A Beitollahi

Arman Beitollahi

B Schweizer

D Mihet

E Pap

EP Klement

O Hadžić

Parvin Azhdari

T Žikić-Došenović

Fixed Point Theory and Applications

Crossref

RESEARCH Open AccessMulti-valued (ψ, , ε, l)-contraction inprobabilistic metric spaceArman Beitollahi1* and Parvin Azhdari2* Correspondence: arman.beitollahi@gmail.com1Department of Statistics,Roudehen Branch, Islamic AzadUniversity, Roudehen, IranFull list of author information isavailable at the end of the articleAbstractIn this article, we present a new definition of a class of contraction for multi-valuedcase. Also we prove some fixed point theorems for multivalued (ψ, , ε, l)-contraction mappings in probabilistic metric space.Keywords: probabilistic metric space, (ψ, φ, ε, λ)-contraction, fixed point1 IntroductionThe class of (ε, l)-contraction as a subclass of B-contraction in probabilistic metricspace was introduced by Mihet [1]. He and other researchers achieved to some inter-esting results about existence of fixed point in probabilistic and fuzzy metric spaces[2-4]. Mihet defined the class of (ψ, , ε, l)-contraction in fuzzy metric spaces [4]. Onthe other hand, Hadzic et al. extended the concept of contraction to the multi valuedcase [5]. They introduced multi valued (ψ - C)-contraction [6] and obtained fixedpoint theorem for multi valued contraction [7]. Also Žikić generalized multi valuedcase of Hick’s contraction [8]. We extended ( - k) - B contraction which introducedby Mihet [9] to multi valued case [10]. Now, we will define the class of (ψ, , ε, l)-contraction in the sense of multi valued and obtain fixed point theorem.The structure of article is as follows: Section 2 recalls some notions and knownresults in probabilistic metric spaces and probabilistic contractions. In Section 3, wewill prove three theorems for multi valued (ψ, , ε, l)- contraction.2 PreliminariesWe recall some concepts from the books [11-13].Definition 2.1. A mapping T : [0, 1] × [0, 1] ® [0, 1] is called a triangular norm (at-norm) if the following conditions are satisfied:(1) T (a, 1) = a for every a Î [0, 1];(2) T (a, b) = T (b, a) for every a, b Î [0, 1];(3) a ≥ b, c ≥ d ⇒ T(a, c) ≥ T(b, d) a, b, c, d Î [0, 1];(4) T(T(a, b), c) = T(a, T(b, c)), a, b, c Î [0, 1].Basic examples are, TL(a, b) = max{a + b - 1, 0}, TP (a, b) = ab and TM (a, b) = min{a, b}.Definition 2.2. If T is a t-norm and (x1, x2 , . . . , xn) ∈ [0, 1]n (n ≥ 1), ∞i=1xi isdefined recurrently by 1i=1xi = x1 and ni=1xi = T(n−1i=1 xi, xn) for all n ≥ 2. T can beBeitollahi and Azhdari Fixed Point Theory and Applications 2012, 2012:10http://www.fixedpointtheoryandapplications.com/content/2012/1/10© 2012 Beitollahi and Azhdari; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.extended to a countable infinitary operation by defining ∞i=1xi for any sequence(xi)i∈N∗ as limn→∞ni=1xi .Definition 2.3. Let Δ+ be the class of all distribution of functions F : [0, ∞] ® [0, 1]such that:(1) F (0) = 0,(2) F is a non-decreasing,(3) F is left continuous mapping on [0, ∞].D+ is the subset of Δ+ which limx®∞F(x) = 1.Definition 2.4. The ordered pair (S, F) is said to be a probabilistic metric space if Sis a nonempty set and F : S × S ® D+ (F(p, q) written by Fpq for every (p, q) Î S × S)satisfies the following conditions:(1) Fuv(x) = 1 for every x > 0 ⇒ u = v (u, v Î S),(2) Fuv = Fvu for every u, v Î S,(3) Fuv (x) = 1 and Fvw(y) = 1 ⇒ Fu,w(x + y) = 1 for every u, v,w Î S, and every x, y ÎR+.A Menger space is a triple (S, F, T) where (S, F) is a probabilistic metric space, T is atriangular norm (abbreviated t-norm) and the following inequality holds Fuv(x + y) ≥ T(Fuw(x), Fwv(y)) for every u, v, w Î S, and every x, y Î R+.Definition 2.5. Let  : (0, 1) ® (0, 1) be a mapping, we say that the t-norm T is-convergent if∀δ ∈ (0, 1)∀λ ∈ (0, 1) ∃s = s(δ,λ) ∈ N ni=1(1 − ϕs+i(δ)) > 1 − λ,∀n ≥ 1.Definition 2.6. A sequence (xn)nÎ N is called a convergent sequence to x Î S if forevery ε > 0 and l Î (0, 1) there exists N = N(ε, l) Î N such thatFxnx(ε) > 1 − λ, ∀n ≥ N.Definition 2.7. A sequence (xn)nÎ N is called a Cauchy sequence if for every ε > 0and l Î (0, 1) there exists N = N(ε, l)Î N such thatFxnxn+m(ε) > 1 − λ, ∀n ≥ N ∀m ∈ N.We also havexn→Fx ⇔ Fxnx(t) → 1 ∀t > 0.A probabilistic metric space (S, F, T) is called sequentially complete if every Cauchysequence is convergent.In the following, 2S denotes the class of all nonempty subsets of the set S and C(S) isthe class of all nonempty closed (in the F-topology) subsets of S.Definition 2.8 [14]. Let F be a probabilistic distance on S and M Î 2S. A mapping f:S ® 2S is called continuous if for every ε > 0 there exists δ > 0, such thatFuv(δ) > 1 − δ ⇒ ∀x ∈ fu ∃y ∈ fv : Fxy(ε) > 1 − ε.Theorem 2.1 [14]. Let (S, F, T) be a complete Menger space, sup 0≤ t < 1T (t, t) = 1and f : S ® C(S) be a continuous mapping. If there exist a sequence (tn)nÎN ⊂ (0, ∞)with∑∞1 tn < ∞ and a sequence (xn) nÎN ⊂ S with the properties:xn+1 ∈ f xn for all n and limn→∞ ∞i=1gn+i−1 = 1,Beitollahi and Azhdari Fixed Point Theory and Applications 2012, 2012:10http://www.fixedpointtheoryandapplications.com/content/2012/1/10Page 2 of 8Where gn := Fxnxn+1(tn), then f has a fixed point.The concept of (ψ, , ε, l) - B contraction has been introduced by Mihet [15]. Wewill consider comparison functions from the class j of all mapping  : (0, 1) ® (0, 1)with the properties:(1)  is an increasing bijection;(2)  (l) <l ∀l Î (0, 1).Since every such a comparison mapping is continuous, it is easy to see that if  Î j,then limn®∞n(l) = 0 ∀l Î (0, 1).Definition 2.9[15]. Let (X, M, *) be a fuzzy Metric space. ψ be a map from (0, ∞) to(0, ∞) and  be a map from (0, 1) to (0, 1). A mapping f: X ® X is called (ψ, , ε, l)-contraction if for any x, y Î X, ε > 0 and l Î (0, 1).M(x, y, ε) > 1 − λ ⇒ M(f (x), f (y), ψ(ε)) > 1 − ϕ(λ).If ψ is of the form of ψ(ε) = kε (k Î (0, 1)), one obtains the contractive mapping con-sidered in [3].3 Main resultsIn this section we will generalize the Definition 2.9 to multi valued case in probabilisticmetric spaces.Definition 3.1. Let S be a nonempty set,  Î j, ψ be a map from (0, ∞) to (0, ∞)and F be a probabilistic distance on S. A mapping f : S ® 2S is called a multi-valued(ψ, , ε, l)-contraction if for every x, y Î S, ε > 0 and for all l Î (0, 1) the followingimplication holds:Fxy(ε) > 1 − λ ⇒ ∀p ∈ fx ∃q ∈ fy : Fpq(ψ(ε)) > 1 − ϕ(λ).Now, we need to define some conditions on the t-norm T or on the contractionmapping in order to be able to prove fixed point theorem. These two conditions areparallel. If one of them holds, Theorem 3.1 will obtain.Definition 3.2[11]. Let (S, F) be a probabilistic metric space, M a nonempty subsetof S and f : M ® 2S - {∅}, a mapping f is weakly demicompact if for every sequence(pn)nÎ N from M such that pn+1Î fpn, for every n Î N and lim Fpn+1,pn(ε) = 1 , for everyε > 0, there exists a convergent subsequence (pnj)j∈N.The other condition is mentioned in the Theorem 3.1.Theorem 3.1. Let (S, F, T) be a complete Menger space with sup 0 ≤ a < 1 T (a, a) =1, M Î C(S) and f : M ® C(M) be a multi-valued (ψ, , ε, l)-contraction, where theseries Σψn(ε) is convergent for every ε > 0 and  Î j. Let there exists x0 Î M and x1Î fx0 such that Fx0x1 ∈ D+ . If f is weakly demicompact orlimn→∞ ∞i=1(1 − ϕn+i−1(ε)) = 1 for every ε > 0 (1)then there exists at least one element x Î M such that x Î fx.Proof. Since there exists x0 Î M and x1 Î fx0 such that Fx0x1 ∈ D+ , hence for every lÎ (0, 1) there exists ε > 0 such that Fx0x1 > 1 − λ . The mapping f is a (ψ, , ε, l)-con-traction and therefore there exists x2 Î fx1 such thatFx2x1(ψ(ε)) > 1 − ϕ(λ)Beitollahi and Azhdari Fixed Point Theory and Applications 2012, 2012:10http://www.fixedpointtheoryandapplications.com/content/2012/1/10Page 3 of 8Continuing in this way we obtain a sequence (xn)nÎN from M such that for every n ≥ 2,xn Î fxn-1andFxn,xn−1 (ψn−1(ε)) > 1 − ϕn−1(λ). (2)Since the series Σψn(ε) is convergent we have limn®∞ψn(ε) = 0 and by assumption Î j, so limn®∞n(l) = 0. We infer for every ε0 > 0 thatlimn→∞ Fxnxn−1 (ε0) = 1. (3)Indeed, if ε0 > 0 and l0 Î (0, 1) are given, and n0 = n0(ε0, l0) is enough large suchthat for every n ≥ n0, ψn(ε) ≤ ε0 and n(l) ≤ l0 thenFxn+1xn(ε0) ≥ Fxn+1xn(ψn(ε)) > 1 − ϕn(λ) > 1 − λ0 for every n ≥ n0.If f is weakly demicompact (3) implies that there exists a convergent subsequence(xnk)k∈N .Suppose that (1) holds and prove that (xn)nÎN is a Cauchy sequence. This means thatfor every ε1 > 0 and every l1 Î (0, 1) there exists n1(ε1, l1) Î N such thatFxn+pxn(ε1) > 1 − λ1 (4)for every n1 ≥ n1(ε1, l1) and every p Î N.Let n2(ε1) Î N such that∑n≥n2(ε1)ψn(ε) < ε1. Since∑∞n=1 ψn(ε) is convergent ser-ies such a natural number n2(ε1) exists. Hence for every p Î N and every n ≥ n2(ε1) wehave thatFxn+p+1,xn(ε1) ≥ p+1i=1 Fxn+i,xn+i−1 (ψn+i−1(ε)),and (2) implies thatFxn+p+1,xn(ε1) ≥ p+1i=1 (1 − ϕn+i−1(λ))for every n ≥ n2(ε1) and every p Î N.For every p Î N and n ≥ n2(ε1)p+1i=1 (1 − ϕn+i−1(λ)) ≥ ∞i=1(1 − ϕn+i−1(λ))and therefore for every p Î N and n ≥ n2(ε1),Fxn+p+1,xn(ε1) ≥ ∞i=1(1 − ϕn+i−1(λ)). (5)From (1) it follows that there exists n3(l1) Î N such that∞i=1(1 − ϕn+i−1(λ)) > 1 − λ1 (6)for every n ≥ n3(l1). The conditions (5) and (6) imply that (4) holds for n1(ε1, l1) =max(n2(ε1), n3(l1)) and every p Î N. This means that (xn)nÎN is a Cauchy sequenceand since S is complete there exists limn®∞xn. Hence in both cases there exists(xnk)k∈N such thatlimk→∞xnk = x.Beitollahi and Azhdari Fixed Point Theory and Applications 2012, 2012:10http://www.fixedpointtheoryandapplications.com/content/2012/1/10Page 4 of 8It remains to prove that x Î fx. Since fx = fx it is enough to prove that x ∈ fx i.e.,for every ε2 > 0 and l2 Î (0, 1) there exists bε2,λ2 ∈ fx such thatFx,bε2,λ2 (ε2) > 1 − λ2. (7)Since supx< 1T(x, x) = 1 for l2 Î (0, 1) there exists δ(l2) Î (0, 1) such that T(1 -δ(l2), 1 - δ(l2)) > 1 - l2.If δ’(l2) is such thatT(1 − δ′(λ2), 1 − δ′(λ2)) > 1 − δ(λ2)and δ’’(l2) = min(δ(l2), δ’(l2)) we have thatT(1 − δ′′(λ2),T((1 − δ′′(λ2), 1 − δ′′(λ2))) ≥ T(1 − δ(λ2),T((1 − δ′(λ2), 1 − δ(λ2)))≥ T(1 − δ(λ2), 1 − δ(λ2))> 1 − λ2.Since limk→∞xnk = x there exists k1 Î N such that Fx,xnk( ε3)> 1 − δ′′(λ2) for everyk ≥ k1. Let k2 Î N such thatFxnk ,xnk+1(ε23)> 1 − δ′′(λ2) for every k ≥ k2.The existence of such a k2 follows by (3). Let ε Î R+ be such that ψ(ε) <ε23and k3Î N such that Fxnk ,x(ε) > 1 − δ′′(λ2) for every k ≥ k3. Since f is a (ψ, , ε, l)-contrac-tion there exists bε2,λ2,k ∈ fx such thatFxnk+1 ,bε2,λ2,k(ψ(ε)) > 1 − ϕ(δ′′(λ2)) for every k ≥ k3.Therefore for every k ≥ k3Fxnk+1, bε2,λ2,k(ε22)≥ Fxnk+1,bε2,λ2,k(ψ(ε))> 1 − ϕ(δ′′(λ2))> 1 − δ′′(λ2)If k ≥ max(k1, k2, k3) we haveFx,bε2,λ2,k(ε2) ≥ T(Fx,xnk(ε23), T(Fxnk ,xnk+1(ε23), Fxnk+1,bε2,λ2,k(ε23)))T(1 − δ′′(λ2), T(1 − δ′′(λ2), 1 − δ′′(λ2)))> 1 − λ2and (7) is proved for bε2,λ2 = bε2,λ2,k, k ≥ max(k1, k2, k3). Hence x ∈ fx = fx, whichmeans x is a fixed point of the mapping f.Now, suppose that instead of Σψn(ε) be convergent series, ψ is increasing bijection.Theorem 3.2. Let (S, F, T) be a complete Menger space with sup 0 ≤ a < 1 T (a, a) =1 and f : S ® C(S) be a multi-valued (ψ, , ε, l)- contraction.If there exist p Î S and q Î fp such that Fpq Î D+, ψ is increasing bijection andlimn→∞∞i=1(1 − ϕn+i−1(λ)) = 1 , for every l Î (0, 1), then, f has a fixed point.Beitollahi and Azhdari Fixed Point Theory and Applications 2012, 2012:10http://www.fixedpointtheoryandapplications.com/content/2012/1/10Page 5 of 8Proof. Let ε > 0 be given and δ Î (0, 1) be such that δ < min{ε, ψ-1(ε)} or ψ(δ) <εsince ψ is increasing bijection. If Fuv(δ) > 1-δ then, due to (ψ, , ε, l)- contraction foreach x Î fu we can find y Î fv such that Fxy(ψ(δ)) > 1 - (δ), from where we obtainthat Fxy(ε) >Fxy(ψ(δ)) > 1 - (δ) > 1 - δ > 1 - ε. So f is continuous. Next, let p0 = p andp1 = q be in fp0. Since Fpq Î D+, hence for every l Î (0, 1) there exist ε > 0 such thatFpq(ε) > 1 - l, namely Fp0p1 (ε) > 1 − λ .Using the contraction relation we can find p2 Î fp1 such thatFp1p2 (ψ(ε)) > 1 − ϕ(λ) , and by induction, pn such that pn Î fpn-1andFpn−1pn(ψn−1(ε)) > 1 − ϕn−1(λ) for all n ≥ 1. Defining tn = ψn(ε), we havegj = Fpjpj+1(tj) ≥ 1 − ϕj(λ) , ∀j, so limn→∞∞i=1gn+i−1 ≥ limn→∞∞i=1(1 − ϕn+i−1(λ)) = 1.On the other hand the sequence (pn) is a Cauchy sequense, that is:∀ε > 0 ∃n0 = n 0(ε) ∈ N : Fpnpn+m(ε) > 1− ∈,∀n ≥ n0,∀m ∈ N.Suppose that ε > 0, then:limn→∞ ∞i=1gn+i+1 = 1 ⇒ ∃n1 = n1(ε) ∈ N : mi=1gn+i−1 > 1 − ε, ∀n ≥ n1, ∀m ∈ N.Since the series∑∞n=1 tn is convergent, there exists n2(= n2(ε)) such that∑∞n=n2 tn < ε .Let n0 = max{n1, n2}, then for all n ≥ n0 and m Î N we have:Fpnpn+m(ε) ≥ Fpnpn+m(n+m−1∑i=nti)≥ mi=1Fpn+i−1pn+1 (tn+i−1)= mi=1gn+i−1 > 1 − ε,as desired.Now we can apply Theorem 2.1 to find a fixed point of f. The theorem is proved. □When ψ is increasing bijection and limn®∞ψn(l) be zero, by using demicompact con-traction we have another theorem.Theorem 3.3. Let (S, F, T) be a complete Menger space, T a t-norm such that sup 0≤ a < 1T (a, a) = 1, M a non-empty and closed subset of S, f : M ® C(M) be a multi-valued (ψ, , ε, l)- contraction and also weakly demicompact. If there exist x0 Î Mand x1 Î fx0 such that Fx0x1 ∈ D+,ψ is increasing bijection and limn®∞ψ (l) = 0 then,f has a fixed point.Proof. We can construct a sequence (pn)nÎ N from M, such that p1 = x1 Î fx0, pn+1Îfpn. Given t > 0 and l Î (0, 1), we will show thatlimn→∞ Fpn+1pn(t) = 1. (11)Indeed, since Fx0x1 ∈ D+ , hence for every ξ > 0 there exist h > 0 such thatFx0x1(η) > 1 − ξ , and by induction Fpn−1pn(ψn(η)) > 1 − ϕn(ξ) for all n Î N. Bychoosing n such that ψn(h) <t and n(ξ) <l, we obtainFpn+1pn(t) > 1 − λ.Since t and l are arbitrary, the proof of (1) is complete.Beitollahi and Azhdari Fixed Point Theory and Applications 2012, 2012:10http://www.fixedpointtheoryandapplications.com/content/2012/1/10Page 6 of 8By Definition 3.2, there exists a subsequence (pnj)j∈N such that limj→∞pnj exists. Weshall prove that x = limj→∞pnj is a fixed point of f. Since fx is closed, fx = fx , andtherefore, it remains to prove that x = fx , i.e., for every ε > 0 and l Î (0, 1), there existb(ε, l) Î fx, such that Fx,b(ε,l)(ε) > 1 - l. From the condition sup 0 ≤ a < 1T (a, a) = 1 itfollows that there exists h(l) Î (0, 1) such thatu > 1 − η(λ) ⇒ T(u, u) > 1 − λ.Let j1(ε, l) Î N be such thatFpnj ,x(ψ−1( ε2))> 1 − η(λ)2for every j ≥ j1(ε,λ).Since x = limj→∞pnj , such a number j1(ε, l) exists. Since f is (ψ, , ε, l)-contractionand ψ is increasing bijection, for pnj+1 ∈ f pnj there exists bj(ε)Î fx such thatFpnj+1,bj(ε)( ε2)> 1 − ϕ(η(λ)2)> 1 − η(λ)2for every j ≥ j1(ε,λ).From (1) it follows that limj→∞pnj+1 = x and therefore, there exists j2(ε, l) Î N suchthat Fx,pnj+1( ε2)> 1 − η(λ)2for every j ≥ j2(ε, l). Let j3(ε, l) = max{j1(ε, l), j2(ε, l)}.Then, for every j ≥ j3(ε, l) we have Fx,bj(ε)(ε) ≥ T(Fx,pnj+1(ε2), Fpnj+1,bj(ε)(ε2))> 1 − λ .Hence, if j >j3(ε, l), then, we can choose b(ε, l) = bj(ε)Î fx. The proof is complete. □Author details1Department of Statistics, Roudehen Branch, Islamic Azad University, Roudehen, Iran 2Department of Statistics, North-Tehran Branch, Islamic Azad University, Tehran, IranAuthors’ contributionsPA defined the definitions and wrote the introduction, preliminaries and abstract. AB proved the theorems. AB hasapproved the final manuscript. Also PA has verified the final manuscriptCompeting interestsThe authors declare that they have no competing interests.Received: 28 October 2011 Accepted: 8 February 2012 Published: 8 February 2012References1. Mihet, D: A class of Sehgal’s contractions in probabilistic metric spaces. An Univ Vest Timisoara Ser Mat Inf. 37, 105–110(1999)2. Hadžić, O, Pap, E: New classes of probabilistic contractions and applications to random operators. In: YJ, Cho, JK, Kim,SM, Kong (eds.) Fixed Point Theory and Application. pp. 97–119. Nova Science Publishers, Hauppauge, New York (2003)3. Mihet, D: A Banach contraction theorem in fuzzy metric spaces. Fuzzy Sets Syst. 144, 431–439 (2004). doi:10.1016/S0165-0114(03)00305-14. Mihet, D: A note on a paper of Hadzic and Pap. In: YJ, Cho, JK, Kim, SM, Kang (eds.) Fixed Point Theory andApplications, vol. 7, pp. 127–133. Nova Science Publishers, New York (2007)5. Hadžić, O, Pap, E: Fixed point theorem for multi-valued probabilistic ψ-contractions. Indian J Pure Appl Math. 25(8),825–835 (1994)6. Pap, E, Hadžić, O, Mesiar, RA: Fixed point theorem in probabilistic metric space and an application. J Math Anal Appl.202, 433–449 (1996). doi:10.1006/jmaa.1996.03257. Hadžić, O, Pap, E: A fixed point theorem for multivalued mapping in probabilistic Metric space and an application infuzzy metric spaces. Fuzzy Sets Syst. 127, 333–344 (2002). doi:10.1016/S0165-0114(01)00144-08. Žikić-Došenović, T: A multivalued generalization of Hicks C-contraction. Fuzzy Sets Syst. 151, 549–562 (2005).doi:10.1016/j.fss.2004.08.0119. Mihet, D: A fixed point theorem in probabilistic metric spaces. The Eighth International Conference on AppliedMathematics and Computer Science, Automat. Comput. Appl. Math 11(1), 79–81 (2002). Cluj-NapocaBeitollahi and Azhdari Fixed Point Theory and Applications 2012, 2012:10http://www.fixedpointtheoryandapplications.com/content/2012/1/10Page 7 of 810. Beitollahi, A, Azhdari, P: Multi-valued contractions theorems in probabilistic metric space. Int J Math Anal. 3(24),1169–1175 (2009)11. Hadžić, O, Pap, E: Fixed point theory in PM spaces. Kluwer Academic Publishers, Dordrecht (2001)12. Klement, EP, Mesiar, R, Pap, E: Triangular Norm. In Trend in Logic, vol. 8,Kluwer Academic Publishers, Dordrecht (2000)13. Schweizer, B, Sklar, A: Probabilistic Metric Spaces. North-Holland, Amesterdam (1983)14. Mihet, D: Multi-valued generalization of probabilistic contractions. J Math Anal Appl. 304, 464–472 (2005). doi:10.1016/j.jmaa.2004.09.03415. Mihet, D: A class of contractions in fuzzy metric spaces. Fuzzy Sets Syst. 161, 1131–1137 (2010). doi:10.1016/j.fss.2009.09.018doi:10.1186/1687-1812-2012-10Cite this article as: Beitollahi and Azhdari: Multi-valued (ψ, , ε, l)-contraction in probabilistic metric space. FixedPoint Theory and Applications 2012 2012:10.Submit your manuscript to a journal and benefi t from:7 Convenient online submission7 Rigorous peer review7 Immediate publication on acceptance7 Open access: articles freely available online7 High visibility within the fi eld7 Retaining the copyright to your article    Submit your next manuscript at 7 springeropen.comBeitollahi and Azhdari Fixed Point Theory and Applications 2012, 2012:10http://www.fixedpointtheoryandapplications.com/content/2012/1/10Page 8 of 8

Multi-valued (ψ, φ, ε, λ)-contraction in probabilistic metric space

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Multi-valued (ψ, φ, ε, λ)-contraction in probabilistic metric space

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