JID:FSS AID:6527 /FLA [m3SC+; v 1.191; Prn:30/04/2014; 10:23] P.1 (1-9)

Available online at www.sciencedirect.com

ScienceDirect

Fuzzy Sets and Systems ••• (••••) •••–•••www.elsevier.com/locate/fss

A note on fuzzy contractive mappings in fuzzy metric spaces

Dorel Mihet

West University of Timisoara, Faculty of Mathematics and Computer Science, Bv. V. Parvan 4, 300223, Timisoara, Romania

Received 24 August 2013; received in revised form 11 April 2014; accepted 12 April 2014

Abstract

We extend the class of fuzzy H-contractive mappings from Wardowski (2013) [23] and establish some fixed point results forfuzzy H contractive mappings in M-complete fuzzy metric spaces in the sense of Kramosil and Michalek under Archimedeant-norms.© 2014 Published by Elsevier B.V.

Keywords: Fuzzy metric space; Archimedean t-norm; Fuzzy contractive mapping

1. Introduction and preliminaries

The study of the fixed point theory in fuzzy metric spaces started with the paper of Grabiec [3]. Later on, theconcept of fuzzy contractive mapping, initiated by Gregori and Sapena in [6], has become of interest for many authorssee, e.g., the papers [4,8,12–14,22,24].

In [6] the authors proved their results by using a strong condition for completeness, namely G-completeness.Recently, Wardowski ([23]) introduced the concept of fuzzy H-contractive mapping, as a generalization of that offuzzy contractive mapping, and established the conditions guaranteeing the existence and the uniqueness of fixedpoints for this type of contractions in M-complete fuzzy metric spaces in the sense of George and Veeramani.

The aim of this paper is to consider a larger class of H-contractive mappings in the setting of M-complete fuzzymetric spaces in the sense of Kramosil and Michalek as well as in the setting of M-complete strong fuzzy metricspaces. Among particular cases of our theorems we mention the results of Sherwood (see [1]) and Tirado ([20]).

For reader’s convenience we recall some basic definitions and the properties from the theory of fuzzy metric spaces,used in the sequel.

Definition 1.1. (See [18].) A binary operation ∗ : [0,1]2 → [0,1] is called a t-norm if the following conditions aresatisfied

(1) a ∗ b = b ∗ a;

E-mail address: mihet@math.uvt.ro.

http://dx.doi.org/10.1016/j.fss.2014.04.0100165-0114/© 2014 Published by Elsevier B.V.

JID:FSS AID:6527 /FLA [m3SC+; v 1.191; Prn:30/04/2014; 10:23] P.2 (1-9)

2 D. Mihet / Fuzzy Sets and Systems ••• (••••) •••–•••

(2) a ∗ b ≤ c ∗ d for a ≤ c, b ≤ d ;(3) (a ∗ b) ∗ c = a ∗ (b ∗ c);(4) a ∗ 1 = a;

for all a, b, c, d ∈ [0,1].

A t-norm ∗ is continuous if for all convergent sequences (xn)n∈N, (yn)n∈N,

limn→∞xn ∗ lim

n→∞yn = limn→∞(xn ∗ yn).

Three basic continuous t-norms are Minimum t-norm ∗M , Product t-norm ∗P , x ∗M y = min{x, y}, x ∗P y = xy

and the Łukasiewicz t-norm ∗L, xT∗y = max(x + y − 1,0).A continuous t-norm ∗ is called Archimedean if x ∗ x < x. It is well known that ∗ is a continuous Archimedean

t-norm iff there exists a continuous and strictly decreasing function f : [0,1] → [0,∞] with f (1) = 0 such thatx ∗ y = f (−1)(f (x) + f (y)), where f (−1) denotes the pseudo-inverse of f , i.e.

f (−1)(y) ={

f −1(y), y ∈ [0, f (0)]0, y ∈ [f (0),∞].

The function f is called an additive generator of ∗. It is unique up to multiplication with positive numbers. Theadditive generator satisfies f (−1)(f (x)) = x (x ∈ [0,1]) and f (f (−1)(y)) = min{y,f (0)} (y ∈ [0,∞]).

Definition 1.2. A t-norm ∗ is said to be strict if it is strictly increasing in each place on (0,1]2 and it is called nilpotentif a ∗ a = 0 for some a ∈ (0,1).

A continuous t-norm ∗ is strict iff its additive generator satisfies f (0) = ∞ and nilpotent iff f (0) < ∞. In thenilpotent (non-strict) case, the additive generator with f (0) = 1 is called the normed generator. It turns out that anycontinuous Archimedean t-norm is either strict or nilpotent.

Any continuous strict t-norm ∗ is positive that is,(∀a, b ∈ (0,1]) a, b > 0 ⇒ a ∗ b > 0.

Example 1.3. The t-norm ∗P is a strict t-norm generated by the function f : [0,1] → [0,∞], f (0) = ∞, g(s) = − ln s

(s = 0).The Hamacher product ∗H ,

∗H (x, y) = xy

x + y − xy

has the additive generator g(t) = 1t− 1.

The nilpotent t-norm ∗L has the normed generator f (x) = 1 − x.

For a1, a2, ..., an ∈ [0,1] and n ∈N, the product a1 ∗a2 ∗ ...∗an will be denoted by (∗)ni=1ai . The following equalitytakes place:

(∗g)ni=1xi = g(−1)

n∑i=1

g(xi).

More details concerning t-norms can be found in the monographs [7] and [10].The notion of fuzzy metric space was introduced by Kramosil and Michalek [11] and later modified by Kaleva and

Seikkala [9] and by George and Veeramani [2]. In this paper we work in fuzzy metric spaces in the sense of Kramosiland Michalek.

Definition 1.4. (See [11].) A fuzzy metric space (in the sense of Kramosil and Michalek) is a triple (X,M,∗), whereX is a nonempty set, � is a continuous t-norm and M : X ×X ×[0,∞) → [0,1] is a mapping satisfying the followingaxioms:

JID:FSS AID:6527 /FLA [m3SC+; v 1.191; Prn:30/04/2014; 10:23] P.3 (1-9)

D. Mihet / Fuzzy Sets and Systems ••• (••••) •••–••• 3

(KM1) M(x,y,0) = 0 ∀x, y ∈ X;(KM2) M(x,y, t) = 1 ∀t > 0 iff x = y;(KM3) M(x,y, t) = M(y,x, t) ∀x, y ∈ X and t > 0;(KM4) M(x,y, ·) : [0,∞) → [0,1] is left continuous ∀x, y ∈ X;(KM5) M(x, z, t + s) ≥ M(x,y, t) ∗ M(y, z, s), ∀x, y, z ∈ X, ∀t, s > 0.

Definition 1.5. A space (X,M,∗) satisfying (KM1)–(KM4) and

(S) M(x, z, t) ≥ M(x,y, t) ∗ M(y, z, t) for all x, y, z ∈ X, t > 0,

is called a strong fuzzy metric space [5].

Every metric space (X,d) induces a fuzzy metric space (X,Md,∗M), with Md(x, y, t) = tt+d(x,y)

.

Definition 1.6. (See [2].) Let (X,M,∗) be a fuzzy metric space. A sequence (xn) in X is said to be Cauchy if for eachε ∈ (0,1) and t > 0 there exists n0 ∈ N such that M(xn, xm, t) > 1 − ε for all m > n ≥ n0. The sequence (xn) in X isconvergent if there exists x ∈ X such that limn→∞ M(xn, x, t) = 1,∀t > 0.

Definition 1.7. An M-complete fuzzy metric space is a fuzzy metric space in which every Cauchy sequence is con-vergent.

There is another (stronger) concept of completeness of a fuzzy metric space, G-completeness, introduced by Gra-biec in [3]. For a comparison between these two types of completeness the reader is referred to the paper [21].

Lemma 1.8. (See [3].) Let (X,M,∗) be a fuzzy metric space and let (xn), (yn) be sequences in X.If xn → x, yn → y, then M(xn, yn, t) → M(x,y, t),∀t > 0.

The following class of fuzzy H-contractive mappings has been recently introduced by Wardowski in [23].

Definition 1.9. (See [23].) Denote by H the family of all onto and strictly decreasing mappings η : (0,1] → [0,∞).Let (X,M,∗) be a fuzzy metric space. A mapping T : X → X is said to be fuzzy H-contractive with respect to η ∈ Hif there exists k ∈ (0,1) satisfying

η(M(T x,T y, t)

) ≤ kη(M(x,y, t)

), ∀x, y ∈ X ∀t > 0.

For η(t) = 1t− 1 one obtains the class of fuzzy contractive mappings introduced by Gregori and Sapena in [6].

In [23] Wardowski also proved the following fixed point theorem for H-contractive mapping in M-complete fuzzymetric spaces in the sense of George and Veeramani:

Theorem 1.10. (See [23].) Let (X,M,∗) be an M-complete fuzzy metric space and let T : X → X be a fuzzyH-contractive mapping with respect to η ∈H such that:

(a) (∗)mM(x,T x, ti) = 0, for all x ∈ X, m ∈ N and any sequence (tn) ⊆ (0,∞), tn ↓ 0;(b) r ∗ s > 0 ⇒ η(r ∗ s) ≤ η(r) + η(s), for all r, s ∈ {M(x,T x, t) : x ∈ X, t > 0};(c) {η(M(x,T x, ti)) : i ∈N} is bounded for all x ∈ X and any sequence (tn) ⊆ (0,∞), tn ↓ 0.

Then T has a unique fixed point x∗ ∈ X and for each x0 ∈ X the sequence (T nx0)n∈N converges to x∗.

2. Main results

The starting point of our paper is the remark that if η ∈ H then η(1) = 0 and η is continuous and the condition (b)in Theorem 1.10 can be written as r ∗ s ≤ η−1(η(r) + η(s)), where η−1 is the inverse of η, which is closely related tothe representation of a strict t-norm.

JID:FSS AID:6527 /FLA [m3SC+; v 1.191; Prn:30/04/2014; 10:23] P.4 (1-9)

4 D. Mihet / Fuzzy Sets and Systems ••• (••••) •••–•••

By removing the requirement “onto” from the definition of the family H we obtain the following definition, natu-rally generalizing Definition 1.9 in connection with the representation of a continuous Archimedean t-norm.

Definition 2.1. Let H be the family of all continuous, strictly decreasing mappings η : [0,1] → [0,∞], with η(1) = 0and (X,M,∗) be a fuzzy metric space. A mapping T : X → X is said to be fuzzy H-contractive with respect to η ∈ Hif there exists k ∈ (0,1) satisfying

η(M(T x,T y, t)

) ≤ kη(M(x,y, t)

), ∀x, y ∈ X ∀t > 0.

If η(0) = ∞, then one gets Wardowski’s definition.As a matter of fact, as one can be seen in the sequel, the removed condition turns out to be restrictive: by using a

similar approach to that in the paper of Wardowski we will be able to obtain more general results, including the caseof nilpotent t-norms.

Example 2.2. Let (X,d) be a metric space and

M(x,y, t) = exp

(−d(x, y)

t

).

Then (X,M,∗P ) is a strong fuzzy metric space and a mapping T : X → X is fuzzy H-contractive with respect toη(t) = − ln t iff it is a Banach contraction in (X,d).

Example 2.3. (See [20].) Let (X,M,∗) be a fuzzy metric space and T be a self-mapping of X with the property thatthere exists k ∈ (0,1) such that 1 − M(f x,fy, t) ≤ k(1 − M(x,y, t)) for all x, y ∈ X and t > 0. Then T is fuzzyH-contractive with respect to η ∈ H, η(t) = 1 − t .

2.1. H-fuzzy contractive mappings in fuzzy metric spaces under strict t-norms

Our first theorem is a counterpart of Theorem 1.10, in the setting of fuzzy metric spaces in the sense of Kramosiland Michalek.

Theorem 2.4. Let ∗g be a strict t-norm and (X,M,∗) be an M-complete fuzzy metric space in the sense ofKramosil and Michalek with ∗ ≥ ∗g . Then every fuzzy g-contractive mapping T : X → X has a fixed point, providedM(x,T x,0+) > 0 for some x ∈ X.

Proof. Let x ∈ X be such that M(x,T x,0+) > 0. Consider the sequence (xn)n∈N of iterates of x, xn = T nx. Then,for all t > 0, n ∈ N,

g(M(xn, xn+1, t)

) ≤ kng(M(x,T x, t)

).

Let now m,n ∈ N,m < n and t > 0 be given and (ak) be a strictly decreasing sequence of positive numbers with∑∞i=1 ai = 1. Then

M(xm,xn, t) ≥ M

(xm,xn,

n−1∑i=m

ait

)≥ (∗)n−1

i=mM(xi, xi+1, ai t)

≥ (∗g)n−1i=mM(xi, xi+1, ai t).

This implies

g(M(xm,xn, t)

) ≤ g((∗)n−1

i=mM(xi, xi+1, ai t)) ≤

n−1∑i=m

g(M(xi, xi+1, ai t)

)

≤n−1∑i=m

kig(M(x,T x, ai t)

) ≤ g(M(x,T x,0+)

) n−1∑i=m

ki,

proving that (xn) is Cauchy.

JID:FSS AID:6527 /FLA [m3SC+; v 1.191; Prn:30/04/2014; 10:23] P.5 (1-9)

D. Mihet / Fuzzy Sets and Systems ••• (••••) •••–••• 5

Let x∗ = limn→∞ xn. As g(M(xn+1, T x∗, t)) ≤ kg(M(xn, x∗, t)) for all n ∈ N and t > 0, it follows that xn+1converges to T x∗, that is, x∗ = T x∗. �Example 2.5. Let X = [0,∞) endowed with the Euclidean distance and (X,M,TM) be the induced fuzzy metricspace. Then (X,M,∗M) is M complete and the mapping T : X → X, T x = x

2 is H-contractive with respect toη(t) = 1

t− 1.

As the strongest t-norm TM is greater than the Hamacher product ∗H having the additive generator g(t) = 1t− 1

and the condition M(x,T x,0+) > 0 holds for x = 0, we can apply Theorem 2.4 to obtain that 0 is (the unique) fixedpoint of T .

An H-fuzzy contractive mapping may have more than a fixed point.

Example 2.6. Let X = [0,∞) and M(x,y, t) ={

0, t≤11, t>1.

Then (X,M,TM) is a complete fuzzy metric space in the sense of Kramosil and Michalek and the mapping T :X → X,T x = x is H-fuzzy contractive mapping with respect to any η ∈H, having any element of X as a fixed point.

It is worth mentioned that the positivity condition

M(T x,T y, t) > 0 (x, y ∈ X, t > 0)

assures the uniqueness of a fixed point for H contraction.

Proposition 2.7. If M(T x,T y, t) > 0 for all x, y ∈ X, t > 0, then any fuzzy H-contractive mapping T has at mostone fixed point.

Indeed, let x∗ and y∗ be fixed points of T . Then

g(M

(x∗, y∗, t

)) = g(M

(T x∗, T y∗, t

)) ≤ kg(M

(x∗, y∗, t

))for any t > 0.

Since M(x∗, y∗, t) = 0, it follows that M(x∗, y∗, t) = 1 ∀t > 0, hence x∗ = y∗.

We are going to emphasize a special subclass of H-fuzzy contractive mappings, having the uniqueness fixed pointproperty.

Example 2.8 (Contraction mappings of Sherwood’s type). (See [19].) As g(M(T x,T y, t)) ≤ g(M(T x,T y, kt)) forany k ∈ (0,1), it follows that if (X,F,∗g) is a probabilistic metric space, then any mapping T : X → X for whichthere exists k ∈ (0,1) such that

g(M(T x,T y, kt)

) ≤ kg(M(x,y, t)

)for all x, y ∈ X and t > 0 is fuzzy H-contractive with respect to g.

We will refer to such a contraction as a Sherwood contraction with respect to g.

A remarkable Sherwood contraction is obtained for g(t) = 1 − t .

Definition 2.9. (See [7], Definition 3.36.) A mapping T : X → X satisfying

1 − M(T x,T y, qt) ≤ q(1 − M(x,y, t)

) (q ∈ (0,1)

)for all x, y ∈ X and t > 0 is called a strong probabilistic q-contraction.

Theorem 2.10. Let ∗g be a strict t-norm and (X,M,∗) be an M-complete fuzzy metric space with ∗ ≥ ∗g , with theproperty that for every x, y ∈ X there exists t > 0 such that M(x,y, t) > 0 and T : X → X be a Sherwood fuzzycontraction with respect to g, such that M(x,T x,0+) > 0 for some x ∈ X. Then T has a unique fixed point.

JID:FSS AID:6527 /FLA [m3SC+; v 1.191; Prn:30/04/2014; 10:23] P.6 (1-9)

6 D. Mihet / Fuzzy Sets and Systems ••• (••••) •••–•••

Proof. We have to prove only the uniqueness of the fixed point. Suppose that p,q ∈ X are such that Tp = p,T q = q .If 0 < M(p,q, t) < 1 for some t > 0, then

g(M(p,q, t)

) = g(M(Tp,T q, t)

) ≤ kg

(M

(p,q,

t

k

))< g

(M(p,q, t)

),

which is a contradiction.Therefore, for every t > 0, either M(p,q, t) = 0 or M(p,q, t) = 1.Let us assume that M(p,q, t) = 0 for some t > 0 and let

α = sup{t > 0 : M(p,q, t) = 0

}.

Then α < ∞ and αk

> α, hence

g(M(p,q,α)

) ≤ kg

(M

(p,q,

α

k

))= kg(1) = 0.

This implies M(p,q,α) = 1, contradicting the left continuity of M(p,q, ·) at α. Therefore M(p,q, t) = 1 ∀t > 0,i.e. p = q . �

In strong fuzzy metric spaces the condition M(x,T x,0+) > 0 can be replaced by a weaker one, M(x,T x, t) > 0,

∀t > 0.

Theorem 2.11. Let ∗g be a strict t-norm and (X,M,∗) be an M-complete strong fuzzy metric space under a t-norm∗ ≥ ∗g .

If T : X → X is an H-contractive mapping with respect to g such that M(x,T x, t) > 0 ∀t > 0 for some x ∈ X,then the sequence (T nx)n∈N converges to a fixed point of T .

Proof. The lines of the proof are the same as those in Theorem 2.4. Let x ∈ X be such that M(x,T x, t) > 0 ∀t > 0and xn = T nx (n ∈ N).

From the triangle inequality (S) and ∗ ≥ ∗g it follows

M(xm,xn, t) ≥ (∗)n−1i=mM(xi, xi+1, t)

≥ (∗g)n−1i=mM(xi, xi+1, t) = g−1

n−1∑i=m

g(M(xi, xi+1, t)

)for all m,n ∈N,m < n and t > 0.

As

g(M(xn, xn+1, t)

) ≤ kng(M(x,T x, t)

)(n ∈ N, t > 0),

it follows that

g(M(xm,xn, t)

) ≤n−1∑i=m

g(M(xi, xi+1, t)

)

≤n−1∑i=m

kig(M(x,T x, t)

) = g(M(x,T x, t)

) n−1∑i=m

ki,

that is, (xn) is Cauchy.Now we can continue as in the proof of the preceding theorem to show that x∗ = limn→∞ xn is a fixed point

for T . �Example 2.12. Let X = [0,∞) and M : X2 × [0,∞) → [0,1], M(x,y, t) = exp(−|x−y|

t). Then (X,M,∗P ) is a

complete strong fuzzy metric space, satisfying the positivity condition M(x,y, t) > 0, ∀t > 0. Besides, the mappingT : X → X, T x = x

2 is fuzzy H-contractive with respect to η(t) = − ln t (see Example 2.2) and limn→∞ T nx = 0 forall x ∈ X. We can apply Theorem 2.11 to conclude that x = 0 is the unique fixed point of T .

JID:FSS AID:6527 /FLA [m3SC+; v 1.191; Prn:30/04/2014; 10:23] P.7 (1-9)

D. Mihet / Fuzzy Sets and Systems ••• (••••) •••–••• 7

2.2. H-fuzzy contractive mappings in fuzzy metric under nilpotent t-norms

Definition 2.1 includes the case of nilpotent t-norms. In this case the condition: M(x,T x,0+) > 0 for some x ∈ X

from Theorem 2.4 can be removed.

Theorem 2.13. Let ∗g be a nilpotent t-norm and (X,M,∗) be an M-complete fuzzy metric space under a t-norm∗ ≥ ∗g . Then every fuzzy g-contractive mapping on X has a fixed point.

Proof. Let x ∈ X and (xn)n∈N, xn = T nx be the sequence of iterates of x. We prove that (xn) is a Cauchy sequence.Indeed, let m,n ∈N,m < n and t > 0 be given and {ai} be a strictly decreasing sequence of positive numbers with∑∞i=1 ai = 1. Then

M(xm,xn, t) ≥ M

(xm,xn,

n−1∑i=m

ait

)≥ (∗)n−1

i=mM(xi, xi+1, ai t)

= g(−1)

(n−1∑i=m

g(M(xi, xi+1, ai t)

))

for all m,n ∈ N,m < n and t > 0, hence

g(M(xm,xn, t)

) ≤ g

(g(−1)

(n−1∑i=m

g(M(xi, xi+1, ai t)

))).

As

n−1∑i=m

g(M(xi, xi+1, ai t)

) ≤n−1∑i=m

kig(M(x,T x, ai t)

)

≤ g(M(x,T x,0+)

) n−1∑i=m

ki →m,n→∞ 0,

it follows that

min

{n−1∑i=m

g(M(xi, xi+1, ai t)

), g(0)

}=

n−1∑i=m

g(M(xi, xi+1, ai t)

),

that is,

g

(g(−1)

(n−1∑i=m

g(M(xi, xi+1, ai t)

)))=

n−1∑i=m

g(M(xi, xi+1, ai t)

),

concluding that (xn) is Cauchy.If x∗ = limn→∞ xn, then from g(M(xn+1, T x∗, t)) ≤ kg(M(xn, x∗, t)) it follows that (xn) converges to T x∗, that

is, x∗ = T x∗. �Corollary 2.14. Let ∗g be a nilpotent t-norm and (X,M,∗) be an M-complete fuzzy metric space with ∗ ≥ ∗g andT : X → X be a Sherwood fuzzy contraction with respect to g. Then T has a unique fixed point.

Corollary 2.15. (See Tirado, [20].) Let (X,M,∗L) be an M-complete fuzzy metric space. If T is a self-mapping ofX with the property that there exists k ∈ (0,1) such that 1 − M(T x,T y, t) ≤ k(1 − M(x,y, t)) for all x, y ∈ X andt > 0, then T has a unique fixed point.

JID:FSS AID:6527 /FLA [m3SC+; v 1.191; Prn:30/04/2014; 10:23] P.8 (1-9)

8 D. Mihet / Fuzzy Sets and Systems ••• (••••) •••–•••

The conclusion of Corollary 2.15 remains true in a fuzzy metric space endowed with a g-convergent t-norm, thatis, a t-norm ∗ with the property:

limn→∞(∗)∞i=n

(1 − qi

) = 1

for some q ∈ (0,1) (see [15]).As a consequence of Corollary 2.16 one obtains:

Corollary 2.16. (See [7].) Let (X,M,∗L) an M-complete fuzzy metric space and T : X → X be a strong probabilisticq-contraction. Then T has a unique fixed point.

In [7] it is shown that Corollary 2.16 still holds if ∗L is replaced by a g-convergent t-norm. For related results thereader is referred to the paper [16].

3. Conclusion

We naturally enlarge the class of an H-fuzzy contractive mappings introduced by Wardowski in [23]. This allows usto work in fuzzy metric spaces endowed with Archimedean t-norms and to investigate the existence and the uniquenessof fixed points for an H-fuzzy contractive mapping in connection with the structure of the t-norm of the space,including the case of nilpotent t-norms.

Our approach can be extended to other classes of contractive mappings (e.g., to quasi-contractions), in fuzzy metricor in fuzzy quasi-metric setting. Further investigations could also refer to the relationship between H-fuzzy contractivemappings and other classes of fuzzy contractions, such as fuzzy ψ -contractive mappings. (α − β − ϕ)-contractivemappings, ψ -contractions of (ε, λ)-type (see [14,15,17]).

Acknowledgements

This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0087.

References

[1] Gh. Constantin, I. Istratescu, Elements of Probabilistic Analysis with Applications, Ed. Academiei Bucureti, Kluwer Academic Publ., 1989.[2] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst. 64 (1994) 395–399.[3] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst. 27 (1988) 385–389.[4] Y.J. Cho, M. Grabiec, V. Radu, On Nonsymmetric Topological and Probabilistic Structures, Nova Publishers, 2006.[5] V. Gregori, S. Morillas, A. Sapena, On a class of completable fuzzy metric spaces, Fuzzy Sets Syst. 161 (2010) 2193–2205.[6] V. Gregori, A. Sapena, On fixed point theorems in fuzzy metric spaces, Fuzzy Sets Syst. 125 (2002) 245–252.[7] O. Hadžic, E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, Mathematics and Its Applications, vol. 536, Kluwer Academic Publish-

ers, Dordrecht, Boston, London, 2001.[8] K. Jha, M. Abbas, I. Beg, R.P. Pant, M. Imdad, Common fixed point theorem for (φ,ψ)-weak contraction in fuzzy metric spaces, Bull. Math.

Anal. Appl. 3 (2011) 149–158.[9] O. Kaleva, S. Seikkala, On fuzzy metric spaces, Fuzzy Sets Syst. 12 (1984) 215–229.

[10] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Trends in Logics, vol. 8, Kluwer Academic Publishers, Dordrecht, Boston, London, 2000.[11] I. Kramosil, J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika 11 (1975) 336–344.[12] D. Mihet, A Banach contraction theorem in fuzzy metric spaces, Fuzzy Sets Syst. 144 (2004) 431–439.[13] D. Mihet, On fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets Syst. 158 (2007) 915–921.[14] D. Mihet, Fuzzy ψ -contractive mappings in non-Archimedean fuzzy metric spaces, Fuzzy Sets Syst. 159 (2008) 739–744.[15] D. Mihet, A class of contractions in fuzzy metric spaces, Fuzzy Sets Syst. 161 (2010) 1131–1137.[16] D. Mihet, A class of probabilistic contractions, in: Y.J. Cho, J.K. Kim, S.M. Kang (Eds.), 8-th International Conference on Nonlinear Func-

tional Analysis and Applications, Gyeongsang Natl. Univ., Chinju, South Korea, 2004, in: Fixed Point Theory and Applications, vol. 6, 2007,pp. 131–135.

[17] P. Salini, C. Vetro, P. Vetro, Some new fixed point results in non-Archimedean fuzzy metric spaces, Nonlinear Anal. 18 (2013) 344–358.[18] B. Schweizer, A. Sklar, Statistical metric spaces, Pac. J. Math. 10 (1960) 313–334.[19] H. Sherwood, Complete probabilistic metric spaces and random variables generated spaces, Ph.D. Thesis, Univ. of Arizona, 1965.

JID:FSS AID:6527 /FLA [m3SC+; v 1.191; Prn:30/04/2014; 10:23] P.9 (1-9)

D. Mihet / Fuzzy Sets and Systems ••• (••••) •••–••• 9

[20] P. Tirado, Contraction mappings in fuzzy quasi-metric spaces and [0,1]-fuzzy posets, Fixed Point Theory 13 (1) (2012) 273–283.[21] R. Vasuki, P. Veeramani, Fixed point theorems and Cauchy sequences in fuzzy metric spaces, Fuzzy Sets Syst. 135 (2003) 415–417.[22] C. Vetro, Fixed points in weak non-Archimedean fuzzy metric spaces, Fuzzy Sets Syst. 162 (2011) 84–90.[23] D. Wardowski, Fuzzy contractive mappings and fixed points in fuzzy metric spaces, Fuzzy Sets Syst. 222 (2013) 108–114.[24] T. Zikic, On fixed point theorems of Gregori and Sapena, Fuzzy Sets Syst. 144 (2004) 421–429.