

andα,β,γ are nonnegative real numbers such thatα+β+ γ< and g has the additional property that if,for each x∈∂C,gx∈C,then g has a unique ﬁxed point in C.

Corollary . Let(X,d)be a complete cone metric space,C a nonempty closed subset of X such that for each x∈C and y∈/C there exists a point z∈∂C such that

d(x,z) +d(z,y) =d(x,y).

Let F,G,S,T:C→X be such that

d(Fx,Gy)αd(Tx,Sy) (.)

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Suppose,further,that F,G,S,T,and C satisfy the following conditions:

(I) ∂C⊆SC∩TC,FC∩C⊆SC,GC∩C⊆TC, (II) Tx⊆∂C⇒Fx∈C,Sx⊆∂C⇒Gx∈C, (III) SCandTC(orFCandGC)are closed inX.

Then

Corollary . Let(X,d)be a complete cone metric space,C a nonempty closed subset of X such that for each x∈C and y∈/C there exists a point z∈∂C such that

d(x,z) +d(z,y) =d(x,y).

d(Fx,Gy)γd(Tx,Fx) +d(Sy,Gy) (.)

for someγ ∈(, /)and for all x,y∈C with x=y.

Then

Corollary . Let(X,d)be a complete cone metric space,C a nonempty closed subset of X such that for each x∈C and y∈/C there exists a point z∈∂C such that

d(x,z) +d(z,y) =d(x,y).

d(Fx,Gy)γd(Tx,Fy) +d(Fx,Sy) (.)

for someγ ∈(, /)and for all x,y∈C with x=y.

Then

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Remark . SettingG=F=fandT=S=gin Corollaries .-., we obtain the following

results.

Corollary . Let(X,d)be a complete cone metric space,C a nonempty closed subset of X such that for each x∈C and y∈/C there exists a point z∈∂C such that

d(x,z) +d(z,y) =d(x,y).

Let f,g:C→X be such that

d(fx,fy)αd(gx,gy) (.)

for someα∈(, )and for all x,y∈C.Suppose,further,that f,g,and C satisfy the following conditions:

(I) ∂C⊆gC,fC∩C⊆gC, (II) gx⊆∂C⇒fx∈C, (III) gCis closed inX.

Corollary . Let(X,d)be a complete cone metric space,C a nonempty closed subset of X such that for each x∈C and y∈/C there exists a point z∈∂C(the boundary of C)such that

d(x,z) +d(z,y) =d(x,y).

d(fx,fy)γd(fx,gx) +d(fy,gy) (.)

for someγ ∈(, /)and for all x,y∈C.Suppose,further,that f,g,and C satisfy the fol-lowing conditions:

Corollary . Let(X,d)be a complete cone metric space,C a nonempty closed subset of X such that for each x∈C and y∈/C there exists a point z∈∂C(the boundary of C)such that

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d(fx,fy)γd(fx,gy) +d(fy,gx) (.)

for someγ ∈(, /)and for all x,y∈C.Suppose,further,that f,g,and C satisfy the fol-lowing conditions:

Remark . Corollaries .-. are the corresponding theorems of Abbas and Jungck from [] in the case thatf,gare non-self-mappings.

There have been a number of papers written about ﬁxed points for non-self-maps. One of the most general papers involving two maps is that in []. For cone metric spaces, the four-function analog of this result would have the contractive condition.

Suppose thatF,G,S,T :C→X are two pairs of non-self-mappings satisfying, for all

x,y∈Cwithx=y,

d(Fx,Gy)hw, (.)

wherew∈ {d(Tx_a,Sy),d(Tx,Fx),d(Sy,Gy),d(Tx,Gy_a)+₊_hd(Sy,Fx)},ais a positive real number satisfy-inga≥ +_+h_h and  <h< .

Note that ifF,G,S, andT satisfy condition (.), thenF,G,S, andTsatisfy condition (.), but the implication is not reversible.

Indeed, there are four cases to consider:

() Ifw=d(Tx_a,Sy) in (.), thend(Fx,Gy)h_ad(Tx,Sy). So setting

α=h_a≤ h

+_+h_h =

h+h

h₊_h_+=_–_(_h–₊h_h_+)<_,β=γ = in (.), it follows that

d(Fx,Gy)αd(Tx,Sy).

() Ifw=d(Tx,Fx)in (.), thend(Fx,Gy)hd(Tx,Fx). So settingβ=h< ,

α=γ = ,u=d(Tx,Fx)in (.), it follows thatd(Fx,Gy)βu, whereu=d(Tx,Fx). () Ifw=d(Sy,Gy)in (.), thend(Fx,Gy)hd(Sy,Gy). So settingβ=h< ,

α=γ = ,u=d(Sy,Gy)in (.), it follows thatd(Fx,Gy)βu, whereu=d(Sy,Gy). () Ifw=d(Tx,Gy_a)+₊_hd(Sy,Fx)in (.), thend(Fx,Gy)_ah₊_h[d(Tx,Gy) +d(Sy,Fx)]. So setting

γ =₍_a₊h_h₎≤ h

+_+h_h+h = h+h

h_+_h_+=_–_(_h–_+h_h_+)<_,α=β= ,

v=d(Tx,Gy) +d(Sy,Fx)in (.), it follows thatd(Fx,Gy)γv, where

v=d(Tx,Gy) +d(Sy,Fx).

Therefore, in all cases we ﬁnd thatF,G,S, andT satisfy condition (.), thenF,G,S, andTsatisfy condition (.).

Now, we give an example to show that condition (.) is more general than condition (.) above.

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ϕ(t) =et_{. Then (}_X_,_d_{) is a complete cone metric space with a non-normal cone having a} nonempty interior. DeﬁneF,G,S, andT:C→Xas

F(x) = x

 +x, G(x) =

x

 +x, T(x) =x, S(x) =x

_, _x_∈_C_. _(.)

Note that for allx,y∈Cwithx=y,

d(Fx,Gy) = x  +x–

y

 +y

ϕ= |x– y

_|_ϕ

( +x)( +y₎=

d(Tx,Sy)



( +x)( +y)

. (.)

Therefore, condition (.) is satisﬁed if we chooseα=   (+x)(+y)∈

[_,_],β=γ = .

Next, we shall see that the inequality (.) is not satisﬁed for all  <h<  anda≥ +_+h_h by takingx=  andy=_.

Indeed, d(F(),G(_)) =| – _|ϕ= _ϕ and w∈ {d(T(),S(

 ))

a ,d(T(),F()),d(S(

 ),G(

 )),

d(T(),G(_))+d(S(_),F())

a+h }={



a

 ϕ, ,

 ϕ,



a+h

 ϕ}.

Since h_a < _ and _a₊h_h <_, d(F(),G(_))hwfor all possible cases ofw,  <h< , and

a≥ +_+h_h, that is, (.) is more general than (.).

So, we can obtain the following corollary.

Corollary . Let(X,d)be a complete cone metric space,C a nonempty closed subset of X such that for each x∈C and y∈/C there exists a point z∈∂C such that

d(x,z) +d(z,y) =d(x,y).

Suppose that F,G,S,T:C→X are two pairs of non-self-mappings satisfying,for all x,y∈C with x=y,

d(Fx,Gy)hw, (.)

where w∈ {d(Tx_a,Sy),d(Tx,Fx),d(Sy,Gy),d(Tx,Gy_a)+₊_hd(Sy,Fx)},a is a positive real number satisfying a≥ +h

+h and <h< .Also assume that

Then

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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Author details

1_{Department of Mathematics, Nanchang University, Nanchang, Jiangxi 330031, P.R. China.}2_{Department of Computer}

Science, Nanchang University, Nanchang, Jiangxi 330031, P.R. China.

Acknowledgements

The authors would like to express their sincere appreciation to the referees for their very helpful suggestions and kind comments. Project was supported by the National Natural Science Foundation of China (11361042 and 11326099) and supported partly by the Provincial Natural Science Foundation of Jiangxi, China (20142BAB201005) and the Science and Technology Project of Educational Commission of Jiangxi Province, China (GJJ11294).

Received: 7 March 2014 Accepted: 27 June 2014 Published: 22 July 2014 References

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doi:10.1186/1687-1812-2014-157