Multi-dimensional coincidence point theorems for weakly compatible mappings with the \(CLR_{g}\)-property in (fuzzy) metric spaces

(1)

R E S E A R C H

Open Access

Multi-dimensional coincidence point

theorems for weakly compatible mappings

with the

CLR

_g

-property in (fuzzy) metric

spaces

J Martínez-Moreno

1

_{, A Roldán}

1

_{, C Roldán}

2

_{and Yeol Je Cho}

3,4*

*_{Correspondence: [email protected]} 3_{Department of Mathematics} Education and RINS, Gyeongsang National University, Jinju, Korea 4_{Department of Mathematics, King} Abdulaziz University, Jeddah, Saudi Arabia

Full list of author information is available at the end of the article

Abstract

In this paper, we extend the concepts of theE.A-property and theCLRg-property in

fuzzy metric spaces to the setting of multi-dimensional fuzzy metric spaces and show the existence of multi-dimensional coincidence point and ﬁxed point theorems for weakly compatible mappings with theE.A-property and theCLRg-property.

Keywords: fuzzy metric space; coincidence point; ﬁxed point; contraction; theE.A property; theCLRg-property

1 Introduction

Since Banach’s fixed point theorem in , many authors have improved, extended and generalized this theorem in many different ways. One of the newest branches of this the-orem is devoted to the existence ofcoupled fixed point, which was introduced by Guo and Lakshmikantham [] in . Later, Berinde and Borcut [] introduced the concept of

tripled ﬁxed pointand proved some tripled ﬁxed point theorems using mixed monotone

mappings (see also [, ]). Recently, Roldánet al.[] proposed the notion ofcoincidence

pointfor nonlinear mappings with any number of variables and showed the existence and

uniqueness theorems that extended the mentioned previous results for nonlinear map-pings, not necessarily permuted or ordered, in the framework of partially ordered com-plete metric spaces by using some weaker contractive condition, which also generalized other works by Berzig and Samet [].

Especially in [], the existence results of coincidence points for the nonlinear mappings in any number of variables in fuzzy metric spaces were presented.

In , Aamri and Moutawakil [] deﬁned the notion of theE.A-property for nonlinear self-mappings which contained the class of non-compatible mappings in metric spaces. It was pointed out that theE.A-property allows replacing the completeness requirement of the space with a more natural condition of closedness of the range as well as relaxes the completeness of the whole space, continuity of one or more mappings and contain-ment of the range of one mapping into the range of another, which is utilized to construct the sequence of some joint iterates. Since Aamri and Moutawakil, many authors have also

(2)

proved common ﬁxed point theorems in fuzzy metric spaces for diﬀerent contractive con-ditions.

Recently, Sintunavarat and Kumam [] deﬁned the notion of theCLRg-property in fuzzy

metric spaces and improved the results of Mihet [] without any requirement of the closedness of the space.

In this paper, we extend the notions of the E.A-property and theCLRg-property for

nonlinear mappings with any number of variables and use these notions to present the existence results of coincidence points for weakly compatible mappings in fuzzy metric spaces. Our results improve, extend and generalize many ﬁxed point theorems in metric spaces and fuzzy metric spaces given by some authors.

2 Preliminaries

Letnbe a positive integer and letn={, , . . . ,n}. Henceforth,Xdenotes a nonempty set

andXn_{denotes the product space}_X_×_X_×_{· · · ×}n _X_{. We represent the identity mapping on}

XasIX.

Throughout this manuscript,mandpdenote non-negative integers,tis a positive real number andi,j,s∈ {, , . . . ,n}. Unless otherwise stated, ‘for allm’ will mean ‘for allm≥’, ‘for allt’ will mean ‘for allt> ’ and ‘for alli’ will mean ‘for alli∈ {, , . . . ,n}’. Let us denote

R+_{= (,}_∞_{) and}_I_{= [, ].}

In the sequel, letF:Xn→Xandg:X→Xbe two mappings. For brevity,g(x) is denoted bygx.

LetF:Xn_→_X_and_g_:_X_→_X_{be two mappings. Henceforth, let}_σ

,σ, . . . ,σn:n→n

benmappings fromninto itself, and letbe then-tuple (σ,σ, . . . ,σn).

Deﬁnition ([]) A point (x,x, . . . ,xn)∈Xnis called a-coincidence pointof the

map-pingsFandgif

F(xσi(),xσi(), . . . ,xσi(n)) =gxi ()

for alli. Ifgis the identity mapping onX, then (x,x, . . . ,xn)∈Xnis called a-ﬁxed point

of the mappingF.

Ifσ:n→nis a mapping, then, from its ordered image,i.e.,σ= (σ(),σ(), . . . ,σ(n)),

we have the following:

() Gnana-Bhaskar and Lakshmikantham’s election [] inn= isσ=τ= (, )and

σ= (, );

() Berinde and Borcut’s election [] inn= isσ=τ= (, , ),σ= (, , )and

σ= (, , );

() Karapınar’s election inn= isσ=τ= (, , , ),σ= (, , , ),σ= (, , , )and

σ= (, , , ).

There exist diﬀerent notions offuzzy metric space(see []). For our purposes, we use the following one.

Deﬁnition (George and Veeramani []) A triple (X,M,∗) is called afuzzy metric space

(brieﬂy, an FMS) if Xis an arbitrary nonempty set, ∗is a continuous t-norm andM:

X×X×R+→Iis a fuzzy set satisfying the following conditions: for allx,y,z∈Xand

(3)

(FM) M(x,y,t) > ;

(FM) M(x,y,t) = if and only ifx=y; (FM) M(x,y,t) =M(y,x,t);

(FM) M(x,y,·) :R+_→_I_{is continuous;}

(FM) M(x,y,t)∗M(y,z,s)≤M(x,z,t+s).

In this case, we also say that (X,M) is anFMSunder∗.

In the sequel, we only consider anFMSsatisfying the following:

(FM) limt→∞M(x,y,t) = for allx,y∈X.

Lemma  (Grabiec []) If (X,M) is an FMS under some t-norm and x,y∈X, then

M(x,y,·)is a non-decreasing function on(,∞).

Let (X,M) be anFMSunder somet-norm. For allt,r> , theopen ballwith centerx∈X

isB(x,r,t) ={y∈X:M(x,y,t) >  –r}. A subsetA⊆Xisopenif, for allx∈X, there exists an open ballB(x,r,t) such thatB(x,r,t)⊆A.

George and Veeramani [] proved that the family of all open sets ofXis a Hausdorﬀ topologyτMonX. In this topology, we may consider the following notions.

Deﬁnition  () A sequence{xm}m≥⊂Xis aCauchy sequenceif, for anyε>  andt> , there existsm∈Nsuch that

M(xm,xm+p,t) >  –ε

for allm≥mandp≥;

() A sequence{xm}m≥⊂Xisconvergentto a pointx∈Xdenoted bylimm→∞xm=xif,

for anyε>  andt> , there existsm∈Nsuch that

M(xm,x,t) >  –ε

for allm≥m;

() AnFMSin which every Cauchy sequence is convergent is said to becomplete.

Lemma (Rodríguez-López and Romaguera []) If(X,M)is an FMS under some t-norm,

then M is a continuous mapping on X×(,∞).

For anyt-norm∗, it is easy to prove that∗ ≤min. Therefore, if (X,M) is anFMSunder

min, then (X,M) is anFMSunder any (continuous or not)t-norm. This is the case in the following examples.

Example  From a metric space (X,d), we can consider an FMS in diﬀerent ways. For all

t>  andx,y∈Xwithx=y, deﬁne

Md(x,y,t) = t

t+d(x,y), M

e₍_x_,_y_,_t_{) = e}–d(x_t,y)_.

(4)

Furthermore, (X,d) is a complete metric space if and only if (X,Md_{) (or (}_X_,_Me_{)) is a} completeFMS. For instance, this is the case for any nonempty closed subset ofRprovided with its Euclidean metric.

The concept of the E.A-property in a metric space has been recently introduced by Aamri and Moutawakil [] and the concept of the CLRg-property by Sintunavarat and

Kumam [] is as follows.

Deﬁnition ([, ]) () Two self-mappingsf andg of a metric space (X,d) are said to satisfy theE.A-propertyif there exists a sequence{xn}inXsuch that

lim

n→∞fxn=nlim→∞gxn=t

for somet∈X;

() Ift∈g(X), thenf andgare said to satisfy theCLRg-property.

Similarly, we say that two self-mappingsf,gof a fuzzy metric space (X,M,∗) satisfy the

E.A-propertyif there exist a sequence{xn}inXandzinXsuch thatfxnandgxnconverge

tozin the sense of Deﬁnition . Similarly, we say that ift∈g(X), thenf andgare said to satisfy theCLRg-property.

3 Main results

Henceforth, ﬁx a partition{A,B}ofn={, , . . . ,n}, that is,A∪B=nandA∩B=∅. We

denote the following:

A,B:=

σ:n→n:σ(A)⊆A,σ(B)⊆B

and

_A_,_B:=σ:n→n:σ(A)⊆B,σ(B)⊆A

.

Ifis a partial order onX(i.e., (X,) is a partially ordered set), then we use the following notation:

xiy ⇐⇒

⎧ ⎨ ⎩

xy ifi∈A,

xy ifi∈B.

Consider on the product spaceXn_{the following partial order:} for all(x,x, . . . ,xn), (y,y, . . . ,yn)∈Xn,

(x,x, . . . ,xn)≤(y,y, . . . ,yn) ⇐⇒ xiiyi

for alli.

Recently, Roldánet al.[] proved the following result.

Theorem ([]) Let{A,B}be a partition ofn={, , . . . ,n}and

A,B:=

(5)

and

_A_,_B:=σ:n→n:σ(A)⊆B,σ(B)⊆A

.

Let(X,M,∗,)be a complete partially ordered FMS such that∗is a t-norm of H-type.

Let= (σ,σ, . . . ,σn)be an n-tuple of mappings from{, , . . . ,n}into itself withσi∈A,B

if i∈A andσi∈A,B if i∈B.Let F:Xn→X and g:X→X be two mappings such that

F has the mixed(g,)-monotone property on X,F(Xn)⊆g(X)and g is continuous and

-compatible with F.Assume that there exists k∈(, )such that

MF(x,x, . . . ,xn),F(y,y, . . . ,yn),kt

≥γ ∗n

i=M(gxi,gyi,t)

()

for all t> and all x, . . . ,xn,y, . . . ,yn∈X with gxi≤igyifor all i,whereγ : [, ]→[, ]

is a continuous mapping such that∗n_γ₍_a₎_≥_{a for each a}_∈_{[, ].}_{Suppose that}

γ ∗n

i=M(gxσj(i),gyσj(i),t)

≥γ ∗n

()

for all j∈ {, , . . . ,n}and x,x, . . . ,xn,y,y, . . . ,yn∈X such that gxiigyifor all i.Suppose

that either F is continuous or(X,τM,)has the sequential g-monotone property.If there

exist x_,x_, . . . ,xn_∈X such that

gxi_iF

xσ_i(),x_σi(), . . . ,xσ_i(n)

for all i,then F and g have at least one-coincidence point in X.

We are going to give a version of the above result using a pair of mappings satisfying theCLRg-property. The following deﬁnitions extend previous considerations from other

authors.

Deﬁnition  Let= (σ,σ, . . . ,σn) be ann-tuple of mappings from{, , . . . ,n}into itself.

The mappingsF:Xn_→_X_and_g_:_X_→_X_{are said to be}_{-weakly compatible}_if

gF(xσi(),xσi(), . . . ,xσi(n)) =F(gxσi(),gxσi(), . . . ,gxσi(n))

whenevergxi=F(xσi(),xσi(), . . . ,xσi(n)) for alliand some (x,x, . . . ,xn)∈Xn.

Deﬁnition  Let (X,M,∗) be an FMS and= (σ,σ, . . . ,σn) be ann-tuple of mappings

from{, , . . . ,n}into itself. Two mappingsF:Xn_→_X_and_g_:_X_→_X_{are said to satisfy}

the-common limitin the range of theg-property (theCLRg-propertyfor short) if there

existnsequences{x_m}m≥,{xm}m≥, . . . ,{xnm}m≥such that

gxi= lim

m→∞gx

i

m=_mlim_→∞F

xσ_mi(),x_mσi(), . . . ,xσ_mi(n)

for alliand somex, . . . ,xn∈X.

Ifσ:n→nis a mapping, then, from its ordered image,i.e.,σ= (σ(),σ(), . . . ,σ(n)),

(6)

() Sintunavarat and Kumam’s election [] inn= ;

() Jain, Tas, Kumar and Gupta’s election [] and Khan and Sumitra’s election [] in

n= isσ=τ= (, )andσ= (, );

() Wairojjana, Sintunavarat and Kumam’s election [] inn= isσ=τ = (, , ),

σ= (, , )andσ= (, , )in abstract metric spaces.

Deﬁnition  Let (X,M,∗) be an FMS and= (σ,σ, . . . ,σn) be an n-tuple of

map-pings from {, , . . . ,n} into itself. Two mappings F :Xn_→_X _and _g_:_X_→_X _{are said}

to satisfy the -E.A-property (shortly, the E.A-property) if there exist n sequences {x

m}m≥,{xm}m≥, . . . ,{xnm}m≥such that

xi= lim

m→∞gx i

m=_mlim_→∞F

xσ_mi(),x_mσi(), . . . ,xσ_mi(n)

Example  Let (R,M,∗) be a fuzzy metric space and∗be a continuoust-norm. Deﬁne

M(x,y,t) =_t₊_|_xt_–_y_|for allx,y∈Randt> . Deﬁne the mappingsF:Rn→Randg:R→R by

F(x, . . . ,xn) = n

i=

(–)i+xi, g(x) =nx

for allx, . . . ,xn,x∈Rand consider the sequences

xi_m=(–)

i+

m

for alli= , . . . ,n. Then we have

Fx_m,x_m, . . . ,xn_m–,xn_m= n

m=g

x_m,

Fx_m,x_m, . . . ,xn_m,x_m=–n

m =g

x_m,

· · ·

Fxn_m,x_m, . . . ,xn_m–,xn_m–=gxn_m.

Therefore, we have

lim

m→∞M

Fx_m,x_m, . . . ,xn_m–,xn_m= lim

m→∞g

x_m=  =g()

and so on. Therefore,Fandgsatisfy theCLRg-property and theE.A-property with=

(σ,σ, . . . ,σn) given by

σ= (, , . . . ,n), σ() = (, , . . . ,n, ), . . . , σn= (n, , . . . ,n– ,n– ).

Note that theE.A-property does not imply theCLRg-property and, in [], there is an

(7)

The following result does not require the conditions on the completeness (or the closed-ness) of the underlying space together with the conditions on continuity and Hadžić’s con-dition oft.

Theorem  Let(X,M,∗)be an FMS and= (σ,σ, . . . ,σn)be an n-tuple of mappings

from{, , . . . ,n}into itself.Let F:Xn_→_{X and g}_:_X_→_{X be two mappings satisfying the}

CLRg-property.Assume that there exists k∈(, )such that

MF(x,x, . . . ,xn),F(y,y, . . . ,yn),kt

≥γ ∗n

()

for all t> ,x, . . . ,xn,y, . . . ,yn∈X,whereγ : [, ]→[, ]is a continuous mapping such

that∗n_γ₍_a₎_≥_{a for each a}_∈_{[, ].}_{Then F and g have at least one}_{-coincidence point}_.

Proof SinceFandgsatisfy theCLRg-property, there existnsequences{xm }m≥,{xm}m≥,

. . . ,{xn

m}m≥such that

gxi= lim

m→∞gx

i

m=_mlim_→∞F

xσi()_m ,xσi()_m , . . . ,xσi(_mn)

for alliand somex, . . . ,xn∈X. Then we have

MFxσ_mi(),x_mσi(), . . . ,xσ_mi(n),F(xσi(),xσi(), . . . ,xσi(n)),kt

≥γ ∗n j=M

gxσ_mi(j),gxσi(j),t

. ()

By lettingm→ ∞, we have

Mgxi,F(xσi(),xσi(), . . . ,xσi(n)),kt

≥γ ∗n

j=M(gxσi(j),gxσi(j),t)

=γ(). ()

Sinceγ veriﬁes ≤ ∗n_γ_()_≤_min₍_γ_{(), . . . ,}_γ_{()) =}_γ_{(), then we have}_γ_{() = . Therefore,}

it follows that

gxi=F(xσi(),xσi(), . . . ,xσi(n))

and so (x,x, . . . ,xn) is a-coincidence point ofFandg. This completes the proof.

Corollary  Under the hypothesis of Theorem,assume also that F and g are weakly compatible.If(x,x, . . . ,xn)∈Xnis a-coincidence point of F and g,then(gx,gx, . . . ,gxn)

is also a-coincidence point of F and g.

Proof Now, we letzi=gxi=F(xσi(),xσi(), . . . ,xσi(n)). SinceFandgare weakly compatible

mappings, we have

gzi=gF(xσi(),xσi(), . . . ,xσi(n))

=F(gxσi(),gxσi(), . . . ,gxσi(n))

=F(zσi(),zσi(), . . . ,zσi(n)).

(8)

Corollary  Let(X,M,∗)be an FMS and= (σ,σ, . . . ,σn)be an n-tuple of mappings

from{, , . . . ,n}into itself.Let F:Xn→X and g:X→X be two mappings satisfying the

E.A-property.Assume that there exists k∈(, )such that inequality()is satisﬁed,where

γ : [, ]→[, ]is a continuous mapping such that∗n_γ₍_a₎_≥_{a for each a}_∈_{[, ].}_{If g}₍_X₎

is a closed subspace of X,then F and g have at least one-coincidence point.

Proof SinceF andgsatisfy theE.A-property, there existnsequences{x_m }m≥,{xm}m≥,

. . . ,{xn_m}m≥such that

xi= lim

m→∞gx i

m=_mlim_→∞F

xσ_mi(),x_mσi(), . . . ,xσ_mi(n)

for alliand somex, . . . ,xn∈X. Sinceg(X) is a closed subspace ofX, it follows thatxi=

g(wi) for somewi∈Xand for alli, and soFandgsatisfy theCLRg-property. Therefore,

by Theorem ,Fandghave at least one-coincidence point.

Remark  Corollary  is a version of Theorem  for theE.A-property. Similarly, we can write a new version by replacing the CLRg-property for theE.A-property and the

closedness ofg(X) for all the result in this paper.

4 The uniqueness of

-coincidence points

Theorem  Under the hypothesis of Corollary,suppose additionally that

γ ∗n

i=M(gxσj(i),gyσj(i),t)

≥γ ∗n

()

for all j∈ {, , . . . ,n}and x,x, . . . ,xn,y,y, . . . ,yn∈X.Then F and g have a unique

-coincidence point(z,z, . . . ,zn)∈Xnsuch that gzi=zifor all i.

Proof From Theorem , the set of-coincidence points ofF andg is nonempty. The

proof is divided in two steps.

Step. We claim that if (x,x, . . . ,xn), (y,y, . . . ,yn)∈Xnare two-coincidence points

ofFandg, then we have

gxi=gyi ()

for alli. Let (x,x, . . . ,xn), (y,y, . . . ,yn)∈Xnbe two-coincidence points ofFandg. For

alltandm, deﬁne

δm(t) = m

∗

j=M

gyj_m,gxj,t

.

Now, we claim thatδm+(kt)≥δm(t) for allmandt> . By inequalities () and (), it

follows that, for allm,tandj,

Mgyj_m_+,gxj,kt

=MFyσmj(),y

σj()

m , . . . ,y

σj(n)

m

,F(xσj(),xσj(), . . . ,xσj(n)),kt

≥γ m∗ i=M

gyσj(mi),gxσj(i),t

≥γ m∗ i=M

gyi_m,gxi,t

=γδm(t)

(9)

Since∗n_γ₍_a₎_≥_a_{for each}_a_∈_{[, ], it follows that}

δm+(kt) =

m

∗

j=M

gyj_m_+,gxj,t

≥ ∗n

j=γ

δm(t)

=∗nγδm(t)

≥δm(t),

i.e.,δm+(kt)≥δm(t)≥ · · · ≥δ(t/km) and, as a consequence,limm→∞δm(t) =  for allt> .

It follows that, for alliandt,

Mgyi_m,gxi,t

= ∗ · · · ∗∗Mgyi_m,gxi,t

∗∗ · · · ∗≥ ∗n

j=M

gyj_m,gxj,t

=δm(t),

which means thatlimm→∞M(gyim,gxi,t) =  for allt,i.e.,

lim

m→∞gy

i

m=gxi ()

for alli. Let (x,x, . . . ,xn)∈Xnbe a-coincidence point ofFandgand deﬁnezi=gxifor

alli. Since (z,z, . . . ,zn) = (gx,gx, . . . ,gxn), from Corollary  it follows that (z,z, . . . ,zn)

is also a-coincidence point ofFandg.

Step. We claim that (z,z, . . . ,zn) is the unique-coincidence point ofFandg such

thatgzi=zifor alli. Indeed, by Step , we observe thatgzi=gxi=zifor alli. Suppose that

(z,z, . . . ,zn)∈Xnis another-coincidence point ofFandg such thatgzi=zifor alli.

Since (z,z, . . . ,zn) and (z,z, . . . ,zn) are two-coincidence points ofFandg, by Step 

it follows thatgzi=gzifor alliand sozi=gzi=gzi=zifor alli. Therefore, (z,z, . . . ,zn) is

the unique-coincidence point ofFandgsuch thatgzi=zifor alli. This completes the

proof.

Corollary  In addition to the hypotheses of Theorem,suppose that∗=min,γ(a) =a

for all a∈Iand(z,z, . . . ,zn)∈Xnis the unique-coincidence point of F and g.Then

z=z=· · ·=zn.In particular,there exists a unique z∈X such that F(z,z, . . . ,z) =z,which

veriﬁes gz=z.

Proof For allt> , deﬁne

λ(t) = ∗n

j,s=M(zj,zs,t) =≤minj,s≤nM(zj,zs,t).

Clearly,λis continuous and non-decreasing onR+_{. Take any}_j_,_s_{∈ {}_{, , . . . ,}_n_}_{. Since}_∗₌

minandγ(a) =afor alla∈I, applying () we have

M(zj,zs,t) =M(gzj,gzs,t)

=MF(zσj(),zσj(), . . . ,zσj(n)),F(zσs(),zσs(), . . . ,zσs(n)),t

≥ min

≤i≤nM(gzσj(i),gzσs(i),t/k)

= min

≤i≤nM(zσj(i),zσs(i),t/k)

for allt> . Therefore, for allt> , we have

λ(t) = min

≤j,s≤nM(zj,zs,t)

≥ min

(10)

≥ min

≤j,s≤nM(zj,zs,t/k)

=λ(t/k).

Repeating this process,λ(t)≥λ(t/k)≥ · · · ≥λ(t/km_{) for all}_m_{. Taking the limit}_m_{→ ∞}_{, we}

deduce thatλ(t) =  for allt> . For anyj,s∈ {, , . . . ,n}, we note thatM(zj,zs,t)≥λ(t) = 

for allt,i.e.,zj=zs. This completes the proof.

5 Some results in metric spaces

It seems natural to introduce the analogous deﬁnitions and results in the setting of metric spaces by using the results in fuzzy metric spaces.

Deﬁnition  Let (X,d) be a metric space and= (σ,σ, . . . ,σn) be ann-tuple of

map-pings from{, , . . . ,n}into itself. Two mappingsF:Xn_→_X_and_g_:_X_→_X_{are said to}

sat-isfy the-common limitin the range of theg-property (shortly,CLRg-property) if there

existnsequences{x_m}m≥,{xm}m≥, . . . ,{xnm}m≥such that

gxi= lim

m→∞gx

i

m=_mlim_→∞F

xσ_mi(),x_mσi(), . . . ,xσ_mi(n)

Remark  Let (X,d) be a metric space,= (σ,σ, . . . ,σn) be ann-tuple of mappings

from{, , . . . ,n}into itself. If we consider the FMS (X,Me,min) induced byd, see Exam-ple , then theCLRg-propertyon (X,d) coincides with theCLRg-propertyon (X,Me,min).

From Theorem , we have the following in metric spaces.

Corollary  Let(X,d)be a metric space and= (σ,σ, . . . ,σn)be an n-tuple of mappings

from{, , . . . ,n}into itself.Let F:Xn→X and g:X→X be two mappings satisfying the

CLRg-property.Assume that there exists k∈[, [such that

dF(x,x, . . . ,xn),F(y,y, . . . ,yn)

≤kmax

≤i≤nd(gxi,gyi) ()

for all i.Then F and g have at least one-coincidence point in X.

Proof It is easy to show that the mappingδ:Xn×Xn→[,∞) deﬁned by

δ(x,x, . . . ,xn), (y,y, . . . ,yn)

=max

≤i≤nd(gxi,gyi)

is a metric onXn_{. The proof follows from Theorem  for}_Me_{induced by}_δ_,_γ _{the identity}

and∗=min.

Ifn= , then we have the following.

Corollary  Let(X,d)be a metric space.Let f :X→X and g:X→X be two mappings

satisfying the CLRg-property.Assume that there exists k∈[, [such that

d(fx,fy)≤kd(gx,gy) ()

(11)

It is well known that in the setting of metric spaces multi-dimensional results only de-pend on their ﬁrst argument. Therefore, multi-dimensional results reduce to the one di-mensional case.

Theorem  Corollariesandare equivalent.

Proof Let= (σ,σ, . . . ,σn),F:Xn→Xandg:X→Xbe two mappings with the

con-ditions of Corollary . It is easy to show that the mappingδ:Xn_×_Xn_→_[,_∞_{) deﬁned}

by

δ(x,x, . . . ,xn), (y,y, . . . ,yn)

=max

≤i≤nd(gxi,gyi)

is a metric onXn_.

Now, deﬁne the mappingsT:Xn_→_Xn_and_G_:_Xn_→_Xn_by

T(x,x, . . . ,xn) =

Fxσ()

 ,x σ()

 , . . . ,x σ(n)



, . . . ,Fxσ_nn(),x_nσn(), . . . ,xσ_nn(n),

G(x,x, . . . ,xn) = (gx,gx, . . . ,gxn),

respectively. It is easy to show that

(a) Fandgsatisfy theCLRg-property if and only ifT andGsatisfy theCLRg-property; (b) (x,x, . . . ,xn)∈Xnis a-coincidence point of the mappingsFandgif and only if

(x,x, . . . ,xn)∈Xnis a coincidence point of the mappingsT andG.

Thus, from () it follows that

δT(x,x, . . . ,xn),T(y,y, . . . ,yn)

≤kδ(x,x, . . . ,xn), (y,y, . . . ,yn)

.

Thus the mappingsT andGsatisfy the conditions of Corollary . This completes the

proof.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, University of Jaén, Jaén, Spain.}2_{Department of Statistics and Operations Research,} University of Granada, Granada, Spain.3_{Department of Mathematics Education and RINS, Gyeongsang National} University, Jinju, Korea. 4_{Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia.}

Acknowledgements

The fourth author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002100).

Received: 24 December 2014 Accepted: 18 March 2015

References

1. Guo, D, Lakshmikantham, V: Coupled ﬁxed points of nonlinear operators with applications. Nonlinear Anal.11, 623-632 (1987)

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

3. Berinde, V: Approximating common ﬁxed points of noncommuting almost contractions in metric spaces. Fixed Point Theory11, 179-188 (2010)

(12)

5. Roldán, A, Martínez-Moreno, J, Roldán, C: Multidimensional ﬁxed point theorems in partially ordered complete metric spaces. J. Math. Anal. Appl.396, 536-545 (2012)

6. Berzig, M, Samet, B: An extension of coupled ﬁxed point’s concept in higher dimension and applications. Comput. Math. Appl.63, 1319-1334 (2012)

7. Rold ´n, A, Martínez-Moreno, J, Roldán, C, Cho, YJ: Multidimensional coincidence point results for compatible mappings in partially ordered fuzzy metric spaces. Fuzzy Sets Syst.251, 71-82 (2014)

8. Aamri, M, El Moutawakil, D: Some new common ﬁxed point theorems under strict contractive conditions. J. Math. Anal. Appl.270, 181-188 (2002)

9. Sintunavarat, W, Kumam, P: Common ﬁxed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces. J. Appl. Math.2011, Article ID 637958 (2011)

10. Mihet, D: Fixed point theorems in fuzzy metric space usingE.A. Nonlinear Anal.73, 2184-2188 (2010) 11. Gnana-Bhaskar, T, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications.

Nonlinear Anal.65, 1379-1393 (2006)

12. Roldán, A, Martínez-Moreno, J, Roldán, C: On interrelationships between fuzzy metric structures. Iran. J. Fuzzy Syst.10, 133-150 (2013)

13. George, A, Veeramani, PV: On some results in fuzzy metric spaces. Fuzzy Sets Syst.64, 395-399 (1994) 14. Grabiec, M: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst.27, 385-389 (1988)

15. Rodríguez-López, J, Romaguera, S: The Hausdorﬀ fuzzy metric on compact sets. Fuzzy Sets Syst.147, 273-283 (2004) 16. Jain, M, Tas, K, Kumar, S, Gupta, N: Coupled ﬁxed point theorems for a pair of weakly compatible maps along with

CLRg-property in fuzzy metric spaces. J. Appl. Math.2012, Article ID 961210 (2012)

17. Khan, MA, Sumitra:CLRgProperty for coupled ﬁxed point theorems in fuzzy metric spaces. Int. J. Appl. Phys. Math.2, 355-358 (2012)