Multidimensional common fixed point theorems under probabilistic φ-contractive conditions in multidimensional Menger probabilistic metric spaces

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R E S E A R C H

Open Access

Multidimensional common ﬁxed point

theorems under probabilistic

ϕ

-contractive

conditions in multidimensional Menger

probabilistic metric spaces

Chuanxi Zhu, Zhe Wei

*

_{, Zhaoqi Wu and Wenqing Xu}

*_{Correspondence:}

[email protected] Department of Mathematics, Nanchang University, Nanchang, 330031, P.R. China

Abstract

In this paper, we introduce the new concepts of multidimensional Menger probabilistic metric spaces and related ﬁxed point for a pair of mappingsT:

X×X× · · · ×X

n

→XandA:X→X. Utilizing the properties of the related triangular

norm and the compatibility ofAwithT, some multidimensional common ﬁxed point problems of hybrid probabilistic contractions with a gauge function

ϕ

are studied. The obtained results generalize some coupled and triple common ﬁxed point theorems in the corresponding literature. Finally, an example is given to illustrate our main results.

Keywords: multidimensional Menger probabilistic metric space; ﬁxed point; hybrid probabilistic contractions; compatible

1 Introduction

Coupled fixed points were studied first by Bhaskar and Lakshmikantham []. Since then, some new results on the existence and uniqueness of coupled fixed points have been pre-sented in partially ordered metric spaces, cone metric spaces, and fuzzy metric spaces [–]. The concept of a probabilistic metric space was initiated and studied by Menger, which is a generalization of the metric space []. Many results for the existence of fixed points or solutions of nonlinear equations under various types of conditions in Menger probabilistic spaces (briefly,PM-spaces) have been extensively considered by many schol-ars [–]. In , Jachymski established a fixed point theorem forϕ-contractions and gave a characterization of a functionϕhaving the property that there exists a probabilistic

ϕ-contraction, which is not a probabilistick-contraction (k∈[, )) []. In , Xiaoet al.obtained some common coupled ﬁxed point results for hybrid probabilistic contrac-tions with a gauge functionϕin Menger probabilistic metric spaces without assuming any continuity or monotonicity conditions forϕ[]. In , Luoet al.introduced the con-cept of generalized Menger probabilistic metric spaces and obtained some tripled com-mon ﬁxed point results with a gauge functionϕwith the same properties in generalized Menger probabilistic metric spaces [].

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The purpose of this paper is to introduce the new concepts of multidimensional Menger probabilistic metric spaces and a related ﬁxed point for a pair of mappings T:

X×X× · · · ×X

n

→X andA:X→X. Utilizing the properties of the related triangular norm and the compatibility of Awith T, some multidimensional common ﬁxed point problems of hybrid probabilistic contractions with a gauge functionϕ are studied. The obtained results generalize some coupled and triple common ﬁxed point theorems in the corresponding literature. Finally, an example is given to illustrate our main results.

2 Preliminaries

Denote bynany given positive integer which is not smaller than ,nthe set{, , . . . ,n},

Xn_{the product}_X_×_X_{× · · · ×}_X

n

,Rthe set of the real numbers,R+ _{the set of the} non-negative real numbers, andZ+_{the set of all positive integers. A mapping}_F_:_R_→_R+_is called a distribution function if it is nondecreasing left-continuous withsup_t_∈_RF(t) =  andinft∈RF(t) = .

We will denote byDthe set of all distribution functions, byD+={F∈D:F(t) = ,∀t≤ }, whileHwill always denote the speciﬁc distribution function deﬁned by

H(t) =

, t≤, , t> .

Ifϕ:R+_→_R+_{is a function such that}_ϕ_{() = , then}_ϕ_{is called a gauge function. If}_t_∈_R+_, thenϕn(t) denotes thenth iteration ofϕ(t) andϕ–({}) ={t∈R+_:_ϕ₍_t_{) = }_}_.

First, we givePM-spaces introduced by Menger with the related triangular norm. Deﬁnition .[] A mapping: [, ]×[, ]→[, ] is called a triangular norm (for short, at-norm) if the following conditions are satisﬁed for anya,b,c,d∈[, ]:

() (a, ) =a; () (a,b) =(b,a);

() (a,c)≥(b,d)fora≥b,c≥d; () (a,(b,c)) =((a,b),c).

Deﬁnition .[] A triplet (X,F,) is called a Menger probabilistic metric space (for short, aMenger PM-space) ifXis a nonempty set,is at-norm, andFis a mapping from

X×XintoD+_{satisfying the following conditions (we denote}_F₍_x_,_y_{) by}_F_x,y):

(MS-) Fx,y(t) =H(t)for allt∈Rif and only ifx=y; (MS-) Fx,y(t) =Fy,x(t)for allt∈R;

(MS-) Fx,y(t+s)≥(Fx,z(t),Fz,y(s))for allx,y,z∈Xandt,s≥.

Then we give the generalized Menger PM-spaces introduced by Luoet al.with the re-lated triangular norm.

Deﬁnition .[] A mapping: [, ]×[, ]×[, ]→[, ] is called a triangular norm (for short, at-norm) if the following conditions are satisﬁed for anya,b,c,d,e,f ∈[, ]:

() (a, , ) =a,(, , ) = ; () (a,b,c) =(a,c,b) =(c,b,a);

() (a,b,c)≥(d,e,f)fora≥d,b≥e,c≥f;

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Deﬁnition .[] A triplet (X,F,) is called a generalized Menger probabilistic metric space (for short, a generalized MengerPM-space) ifXis a nonempty set,is at-norm, andF is a mapping fromX×XintoD+_{satisfying the following conditions (we denote}

F(x,y) byFx,y):

(GPM-) Fx,y(t) =H(t)for allt∈Rif and only ifx=y; (GPM-) Fx,y(t) =Fy,x(t)for allt∈R;

(GPM-) Fx,w(t+t+t)≥(Fx,y(t),Fy,z(t),Fz,w(t))for allx,y,z,w∈Xand

t,t,t≥.

Now, we introduce the deﬁnition of multidimensional Menger probabilistic metric spaces with the related triangular norm.

Deﬁnition . A mapping : [, ] ×[, ]× · · · ×[, ] n

→[, ] is called a triangular norm (for short, at-norm) if the following conditions are satisﬁed for anya,a, . . . ,an,

an+, . . . ,an∈[, ]:

() (a, , . . . , ) =a,(, , . . . , ) = ;

() (a,a, . . . ,an–,an–,an) =(a,an, . . . ,an–,an–) =(a,an,an–, . . . ,an–) =· · ·=

(a,an,an–,an–, . . . ,a) =(an,an–,an–, . . . ,a,a);

() (a,a, . . . ,an)≥(an+,an+, . . . ,an)fora≥an+,a≥an+, . . . ,an≥an;

() ((a,a, . . . ,an),an+, . . . ,an–) =(a,(a, . . . ,an+),an+· · ·,an–) =· · ·=

(a, . . . ,an–,(an,an+, . . . ,an–)).

Two typical examples oft-norm areM(a,a, . . . ,an) =min{a,a, . . . ,an}andP(a,a, . . . ,an) =aa· · ·anfor alla,a, . . . ,an∈[, ].

Deﬁnition . A triplet (X,F,) is called a multidimensional Menger probabilistic met-ric space (for short, a multidimensional MengerPM-space) ifXis a nonempty set,is a

t-norm andF is a mapping fromX×XintoD+_{satisfying the following conditions (we} denoteF(x,y) byFx,y):

(MPM-) Fx,y(t) =H(t)for allt∈Rif and only ifx=y;

(MPM-) Fx,y(t) =Fy,x(t)for allt∈R;

(MPM-) Fx,xn+(t+t+· · ·+tn)≥(Fx,x(t),Fx,x(t), . . . ,Fxn,xn+(tn))for all x,x, . . . ,xn+∈Xandt,t, . . . ,tn≥.

Remark . If n= , the multidimensional MengerPM-space is a Menger PM-space. Whilen= , the multidimensional MengerPM-space is a generalized MengerPM-space. Remark . If=M, the multidimensional MengerPM-space is a MengerPM-space. In fact, letx=x,x=z, . . . ,xn=z,xn+=yin (MPM-), then for anyt,s,δ≥, (n– )δ≤s, we have

Fx,y(t+s)≥min(Fx,z(t),Fz,z(δ), . . . ,Fz,z(δ),Fz,y

s– (n– )δ .

Thus we have

Fx,y(t+s)≥min(Fx,z(t),Fz,y

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Takingδ→, we obtain

Fx,y(t+s)≥min

(Fx,z(t),Fz,y(s)

.

Therefore, if=M, the multidimensional MengerPM-space is a MengerPM-space. Example . Suppose thatX= [–, ]. DeﬁneF :X×X→D+by

Fx,y(t) =Fx,y(t) =

t

t+|x–y|, t> ,

, t≤,

for allx,y∈X. It is easy to verify that (X,F,M) satisﬁes (MPM-) and (MPM-). Now we prove it also satisﬁes (MPM-). Assume thatt,t, . . . ,tn≥ andx,x, . . . ,xn+∈X. Then we have

Fx,xn+(t+· · ·+tn) =

t+· · ·+tn

t+· · ·+tn+|x–xn+|

≥ t+· · ·+tn

t+· · ·+tn+|x–x|+· · ·+|xn–xn+|

≥min

t

t+|x–x|

, . . . , tn

tn+|xn–xn+|

=M

Fx,x(t), . . . ,Fxn,xn+(tn) .

Hence (X,F,M) a multidimensional MengerPM-space.

Proposition . Let(X,F,)be a multidimensional Menger PM-space andbe a con-tinuous t-norm.Then(X,F,)is a Hausdorﬀ topological space in the(,λ)-topologyT,

i.e.,the family of sets

Ux(,λ) :> ,λ∈(, ],x∈X

is a base of neighborhoods of a point x forF,where

Ux(,λ) =y∈X:Fx,y() >  –λ.

Proof It suﬃces to prove that:

(i) for anyx∈X, there exists anU=Ux(,λ)such thatx∈U;

(ii) for any givenUx(,λ)andUx(,λ), there exist> andλ> , such that Ux(,λ)⊂Ux(,λ)∩Ux(,λ);

(iii) for anyy∈Ux(,λ), there exist > andλ > , such thatUy(,λ)⊂Ux(,λ); (iv) for anyx,y∈X,x=y, there existUx(,λ)andUy(,λ), such that

Ux(,λ)∩Uy(,λ) =∅.

It is easy to check that (i)-(iii) are true. Now we prove that (iv) is also true. In fact, suppose thatx,y∈Xandx=y. Then there existt>  and  <a< , such thatFx,y(t) =a. Let

Ux=

r:Fx,r

t

n

>b

, Uy=

r:Fy,r

t

n

>b

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where  <b<  and(b, , . . . ,  n–

,b) >a(sinceis continuous and(, . . . , ) = , suchb

exists). Now suppose that there exists a pointv∈Ux∩Uy, which implies thatFx,v(t_n) >b andFy,v(t_n) >b. Then we have

a=Fx,y(t)≥

Fx,v

t

n

,Fv,v

t

n

, . . . ,Fv,v

t

n

n–

,Fv,y

t

n

≥(b, , . . . ,  n–

,b) >a,

which is a contradiction. Thus the conclusion (iv) is proved. This completes the proof.

Deﬁnition . Let (X,F,) be a multidimensional MengerPM-space,be a continu-oust-norm.

(i) A sequence{xm}inXis said to beT-convergent tox∈Xiflimm→∞Fxm,x= for

allt> ;

(ii) a sequence{xm}inXis said to be aT-Cauchy sequence, if for any given> and

λ∈(, ], there exists a positive integerN=N(,λ), such thatFxm,xk() >  –λ,

wheneverm,k≥N;

(iii) (X,F,)is said to beT-complete, if eachT-Cauchy sequence inXis T-convergent to some point inX.

Deﬁnition . At-normis said to beH-type if the family of functions{m(t)}∞_m=is equi-continuous att= , where

(t) =(t, . . . ,t), m+(t) =t, . . . ,t

n–

,m(t), m= , , . . . ,t∈[, ].

Deﬁnition . LetXbe a nonempty set,T:Xn_→_X_and_A_:_X_→_X_{be two mappings.}

Ais said to be commutative withT, ifAT(x, . . . ,xn) =T(Ax, . . . ,Axn) for allx, . . .xn∈X. A pointu∈Xis called a multidimensional common ﬁxed point ofT andA, ifu=Au=

T(u, . . . ,u).

Deﬁnition . LetXbe a nonempty set,T:Xn_→_X_and_A_:_X_→_X_{be two mappings.} Let{x

m}, . . . ,{xnm}be n sequences inXandσ, . . . ,σnbenpermutations ofn.AandTare said to be compatible in (X,F,) if

lim

m→∞FAT(xσmi(),...,xσmi(n)),T(Axσmi(),...,Axσmi(n))(t) = 

for alli= , . . . ,nandt> , whenever

lim

m→∞T

xσi()

m , . . . ,x σi(n)

m =_mlim_→∞Axim∈X for alli= , . . . ,n;

AandT are said to be compatible in (X,d) where (X,d) is a usual metric space if

lim

m→∞d

ATxσi()

m , . . . ,x σi(n)

m ,T

Axσi()

m , . . . ,Ax σi(n)

m

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for alli= , . . . ,nandt> , whenever

lim

m→∞T

xσi()

m , . . . ,x σi(n)

m =_mlim_→∞Axim∈X for alli= , . . . ,n.

Obviously, ifTandAare commutative, then they are compatible, but the converse does not hold.

The following lemmas play an important role in proving our main results in Section . Lemma .[] Suppose that F∈D+.For every m∈Z+,let Fm:R→[, ]be

nondecreas-ing and gm: (, +∞)→(, +∞)satisfylimm→∞gm(t) = for any t> .If Fm(gm(t))≥F(t)

for any t> ,thenlimm→∞Fm(t) = for any t> .

Lemma . Let X be a nonempty set,and T:Xn_→_{X and A}_:_X_→_{X be two mappings}_.

If T(Xn₎_⊂_A₍_X_),_{then there exist n sequences}_{_x

m}∞m=, . . . ,{xnm}∞m=in X,such that Axm+=

T(x

m,xm, . . . ,xnm),Axm+ =T(xm,xm, . . . ,xnm,xm), . . . ,Axnm+=T(xnm,xm, . . . ,xn–m ).

Proof Let x_,x_, . . . ,xn_ be any given points in X. SinceT(Xn)⊂A(X), we can choose

x

,x, . . . ,xn ∈ X such that Ax = T(x,x, . . . ,xn),Ax = T(x,x, . . . ,xn,x), . . . ,Axn =

T(xn

,x, . . . ,xn– ). Continuing this process, we can constructn sequences {xm}∞m=, . . . ,

{xn

m}∞m=inX, such that

Ax_m+=Tx_m,x_m, . . . ,xn_m , Ax_m+ =Tx_m,x_m, . . . ,xn_m,x_m , . . . ,

Axn_m+=Txn_m,x_m, . . . ,xn–_m . Lemma .[] Let(X,d)is a usual metric space.DeﬁneF :X×X→D+_by

Fx,y=H

t–d(x,y) , for x,y∈X and t> .

Then (X,F,M)is a Menger PM-space and is called the induced Menger PM-space by

(X,d).It is complete if(X,d)is complete.

Lemma .[] Letϕ(t) :R+_→_R+_{be a function}_._{Let a}_,_b_,_t_∈_R+_._{Then we have}

H(t–a)≥Hϕ(t) –b if and only if ϕ(b)≤a.

3 Main results

In this section, we shall give the main results of this paper.

Theorem . Let(X,F,)be a complete multidimensional Menger PM-space witha continuous related t-norm of H-type,ϕ:R+_→_R+_{be a gauge function such that}_ϕ–₍_{__}_{) =}

{},ϕ(t) <t,andlim_m_→₊_∞ϕm(t) = for any t> .Let T:Xn_→_{X and A}_:_X_→_{X be two}

mappings satisfying the following conditions:

FT(x,x,...,xn),T(y,y,...,yn)

ϕ(t) ≥FAx,Ay(t)FAx,Ay(t)· · ·FAxn,Ayn(t)



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for all x,x, . . . ,xn,y,y, . . . ,yn∈X,and t> ,where T(Xn)⊂A(X),A is continuous and

compatible with T.Then T and A have a unique multidimensional common ﬁxed point in X.

Proof By Lemma ., we can constructnsequences{x_m}∞_m=, . . . ,{xn_m}∞_m=inX, such that

Ax

m+=T(xm,xm, . . . ,xnm),Axm+=T(xm,xm, . . . ,xnm,xm), . . . ,Axnm+=T(xnm,xm, . . . ,xn–m ). From (.), for allt> , we have

F_Ax m,Axm+

ϕ(t) =F_T(x

m–,xm–,...,xnm–),T(xm,xm,...,xnm)

ϕ(t)

≥F_Ax

m–,Axm(t)FAxm–,Axm(t)· · ·FAxnm–,Axnm(t)

 n_, F_Ax

m,Axm+

ϕ(t) =F_T(x

m–,xm–,...,xm–),T(xm,xm,...,xm)

ϕ(t)

≥F_Ax

m–,Axm(t)FAxm–,Axm(t)· · ·FAxm–,Axm(t)

 n_,

.. .

FAxnm,Axnm+

ϕ(t) =F_T(xn

m–,xm–,...,xnm––),T(xnm,xm,...,xnm–)

ϕ(t)

≥FAxn_m_–,Axmn(t)FAxm–,Axm(t)· · ·FAxmn––,Axnm–(t)

 n_.

(.)

DenotePm(t) = [F_Ax

m–,Axm(t)FAxm–,Axm(t)· · ·FAxnm–,Axnm(t)] 

n_{. From (.), we have}

Pm+

ϕ(t) =F_Ax m,Axm+

ϕ(t) F_Ax m,Axm+

ϕ(t) · · ·FAxn m,Axnm+

ϕ(t)  n

≥Pm(t)Pm(t)· · ·Pm(t) n

 n₌_P

m(t),

which implies that

ϕm(t) ≥Pm

ϕm–(t) ≥ · · ·P(t),

F_Ax m,Axm+

ϕm(t) ≥Pm

ϕm–(t) ≥ · · ·P(t), ..

.

F_Ax m,Axm+

ϕm(t) ≥Pm

ϕm–(t) ≥ · · ·P(t).

(.)

SinceP(t) = [F_Ax

,Ax(t)FAx,Ax(t)· · ·FAx n ,Axn(t)]



n ∈D+_and_lim_m_→∞_ϕm₍_t_{) =  for each} t> , using Lemma ., we have

lim

m→∞FAxm,Axm+(t) = , FAxm,Axm+(t) = , . . . , FAx n

m,Axnm+(t) = . (.) Thus

lim

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We claim that, for anyk∈Z+_and_t_{> ,}

F_Ax

m,Axm+k(t)≥

k

Pm

t–ϕ(t)

n– 

,

F_Ax

m,Axm+k(t)≥

k

Pm

t–ϕ(t)

n– 

, ..

.

FAxn

m,Axnm+k(t)≥

k

Pm

t–ϕ(t)

n– 

.

(.)

In fact, by (.) and ϕ(t) < t, we can conclude that (.) holds for k =  since

F_Ax

m,Axm+(t)≥FAxm,Axm+(ϕ(t))≥Pm(t)≥Pm( t–ϕ(t)

n– )≥ ₍_P

m(t–_n–ϕ(t))). Assume that (.) holds for somek. Sinceϕ(t) <t, by the ﬁrst inequality of (.), we haveF_Ax

m,Axm+(t)≥ F_Ax

m,Axm+(ϕ(t))≥Pm(t). By (.) and (.), we have F_Ax

m+,Axm+k+

ϕ(t) ≥F_Ax

m,Axm+k(t)FAxm,Axm+k(t)· · ·FAx n

m,Axnm+k(t)

 n

≥k

Pm

t–ϕ(t)

n– 

.

Hence, by the monotonicity of, we have

F_Ax

m,Axm+k+(t) =FAxm,Axm+k+

t–ϕ(t) +ϕ(t)

≥

t–ϕ(t)

n– 

, . . . ,F_Ax m,Axm+

t–ϕ(t)

n– 

,

F_Ax

m+,Axm+k+

ϕ(t)

≥

Pm

t–ϕ(t)

n– 

, . . . ,Pm

t–ϕ(t)

n– 

,k

Pm

t–ϕ(t)

n– 

=k+

Pm

t–ϕ(t)

n– 

.

Similarly, we haveF_Ax

m,Axm+k+(t)≥ k+₍_P

m(t–_n–ϕ(t))), . . . ,FAxnm,Axnm+k+(t)≥

k+₍_P_m(t–ϕ(t) n– )). Therefore, by induction, (.) holds for allk∈Z+_and_t_{> .}

Suppose thatλ∈(, ] is given. Sinceis at-norm ofH-type, there existsδ>  such that

k(s) >  –λ, s∈( –δ, ],k∈Z+. (.)

By (.), there existsM∈Z+_{, such that}_P_m(t–ϕ(t)

n– ) >  –δfor allm≥M. Hence, from (.) and (.), we getF_Ax

m,Axm+k(t) >  –λ,FAxm,Axm+k(t) >  –λ, . . . ,FAx n

m,Axnm+k(t) >  –λfor all m≥M,k∈Z+_{. Therefore}_{_Ax

m},{Axm}, . . . ,{Axnm}arenCauchy sequences. Since (X,F,) is complete, there existu,u, . . . ,un∈X, such that

lim

m→∞Ax 

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By the continuity ofA, we have

lim

m→∞AAx



m=Au, _mlim_→∞AAxm=Au, . . . , _mlim_→∞AAxnm=Aun. The compatibility ofAwithTimplies that

lim

m→∞FAT(xm,xm,...,xnm),T(Axm,Axm,...,Axnm)(t) = , . . . ,

lim

m→∞FAT(xnm,xm,...,xnm–),T(Axnm,Axm,...,Axnm–)(t) = ,

whereσ= (, , . . . ,n),σ= (, , . . . , ), . . . ,σn= (n, , . . . ,n– ). From (.) andϕ(t) <t, we obtain

F_AAx

m+,T(u,u,...,un)(t) =FAAxm+,T(u,u,...,un)

t–ϕ(t) +ϕ(t)

≥

F_AAx

m+,T(Axm,Axm,...,Axnm)

t–ϕ(t)

n– 

,

F_T(Ax

m,Axm,...,Axnm),T(Axm,Axm,...,Axnm)

t–ϕ(t)

n– 

, . . . ,

F_T(Ax

m,Axm,...,Axnm),T(Axm,Axm,...,Axnm)

t–ϕ(t)

n– 

,

F_T(Ax

m,Axm,...,Axnm),T(u,u,...,un)

ϕ(t)

=

F_AAx

m+,T(Axm,Axm,...,Axnm)

t–ϕ(t)

n– 

, , . . . , ,

F_T(Ax

ϕ(t) . (.)

From (.), we have

F_T(Ax

ϕ(t) ≥F_AAx

m,Au(t)FAAxm,Au(t)· · ·FAAxnm,Aun(t)

 n_{. (.)}

Combining (.) with (.) and letting m→ ∞, we obtain limm→∞AAxm =T(u,u, . . . ,un_{). Hence} _T₍_u_,_u_{, . . . ,}_un_{) =}_Au_{. Similarly, we can show that} _T₍_u_,_u_{, . . . ,}_u_{) =}

Au,T(u,u, . . . ,u) =Au, . . . ,T(un,u, . . . ,un–) =Aun.

Next we show thatAu₌_u_,_Au₌_u_{, . . . ,}_Aun₌_un_{. In fact, from (.), for all}_t_{> , we} have

F_Au_,Ax m

ϕ(t) =F_T(u_,u_,...,un_),T(x

m–,xm–,...,xnm–)

ϕ(t)

≥F_Au_,Ax

m–(t),FAu,Axm–(t), . . . ,FAun,Ax n m–(t)

 n_, F_Au_,Ax

m

ϕ(t) =F_T(u_,u_,...,u_),T(x

m–,xm–,...,xm–)

ϕ(t)

≥F_Au_,Ax

m–(t),FAu,Axm–(t), . . . ,FAu,Axm–(t) 

n_,

..

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FAun_,Axn m

ϕ(t) =F_T(un_,u_,...,un–_),T(xn

m–,xm–,...,xnm––)

ϕ(t)

≥FAun_,Axn

m–(t),FAu,Axm–(t), . . . ,FAun–,Axnm––(t) 

n_.

DenoteQm(t) = [FAu_,Ax

m(t),FAu,Axm(t), . . . ,FAun,Axnm(t)] 

n. By (.), we haveQ_m(_ϕ(t))≥ Qm–(t), and hence for allt> 

Qm

ϕm(t) ≥Qm–

ϕm–(t) ≥ · · · ≥Q(t).

Thus, for allt> , we have

F_Au_,A m

ϕm(t) ≥Q(t), F_Au_,A m

ϕm(t) ≥Q(t), . . . ,

FAun_,An m

ϕm(t) ≥Q(t).

SinceQ(t)∈D+andlimm→∞(ϕm(t)) =  for allt> , by Lemma ., we conclude that

lim

m→∞Ax



m=Au, _mlim_→∞Axm=Au, . . . , _mlim_→∞Axnm=Aun. (.) This shows that Au ₌_u_,_Au ₌_u_{, . . . ,}_Aun₌_un_{. Hence} _u ₌_T₍_u_,_u_{, . . . ,}_un_),_u ₌

T(u,u, . . . ,u), . . . ,un=T(un,u, . . . ,un–). Finally, we prove thatu=u=· · ·=un.

F_u_,u

ϕ(t) =F_T(u_,u_,...,un–_,un_),T(u_,u_,...,un_,u₎

ϕ(t)

≥FAu_,Au(t),F_Au_,Au(t), . . . ,F_Aun–_,Aun(t),F_Aun_,Au(t) 

n

=Fu_,u(t),F_u_,u(t), . . . ,F_un–_,un(t),F_un_,u(t) 

n_, Fu_,u

ϕ(t) =FT(u_,u_,...,un_,u_),T(u_,u_,...,u_,u₎

ϕ(t)

≥FAu_,Au(t),F_Au_,Au(), . . . ,F_Aun_,Au(t),F_Au_,Au(t)  n

=Fu_,u(t),F_u_,u(t), . . . ,F_un–_,un(t),F_un_,u(t)  n_,

.. .

Fun_,u

ϕ(t) =FT(un_,u_,...,un–_,un–_),T(u_,u_,...,un–_,un₎

ϕ(t)

≥FAun_,Au(t),F_Au_,Au(t), . . . ,F_Aun–_,Aun–(t),F_Aun–_,Aun(t)  n

=Fu_,u(t),F_u_,u(t), . . . ,F_un–_,un(t),F_un_,u(t)  n_.

(.)

DenoteR(t) = [Fu_,u(t),F_u_,u(t), . . . ,F_un–_,un(t),F_un_,u(t)] 

n_{. From (.), we have} Rϕm(t) ≥Rϕm–(t) ≥ · · · ≥R(t).

SinceR(t)∈D+_{, by Lemma ., we get}_u₌_u₌_{· · ·}₌_un_{. Hence, there exists}_u_∈_X_{, such} thatu=Au=T(u, . . . ,u).

Finally, we show the uniqueness of the multidimensional common ﬁxed point of T

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i.e.,v=Av=T(v, . . . ,v). By (.), for allt> , we have

Fu,v

ϕ(t) =FT(u,u,...,u),T(v,v,...,v)

ϕ(t)

≥FAu,Av(t)FAu,Av(t)· · ·FAu,Av(t)  n

=FAu,Av(t) =Fu,v(t), (.)

which implies thatFu,v(ϕm(t))≥Fu,v(t) for allt> . Using Lemma ., we haveFu,v(t) =  for allt> ,i.e.,u=v. This completes the proof.

Remark . Ifn= , Theorem . generalizes Theorem . in []. Whilen= , Theo-rem . generalizes TheoTheo-rem . in [].

From Theorem ., we can obtain the following corollaries.

Corollary . Let(X,F,)be a complete multidimensional Menger PM-space witha continuous related t-norm of H-type,ϕ:R+→R+be a gauge function such thatϕ–({}) =

{},ϕ(t) <t,andlim_m_→∞ϕm(t) = for any t> .Let T:Xn_→_{X and A}_:_X_→_{X be two}

FT(x,x,...,xn),T(y,y,...,yn)

ϕ(t) ≥FAx,Ay(t)FAx,Ay(t)· · ·FAxn,Ayn(t)



n _(.)

commutative with T.Then T and A have a unique multidimensional common ﬁxed point in X.

Ifϕ:R+→R+be a gauge function such thatlimm→∞ ∞

m=ϕm(t) <∞for anyt> , we can obtainlim_m_→∞ϕm(t) = . Hence we have Corollary . as follows.

Corollary . Let(X,F,)be a complete multidimensional Menger PM-space witha continuous related t-norm of H-type,and≥P,ϕ:R+_→_R+_{be a gauge function such}

thatϕ–({}) ={},ϕ(t) <t,andlim_m_→∞∞_m=ϕm(t) <∞for any t> .Let T:Xn_→_{X and}

A:X→X be two mappings satisfying the following conditions:

FT(x,x,...,xn),T(y,y,...,yn)

ϕ(t) ≥FAx,Ay(t),FAx,Ay(t), . . . ,FAxn,Ayn(t) 

n _(.)

LetA=I(Iis the identity mapping) in Corollary ., we can obtain the following corol-lary.

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thatϕ–₍_{__}_{) =}_{__}_,_ϕ₍_t_{) <}_t_,_and_lim

m→∞∞m=ϕm(t) <∞for any t> .Let T:Xn→X be

a mapping satisfying the following conditions:

FT(x,x,...,xn),T(y,y,...,yn)

ϕ(t) ≥Fx,y(t),Fx,y(t), . . . ,Fxn,yn(t) 

n _(.)

for all x,x, . . . ,xn,y,y, . . . ,yn∈X,and t> .Then T has a unique multidimensional ﬁxed

point in X.

Lettingϕ(t) =αt( <α< ) in Corollary ., we can obtain the following corollary. Corollary . Let(X,F,)be a complete multidimensional Menger PM-space witha continuous related t-norm of H-type,and≥P.Let T:Xn_→_{X and A}_:_X_→_{X be two}

FT(x,x,...,xn),T(y,y,...,yn)(αt)≥

FAx,Ay(t),FAx,Ay(t), . . . ,FAxn,Ayn(t) 

n _(.)

From the proof of Theorem ., we can similarly prove the following result.

Theorem . Let(X,F,)be a complete multidimensional Menger PM-space witha continuous related t-norm of H-type,ϕ:R+_→_R+_{be a gauge function such that}_ϕ–₍_{__}_{) =}

{},ϕ(t) >t,andlimm→∞ϕm(t) = +∞for any t> .Let T:Xn_→_{X and A}_:_X_→_{X be two}

FT(x,x,...,xn),T(y,y,...,yn)(t)≥min

FAx,Ay

ϕ(t) ,FAx,Ay

ϕ(t) , . . . ,FAxn,Ayn

ϕ(t) (.)

for all x,x, . . . ,xn,y,y, . . . ,yn∈X,and t> ,where T(Xn)⊂A(X)and A is continuous

and compatible with T.Then T and A have a unique multidimensional common ﬁxed point in X.

Remark . Ifn= , Theorem . generalizes Theorem . in []. Whilen= , Theo-rem . generalizes TheoTheo-rem . in [].

LettingA=I(Iis the identity mapping) in Theorem ., we can obtain the following corollary.

Corollary . Let(X,F,)be a complete multidimensional Menger PM-space witha continuous related t-norm of H-type,ϕ:R+_→_R+_{be a gauge function such that}_ϕ–₍_{__}_{) =}

{},ϕ(t) >t,andlim_m_→∞ϕm(t) =∞for any t> .Let T:Xn_→_{X be a mapping satisfying}

the following conditions:

FT(x,x,...,xn),T(y,y,...,yn)(t)≥min

Fx,y

ϕ(t),Fx,y

ϕ(t), . . . ,Fxn,yn

ϕ(t) (.)

for all x,x, . . . ,xn,y,y, . . . ,yn∈X,and t> .Then T and A have a unique

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Theorem . Let(X,d)be a complete metric space,ϕ:R+_→_R+_{be a gauge function such}

thatϕ–({}) ={},ϕ(t) >t,andlim_m_→∞ϕm(t) = +∞for any t> .Let T:Xn_→_{X and A}_:

X→X be two mappings satisfying the following conditions:

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

≤maxd(Ax,Ay),d(Ax,Ay), . . . ,d(Axn,Ayn)

(.)

compatible with T.Then T and A have a unique multidimensional common ﬁxed point in X.

Proof Take =M andFx,y(t) =H(t–d(x,y)). Then by Lemma . and Remark ., (X,F,M) is a complete multidimensional MengerPM-space (or a MengerPM-space). From Lemma . and (.), we have

FT(x,x,...,xn),T(y,y,...,yn)(t) =H(t–d

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

≥Hϕ(t) –maxd(Ax,Ay),d(Ax,Ay), . . . ,d(Axn,Ayn) =minHϕ(t) –d(Ax,Ay), . . . ,H

ϕ(t) –d(Axn,Ayn) =minFAx,Ay

ϕ(t) , . . . ,FAxn,Ayn

ϕ(t) . (.)

Hence the conclusion follows from Theorem ..

4 An application

In this section, we will provide an example to exemplify the validity of the main result of this paper.

Example . Suppose thatX∈[–, ]⊂R,=M. ThenMis at-norm ofH-type and

M≥P. DeﬁneF:X×X→Dby

Fx,y(t) =Fx,y(t) = ⎧ ⎨ ⎩

e–|x–_ty|_, _t_{> ,}_x_,_y_∈_X_, , t≤,x,y∈X.

We claim that (X,F,M) is a multidimensional MengerPM-space. In fact, it is easy to verify (MPM-) and (MPM-). Assume that for anyt,t, . . . ,tn> , andx,x, . . . ,xn+∈X,

M

Fx,x(t),Fx,x(t), . . . ,Fxn,xn+(tn) =min

e– |x_–x_|

t _,_e– |x_–x_|

t _,_e– |xn–_xn_+|

tn ₌_e– |x_–x_|

t _.

Then we havet|x–x| ≤t|x–x|,t|x–x| ≤t|x–x|, . . . ,t|xn–xn+| ≤tn|x–x|, and so t+t+···+tn

t |x–x| ≥ |x–x|+|x–x|+· · ·+|xn–xn+| ≥ |x–xn+|. It follows that

Fx,xn+(t+t+· · ·+tn) =e

– |x–xn+| t+t+···+tn ≥_e–|

x–x| t

=M

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Hence (MPM-) holds. It is obvious that (X,F,M) is complete. Suppose thatϕ(t) =_nt, then it is easy to verify thatϕ–₍_{__}_{) =}_{__}_,_ϕ₍_t_{) <}_t_{, and}_lim

m→∞∞m=ϕm(t) <∞for any

t> . Forx,x, . . . ,xn∈X, deﬁneT:Xn→Xas follows:

T(x,x, . . . ,xn) = 

n–

x 

n–

x 

n–· · ·–

x n–

n –

|xn|

n . Then, for eacht>  andx,x, . . . ,xn,y,y, . . . ,yn∈X, we have

x_–y_ +· · ·+x_n– –y_n– +n|xn|–|yn|

≤ |x–y|

|x|+|y| +· · ·+|xn––yn–|

|xn–|+|yn–| +n

|xn|–|yn|

≤nmax|x–y|, . . . ,|xn–yn|

,

and so

FT(x,x,...,xn–,xn),T(y,y,...,yn–,yn)

ϕ(t) =FT(x,x,...,xn–,xn),T(y,y,...,yn–,yn)

t n

=e–

|(x_{ –}y_{ )+}···+(x_n_––y_n_–)+n(|xn|–yn)| nt

≥mine–|x–nty|_,_e– |x_–y_|

nt _{, . . . ,}_e– |xn–yn|

nt

=M

Fx,y(t),Fx,y(t), . . . ,Fxn,yn(t)  n_.

Thus, all conditions of Corollary . are satisﬁed. Therefore,T has a unique ﬁxed point inX.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally. All authors read and approved the ﬁnal manuscript.

Acknowledgements

The authors would like to thank the editor and the referees for their constructive comments and suggestions. The research was supported by the National Natural Science Foundation of China (11361042, 11326099, 11461045, 11071108) and the Provincial Natural Science Foundation of Jiangxi, China (20132BAB201001, 20142BAB211016, 2010GZS0147). Received: 5 May 2015 Accepted: 7 October 2015

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