Some coincidence point results for generalized (ψ,φ)-weakly contractive mappings in ordered G-metric spaces

R E S E A R C H

Open Access

Some coincidence point results for

generalized

(

ψ

,

ϕ

)

-weakly contractive

mappings in ordered

G-metric spaces

Zead Mustafa

_{, Vahid Parvaneh}

_{, Mujahid Abbas}

_{and Jamal Rezaei Roshan}

*_{Correspondence:} [email protected] 3_{Young Researchers and Elite Club,} Kermanshah Branch, Islamic Azad University, Kermanshah, Iran Full list of author information is available at the end of the article

Abstract

The aim of this paper is to present some coincidence point results for six mappings satisfying the generalized (

ψ

ϕ

MSC: Primary 47H10; secondary 54H25

Keywords: coincidence point; common fixed point; generalized weak contraction; generalized metric space; partially weakly increasing mapping; altering distance function

1 Introduction and mathematical preliminaries

Let (X,d) be a metric space andfbe a self-mapping onX. Ifx=fxfor somexinX, thenxis called a fixed point off. The set of all fixed points off is denoted byF(f). IfF(f) ={z}, and for eachxin a complete metric spaceX, the sequencexn+=f(xn) =fn(x),n= , , , . . . , converges toz, thenf is called a Picard operator.

The functionϕ: [,∞)→[,∞) is called an altering distance function ifϕis continuous and nondecreasing andϕ(t) =  if and only ift=  [].

A self-mappingfonXis a weak contraction if the following contractive condition holds:

d(fx,fy)≤d(x,y) –ϕd(x,y),

for allx,y∈X, whereϕis an altering distance function.

The concept of a weakly contractive mapping was introduced by Alber and Guerre-Delabrere [] in the setup of Hilbert spaces. Rhoades [] considered this class of mappings in the setup of metric spaces and proved that a weakly contractive mapping is a Picard operator.

Letf andgbe two self-mappings on a nonempty setX. Ifx=fx=gxfor somexinX, thenxis called a common fixed point off andg. Sessa [] defined the concept of weakly commutative maps to obtain common fixed point for a pair of maps. Jungck generalized this idea, first to compatible mappings [] and then to weakly compatible mappings []. There are examples which show that each of these generalizations of commutativity is a proper extension of the previous definition.

Zhang and Song [] introduced the concept of a generalizedϕ-weak contractive map-ping as follows.

Self-mappingsf andgon a metric spaceXare called generalizedϕ-weak contractions if there exists a lower semicontinuous functionϕ: [,∞)→[,∞) withϕ() =  andϕ(t) >  for allt>  such that for allx,y∈X,

d(fx,gy)≤N(x,y) –ϕN(x,y), where

N(x,y) =max

d(x,y),d(x,fx),d(y,gy), 

d(x,gy) +d(y,fx).

Based on the above definition, they proved the following common fixed point result.

Theorem .[] Let(X,d)be a complete metric space.If f,g:X→X are generalized ϕ-weak contractive mappings,then there exists a unique point u∈X such that u=fu=gu.

For further works in this direction, we refer the reader to [–].

Recently, many researchers have focused on different contractive conditions in complete metric spaces endowed with a partial order and studied fixed point theory in the so-called bistructural spaces. For more details on fixed point results, its applications, comparison of different contractive conditions and related results in ordered metric spaces, we refer the reader to [–] and the references mentioned therein.

Mustafa and Sims [] generalized the concept of a metric, in which to every triplet of points of an abstract set, a real number is assigned. Based on the notion of generalized met-ric spaces, Mustafaet al.[–] obtained some fixed point theorems for mappings satis-fying different contractive conditions. Chughet al.[] obtained some fixed point results for maps satisfying propertyPinG-metric spaces. Saadatiet al.[] studied fixed point of contractive mappings in partially orderedG-metric spaces. Shatanawi [] obtained fixed points of-maps inG-metric spaces. For more details, we refer to [, –].

Very recently, Jleli and Samet [] and Sametet al.[] noticed that some fixed point theorems in the context of aG-metric space can be concluded by some existing results in the setting of a (quasi-)metric space. In fact, if the contraction condition of the fixed point theorem on aG-metric space can be reduced to two variables instead of three variables, then one can construct an equivalent fixed point theorem in the setting of a usual metric space. More precisely, in [, ], the authors noticed thatd(x,y) =G(x,y,y) forms a quasi-metric. Therefore, if one can transform the contraction condition of existence results in aG-metric space in such terms,G(x,y,y), then the related fixed point results become the known fixed point results in the context of a quasi-metric space.

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

Definition .[] LetXbe a nonempty set. Suppose that a mappingG:X×X×X→R+

satisfies:

(G) G(x,y,z) = ifx=y=z;

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

ThenGis called aG-metric onXand (X,G) is called aG-metric space.

Definition .[] A sequence{xn}in aG-metric spaceXis:

(i) aG-convergent sequence if there isx∈Xsuch that for anyε> , andn∈N, for all

n,m≥n,G(xn,xm,x) <ε.

(ii) aG-Cauchysequence if, for everyε> , there is a natural numbernsuch that for alln,m,l≥n,G(xn,xm,xl) <ε.

AG-metric space onX is said to beG-complete if everyG-Cauchy sequence inXis

G-convergent inX. It is known that{xn}G-converges tox∈Xif and only ifG(xm,xn,x)→

 asn,m→ ∞.

Lemma .[] Let X be a G-metric space.Then the following are equivalent: () The sequence{xn}isG-convergent tox.

() G(xn,xn,x)→asn→ ∞.

() G(xn,x,x)→asn→ ∞.

Lemma .[] Let X be a G-metric space.Then the following are equivalent:

The sequence{xn}isG-Cauchy.

For everyε> ,there existsn∈Nsuch that for alln,m≥n,G(xn,xm,xm) <ε;that is,

ifG(xn,xm,xm)→asn,m→ ∞.

Definition .[] Let (X,G) and (X,G) be twoG-metric spaces. Then a functionf :

X→XisG-continuous at a pointx∈Xif and only if it isG-sequentially continuous atx; that is, whenever{xn}isG-convergent tox,{f(xn)}isG-convergent tof(x).

Definition . A G-metric onX is said to be symmetric ifG(x,y,y) =G(y,x,x) for all

x,y∈X.

Proposition . Every G-metric on X defines a metric dGon X by

dG(x,y) =G(x,y,y) +G(y,x,x), ∀x,y∈X. (.)

For a symmetric G-metric space,one obtains

dG(x,y) = G(x,y,y), ∀x,y∈X. (.) However,if G is not symmetric,then the following inequality holds:



G(x,y,y)≤dG(x,y)≤G(x,y,y), ∀x,y∈X. (.)

Definition . A partially orderedG-metric space (X,,G) is said to have the sequential limit comparison property if for every nondecreasing sequence (nonincreasing sequence)

Definition . Letf andgbe two self-maps on a partially ordered setX. A pair (f,g) is said to be

(i) weakly increasing iffxgfxandgxfgxfor allx∈X[], (ii) partially weakly increasing iffxgfxfor allx∈X[].

LetXbe a nonempty set andf:X→Xbe a given mapping. For everyx∈X, letf–_{(x) =}

{u∈X:fu=x}.

Definition . Let (X,) be a partially ordered set, and letf,g,h:X→Xbe mappings such thatfX⊆hXandgX⊆hX. The ordered pair (f,g) is said to be: (a) weakly increasing with respect tohif and only if for allx∈X,fxgyfor ally∈h–_{(fx), andgxfyfor all}

y∈h–_{(gx) [], (b) partially weakly increasing with respect tohiffxgyfor ally∈h}–_(fx) [].

Remark . In the above definition: (i) iff =g, we say thatfis weakly increasing (partially weakly increasing) with respect toh, (ii) ifh=I(the identity mapping onX), then the above definition reduces to a weakly increasing (partially weakly increasing) mapping (see [, ]).

The following is an example of mappingsf,g,h,R,SandTfor which all ordered pairs (f,g), (g,h) and (h,f) are partially weakly increasing with respect toR,SandT but not weakly increasing with respect to them.

Example . LetX= [,∞). We define functionsf,g,h,R,S,T:X→Xby

f(x) =

x, ≤x≤,

, ≤x≤ ∞, g(x) =

_√

x, ≤x≤, , ≤x≤ ∞,

h(x) =

x, ≤x≤, , ≤x≤ ∞, and

R(x) =

x_{, ≤x≤,}

, ≤x≤ ∞, S(x) =



√

x, ≤x≤, , ≤x≤ ∞,

T(x) =



√

x, ≤x≤, , ≤x≤ ∞.

Definition .[, ] LetXbe aG-metric space andf,g:X→X. The pair (f,g) is said to be compatible if and only iflimn→∞G(fgxn,fgxn,gfxn) = , whenever{xn}is a sequence

inXsuch thatlimn→∞fxn=limn→∞gxn=tfor somet∈X.

Definition .(see,e.g., []) A quasi-metric on a nonempty setXis a mappingp:X× X→[,∞) such that (p)x=yif and only ifp(x,y) = , (p)p(x,y)≤p(x,z) +p(z,y) for all

x,y,z∈X. A pair (X,p) is said to be a quasi-metric space.

point theorems in generalized metric spaces was initiated by Abbas and Rhoades [] (see also [, , ]). Motivated by the work in [, , , , , ] and [], we prove some coincidence point results for nonlinear generalized (ψ,ϕ)-weakly contractive mappings in partially orderedG-metric spaces.

2 Main results

Let (X,,G) be an orderedG-metric space, and let f,g,h,R,S,T :X→X be six self-mappings. Throughout this paper, unless otherwise stated, for allx,y,z∈X, let

M(x,y,z) =max

G(Tx,Ry,Sz),

G(Tx,fx,fx),G(Ry,gy,gy),G(Sz,hz,hz),

G(Tx,Tx,fx) +G(Ry,Ry,gy) +G(Sz,Sz,hz) 

.

LetXbe any nonempty set andf,g,h,R,S,T:X→Xbe six mappings such thatf(X)⊆

R(X),g(X)⊆S(X) andh(X)⊆T(X). Letxbe an arbitrary point ofX. Choosex∈Xsuch thatfx=Rx,x∈Xsuch thatgx=Sxandx∈Xsuch thathx=Tx. This can be done asf(X)⊆R(X),g(X)⊆S(X) andh(X)⊆T(X).

Continuing in this way, we construct a sequence{zn}defined by:zn+=Rxn+=fxn, zn+=Sxn+=gxn+, andzn+=Txn+=hxn+for alln≥. The sequence{zn}inXis

said to be a Jungck-type iterative sequence with initial guessx.

Theorem . Let(X,,G)be a partially ordered G-complete G-metric space.Let f,g,h,R,

S,T:X→X be six mappings such that f(X)⊆R(X),g(X)⊆S(X)and h(X)⊆T(X). Sup-pose that for every three comparable elements Tx,Ry,Sz∈X,we have

ψG(fx,gy,hz)≤ψM(x,y,z)–ϕM(x,y,z), (.)

whereψ,ϕ: [,∞)→[,∞)are altering distance functions. Let f,g, h,R, S and T be continuous,the pairs(f,T), (g,R)and(h,S)be compatible and the pairs(f,g), (g,h)and

(h,f)be partially weakly increasing with respect to R,S and T,respectively.Then the pairs

(f,T), (g,R)and(h,S)have a coincidence point z∗in X.Moreover,if Rz∗,Sz∗and Tz∗are comparable,then z∗is a coincidence point of f,g,h,R,S and T.

Proof Let{zn}be a Jungck-type iterative sequence with initial guessxinX; that is,zn+=

Rxn+=fxn,zn+=Sxn+=gxn+andzn+=Txn+=hxn+for alln≥.

Asx∈R–(fx),x∈S–(gx) andx∈T–(hx), and the pairs (f,g), (g,h) and (h,f) are partially weakly increasing with respect toR,SandT, so we have

Rx=fxgx=Sxhx=Txfx=Rx.

Continuing this process, we obtainRxn+Sxn+Txn+forn≥. We will complete the proof in three steps.

Define Gk=G(zk,zk+,zk+). SupposeGk=  for somek. Thenzk =zk+=zk+. If

k= n, thenzn=zn+=zn+giveszn+=zn+=zn+. Indeed,

ψG(zn+,zn+,zn+)

=ψG(fxn,gxn+,hxn+)

≤ψM(xn,xn+,xn+)

–ϕM(xn,xn+,xn+)

,

where

M(xn,xn+,xn+)

=max

G(Txn,Rxn+,Sxn+),G(Txn,fxn,fxn),

G(Rxn+,gxn+,gxn+),G(Sxn+,hxn+,hxn+),

G(Txn,Txn,fxn) +G(Rxn+,Rxn+,gxn+) +G(Sxn+,Sxn+,hxn+) 

=max

G(zn,zn+,zn+),G(zn,zn+,zn+),

G(zn+,zn+,zn+),G(zn+,zn+,zn+),

G(zn,zn,zn+) +G(zn+,zn+,zn+) +G(zn+,zn+,zn+) 

=max

, , ,G(zn+,zn+,zn+),

 +  +G(zn+,zn+,zn+) 

=G(zn+,zn+,zn+)

≤G(zn+,zn+,zn+) = G(zn+,zn+,zn+).

Thus,

ψG(zn+,zn+,zn+)

≤ψG(zn+,zn+,zn+)

–ϕG(zn+,zn+,zn+)

,

which implies thatϕ(G(zn+,zn+,zn+)) = ; that is,zn+=zn+=zn+. Similarly, ifk= n+ , thenzn+=zn+=zn+giveszn+=zn+=zn+. Also, ifk= n+ , thenzn+=

zn+=zn+implies thatzn+=zn+=zn+. Consequently, the sequence{zk}becomes

constant fork≥k. Suppose that

Gk=G(zk,zk+,zk+) >  (.)

for eachk. We now claim that the following inequality holds:

G(zk+,zk+,zk+)≤G(zk,zk+,zk+) =M(xk,xk+,xk+) (.)

Letk= nand forn≥,G(zn+,zn+,zn+)≥G(zn,zn+,zn+) > . Then, asTxn Rxn+Sxn+, using (.), we obtain that

ψG(zn+,zn+,zn+)

≤ψG(zn+,zn+,zn+)

=ψG(fxn,gxn+,hxn+)

≤ψM(xn,xn+,xn+)

–ϕM(xn,xn+,xn+)

, (.)

where

M(xn,xn+,xn+)

=max

G(Txn,Rxn+,Sxn+),

G(Txn,fxn,fxn),G(Rxn+,gxn+,gxn+),G(Sxn+,hxn+,hxn+),

G(Txn,Txn,fxn) +G(Rxn+,Rxn+,gxn+) +G(Sxn+,Sxn+,hxn+) 

=max

G(zn,zn+,zn+),

G(zn,zn+,zn+),G(zn+,zn+,zn+),G(zn+,zn+,zn+),

G(zn,zn,zn+) +G(zn+,zn+,zn+) +G(zn+,zn+,zn+) 

≤max

G(zn,zn+,zn+),G(zn+,zn+,zn+),

G(zn,zn+,zn+) +G(zn+,zn+,zn+) 

=G(zn+,zn+,zn+).

Hence, (.) implies that

ψG(zn+,zn+,zn+)

≤ψG(zn+,zn+,zn+)

–ϕM(xn,xn+,xn+)

,

which is possible only ifM(xn,xn+,xn+) = ; that is,G(zn,zn+,zn+) = , a contradic-tion to (.). Hence,G(zn+,zn+,zn+)≤G(zn,zn+,zn+) and

M(xn,xn+,xn+) =G(zn,zn+,zn+).

Therefore, (.) is proved fork= n. Similarly, it can be shown that

G(zn+,zn+,zn+)≤G(zn+,zn+,zn+) =M(xn+,xn+,xn+), (.)

and

Hence,{G(zk,zk+,zk+)}is a nondecreasing sequence of nonnegative real numbers. There-fore, there isr≥ such that

lim

k→∞G(zk,zk+,zk+) =r. (.)

Since

G(zk+,zk+,zk+)≤M(xk,xk+,xk+)≤G(zk,zk+,zk+), (.) taking limit ask→ ∞in (.), we obtain

lim

k→∞M(xk,xk+,xk+) =r. (.)

Taking limit asn→ ∞in (.), using (.), (.) and the continuity ofψ andϕ, we have

ψ(r)≤ψ(r) –ϕ(r)≤ψ(r). Thereforeϕ(r) = . Hence,

lim

k→∞G(zk,zk+,zk+) = , (.)

from our assumptions aboutϕ. Also, from Definition ., part (G), we have

lim

k→∞G(zk,zk+,zk+) = , (.)

and, sinceG(x,y,y)≤G(x,x,y) for allx,y∈X, we have

lim

k→∞G(zk,zk,zk+) = . (.)

Step II.We now show that{zn}is aG-Cauchy sequence in X. Because of (.), it is

sufficient to show that{zn}isG-Cauchy.

We assume on contrary that there exists ε>  for which we can find subsequences

{zm(k)}and{zn(k)}of{zn}such thatn(k) >m(k)≥kand

G(zm(k),zn(k),zn(k))≥ε, (.) andn(k) is the smallest number such that the above statement holds;i.e.,

G(zm(k),zn(k)–,zn(k)–) <ε. (.)

From the rectangle inequality and (.), we have

G(zm(k),zn(k),zn(k))

≤G(zm(k),zn(k)–,zn(k)–) +G(zn(k)–,zn(k),zn(k)) <ε+G(zn(k)–,zn(k),zn(k))

<ε+G(zn(k)–,zn(k)–,zn(k)–) +G(zn(k)–,zn(k)–,zn(k)–)

Taking limit ask→ ∞in (.), from (.) and (.) we obtain that

lim

k→∞G(zm(k),zn(k),zn(k)) =ε. (.)

Using the rectangle inequality, we have

G(zm(k),zn(k)+,zn(k)+)

≤G(zm(k),zn(k),zn(k)) +G(zn(k),zn(k)+,zn(k)+)

≤G(zm(k),zn(k)+,zn(k)+) +G(zn(k)+,zn(k),zn(k)) +G(zn(k),zn(k)+,zn(k)+)

≤G(zm(k),zn(k)+,zn(k)+) +G(zn(k)+,zn(k)+,zn(k)+) +G(zn(k)+,zn(k),zn(k)) +G(zn(k),zn(k)+,zn(k)+). (.)

Taking limit ask→ ∞in (.), from (.), (.), (.) and (.) we have

lim

k→∞G(zm(k),zn(k)+,zn(k)+) =ε. (.)

Finally,

G(zm(k)+,zn(k)+,zn(k)+)

≤G(zm(k)+,zm(k),zm(k)) +G(zm(k),zn(k)+,zn(k)+)

≤G(zm(k)+,zm(k),zm(k)) +G(zm(k),zn(k),zn(k)) +G(zn(k),zn(k)+,zn(k)+)

≤G(zm(k)+,zm(k),zm(k)) +G(zm(k),zn(k),zn(k))

+G(zn(k),zn(k)+,zn(k)+) +G(zn(k)+,zn(k)+,zn(k)+). (.)

Taking limit ask→ ∞in (.) and using (.), (.), (.) and (.), we have

lim

k→∞G(zm(k)+,zn(k)+,zn(k)+)≤ε.

Consider

G(zm(k),zn(k)+,zn(k)+)

≤G(zm(k),zm(k)+,zm(k)+) +G(zm(k)+,zn(k)+,zn(k)+)

≤G(zm(k),zm(k)+,zm(k)+) +G(zm(k)+,zn(k)+,zn(k)+) +G(zn(k)+,zn(k)+,zn(k)+)

≤G(zm(k),zm(k)+,zm(k)+) +G(zm(k)+,zn(k)+,zn(k)+)

+G(zn(k)+,zn(k)+,zn(k)+). (.)

Taking limit ask→ ∞and using (.), (.) and (.), we have

ε≤ lim

Therefore,

lim

k→∞G(zm(k)+,zn(k)+,zn(k)+) =ε. (.)

AsTxm(k)Rxn(k)+Sxn(k)+, so from (.) we have

ψG(zm(k)+,zn(k)+,zn(k)+)

=ψG(fxm(k),gxn(k)+,hxn(k)+)

≤ψM(xm(k),xn(k)+,xn(k)+)

–ϕM(xm(k),xn(k)+,xn(k)+)

, (.)

where

M(xm(k),xn(k)+,xn(k)+)

=max

G(Txm(k),Rxn(k)+,Sxn(k)+),G(Txm(k),fxm(k),fxm(k)),

G(Rxn(k)+,gxn(k)+,gxn(k)+),G(Sxn(k)+,hxn(k)+,hxn(k)+),

G(Txm(k),Txm(k),fxm(k)) +G(Rxn(k)+,Rxn(k)+,gxn(k)+) +G(Sxn(k)+,Sxn(k)+,hxn(k)+)



=max

G(zm(k),zn(k)+,zn(k)+),G(zm(k),zm(k)+,zm(k)+),

G(zn(k)+,zn(k)+,zn(k)+),G(zn(k)+,zn(k)+,zn(k)+),

G(zm(k),zm(k),zm(k)+) +G(zn(k)+,zn(k)+,zn(k)+) +G(zn(k)+,zn(k)+,zn(k)+)



.

Taking limit ask→ ∞and using (.), (.), (.) and (.) in (.), we have

ψ(ε)≤ψ(ε) –ϕ(ε) <ψ(ε),

a contradiction. Hence,{zn}is aG-Cauchy sequence.

Step III.We will show thatf,g,h,R,SandThave a coincidence point.

Since{zn}is aG-Cauchy sequence in the G-completeG-metric spaceX, there exists z∗∈Xsuch that

lim

n→∞G

zn+,zn+,z∗

= lim

n→∞G

Rxn+,Rxn+,z∗

= lim

n→∞G

fxn,fxn,z∗

= , (.)

lim

n→∞G

zn+,zn+,z∗

= lim

n→∞G

Sxn+,Sxn+,z∗

= lim

n→∞G

gxn+,gxn+,z∗

and

lim

n→∞G

zn+,zn+,z∗

= lim

n→∞G

Txn+,Txn+,z∗

= lim

n→∞G

hxn+,hxn+,z∗

= . (.)

Hence,

Txn→z∗ and fxn→z∗ asn→ ∞. (.)

As (f,T) is compatible, so

lim

n→∞G(Tfxn,fTxn,fTxn) = . (.)

Moreover, fromlim_n→∞G(fxn,fxn,z∗) = ,limn→∞G(Txn,z∗,z∗) = , and the continuity

ofTandf, we obtain

lim

n→∞G

Tfxn,Tfxn,Tz∗

=  = lim

n→∞G

fTxn,fz∗,fz∗

. (.)

By the rectangle inequality, we have

GTz∗,fz∗,fz∗

≤GTz∗,Tfxn,Tfxn

+GTfxn,fz∗,fz∗

≤GTz∗,Tfxn,Tfxn

+G(Tfxn,fTxn,fTxn) +G

fTxn,fz∗,fz∗

. (.)

Taking limit asn→ ∞in (.), we obtain

GTz∗,fz∗,fz∗≤,

which implies thatfz∗=Tz∗, that is,z∗is a coincidence point off andT.

Similarly,gz∗=Rz∗andhz∗=Sz∗. Now, letRz∗,Sz∗andTz∗be comparable. By (.) we have

ψGfz∗,gz∗,hz∗

≤ψMz∗,z∗,z∗–ϕMz∗,z∗,z∗, (.)

where

Mz∗,z∗,z∗=max

GTz∗,Rz∗,Sz∗,

GTz∗,fz∗,fz∗,GRz∗,gz∗,gz∗,GSz∗,hz∗,hz∗,

G(Tz∗,Tz∗,fz∗) +G(Rz∗,Rz∗,gz∗) +G(Sz∗,Sz∗,hz∗) 

Hence (.) gives

ψGfz∗,gz∗,hz∗≤ψGfz∗,gz∗,hz∗–ϕGfz∗,gz∗,hz∗.

Therefore,fz∗=gz∗=hz∗=Tz∗=Rz∗=Sz∗.

In the following theorem, the continuity assumption on the mappingsf,g,h,R,Sand

T is in fact replaced by the sequential limit comparison property of the space, and the compatibility of the pairs (f,T), (g,R) and (h,S) is in fact replaced by weak compatibility of the pairs.

Theorem . Let(X,,G)be a partially ordered G-complete G-metric space with the

sequential limit comparison property,let f,g,h,R,S,T:X→X be six mappings such that f(X)⊆R(X), g(X)⊆S(X),and let h(X)⊆T(X),RX,SX and TX be G-complete subsets of X.Suppose that for comparable elements Tx,Ry,Sz∈X,we have

ψG(fx,gy,hz)≤ψM(x,y,z)–ϕM(x,y,z), (.)

whereψ,ϕ: [,∞)→[,∞)are altering distance functions.Then the pairs(f,T), (g,R)

and(h,S)have a coincidence point z∗in X provided that the pairs(f,T), (g,R)and(h,S)

are weakly compatible and the pairs(f,g), (g,h)and(h,f)are partially weakly increasing with respect to R,S and T,respectively.Moreover,if Rz∗,Sz∗and Tz∗are comparable,then z∗∈X is a coincidence point of f,g,h,R,S and T.

Proof Following the proof of Theorem ., there existsz∗∈Xsuch that

lim

k→∞G

zk,zk,z∗

= . (.)

SinceR(X) isG-complete and{zn+} ⊆R(X), thereforez∗∈R(X). Hence, there existsu∈

Xsuch thatz∗=Ruand

lim

n→∞G(zn+,zn+,Ru) =nlim→∞G(Rxn+,Rxn+,Ru) = . (.)

Similarly, there existsv,w∈Xsuch thatz∗=Sv=Twand

lim

n→∞G(Sxn+,Sxn+,Sv) =nlim→∞G(Txn,Txn,Tw) = . (.)

Now, we prove thatwis a coincidence point off andT. For this purpose, we show that

fw=gu. Suppose oppositefw=gu. SinceSxn+→z∗=Tw=Ruasn→ ∞, soSxn+

Tw=Ru.

Therefore, from (.), we have

ψG(fw,gu,hxn+)

≤ψM(w,u,xn+)

–ϕM(w,u,xn+)

where

M(w,u,xn+)

=max

G(Tw,Ru,Sxn+),G(Tw,fw,fw),

G(Ru,gu,gu),G(Sxn+,hxn+,hxn+),

G(Tw,Tw,fw) +G(Ru,Ru,gu) +G(Sxn+,Sxn+,hxn+) 

.

Taking limit as n→ ∞ in (.), as G(z∗,z∗,z∗) =  and from (G) and the fact that

G(x,x,y)≤G(x,y,y) , we obtain that

ψGfw,gu,z∗

≤ψ max

Gz∗,fw,fw,Gz∗,gu,gu,G(z

∗_,z∗_{,fw) +G(z}∗_,z∗_,gu)



–ϕ max

Gz∗,fw,fw,Gz∗,gu,gu,G(z

∗_,z∗_{,fw) +G(z}∗_,z∗_,gu)



≤ψGfw,gu,z∗

–ϕ max

Gz∗,fw,fw,Gz∗,gu,gu,G(z

∗_,z∗_{,fw) +G(z}∗_,z∗_,gu)



,

which implies thatfw=z∗=gu, a contradiction, sofw=gu. Again from the above inequal-ity it is easy to see thatfw=z∗. So, we havefw=z∗=Tw.

Asf andTare weakly compatible, we havefz∗=fTw=Tfw=Tz∗. Thusz∗is a coinci-dence point off andT.

Similarly it can be shown thatz∗is a coincidence point of the pairs (g,R) and (h,S). The rest of the proof can be obtained from the same arguments as those in the proof of

Theorem ..

Remark . Let (X,G) be aG-metric space. Letf,R,S,T :X→X be mappings. If we define functionsp,q:X×X→[,∞) in the following way:

p(x,y) =G(Tx,Ry,Sy)

and

q(x,y) =G(Tx,fy,fy)

for allx,y∈X, it is easy to see that both mappingspandqdo not satisfy the conditions of Definition .. Hence, Theorem . and Theorem . cannot be characterized in the context of quasi-metric as it is suggested in [, ].

Corollary . Let(X,,G)be a partially ordered G-complete G-metric space,and let f,g,h,R:X→X be four mappings such that f(X)∪g(X)∪h(X)⊆R(X).Suppose that for every three comparable elements Rx,Ry,Rz∈X,we have

ψG(fx,gy,hz)≤ψM(x,y,z)–ϕM(x,y,z), (.)

where

M(x,y,z) =max

G(Rx,Ry,Rz),

G(Rx,fx,fx),G(Ry,gy,gy),G(Rz,hz,hz),

G(Rx,Rx,fx) +G(Ry,Ry,gy) +G(Rz,Rz,hz) 

andψ,ϕ: [,∞)→[,∞)are altering distance functions.Then f,g,h and R have a co-incidence point in X provided that the pairs(f,g), (g,h)and(h,f)are partially weakly increasing with respect to R and either

a. f is continuous and the pair(f,R)is compatible,or

b. gis continuous and the pair(g,R)is compatible,or

c. his continuous and the pair(h,R)is compatible.

TakingR=S=T andf =g=hin Theorem ., we obtain the following coincidence point result.

Corollary . Let(X,,G)be a partially ordered G-complete G-metric space,and let f,R:

X→X be two mappings such that f(X)⊆R(X).Suppose that for every three comparable elements Rx,Ry,Rz∈X,we have

ψG(fx,fy,fz)≤ψM(x,y,z)–ϕM(x,y,z), (.)

where

M(x,y,z) =max

G(Rx,Ry,Rz),

G(Rx,fx,fx),G(Ry,fy,fy),G(Rz,fz,fz),

G(Rx,Rx,fx) +G(Ry,Ry,fy) +G(Rz,Rz,fz) 

andψ,ϕ: [,∞)→[,∞)are altering distance functions.Then the pair(f,R)has a coin-cidence point in X provided that f and R are continuous,the pair(f,R)is compatible and f is weakly increasing with respect to R.

Example . LetX= [,∞) andGonXbe given byG(x,y,z) =|x–y|+|y–z|+|x–z|for allx,y,z∈X. We define an ordering ‘’ onXas follows:

Define self-mapsf,g,h,S,TandRonXby

fx=ln√x_{+  +x=sinh}–_{x, Rx=sinh(x),}

gx=sinh– x



, Sx=sinh(x),

hx=sinh– x



, Tx=sinh(x).

To prove that (f,g) are partially weakly increasing with respect toR, letx,y∈Xbe such thaty∈R–_{fx; that is,Ry=fx. By the definition off andR, we havesinh}–_{x=sinhyand}

y=sinh–(sinh_ –x). Assinhx≥(sinh–x) for allx≥, therefore x≥sinh–(sinh–x), or

fx=sinh–x≥sinh– 

sinh

–_sinh–_x=sinh–  y

=gy.

Therefore,fxgy. Hence (f,g) is partially weakly increasing with respect toR. Similarly, one can show that (g,h) and (h,f) are partially weakly increasing with respect toSandT, respectively.

Furthermore,fX=TX=gX=SX=hX =RX= [,∞) and the pairs (f,T), (g,R) and (h,S) are compatible. Indeed, if {xn} is a sequence in X such that for some t ∈ X,

limn→∞G(t,fxn,fxn) =limn→∞G(t,Txn,Txn) = . Therefore, we have

lim

n→∞sinh

–_x

n–t= lim

n→∞|sinhxn–t|= .

Continuity ofsinh–andsinhimplies that

lim

n→∞|xn–sinht|=nlim→∞ xn–

sinh–t



= ,

and the uniqueness of the limit gives thatsinht=sinh–t

 . But

sinht=sinh –_t

 ⇐⇒ t= .

So, we havet= . Sincef andT are continuous, we have

lim

n→∞G(fTxn,fTxn,Tfxn) = nlim→∞|fTxn–Tfxn|= .

Defineψ,ϕ: [,∞)→[,∞) asψ(t) =btandϕ(t) = (b– )tfor allt∈[,∞), where  <

b≤.

Using the mean value theorem simultaneously for the functionssinh–andsinh, we have

ψG(fx,gy,hz)= b|fx–gy|+|fx–hz|+|gy–hz|

= b sinh–x–sinh– y



+sinh–(x) –sinh– z



+sinh– y



–sinh– z



≤b 

|x–y|+ 

|x–z|+ 

|y– z| =b(|x– y|+|x– z|+|y– z|)



≤ b



|sinhx–sinhy|+|sinhy–sinhz|+|sinhz–sinhx|

≤ |Tx–Ry|+|Ry–Sz|+|Sz–Tx|

=G(Tx,Ry,Sz)≤M(x,y,z) =ψM(x,y,z)–ϕM(x,y,z).

Thus, (.) is satisfied for allx,y,z∈X. Therefore, all the conditions of Theorem . are satisfied. Moreover,  is a coincidence point off,g,h,R,SandT.

The following example supports the usability of our results for non-symmetricG-metric spaces.

Example . LetX={, , , }be endowed with the usual order. Let

A=(, , ), (, , ), (, , )

and

B=(, , ), (, , ), (, , ).

DefineG:X_→R+_by

G(x,y,z) =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

 if (x,y,z)∈A,  if (x,y,z)∈B,

 if (x,y,z)∈X–A∪B,  ifx=y=z.

It is easy to see that (X,G) is a non-symmetricG-metric space.

Also, (X,G) has the sequential limit comparison property. Indeed, for each{xn}inX

such thatG(xn,x,x)→ for anx∈X, there isk∈Nsuch that for eachn≥k,xn=x.

Define the self-mapsf,g,h,R,SandT by

f =

       

, R=

       

,

g=

       

, S=

       

,

h=

       

, T=

       

.

We see that

fX⊆RX=X,

and

hX⊆TX=X.

Also,RX,SXandTXareG-complete. The pairs (f,T), (g,R) and (h,S) are weakly com-patible.

On the other hand, one can easily check that the pairs (f,g), (g,h) and (h,f) are partially weakly increasing with respect toR,SandT, respectively.

Defineψ,ϕ: [,∞)→[,∞) byψ(t) =_tandϕ(t) =_t.

According to the definition off,g,handGfor each three elementsx,y,z∈X, we see that

G(fx,gy,hz)∈ {, , }. But

G(Tx,Ry,Sz),G(Tx,fx,fx),G(Ry,gy,gy),G(Sz,hz,hz)∈ {, , , } and

G(Tx,Tx,fx),G(Ry,Ry,gy),G(Sz,Sz,hz)∈ {, , , }. Hence, we have

ψG(fx,gy,hz)≤ =M(x,y,z) =ψM(x,y,z)–ϕM(x,y,z).

Therefore, all the conditions of Theorem . are satisfied. Moreover,  is a coincidence point off,g,h,R,SandT.

Letbe the set of all functionsμ: [, +∞)→[, +∞) satisfying the following condi-tions:

(I) μis a positive Lebesgue integrable mapping on each compact subset of[, +∞). (II) For allε> ,_εμ(t)dt> .

Remark . Suppose that there existsμ∈such that mappingsf,g,h,R,SandTsatisfy the following condition:

ψ(G(fx,gy,hz)) 

μ(t)dt≤

ψ(M(x,y,z)) 

μ(t)dt–

ϕ(M(x,y,z)) 

μ(t)dt. (.)

Thenf,g,h,R,SandT have a coincidence point if the other conditions of Theorem . are satisfied.

For this, define the function(x) =_xμ(t)dt. Then (.) becomes

ψG(fx,gy,hz)≤ψM(x,y,z)–ϕM(x,y,z).

Takingg=h,T=R=S=IXandy=zin Theorems . and ., we obtain the following

common fixed point result.

Theorem . Let(X,,G)be a partially ordered G-complete G-metric space,and let f and g be two self-mappings on X.Suppose that for every comparable elements x,y∈X,

ψG(fx,gy,gy)≤ψM(x,y,y)–ϕM(x,y,y), (.)

where

M(x,y,y) =max

G(x,y,y),G(x,fx,fx),G(y,gy,gy),

G(x,x,fx) + G(y,y,gy) 

,

andψ,ϕ: [,∞)→[,∞)are altering distance functions.Then the pair(f,g)has a com-mon fixed point z in X provided that the pair(f,g)is weakly increasing and either

a. f orgis continuous,or

b. Xhas the sequential limit comparison property.

Takingf =gin the above, we obtain the following common fixed point result.

Theorem . Let(X,,G)be a partially ordered complete G-metric space,and let f be a self-mapping on X.Suppose that for every comparable elements x,y∈X,

ψG(fx,fy,fy)≤ψM(x,y,y)–ϕM(x,y,y), (.)

where

M(x,y,y) =max

G(x,y,y),G(x,fx,fx),G(y,fy,fy),

G(x,x,fx) + G(y,y,fy) 

andψ,ϕ: [,∞)→[,∞)are altering distance functions.Then f has a fixed point z in X provided that f is weakly increasing and either

a. f is continuous,or

b. Xhas the sequential limit comparison property.

3 Existence of a common solution for a system of integral equations

Motivated by the work in [] and [], we consider the following system of integral equa-tions:

x(t) =

b

a K

t,s,x(s)ds+k(t),

x(t) =

b

a K

t,s,x(s)ds+k(t),

x(t) =

b

a K

t,s,x(s)ds+k(t),

whereb>a≥. The aim of this section is to prove the existence of a solution for (.) which belongs toX=C[a,b] (the set of all continuous real-valued functions defined on [a,b]) as an application of Corollary ..

The considered problem can be reformulated as follows. Letf,g,h:X→Xbe defined by

fx(t) =

b

a K

t,s,x(s)ds,

gx(t) =

b

a K

t,s,x(s)ds,

and

hx(t) =

b

a K

t,s,x(s)ds

for allx∈Xand for allt∈[a,b]. Obviously, the existence of a solution for (.) is equivalent to the existence of a common fixed point off,gandh.

Let

d(u,v) = max

t∈[a,b]

u(t) –v(t).

EquipXwith theG-metric given by

G(u,v,w) =maxd(u,v),d(v,w),d(w,u)

for allu,v,w∈X. Evidently, (X,G) is a completeG-metric space. We endowXwith the partial orderedgiven by

xy ⇐⇒ x(t)≤y(t)

for allt∈[a,b]. It is known that (X,) has the sequential limit comparison property []. Now, we will prove the following result.

Theorem . Suppose that the following hypotheses hold:

(i) K,K,K: [a,b]×[a,b]×R→Randk: [a,b]→Rare continuous; (ii) For allt,s∈[a,b]and for allx∈X,we have

K

t,s,x(s)≤K t,s,

b

a K

t,s,x(s)ds+k(t)

,

K

t,s,x(s)≤K t,s,

b

a K

t,s,x(s)ds+k(t)

,

and

K

t,s,x(s)≤K t,s,

b

a K

t,s,x(s)ds+k(t)

(iii) For alls,t∈[a,b]and for allx,y∈Xwithxy,we have Ki

t,s,x(s)–Kj

t,s,y(s)≤

p(t,s)ln +x(s) –y(s),

wherep: [a,b]×[a,b]→[,∞)is a continuous function satisfying

sup

b

a

p(s,t)ds<  (b–a).

Then system(.)has a solution x∈X.

Proof From condition (ii), the ordered pairs (f,g), (g,h) and (h,f) are partially weakly in-creasing.

Now, letx,y∈Xbe such thatxy. From condition (iii), for allt∈[a,b], we have

fx(t) –gy(t)≤

b

a K

t,s,x(s)–K

t,s,x(s)ds 

≤

b

a

ds

b

a K

t,s,x(s)–K

t,s,x(s)ds

≤(b–a)

b

a

p(t,s)ln +x(s) –y(s)ds

≤(b–a)

b

a

p(t,s)ln +d(x,y)ds

≤(b–a)

b

a

p(t,s)ln +G(x,y,z)ds

= (b–a)

b

a

p(t,s)ds

ln +G(x,y,z)

= (b–a)

b

a

p(t,s)ds

ln +M(x,y,z)

<ln +M(x,y,z)

=M(x,y,z)–M(x,y,z)–ln +M(x,y,z).

Hence,

d(fx,gy)=

 sup

t∈[a,b]

fx(t) –gy(t)

≤M(x,y,z)–M(x,y,z)–ln +M(x,y,z). (.)

Similarly, we can show that

d(gy,hz)=

 sup

t∈[a,b]

gy(t) –hz(t)

and

d(hz,fx)= sup

t∈[a,b]

hz(t) –fx(t)

≤M(x,y,z)–M(x,y,z)–ln +M(x,y,z). (.)

Therefore, from (.), (.) and (.), we have

G(fx,gy,hz)=maxd(fx,gy),d(gy,hz),d(hz,fx)

=maxd(fx,gy),d(gy,hz),d(hz,fx)

≤M(x,y,z)–M(x,y,z)–ln +M(x,y,z).

Putting,ψ(t) =t_{,ϕ(t) =t}_{–ln( +t}_{) andR=I}

X in Corollary ., there existsx∈X, a

common fixed point off andgandh, which is a solution of (.).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, Statistics and Physics, Qatar University, Doha, Qatar.}2_{Department of Mathematics, The} Hashemite University, P.O. Box 150459, Zarqa, 13115, Jordan.3_{Young Researchers and Elite Club, Kermanshah Branch,} Islamic Azad University, Kermanshah, Iran. 4_{Department of Mathematics and Applied Mathematics, University Pretoria,} Lynnwood Road, Pretoria 0002, South Africa. 5_{Department of Mathematics, Qaemshahr Branch, Islamic Azad University,} Qaemshahr, Iran.

Received: 7 July 2013 Accepted: 28 October 2013 Published:02 Dec 2013

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