R E S E A R C H
Open Access
Further generalization of fixed point
theorems in Menger PM-spaces
Marwan Amin Kutbi
1, Dhananjay Gopal
2, Calogero Vetro
3and Wutiphol Sintunavarat
4**Correspondence:
[email protected]; [email protected] 4Department of Mathematics and
Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani, 12121, Thailand
Full list of author information is available at the end of the article
Abstract
In this work, we establish some fixed point theorems by revisiting the notion of
ψ
-contractive mapping in Menger PM-spaces. One of our results (namely,Theorem 2.3) may be viewed as a possible answer to the problem of existence of a fixed point for generalized type contractive mappings inM-complete Menger PM-spaces under arbitraryt-norm. Some examples are furnished to demonstrate the validity of the obtained results.
MSC: 47H10; 45D05
Keywords: fixed point; Menger PM-spaces;
ψ
-contractive mapping1 Introduction and preliminaries
In Menger [] initiated the study of probabilistic metric spaces; see also [–]. Suc-cessively, Sehgal and Bharucha-Reid [, ] established fixed point theorems in probabilis-tic metric spaces (for short, PM-spaces). Indeed, by using the notion of probabilisprobabilis-tic q-contraction, they proved a unique fixed point result, which is an extension of the cele-brated Banach contraction principle []. For the interested reader, a comprehensive study of fixed point theory in the probabilistic metric setting can be found in the book of Hadžić and Pap [], see also [] for further discussion on generalizations of metric fixed point theory. Recently, Choudhury and Das [] gave a generalized unique fixed point theorem by using an altering distance function, which was originally introduced by Khanet al.[]. For other results in this direction, we refer to [–]. In particular, Duttaet al.[] de-fined nonlinear generalized contractive type mappings involving altering distances (say,
ψ-contractive mappings) in Menger PM-spaces and proved their theorem for such kind of mappings in the setting ofG-complete Menger PM-spaces.
On contributing to this study, we weaken the notion ofψ-contractive mapping and es-tablish some fixed point theorems inG-complete andM-complete Menger PM-spaces, besides discussing some related results and illustrative examples. Indeed, we not only de-rive the result of Duttaet al.[], Theorem , as a particular case of our result, but also we notice that our Theorem . may be viewed as a possible answer to the problem of ex-istence of a fixed point for generalized type contractive mappings inM-complete Menger PM-spaces under arbitraryt-norm.
Here, we state some allied definitions and results which are needed for the development of the present topic. We denote byRthe set of real numbers, byR+the set of non-negative real numbers and byNthe set of positive integers.
Definition .([, ]) A mappingF:R→R+ is called a distribution function if it is
non-decreasing and left continuous withinft∈RF(t) = andsupt∈RF(t) = .
We shall denote byD+the set of all distribution functions, while H∈D+ will always
denote the specific distribution function defined by
H(t) = ⎧ ⎨ ⎩
ift≤,
ift> .
Definition .([]) A binary operationT: [, ]×[, ]→[, ] is a continuoust-norm if the following conditions hold:
(a) Tis commutative and associative, (b) Tis continuous,
(c) T(a, ) =afor alla∈[, ],
(d) T(a,b)≤T(c,d), whenevera≤candb≤d, fora,b,c,d∈[, ].
The following are three basic continuoust-norms from the literature: (i) The minimumt-norm, sayTM, defined byTM(a,b) =min{a,b}.
(ii) The productt-norm, sayTp, defined byTp(a,b) =a·b.
(iii) The Lukasiewiczt-norm, sayTL, defined byTL(a,b) =max{a+b– , }.
Theset-norms are related in the following way:TL≤Tp≤TM.
Definition . A Menger PM-space is a triple (X,F,T) whereXis a nonempty set,Tis 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=yfor allt∈R+,
(PM) Fx,y(t) =Fy,x(t)for allx,y∈Xandt∈R+,
(PM) Fx,y(t+s)≥T(Fx,z(t),Fz,y(s))for allx,y,z∈Xands,t∈R+. Definition . Let (X,F,T) be a Menger PM-space. Then
(i) A sequence{xn}inXis said to be convergent tox∈Xif, for every> andλ> , there exists a positive integerNsuch thatFxn,x() > –λwhenevern≥N.
(ii) A sequence{xn}inXis called Cauchy sequence if, for every> andλ> , there exists a positive integerNsuch thatFxn,xm() > –λwhenevern,m≥N.
(iii) A Menger PM-space is said to beM-complete if every Cauchy sequence inXis convergent to a point inX.
(iv) A sequence{xn}is calledG-Cauchy iflimn→∞Fxn,xn+m(t) = for eachm∈Nand
t> .
(v) The space(X,F,T)is calledG-complete if everyG-Cauchy sequence inXis convergent.
According to [], the (,λ)-topology in a Menger PM-space (X,F,T) is introduced by the family of neighborhoodsNxof a pointx∈Xgiven by
Nx=
Nx(,λ) :> ,λ∈(, )
where
Nx(,λ) =
y∈X:Fx,y() > –λ
.
The (,λ)-topology is a Hausdorff topology. In this topology a functionf is continuous in x∈Xif and only iff(xn)→f(x), for every sequencexn→x, asn→ ∞.
The following class of functions was introduced in [] and will be used in proving our results in the next section.
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 .([]) Let (X,F,T) be a Menger PM-space. The probabilistic metricFis triangular if it satisfies the condition
Fx,y(t)
– ≤
Fx,z(t)
–
+
Fz,y(t)
–
for everyx,y,z∈Xand eacht> .
In the sequel, the class of allφ-functions will be denoted by. Also we denote by
the class of all continuous non-decreasing functionsψ:R+→R+such thatψ() = and
ψn(an)→, wheneveran→, asn→ ∞.
We conclude this section recalling the following fixed point theorem of Duttaet al., see [], which is the main inspiration of our paper.
Theorem . Let(X,F,T)be a G-complete Menger space and f :X→X be a mapping satisfying the following inequality:
Ffx,fy(φ(ct))
– ≤ψ
Fx,y(φ(t))
–
, (.)
where x,y∈X, c∈(, ),φ∈,ψ ∈ and t> such that Fx,y(φ(t)) > .Then f has a
unique fixed point.
A mappingf:X→Xsatisfying condition (.) is usually calledψ-contractive mapping. However, for some discussion on this notion and Theorem ., the reader can refer to the recent paper of Gopalet al.[], where analogous results are proved by using some control functions.
2 Main results
In this section, firstly we weaken the class of functionsby assuming the continuity only at pointt= . Precisely, we denote by the class of all non-decreasing functionsψ :
R+→R+such thatψ is continuous at ,ψ() = andψn(a
n→ ∞; then we utilize this class to prove some fixed point theorems. We start with a revised version of Theorem . useful to obtaining an affirmative answer to an existence problem of a fixed point in aG-complete Menger space.
Theorem . Let(X,F,T)be a G-complete Menger space and f :X→X be a mapping satisfying the following inequality:
Ffx,fy(φ(ct))
– ≤ψ
Fx,y(φ(t))
–
, (.)
where x,y∈X,c∈(, ),φ∈,ψ ∈and t> such that Fx,y(φ(t)) > .Then f has a
unique fixed point.
Proof Letx∈X. Define a sequence{xn}inXso thatxn+=fxnfor alln∈N∪ {}. We
supposexn+=xnfor alln∈N, otherwisef has trivially a fixed point.
Notice that in view of the fact thatsupt∈RFx,x(t) = and by (ii) of Definition ., one
can findt> such thatFx,x(φ(t)) > . SinceFx,x(φ(t)) > implies thatFx,x(φ( t c)) > ,
therefore (.) gives that
Fx,x(φ(t))
– =
Ffx,fx(φ( ct c)) – ≤ψ Fx,x(φ(
t c))
–
. (.)
From (.) we deduce thatFx,x(φ(t)) > and soFx,x(φ( t
c)) > . Again, by applying (.),
we get
Fx,x(φ(t))
– =
Ffx,fx(φ(t))
–
≤ψ
Fx,x(φ(
t c)) – , that is, Fx,x(φ(t))
– ≤ψ
Fx,x(φ(
t c))
–
.
On using (.) and the hypothesis thatψis non-decreasing, the above expression becomes
Fx,x(φ(t))
– ≤ψ
Fx,x(φ(
t c))
–
. (.)
Repeating the above procedure successivelyntimes, we obtain
Fxn,xn+(φ(t))
– ≤ψn
Fx,x(φ(
t cn))
–
.
If we changexwithxrin the previous inequalities, then for alln>rwe get
Fxn,xn+(φ(crt))
– ≤ψn–r
Fxr,xr+(φ(
crt cn–r))
–
Sinceψn(a
n)→ wheneveran→ asn→ ∞, therefore the above inequality implies that
lim
n→∞Fxn,xn+ φ c
rt= . (.)
Now, let> be given, then by using the properties (i) and (iv) of a functionφwe can find r∈Nsuch thatφ(crt) <. It follows from (.) that
lim
n→∞Fxn,xn+()≥n→∞lim Fxn,xn+ φ c
rt= . (.)
By using a triangle inequality, we obtain
Fxn,xn+p()≥T Fxn,xn+(/p),T Fxn+,xn+(/p), . . . , Fxn+p–,xn+p(/p)
· · ·
p-times
.
Thus, lettingn→ ∞and making use of (.), for any integerp, we get
lim
n→∞Fxn,xn+p() = for every> . (.)
Hence{xn}is aG-Cauchy sequence. Since (X,F,T) isG-complete, thereforexn→u, as
n→ ∞, for someu∈X.
Now we show thatuis a fixed point off. Since
Ffu,u()≥T Ffu,xn+(/),Fxn+,u(/)
, (.)
by using the properties (i) and (iv) of a functionφ, we can finds> such thatφ(s) <
.
Again, sincexn→uasn→ ∞, then there existsn∈Nsuch that, for alln>n, we have
Fxn,u(φ(s)) > .
Therefore, forn>n, we obtain
Fxn+,fu(
)
– ≤
Ffxn,fu(φ(s))
–
≤ψ
Fxn,u(φ(sc))
–
.
Sinceψis continuous at andψ() = , we obtain
lim
n→∞Fxn+,fu(/) = . (.)
From (.) and (.), we getFfu,u() = for every> , which in turn yields thatfu=u.
The following example illustrates our Theorem ..
Example . LetX= [, ] anddbe the usual metric onX. Definef :X→Xasfx= sinx
and
Fx,y(t) =
⎧ ⎨ ⎩
t
t+d(x,y) ift> ,
for allx,y∈X. Then (X,F,T) is a complete Menger PM-space withTp(a,b) =a·b. Define
φ∈byφ(s) = scfor alls∈R+, withc=
, andψ∈by
ψ(s) = ⎧ ⎨ ⎩
s ifs∈Q∩R+,
s
otherwise.
By the mean valued theorem with functionsin, we obtain that
Ffx,fy(φ(ct))
– = d(fx,fy) t
=
t|sinx–siny|
≤ |x–y|
t
= d(x,y)
t ≤ψ
Fx,y(φ(t))
–
.
Thus,f satisfies all the hypotheses of Theorem .; hereu= is a fixed point off.
Next we consider the uniqueness problem of a fixed point in aG-complete Menger space; to this aim we give the following condition:
(∗) Fu,v() = ifu,v∈Fix(f), whereFix(f)denotes the set of all fixed points of a mappingf,
that is,Fix(f) :={x∈X:x=fx}.
Theorem . Adding condition(∗)to the hypotheses of Theorem.,we obtain uniqueness of the fixed point.
Proof We prove uniqueness of the fixed point. Letuandvbe two fixed points off, that is, u=fuandv=fv. First, we prove thatFu,v(φ(s)) > for alls> . By condition (ii) of
Definition ., we haveφ(s/cn)→ ∞asn→ ∞. Sincesup
n∈NFu,v(φ(s/cn)) = , we deduce
that there existsn∈Nsuch thatFu,v(φ(s/cn)) > . Now, by using (.), we obtain
Fu,v(φ(cns–))
– =
Ffu,fv(φ(ccsn))
–
≤ψ
Fu,v(φ(csn))
–
,
that implies Fu,v(φ(cns–)) > . By repeating a similar reasoningntimes, we deduce that
Fu,v(φ(s)) > for alls> .
Next, we show thatFu,v(φ(s)) = . In fact, for everys> , we have thatFu,v(φ(csi)) > for
all ≤i≤nand for alln∈N. Therefore, by using (.), we get
Fu,v(φ(s))
– ≤ψ
Fu,v(φ(sc))
–
≤ · · · ≤ψn
Fu,v(φ(csn))
–
.
Thus, sinceψn(a
It follows thatFu,v(t) =H(t) for allt> . In fact, iftis not in range ofφ, sinceφis
con-tinuous at , then there existss> such thatφ(s) <t. This impliesFu,v(t)≥Fu,v(φ(s)) = ,
yielding therebyu=v.
Our next step is to furnish a fixed point theorem in anM-complete Menger PM-space.
Theorem . Let(X,F,T)be an M-complete Menger PM-space and f :X→X be aψ
-contractive mapping,where the functionψ:R+→R+is non-decreasing,continuous at,
ψ() = and∞n=ψn(a
n) <∞,whenever an→as n→ ∞.Then f has a fixed point
provided that F is triangular.
Proof In view of the assumptions on the functionψ, it is clear thatψ∈. Then, following
similar arguments to those given in Theorem ., one obtainsFxn,xn+()→ asn→ ∞.
Now, we shall show that{xn}is anM-Cauchy sequence. By the properties ofφ, given> , we can finds> such that>φ(s) > . Therefore,
Fxn,xn+p()
– ≤
Fxn,xn+p(φ(s))
– .
Now, sinceFis triangular, we get
Fxn,xn+p()
– ≤
Fxn,xn+()
– +
Fxn+,xn+()
– +· · ·+ Fxn+p–,xn+p()
–
≤
Fxn,xn+(φ(s))
– +
Fxn+,xn+(φ(s))
–
+· · ·+ Fxn+p–,xn+p(φ(s))
–
≤ψn
Fxn,xn+(φ(
s cn))
–
+ψn+
Fxn+,xn+(φ(
s cn+))
–
+· · ·+ψn+p
Fxn+p–,xn+p(φ(
s cn+p))
–
≤ ∞
k=n
ψk
Fxk,xk+(φ(
s ck))
–
.
Since ∞n=ψn(a
n) <∞, where an = (F
xn,xn+(φ(cns)) – ) → as n → ∞, we obtain
Fxn,xn+p()→ asn→ ∞. Thus{xn}is anM-Cauchy sequence inX. The rest of the proof
of this theorem can be completed on the lines of Theorem .. This concludes the proof.
Clearly, on the same lines of Theorem . one can solve the uniqueness problem of a fixed point in anM-complete Menger space. To avoid repetition, we give the statement of this theorem without the proof.
Remark . Our Theorem . is proved in anM-complete MengerPM-space under arbi-traryt-norm, therefore Theorem . can be realized as a possible answer to the problem of existence of a fixed point for generalized type contractive mappings in MengerPM-spaces.
As an application of Theorems . and ., we prove the following common fixed point theorem for a finite family of mappings which runs as follows.
Theorem . Let(X,F,T)be a G-complete Menger PM-space,{fi}m
be a finite family of
self-mappings defined on X and denote f =fff· · ·fm.If f:X→X satisfies all the
hypothe-ses of Theorem.,then the family{fi}m has a unique common fixed point provided that fifj=fjfiwhenever i=j,with i,j∈ {, , . . . ,m}.
Proof Notice that all the hypotheses of Theorems . and . are satisfied in respect of the mappingf, therefore there exists a uniquex∈Xsuch thatfx=x. Now
f(fix) = (ff· · ·fm)fi
x
= (ff· · ·fm–)(fmfi)x
= (ff· · ·fm–)(fifmx)
=· · ·
=ffi(ff· · ·fmx)
=fif(ff· · ·fmx) =fi(fx) =fix,
which shows thatfixis also a fixed point off. Sincexis the unique fixed point off, therefore
fix=xand hencexis also a fixed point of all mappingsfifori∈ {, , . . . ,m}.
By setting f =f=· · ·=fm =g in Theorem ., we deduce the following fixed point
theorem formth iterates of a mappingg.
Corollary . Let(X,F,T)be a G-complete Menger PM-space and g:X→X be a map-ping such that gm satisfies all the hypotheses of Theorem..Then g has a unique fixed
point.
Remark . Results similar to Theorem . and Corollary . can be outlined in respect of Theorems . and ..
Finally, by using the following example, we show that Corollary . can be situationally more useful than Theorems . and ..
Example . LetX= [, ] be equipped with the usual metricdonX. Definef :X→X as follows:
fx= ⎧ ⎨ ⎩
ifx∈ {,, }, ifx∈(,
)∪( , ).
Also define
Fx,y(t) =
⎧ ⎨ ⎩
t
t+d(x,y) ift> ,
for allx,y∈X. Then (X,F,T) is a complete Menger PM-space withTp(a,b) =a·b. Notice
thatfx= for everyx∈Xand hence the condition
Ffx,fy(φ(ct))– ≤
ψ
Fx,y(φ(t))
–
is always satisfied for every choice of functions φ∈andψ as in Theorem ., with any constantc∈(, ). On the other hand,f does not satisfy condition (.). In fact, for instance, puttingx= andy∈(, ) the inequality
Ffx,fy(ct)≥Fx,y(t)
does not hold true and consequently condition (.) is not satisfied forφ(s) =ψ(s) =sfor alls∈R+. Moreover, if we choose againx= and makey→, thenf does not satisfy the
condition
Ffx,fy(φ(ct))
– ≤ψ
Fx,y(φ(t))
–
for any choice ofφ∈,ψ∈andc∈(, ).
Thus we conclude thatf does not meet the requirements of Theorem ., whereas the power mappingfsatisfies all the conditions of Corollary . substantiating the utility of
Corollary . (and hence Theorem .) over Theorem ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.2Department of Applied Mathematics & Humanities, S. V. National Institute of Technology, Surat, Gujarat 395007, India.3Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Via Archirafi 34, Palermo, 90123, Italy.4Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani, 12121, Thailand.
Authors’ information
Calogero Vetro is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
Acknowledgements
We would like to acknowledge the anonymous reviewers for their helpful comments and detailed suggestions. The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. The fourth author would like to thank the Thailand Research Fund and Thammasat University under Grant No. TRG5780013 for financial support during the preparation of this manuscript.
Received: 28 September 2014 Accepted: 4 February 2015 References
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