R E S E A R C H
Open Access
New multipled common fixed point
theorems in Menger PM-spaces
Xiaohuan Mu, Chuanxi Zhu
*and Zhaoqi Wu
*Correspondence:
[email protected] Department of Mathematics, Nanchang University, Nanchang, 330031, P.R. China
Abstract
In this work, we introduce a new
ϕ
-contractive mapping; following that, we obtain some multipled common fixed point theorems for a pair of mappingsT:X×X× · · · × X
m-times
→XandA:X→X. The main results of this paper are
generalization of the main results of Kutbiet al.(Fixed Point Theory Appl. 2015(1):32, 2015). As an illustration, we give an example to demonstrate the validity of the obtained results.
MSC: Primary 47H10; secondary 46S10
Keywords: multipled common fixed point; Menger PM-spaces;
ϕ
-contractive mapping1 Introduction
In , Menger [] initiated the study of probabilistic metric spaces. Since then, many scholars have studied the existence of fixed points or solutions of nonlinear equations under various types of conditions in Menger spaces (see [–]). Precisely, Sehgal and Bharucha-Reid [] introduced probabilistic q-contractions and proved corresponding unique fixed point results by giving a generalization of the classical Banach fixed point principle. Then, we point out an important theoretical development in the way of defin-ing the concept of contractive mappdefin-ing in Menger spaces. In , Khanet al.[] intro-duced the concept of altering distance function. Choudhury and Das [] defined a gen-eralized contractive condition with the help of such functions and established an unique fixed point result. In , Jachymski [] established a fixed point theorem for probabilis-ticϕ-contractions. Duttaet al.[] defined nonlinear generalized contractive type map-pings in Menger PM-spaces and proved their theorems under the mapping inG-complete Menger PM-spaces. Recently, [–] have studied some new fixed point theorems in Menger PM-spaces and fuzzy metric spaces.
Coupled and tripled fixed point results were studied in [–]. In this paper, from the idea ofψ-contractive type mappings in [], we introduce a newϕ-contractive mapping. Following this, we obtain some multipled common fixed point theorems for a pair of map-pingsT :X ×X× · · · × X
m-times
→XandA:X→X, which is a generalization of []. As an
illustration, we give an example to demonstrate the validity of the obtained results.
2 Preliminaries
LetRdenote the set of reals,R+the nonnegative reals andZ+be the set of all positive integers. A mappingF:R→R+ is called a distribution function if it is nondecreasing and left-continuous withinft∈RF(t) = andsupt∈RF(t) = . We will denote byDthe set of all distribution functions, whileHwill always denote the specific distribution function defined by
H(t) =
, t≤, , t> .
A mapping: [, ]×[, ]→[, ] is called a triangular norm (for short, at-norm) if the following conditions are satisfied: (a, ) =a;(a,b) =(b,a); a≥b, c≥d ⇒
(a,c)≥(b,d);(a,(b,c)) =((a,b),c).
Three examples oft-norm arem(a,b) =min{a,b},p(a,b) =abandL=max{a+b– , }, theset-norms are related in the following way:L≤p≤m.
Definition .[] A Menger PM-space is a triple (X,F,) whereXis a nonempty set,
is a continuoust-norm andFis a mapping fromX×XintoD+such that, ifF
x,ydenotes the value ofFat the pair (x,y), the following conditions hold:
(PM-) Fx,y(t) =H(t)if and only ifx=y,t> ;
(PM-) Fx,y=Fy,xfor allx,y∈X;
(PM-) Fx,y(t+s)≥(Fx,z(t),Fz,y(s))for allx,y,z∈Xands,t≥. Definition .[] Let (X,F,) be a Menger PM-space. Then
(i) a sequence{xn}is said to be convergent tox∈Xif for every> andλ> , there
exists a positive integerZ+such thatFxn,x() > –λwhenevern≥Z+;
(ii) a sequence{xn}inXis called aCauchysequence if for every> andλ> , there
exists a positive integerZ+such thatF
xn,xm() > –λwhenevern,m≥Z+; (iii) a Menger PM-space is said to beM-complete if everyCauchysequence inXis
convergent to a point inX;
(iv) a sequence{xn}is said to be aG-Cauchysequence iflimn→∞Fxn,xn+m(t) = for each
m∈Z+andt> ;
(v) the space(X,F,)is calledG-complete if everyG-Cauchysequence inXis convergent to a point inX.
According to [], the (,λ)-topology in a Menger PM-space (X,F,) is introduced by the family of neighborhoodsNxof a pointx∈Xgiven byNx={Nx(,λ) :> ,λ∈(, )}, whereNx(,λ) ={y∈X:Fx,y() > –λ}. Then (,λ)-topology is a Hausdorff topology. Definition . [] A functionφ:R+→R+ is said to be aφ-function if it satisfies the following conditions:
(i) φ(t) = if and only ift= ;
(ii) φ(t)is strictly increasing andφ(t)→ ∞ast→ ∞; (iii) φis left-continuous in(,∞);
(iv) φis continuous at .
Definition . LetXbe a nonempty set. LetT:X ×X× · · · × X
m-times
→XandA:X→Xbe
allx,y, . . . ,z∈X. A pointu∈Xis called a multipled common fixed point ofT andAif
u=Au=T(u,u, . . . ,u).
3 Main results
In this section, we denote bythe class of all nondecreasing functionsϕ:R+→R+such thatϕis continuous at ,ϕ() = andϕn(a
n)→ wheneveran→ asn→ ∞.
Theorem . Let(X,F,)be a G-complete Menger space witha continuous t-norm.
Let T:X ×X× · · · × X
m-times
→X and A:X→X be two mappings satisfying the following
in-equality:
FT(x,y,...,z),T(p,q,...,r)(φ(ct)) –
≤ϕ
(
FAx,Ap(φ(t))– ) + (
FAy,Aq(φ(t))– ) +· · ·+ (
FAz,Ar(φ(t))– )
m
(.)
for all x,y, . . . ,z,p,q, . . . ,r∈ X, c∈ (, ), ϕ ∈ , φ is a φ-function, t > , such that FAx,Ap(φ(t)) > ,FAy,Ap(φ(t)) > , . . . ,FAz,Ar(φ(t)) > ,where T(X×X× · · · ×X)⊂A(X),A is
continuous and commutative with T.Then there exists a unique multipled common fixed point of A and T,i.e.,there exists a unique u∈X such that u=Au=T(u,u, . . . ,u).
Proof Let{xn}∞n=,{yn}∞n=, . . . ,{zn}∞n=bem-times sequences inXsuch thatAxn+=T(xn,yn, . . . ,zn) and Ayn+ = T(yn, . . . ,zn,xn), Azn+ =T(zn,xn,yn, . . .). From supt∈RFAx,Ax(t) =
,supt∈RFAy,Ay(t) = , . . . ,supt∈RFAz,Az(t) = and the definition ofφ, one can findt>
such thatFAx,Ax(φ(
t
c)) > ,FAy,Ay(φ(
t
c)) > , . . . ,FAz,Az(φ(
t
c)) > . From (.), we have
FAx,Ax(φ(t))
–
=
FT(x,y,...,z),T(x,y,...,z)(φ(t))
–
≤ϕ
( FAx,Ax(φ(tc))
– ) + ( FAy,Ay(φ(tc))
– ) +· · ·+ ( FAz,Az(φ(tc))
– )
m
. (.)
Similarly, we have
FAy,Ay(φ(t))
–
≤ϕ
( FAy,Ay(φ(
t
c))– ) +· · ·+ ( FAz,Az(φ(
t
c))– ) + ( FAx,Ax(φ(
t c))– )
m
, (.)
. . . ,
FAz,Az(φ(t))
–
≤ϕ
( FAz,Az(φ(tc))
– ) + ( FAx,Ax(φ(tc))
– ) + ( FAy,Ay(φ(tc))
– ) +· · ·
m
Suppose that P(t) = (FAx
,Ax(φ(t))–)+(
FAy,Ay(φ(t))–)+···+( FAz,Az(φ(t))–)
m , from (.), (.) and
(.) we deduce that FAx,Ax(φ(t)) > ,FAy,Ay(φ(t)) > , . . . ,FAz,Az(φ(t)) > , and so FAx,Ax(φ(
t
c)) > ,FAy,Ay(φ(
t
c)) > , . . . ,FAz,Az(φ(
t
c)) > , then we have
FAx,Ax(φ(t))
–
=
FT(x,y,...,z),T(x,y,...,z)(φ(t))
–
≤ϕ
( FAx,Ax(φ(tc))
– ) + ( FAy,Ay(φ(ct))
– ) +· · ·+ ( FAz,Az(φ(tc))
– )
m
≤ϕ
ϕ(P
(ct)) +ϕ(P(ct)) +· · ·+ϕ(P(ct)) m
=ϕ
P
t c
.
Similarly, we have
FAy,Ay(φ(t))
– ≤ϕ
P
t c
,
. . . ,
FAz,Az(φ(t))
– ≤ϕ
P
t c
.
Repeating the above procedure, we get
FAxn,Axn+(φ(t))
– ≤ϕn
P
t cn
. (.)
If we changeAx,Ay, . . . ,AzwithAxr,Ayr, . . . ,Azrin (.), then for alln>rwe get
FAxn,Axn+(φ(crt))
– ≤ϕn–r
Pr
crt
cn–r
.
Sinceϕn(an)→ wheneveran→ asn→ ∞, therefore the above inequality implies that
lim
n→∞FAxn,Axn+
φcrt= . (.)
Now, let be given, using the properties ofφ-function, we can findr∈Z+ such that
φ(crt) <. Then we have
lim
n→∞FAxn,Axn+()≥ lim
n→∞FAxn,Axn+
φcrt= . (.)
By using a triangle inequality, we obtain
FAxn,Axn+p()≥ FAxn,Axn+
p
, FAxn+,Axn+
p
, . . . ,FAxn+p–,Axn+p
p
p-times
.
Letn→ ∞and make use of (.), for any integerp, we get
lim
Hence{Axn}is aG-Cauchysequence. Similarly, we can obtain that{Ayn}, . . . ,{Azn}areG
-Cauchysequences. Since (X,F,) isG-complete, thereforelimn→∞Axn=u,limn→∞Ayn=
v, . . . ,limn→∞Azn=wfor someu,v, . . . ,w∈X. Now we show thatAu=T(u,v, . . . ,w).
By the continuity ofA, we can obtain thatlimn→∞AAxn=Au,limn→∞AAyn=Av, . . . ,
limn→∞AAzn=Aw. Then the commutativity ofAwithT implies thatAAxn+=T(Axn,
Ayn, . . . ,Azn). From (.), we obtain
FAAxn+,T(u,v,...,w)(φ(t))
–
=
FT(Axn,Ayn,...,Azn),T(u,v,...,w)(φ(t)) –
≤ϕ
(
FAAxn,Au(φ(tc))– ) + (
FAAyn,Av(φ(tc))– ) +· · ·+ (
FAAzn,Aw(φ(tc))– )
m
.
Lettingn→ ∞, sinceϕ() = , we havelimn→∞AAxn+=T(u,v, . . . ,w), from the above inequality, we get Au=T(u,v, . . . ,w). Similarly, we have Av=T(v, . . . ,w,u), . . . ,Aw=
T(w,u,v, . . .).
Next we will show thatAu=u. From (.), we have
FAx,Au(φ(t))
–
=
FT(x,y,...,z),T(u,v,...,w)(φ(t))
–
≤ϕ
( FAx,Au(φ(tc))
– ) + ( FAy,Av(φ(tc))
– ) +· · ·+ ( FAz,Aw(φ(ct))
– )
m
, (.)
FAy,Av(φ(t))
–
≤ϕ
(F
Ay,Av(φ(tc))– ) +· · ·+ (
FAz,Aw(φ(tc))– ) + (
FAx,Au(φ(ct))– )
m
, (.)
. . . ,
FAz,Aw(φ(t)) –
≤ϕ
( FAz,Aw(φ(tc))
– ) + ( FAx,Au(φ(tc))
– ) + ( FAy,Av(φ(tc))
– ) +· · ·
m
. (.)
Suppose that Q(t) = (FAx
,Au(φ(t))
–)+(FAy
,Av(φ(t))
–)+···+(FAz
,Aw(φ(t))
–)
m . Combining (.), (.)
with (.), we obtain
FAx,Au(φ(t))
–
≤ϕ
(
FAx,Au(φ(tc))– ) + (
FAy,Av(φ(ct))– ) +· · ·+ (
FAz,Aw(φ(tc))– )
m
≤ϕ
ϕ(Q
(ct)) +ϕ(Q(ct)) +· · ·ϕ(Q(ct))+ m
=ϕ
Q
t c
,
FAy,Av(φ(t))
– ≤ϕ
Q
t c
,
. . . ,
FAz,Aw(φ(t))
– ≤ϕ
Q
t c
.
Repeating the above procedure, we obtain
FAxn,Au(φ(t))
– ≤ϕn
Q
t cn
. (.)
Sinceϕn(an)→ wheneveran→ asn→ ∞, we have limn→∞Axn=Au, which im-plies thatAu=u=T(u,v, . . . ,w). Similarly, we haveAv=v=T(v, . . . ,w,u), . . . ,Aw=w=
T(w,u,v, . . .).
Finally, we show thatu=v=· · ·=w.
For a better expression, we denoteu=e,v=e, . . . ,w=em, thenAe=e=T(e,e,e, . . . ,em–,em),Ae=e=T(e,e, . . . ,em–,em,e), . . . ,Aem=em=T(em,e,e,e, . . . ,em–).
First, we prove that Fe,e(φ(s)) > for all s> . By the definition of φ, we have
φ(csn) → ∞ as n → ∞. Since supn∈Z+Fe,e(φ(
s
cn)) = ,supn∈Z+Fe,e(φ(
s
cn)) = , . . . , supn∈Z+Fem,e(φ(
s
cn)) = , we deduce that there exists n∈Z+ such that Fe,e(φ(
s cn)) > ,Fe,e(φ(
s
cn)) > , . . . ,Fem,e(φ(
s
cn)) > . Using (.), we obtain
Fe,e(φ(
s cn–))
–
=
FT(e,e,...,em–,em),T(e,...,em–,em,e)(φ(
s cn–))
–
≤ϕ
(
Fe,e(φ(cns))– ) +· · ·+ (
Fem–,em(φ(cns))– ) + (
Fem,e(φ(cns))– )
m
,
which implies thatFe,e(φ(
s
cn–)) > . Similarly, we can obtain thatFe,e(φ(
s
cn–)) > , . . . , Fem,e(φ(
s
cn–)) > . By repeating a similar reasoningn-times, we deduce thatFe,e(φ(s)) >
,Fe,e(φ(s)) > , . . . ,Fem,e(φ(s)) > for alls> .
Second, we show thatFe,e(φ(s)) = . In fact, for everys> , we haveFe,e(φ(
s
ci)) > for all ≤i≤nandn∈Z+. Then, by using (.), we get
Fe,e(φ(s))
–
=
FT(e,e,...,em–,em),T(e,...,em–,em,e)(φ(s))
–
≤ϕ
(
Fe,e(φ(sc))– ) +· · ·+ (
Fem–,em(φ(sc))– ) + (
Fem,e(φ(sc))– )
m
,
Fe,e(φ(s))
–
=
FT(e,e,...,em,e),T(e,e,...,e,e)(φ(s))
≤ϕ
(
Fe,e(φ(cs))– ) +· · ·+ (
Fem,e(φ(sc))– ) + (
Fe,e(φ(sc))– )
m
,
. . . ,
Fem,e(φ(s))
–
=
FT(em,e,...,em–,em–),T(e,e,...,em–,em)(φ(s)) –
≤ϕ
(
Fem,e(φ(cs))– ) + (
Fe,e(φ(sc))– ) +· · ·+ (
Fem–,em(φ(sc))– )
m
.
Suppose thatE(s) =(
Fe,e (φ(s))–)+(
Fe,e (φ(s))–)+···+( Fem,e (φ(s))–)
m , thenE(s)≤ϕ{E(
s
c)}. Byn -iter-ations we get
Fe,e(φ(s))
– ≤ϕ
E s
c
≤ϕ
E s
c
≤ · · · ≤ϕn
E s
cn
. (.)
Thus, sinceϕn(an)→ wheneveran→ asn→ ∞, we getFe,e(φ(s)) = . It follows
that Fe,e(t) =H(t) for allt> . In fact, ift is not in range ofφ, since φ is continuous
at , there existss> such thatφ(s) <t. This implies thatFe,e(t)≥Fe,e(φ(s)) = , then e=e. Similarly, we havee=e, . . . ,em=e,i.e.,u=v=· · ·=w. Thus,u∈Xis the unique
multipled common fixed point ofAandT.
Takingm= in Theorem ., thenT:X→X,A=Ix(the identity mapping onX). It is obvious thatT(X)⊂A(X),Ais continuous and commutative withT, which also satisfies the conditions in Theorem ., then we have the following consequence.
Corollary . Let(X,F,)be a G-complete Menger space witha continuous t-norm.
Let T:X→X satisfying the following inequality:
FTx,Ty(φ(ct)) – ≤ϕ
Fx,y(φ(t)) –
for all x,y∈X,c∈(, ),ϕ∈,φ is aφ-function,t> ,such that Fx,y(φ(t)) > .Then T
has a unique fixed point,i.e.,there exists u∈X such that u=Au=Tu.
Remark . Corollary . is Theorem . of [].
Theorem . Let(X,F,)be a G-complete Menger space witha continuous t-norm and≤p.Let T:X ×X× · · · × X
m-times
→X and A:X→X be two mappings satisfying the
following inequality:
FT(x,y,...,z),T(p,q,...,r)(φ(ct)) –
≤ϕ
m
FAx,Ap(φ(t))
– ,
FAy,Aq(φ(t))
– , . . . ,
FAz,Ar(φ(t)) –
for all x,y, . . . ,z,p,q, . . . ,r∈ X, c∈ (, ), ϕ ∈ , φ is a φ-function, t > , such that FAx,Ap(φ(t)) > ,FAy,Ap(φ(t)) > , . . . ,FAz,Ar(φ(t)) > ,where T(X×X× · · · ×X)⊂A(X),A is
continuous and commutative with T.Then there exists a unique multipled common fixed point of A and T,i.e.,u∈X such that u=Au=T(u,u, . . . ,u).
Proof Since≤p, we get
FT(x,y,...,z),T(p,q,...,r)(φ(ct)) –
≤ϕ
m
FAx,Ap(φ(t))
– ,
FAy,Aq(φ(t))
– , . . . ,
FAz,Ar(φ(t)) –
≤ϕ
m
FAx,Ap(φ(t))
–
FAy,Aq(φ(t)) –
· · ·
FAz,Ar(φ(t)) –
≤ϕ
(
FAx,Ap(φ(t))– ) + (
FAy,Aq(φ(t))– ) +· · ·+ (
FAz,Ar(φ(t))– )
m
.
Then we can complete the proof by Theorem ..
4 An illustration
Example . LetX= [, ],dbe the usual metric onX. DefineT:X ×X× · · · × X
m-times
→X
asT(x,x, . . . ,xm) =x+xm+···+xm,A:X→XasAx=x and
Fx,y(t) =
t
t+d(x,y), t> ,
, t=
for allx,x, . . . ,xm,x,y∈X, whereT(X×X× · · · ×X)⊂A(X). Then (X,F,m) is a com-plete Menger PM-space, m is a continuous t-norm. Defineϕ ∈by ϕ(t) = t and
φ(t) = t
for allt> ,c=
. We obtain
FT(x,x,...,xm),T(y,y,...,ym)(φ(ct))
– =|T(x,x, . . . ,xm) –T(y,y, . . . ,ym)|
φ(ct)
=|(x+x+· · ·+xm) – (y+y+· · ·+ym)|
mt ,
ϕ
(
FAx,Ay(φ(t))– ) + (
FAx,Ay(φ(t))– ) +· · ·+ (
FAxm,Aym(φ(t))– )
m
=ϕ
|Ax–Ay|+|Ax–Ay|+· · ·+|Axm–Aym|
mφ(t)
=(|x–y|+|x–y|+· · ·+|xm–ym|)
mt .
It is obvious that
FT(x,x,...,xm),T(y,y,...,ym)(φ(ct)) –
≤ϕ
(
FAx,Ay(φ(t))– ) + (
FAx,Ay(φ(t))– ) +· · ·+ (
FAxm,Aym(φ(t))– )
m
for allt> . Thus all the conditions of Theorem . are satisfied. Therefore, is the unique multipled common fixed point ofAandT.
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: 17 May 2015 Accepted: 23 July 2015
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