R E S E A R C H
Open Access
(ψ
,
ϕ
,
,
λ)-Contraction theorems in
probabilistic metric spaces for single valued
case
Arman Beitollahi
1*and Parvin Azhdari
2*Correspondence:
arman.beitollahi@gmail.com
1Department of Statistics,
Roudehen Branch, Islamic Azad University, Roudehen, Iran Full list of author information is available at the end of the article
Abstract
In this article, we prove some fixed-point theorems for (
ψ
,ϕ
,,
λ
)-contraction in probabilistic metric spaces for single valued case. We will generalize the definition of (ψ
,ϕ
,,
λ
)-contraction and present fixed-point theorem in the generalized(
ψ
,ϕ
,,
λ
)-contraction.Keywords: probabilistic metric spaces; (
ψ
,ϕ
,,
λ
)-contraction; fixed-point1 Introduction
The probabilistic metric space was introduced by Menger []. Mihet presented the class of (ψ,ϕ,,λ)-contraction for a single valued case in fuzzy metric spaces [, ]. This class is a generalization of the (,λ)-contraction which was introduced in []. We defined the class of (ψ,ϕ,,λ)-contraction for the multi-valued case in a probabilistic metric space before []. Now, we obtain two fixed-point theorems of (ψ,ϕ,,λ)-contraction for sin-gle valued case. Also, we extend the concept of (ψ,ϕ,,λ)-contraction to the generalized (ψ,ϕ,,λ)-contraction.
The structure of this paper is as follows: Section is a review of some concepts in prob-abilistic metric spaces and probprob-abilistic contractions. In Section , we will show two the-orems for (ψ,ϕ,,λ)-contraction in the single-valued case and explain the generalized (ψ,ϕ,,λ)-contraction.
2 Preliminary notes
We recall some concepts from probabilistic metric space, convergence and contraction. For more details, we refer the reader to [–].
Let D+ be the set of all distribution of functions F such that F() = (F is a non-decreasing, left continuous mapping fromRinto [, ] such thatlimx→∞F(x) = ).
The ordered pair (S,F) is said to be a probabilistic metric space ifSis a nonempty set andF:S×S→D+(F(p,q) written byFpqfor every (p,q)∈S×S) satisfies the following conditions:
() Fuv(x) = for everyx> ⇔u=v(u,v∈S), () Fuv=Fvufor everyu,v∈S,
() Fuv(x) = andFvw(y) = ⇒Fu,w(x+y) = for everyu,v,w∈S, and everyx,y∈R+.
A Menger space is a triple (S,F,T) where (S,F) is a probabilistic metric space,T is a tri-angular norm (abbreviatedt-norm) and the following inequality holds:
Fuv(x+y)≥T
Fuw(x),Fwv(y)
for everyu,v,w∈S, and everyx,y∈R+.
Recall the mappingT: [, ]×[, ]→[, ] is called a triangular norm (at-norm) if the following conditions are satisfied:T(a, ) =afor everya∈[, ];T(a,b) =T(b,a) for every
a,b∈[, ];a≥b,c≥d⇒T(a,c)≥T(b,d),a,b,c,d∈[, ];T(T(a,b),c) =T(a,T(b,c)),
a,b,c∈[, ]. Basic examples oft-norms areTL(Lukasiewiczt-norm),TPandTM, defined byTL(a,b) =max{a+b– , },TP(a,b) =abandTM(a,b) =min{a,b}. IfT is at-norm and (x,x, . . . ,xn)∈[, ]n(n∈N∗), one can define recurrently ni=xi=T( n–i=xi,xn) for all n≥. One can also extendTto a countable infinitary operation by defining ∞i=xifor any sequence (xi)i∈N∗aslimn→∞ ni=xi.
Ifq∈(, ) is given, we say that thet-normT isq-convergent iflimn→∞ ∞i=n( –qi) = . We remark that ifT isq-convergent, then
∀λ∈(, )∃s=s(λ)∈N ni= –qs+i> –λ, ∀n∈N.
Also, note that if thet-normT isq-convergent, thensup≤t<T(t,t) = .
Proposition . Let(S,F,T) be a Menger space. Ifsup≤t<T(t,t) = ,then the family
{U}>,where
U=
(x,y)∈S×S|Fx,y() > –
is a base for a metrizable uniformity on S,called the F-uniformity[–].The F-uniformity naturally determines a metrizable topology on S,called the strong topology or F-topology
[],a subset O of S is F-open if for every p∈O there exists t> such that Np={q∈S|Fpq(t) > –t} ⊂O.
Definition .[] A sequence (xn)n∈Nis called an F-convergent sequence tox∈Sif for every> andλ∈(, ) there existsN=N(,λ)∈N such thatFxnx() > –λ,∀n≥N.
Definition . [] Letϕ: (, )→(, ) be a mapping, we say that thet-normT isϕ -convergent if
∀δ∈(, ),∀λ∈(, )∃s=s(δ,λ)∈N ni= –ϕs+i(δ)> –λ, ∀n≥.
Definition .[] A sequence (xn)n∈Nis called a convergent sequence tox∈Sif for every
> andλ∈(, ) there existsN=N(,λ)∈N such thatFxnx() > –λ,∀n≥N.
Definition .[] A sequence (xn)n∈Nis called a Cauchy sequence if for every> and
λ∈(, ) there existsN=N(,λ)∈N such thatFxnxn+m() > –λ,∀n≥N,∀m∈N. We also have
A probabilistic metric space (S,F,T) is called sequentially complete if every Cauchy se-quence is convergent.
The concept of (ψ,ϕ,,λ)-contraction has been introduced by Mihet [].
We will consider comparison functions from the classφof all mappingϕ: (, )→(, ) with the properties:
() ϕis an increasing bijection; () ϕ(λ) <λ∀λ∈(, ).
Since every such a comparison mapping is continuous, ifϕ∈φ, thenlimn→∞ϕn(λ) =
∀λ∈(, ).
Definition .[] Let (S,F) be a probabilistic space,ϕ∈φandψbe a map from (,∞) to (,∞). A mappingf :S→Sis called a (ψ,ϕ,,λ)-contraction onSif it satisfies in the following condition:
x,y∈S,> ,λ∈(, ), Fx,y() > –λ ⇒ Ff(x),f(y)
ψ()> –ϕ(λ). In the rest of paper we suppose thatψ is increasing bijection.
Example . LetS={, , , . . .}and (forx=y)
Fx,y(t) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ift≤–min(x,y),
– –min(x,y) if –min(x,y)<t≤,
ift> .
Suppose thatf :S→S,f(r) =r+ .
Then (S,F,TL) is a probabilistic metric space []. Letx,y,,λbe such thatFx,y() > –λ.
(i) If–min(x,y)<≤, then – –min(x,y)> –λ. This implies – –min(x+,y+)> –
λ, that is, Ffx,fy() > –
λ.
(ii) If> thenFfx,fy() = , hence againFfx,fy() > –λ. Thus, the mappingf is a (ψ,ϕ,,λ)-contraction onSwithψ() =andϕ(λ) = λ.
3 Main results
In this section, we will show (ψ,ϕ,,λ)-contraction is continuous. By using this assump-tion, we will also prove two theorems.
Definition . LetFbe a probabilistic distance onS. A mappingf :S→Sis called con-tinuous if for every> there existsδ> such that
Fu,v(δ) > –δ ⇒ Ffu,fv() > –.
Lemma . Every(ψ,ϕ,,λ)-contraction is continuous.
Proof Suppose that> be given andδ∈(, ) be such thatδ<min{,ψ–()}and since
ψ is increasing bijection thenψ(δ) <. IfFx,y(δ) > –δ then, by (ψ,ϕ,,λ)-contraction we haveFfx,fy(ψ(δ)) > –ϕ(δ), from where we obtain thatFfx,fy() >Ffx,fy(ψ(δ)) > –ϕ(δ) >
–δ> –. Sof is continuous.
Theorem . Let (S,F,T) be a complete Menger space and T a t-norm satisfies in
sup≤a<T(a,a) = .Also,f :S→S a(ψ,ϕ,,λ)-contraction wherelimn→∞ψn(δ) = for
everyδ∈(,∞).Iflimt→∞Fx,fmx(t) = for some x∈S and all m∈N,then there exists a
unique fixed point x of the mapping f and x=limn→∞fn(x).
Proof Letxn=fnx,n∈N. We shall prove that (xn)n∈Nis a Cauchy sequence.
Letn,m∈N,> ,λ∈(, ). Sincelimt→∞Fx,fmx(t)(t) = , it follows that for everyξ∈
(, ) there existsη> such thatFx,fm(x)(η) > –ξand by inductionFfmx,fn+mx(ψ
n(η)) > –ϕn(ξ) for alln∈N. By choosingnsuch thatψn(η) <andϕn(ξ) <λ, we obtain
Fxn,xn+m() > –λ.
Hence, (xn)n∈N is a Cauchy sequence and sinceSis complete, it follows the existence of
x∈Ssuch that x=limn→∞xn. By continuity off andxn+ =fxnfor everyn∈N, when
n→ ∞, we obtain thatx=fx.
Example . Let (S,F,T) be a complete Menger space whereS={x,x,x,x},T(a,b) =
min{a,b}andFxy(t) is defined as
Fx,x(t) =Fx,x(t) =
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ift≤,
. if <t≤,
ift>
and
Fx,x(t) =Fx,x(t) =Fx,x(t) =Fx,x(t) =Fx,x(t) =Fx,x(t) =Fx,x(t)
=Fx,x(t) =Fx,x(t) =Fx,x(t) =
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ift≤,
. if <t< ,
if ≤t
f : S →S is given by f(x) =f(x) =x and f(x) =f(x) =x. If we take ϕ(λ) = λ,
ψ() =
, thenf is a (ψ,ϕ,,λ)-contraction wherelimn→∞ψn(δ) =limn→∞ δ
n= for ev-ery δ∈(,∞) and if we setx=x, then for all m∈N, we havefmx=fmx=x and
limt→∞Fxx(t) = , soxis the unique fixed point forf.
Theorem . Let (S,F,T) be a complete Menger space, T be a t-norm such that
sup≤a<T(a,a) = and f :S→S a(ψ,ϕ,,λ)-contraction where the seriesiψi(δ)is convergent for allδ> and suppose that for some p∈S and j>
sup
x>
xj –Fp,fp(x)
If t-norm T isϕ-convergent,then there exist a unique fixed point z of mapping f and z=
liml→∞flp.
Proof Choose> andλ∈(, ). Letzl=flp,l∈N. We shall prove that (zl)l∈Nis a Cauchy sequence. It means we prove that there existsn(,λ)∈Nsuch that
Fflp,fl+mp() > –λ for everyl≥n(,λ) and everym∈N. Suppose thatμ∈(, ),M> are such that
xj –Fp,fp(x)
≤M ∀x> . ()
Letnbe such that
–Mμjn∈[, ). From (), it follows that
Fp,fp
μn
> –Mμjn ∀n∈Nspecially forn=n.
Sincef is (ψ,ϕ,,λ)-contraction, we derived by inductionFflp,fl+p(ψl(μn)) > –ϕl( –
M(μj)n)∀l> . Since the series∞
i=ψi(δ) is convergent, there existsn=n()∈Nsuch that∞i=lψi(δ)≤∀l≥n
. We know
∞
i=lψi(μn)≤for everyl>max{n,n}.
Now
Fflp,fl+mp()≥Fflp,fl+mp
∞ i=l ψi
μn
≥Fflp,fl+mp l+m– i=l ψi
μn
≥T
T
· · ·T
Fflp,fl+p
ψl
μn
,Ffl+p,fl+p
ψl+
μn
, . . . ,
Ffl+m–p,fl+mp
ψl+m–
μn
≥TT· · ·T –ϕl –Mμjn, –ϕl+ –Mμjn, . . . ,
–ϕl+m– –Mμjn
≥ ∞
i=l
–ϕi –Mμjn. SinceT isϕ-convergent, we conclude that (flp)
l∈N is a Cauchy sequence. On the other
hand,Sis complete, therefore, there is az∈Ssuch thatz=liml→∞flp. By the continuity
of the mappingf andzl+=fzlwhenl→+∞, it follows thatfz=z.
Example . Let (S,F,T) and the mappingsf,ψ andϕ be the same as in Example .. Sinceiψi(δ) =iδi=δfor allδ> and if we setp=x∈S,j> thentj( –Fxx(t)) =
for everyt> orsupt>tj( –Fxx(t)) <∞, soxis the unique fixed point forf.
Letℵbe the family of all the mappingsm:R→Rsuch that the following conditions are satisfied:
() ∀t,s≥ :m(t+s)≥m(t) +m(s); () m(t) = ⇔t= ;
() mis continuous.
Definition . Let (S,F) be a probabilistic metric space andf :S→S. The mappingf
is a generalized (ψ,ϕ,,λ)-contraction if there exist a continuous, decreasing function
h: [, ]→[,∞] such thath() = ,m,m∈ ℵ, andλ∈(, ) such that the following implication holds for everyp,q∈Sand for every> :
hoFp,q
m()
<m(λ) ⇒ hoFf(p),f(q)
m
ψ()<m
ϕ(λ).
Ifm(a) =m(a) =a, andh(a) = –afor everya∈[, ], we obtain the Mihet definition.
Theorem . Let (S,F,T) be a complete Menger space with t-norm T such that
sup≤a<T(a,a) = and f :S→S be a generalized(ψ,ϕ,,λ)-contraction such thatψ is continuous on(,∞)andlimn→∞ψn(δ) = for everyδ∈(,∞).Suppose that there exists
λ∈(, )such that h() <m(λ)andϕ,ψ satisfyϕ() =ψ() = .Then x=limn→∞fn(p)
is the unique fixed point of the mapping f for an arbitrary p∈S.
Proof First we shall prove thatf is uniformly continuous. Letζ> andη∈(, ). We have to prove that there existsN(ζ,η) ={(p,q)|(p,q)∈S×S,Fp,q(ζ) > –η}such that
(p,q)∈N(ζ,η) ⇒ Ff(p),f(q)(ζ) > –η.
Letbe such thatm(ψ()) <ζ andλ∈(, ) such that m
ϕ(λ)<h( –η). () Sincem andm are continuous at zero, andm() =m() = such numbers andλ exist. We prove thatζ =m(),η= –h–(m(λ)). If (p,q)∈N(ζ,η), we have
Fp,q
m()
> – –h–m(λ)
=h–m(λ)
.
Sincehis decreasing, it follows thathoFp,q(m()) <m(λ). Hence, hoFf(p),f(q)
m
ψ()<m
ϕ(λ). Using (), we conclude that
hoFf(p),f(q)
m
ψ()<h( –η) and sincehis decreasing we have
Ff(p),f(q)(ζ)≥Ff(p),f(q)
m
Therefore, (f(p),f(q))∈N(ζ,η) if (p,q)∈N(ζ,η). We prove that for everyζ > andη∈
(, ) there existsn(ζ,η)∈Nsuch that for everyp,q∈S
n>n(ζ,η) ⇒ Ffn(p),fn(q)(ζ) > –η. () By assumption, there is aλ∈(, ) such thath() <m(λ). FromFp,q(m())≥, it follows that
hoFp,q
m()
≤h() <m(λ)
which implies thathoFf(p),f(q)(m(ψ())) <m(ϕ(λ)), and continuing in this way we obtain that for everyn∈N
hoFfn(p),fn(q)mψn()<mϕn(λ).
Letn(ζ,η) be a natural number such thatm(ψn()) <ζ andm(ϕn(λ)) <h( –η), for everyn≥n(ζ,η). Thenn>n(ζ,η) implies that
Ffn(p),fn(q)(ζ)≥Ffn(p),fn(q)mψn()> –η. Ifq=fm(p), from () we obtain that
Ffn(p),fn+m(p)(ζ) > –η for everyn>n(ζ,η) and everym∈N. () Relation () means that (fn(p))n∈N is a Cauchy sequence, and sinceSis complete there existsx=limn→∞fn(p), which is obviously a fixed point off sincef is continuous.
For everyp∈Sandq∈Ssuch thatf(p) =pandf(q) =qwe have for everyn∈N that
fn(p) =p,fn(q) =qand, therefore, from () we haveF
p,q(ζ) > –ηfor everyη∈(, ) and
ζ > . This implies thatFp,q(ζ) = for everyζ> and, therefore,p=q.
Example . Let (S,F,T) and the mappingsf,ψandϕbe the same as in Example .. Set
h(a) =e–a–e–for everya∈[, ] andm(a) =m(a) =a. The mappingf is generalized (ψ,ϕ,,λ)-contraction andlimn→∞ψn(δ) =limn→∞δn = for every δ∈(,∞). On the other hand, there existsλ∈(, ) such thath() = –e <λandψ() =ϕ() = . Soxis the unique fixed point forf.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
PA defined the definitions and wrote the Introduction, preliminaries and abstract. AB proved the theorems. AB has approved the final manuscript. Also, PA has verified the final manuscript.
Author details
1Department of Statistics, Roudehen Branch, Islamic Azad University, Roudehen, Iran.2Department of Statistics, North
Tehran Branch, Islamic Azad University, Tehran, Iran.
Acknowledgements
The authors thank the reviewers for their useful comments.
References
1. Menger, K: Statistical metrics. Proc. Natl. Acad. Sci. USA28, 535-537 (1942)
2. Mihet, D: A class of contractions in fuzzy metric spaces. Fuzzy Sets Syst.161, 1131-1137 (2010) 3. Mihet, D: A note on a paper of Hadži´c and Pap. In: Cho, YJ, Kim, JK, Kang, SM (eds.) Fixed Point Theory and
Applications, vol. 7, pp. 127-133. Nova Science Publishers, New York (2007)
4. Mihet, D: A class of Sehgal’s contractions in probabilistic metric spaces. An. Univ. Timi¸soara Ser. Mat.-Inform.37, 105-110 (1999)
5. Beitollahi, A, Azhdari, P: Multi-valued (ψ,ϕ,,λ)-contraction in probabilistic metric space. Fixed Point Theory Appl. 2012, 1-10 (2012)
6. Hadži´c, O, Pap, E: Fixed Point Theory in PM Spaces. Kluwer Academic, Dordrecht (2001) 7. Schweizer, B, Sklar, A: Probabilistic Metric Spaces. North-Holland, Amsterdam (1983) 8. Schweizer, B, Sklar, A, Throp, E: The metrization of SM-spaces. Pac. J. Math.10, 673-675 (1960)
9. Radu, V: Lectures on probabilistic analysis. In: Surveys Lectures Notes and Monographs Series on Probability Statistics and Applied Mathematics, vol. 2. University of Timisoara (1994)
10. Schweizer, B, Sherwood, H, Tardif, RM: Contractions on PM-spaces:examples and counterexamples. Stochastica12(1), 5-17 (1988)
doi:10.1186/1687-1812-2013-109
