A class of contractions in fuzzy metric spaces

Fuzzy Sets and Systems 161 (2010) 1131 – 1137 www.elsevier.com/locate/fss

A class of contractions in fuzzy metric spaces Dorel Mihe¸t West University of Timi¸soara, Faculty of Mathematics and Computer Science, Bv. V. Parvan 4, 300223 Timi¸soara, Romania Received 15 September 2008; received in revised form 14 September 2009; accepted 21 September 2009 Available online 26 September 2009

Abstract Using the notion of geometrically convergent t-norms, a fixed point theorem in fuzzy metric spaces in the sense of Kramosil and Michalek for a class of contractions, larger than the class of (, )- contraction mappings, has been proved. © 2009 Elsevier B.V. All rights reserved. MSC: 54E70; secondary 54H25 Keywords: Fuzzy metric space; Probabilistic B-contraction; Fixed point; Geometrically convergent t-norm; Dombi, Aczel-Alsina and Sugeno-Weber families of t-norms

1. Introduction Fixed point theory in probabilistic and fuzzy metric spaces has been developing since the paper by Sehgal and Bharucha-Reid [22], where the notion of contraction mapping on probabilistic metric spaces (now known as a probabilistic B contraction) has been introduced. In their paper Sehgal and Bharucha-Reid proved that every B-contraction on a complete Menger space under the strongest triangular norm TM has a unique fixed point. Subsequently, Sherwood [23] showed that it is possible to construct a complete Menger space under the Łukasiewics t-norm TL and a contraction mapping on this space which has no fixed point. Later on, Radu [17] proved that the only continuous t-norms which could replace the t-norm TM in the theorem of Sehgal and Bharucha-Reid are the so-called t-norms of Hadži´c-type. Therefore, it is interesting to investigate classes of contractions having fixed points on every complete Menger space (S, F, TL ). Such a class was given in [12], where the notion of (, )-contraction on probabilistic metric spaces was introduced. The class of (, )-contraction mappings is a subclass of probabilistic B-contractions. Some fixed point theorems for this class of contractions in probabilistic Menger spaces or in fuzzy metric spaces in the sense of Kramosil and Michalek were obtained in [16,13,9]. Thus, in order to obtain a generalization of the result in [12], some results on the countable extension of t-norms was used in [9]. It turned out that the notion of countable extension of a triangular norm has important applications to fixed point theory in probabilistic and fuzzy metric spaces see, e.g., [3,5–9,13–15,21,25]. The aim of this paper is to show that the contractive condition in the abovementioned theorems can be weakened. Particularly, we prove the existence of a fixed point for this weaker contractions in complete fuzzy metric spaces E-mail address: [email protected]. 0165-0114/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2009.09.018

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under geometrically convergent t-norms. It should be noted that Łukasiewics t-norm, t-norms of Hadži´c-type and some subclasses of Dombi, Aczel-Alsina and Sugeno-Weber families of t-norms are geometrically convergent, see [6]. 2. Preliminaries The terminology used in the paper follows the books [5,19]. A triangular norm (shorter t-norm) is a binary operation T : [0, 1] × [0, 1] → [0, 1] which is commutative, associative, non-decreasing in each variable and has 1 as the unit element. Basic examples are the Łukasiewicz t-norm TL , TL (a, b) = Max(a + b − 1, 0) ∀a, b ∈ [0, 1] and the t-norms TP , TM , TD , where TP (a, b) := ab, TM (a, b) := Min{a, b},  min(x, y) if max(x, y) = 1, TD (a, b) := 0 otherwise. (n)

(n−1)

If T is a t-norm then x T is defined for every x ∈ [0, 1] and n ∈ N ∪ 0 by 1, if n = 0 and T (x T , x), if n ≥ 1. A (n) t-norm T is said to be of Hadži´c-type (we denote T ∈ H) if the family (x T )n∈N is equicontinuous at x = 1 [4]. Other important triangular norms are (see [6]): SW = T , T SW = T and • the Sugeno-Weber family {TSW }∈[−1,∞] is defined by T−1 D ∞ P   x + y − 1 + x y ( ∈ (−1, ∞)); TSW = max 0, 1+

• the Domby family {TD }∈[0,∞] , defined by TD , if  = 0, TM , if  = ∞ and TD (x, y) =

 1+

1 1−x x



 +

 1/ 1−y  y

( ∈ (0, ∞));

• the Aczel-Alsina family {TA A }∈[0,∞] , defined by TD , if  = 0, TM , if  = ∞ and TA A (x, y) = e−(| log x|

 +| log y| )1/

( ∈ (0, ∞)).

A t-norm T can be extended (by associativity) in a unique way to an n-ary operation taking for (x1 , . . . , xn ) ∈ [0, 1]n the value T (x1 , . . . , xn ) defined by n−1 0 n xi = 1, Ti=1 xi = T (Ti=1 xi , xn ) = T (x1 , . . . , xn ). Ti=1

T can also be extended to a countable operation taking for any sequence (xn )n∈N in [0,1] the value ∞ n xi = lim Ti=1 xi . Ti=1 n→∞

Proposition 2.1 (Hadži´c et al. [6]). (i) For T ≥ TL the following implication holds: ∞ lim Ti=1 xn+i = 1 ⇐⇒

n→∞

∞ 

(1 − xn ) < ∞.

n=1

(ii) If T is of Hadži´c-type then ∞ xn+i = 1 lim Ti=1

n→∞

for every sequence (xn )n∈N in [0,1] such that limn→∞ xn = 1.

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(iii) If T ∈ {TA A }∈(0,∞) ∪ {TD }∈(0,∞) , then ∞ xn+i = 1 ⇐⇒ lim Ti=1

n→∞

(iv) If T ∈

∞ 

(1 − xn ) < ∞.

n=1

{T SW } 

∈[−1,∞) ,

lim T∞ xn+i n→∞ i=1

then

= 1 ⇐⇒

∞ 

(1 − xn ) < ∞.

n=1

The notion of g-convergent t-norm has been introduced by Hadži´c et al. [6]. Definition 2.2. A t-norm T is said to be geometrically convergent (or g-convergent) if, for all q ∈ (0, 1), lim T ∞ (1 − q i ) n→∞ i=n

= 1.

Taking into account Proposition 1.1, we obtain that every member of the family:    {T D} {TA A } {TSW } G=H ∈(0,∞)

∈(0,∞)

∈(−1,∞)

is g-convergent. Various concepts of fuzzy metric spaces were considered in the literature see, e.g., [1,2,11]. A very general definition of a fuzzy metric space has been given by Kaleva and Seikkala [10]. In this paper we work in fuzzy metric spaces in the sense of Kramosil and Michalek, which are closely related to probabilistic metric spaces. Definition 2.3 (Kramosil and Michalek [11]). A fuzzy metric space (in the sense of Kramosil and Michalek) is a triple (X, M, T ) where X is a non-empty set, T is a continuous t-norm and M is a fuzzy set on X 2 × [0, ∞), satisfying the following properties: (FM-1) (FM-2) (FM-3) (FM-4) (FM-5)

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

Let (X, M, ∗) be a fuzzy metric space. A sequence (xn )n∈N in X is said to be convergent if there exists x ∈ X such that lim M(x, xn , t) = 1 ∀t > 0.

n→∞

A sequence {xn }n∈N in a fuzzy metric space (X, M, ∗) is called Cauchy (cf. [2,19]) if for each  ∈ (0, 1) and t > 0 there exists n 0 ∈ N such that M(x n , xm , t) > 1 −  for all m, n ≥ n 0 . The space (X, M, ∗) is called complete if every Cauchy sequence is convergent. Definition 2.4 (Sehgal and Bharucha-Reid [22]). Let (X, M, ∗) be a fuzzy metric space. A mapping f : X → X is called a probabilistic B-contraction if there exists q in (0,1) such that M( f (x), f (y), qt) ≥ M(x, y, t) for every x, y ∈ X and t > 0. The subclass of B-contractions in the next definition has been introduced in [12] in order to obtain a Banach fixed point theorem in a complete fuzzy metric space under the t-norm TL .

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Definition 2.5 (Mihe¸t [12]). Let (X, M, ∗) be a fuzzy metric space. A mapping f : X → X is called an (, )contraction if there is k ∈ (0, 1) such that the following implication holds for every  > 0,  ∈ (0, 1) and x, y ∈ X : M(x, y, ) > 1 −  ⇒ M( f (x), f (y), k) > 1 − k. We also mention two related definitions: Definition 2.6 (Hadži´c and Pap [9]). Let (X, M, ∗) be a fuzzy metric space and  : R+ → R+ . A mapping f : X → X is said to be a (, , )-contraction if for any  > 0,  ∈ (0, 1) and x, y ∈ X , following implication holds for every  > 0,  ∈ (0, 1) and x, y ∈ X : M(x, y, ) > 1 −  ⇒ M( f (x), f (y), ()) > 1 − (). We note that if (t) = kt (t > 0), then the notion of a (, , )-contraction reduces to that of (, )-contraction. Definition 2.7 (Mihe¸t [16]). Let (X, M, ∗) be a fuzzy metric space,  be a map from (0, ∞) to (0, ∞) and  be a map from (0,1) to (0,1). A mapping f : X → X is called a (, , , )-contraction if, for any x, y ∈ X,  > 0 and  ∈ (0, 1), M(x, y, ) > 1 −  ⇒ M( f (x), f (y), ()) > 1 − (). in [13]. If  is of the form (t) = kt (k ∈ (0, 1)), one obtains the contractive mappings considered Every (, , )-contraction mapping on a complete fuzzy metric space such that n∈N n (t) < ∞ ((t > 0), ∞ (1 − i ()) = 1 for some  ∈ (0, 1) has a unique fixed point [5, Theorem 19]. ((0, 1)) ⊂ (0, 1) and limn→∞ Ti=n We also mention a similar theorem for (, , , )-contractions: Theorem 2.8 (cf. Mihe¸t [16, Theorem 3.7]). Let (S, M, T ) be a complete fuzzy metric space and f : S → S be a (, the property that there are p ∈ S and t > 0 such that M( p, f ( p), t) > 0. If , , n)-contraction mapping with ∞ (1 − i ()) = 1 for some  ∈ (0, 1) then f has a fixed point.  (t) < ∞ and lim T n→∞ n∈N i=n In this paper we define a larger class of contractions and investigate the existence and the uniqueness of fixed points for these mappings. 3. Main results In [13] it was pointed out that the conditions on the t-norm in Theorem 2.8 can be relaxed. Namely (see [13, Theorem 3.11]), if (t) = kt (t > 0) for some k ∈ (0, 1), then Theorem 2.8 remains true even if T is an arbitrary continuous t-norm. Thus, it was natural to ask whether one may weaken the contractive conditions in Theorem 2.8. In the following we show that this is indeed the case: a fixed point theorem similar to Theorem 2.8 and to Theorem 19 in [9] can be proved even for a larger class of contractions, which is introduced in the following definition. Definition 3.1. Let (X, M, ∗) be a fuzzy metric space and  : (0, 1) → (0, 1) be a mapping. A mapping f : X → X is called a fuzzy -contraction of (, )-type if the following implication holds for every  > 0,  ∈ (0, 1) and x, y ∈ X : M(x, y, ) > 1 −  ⇒ M( f (x), f (y), ) > 1 − (). If (t) = qt (t ∈ (0, 1)) for some q ∈ (0, 1), then f will be called a fuzzy q-contraction of (, )-type. We note that every fuzzy -contraction of (, )-type with (t) < t ∀t ∈ (0, 1) is contractive, that is, it satisfies the relation M( f (x), f (y), t) ≥ M(x, y, t) for all x, y ∈ X and t > 0. Indeed, if we suppose that M( f (x), f (y), t)0, then there is ∈(0, 1) such that M( f (x), f (y), t) < 1 −  < M(x, y, t), that is M(x, y, t) > 1 −  and M( f (x), f (y), t) < 1 −  < 1 − (), which is a contradiction.

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Example 3.2. Every (, , )-contraction with (t) < t for all t ∈ (0, 1) is a fuzzy -contraction of (, )-type. Indeed, if f is a (, , )-contraction with (t) < t ∀t ∈ (0, 1) and  > 0,  ∈ (0, 1), x, y ∈ X are such that M(x, y, ) > 1 − , then M( f (x), f (y), ()) > 1 − (), which implies M( f (x), f (y), ) > 1 − (). Example 3.3 (Compare with Hadži´c and Pap [5, Example 3.39]). Let (X, M, ∗) be a fuzzy metric space and f : X → X be a mapping such that, for some q ∈ (0, 1), 1 − M( f (x), f (y), t) ≤ q(1 − M(x, y, t)) for all x, y ∈ X and t > 0. Then f is a fuzzy q-contraction of (, )-type. Indeed, if M(x, y, u) > 1 −  then, for every q ∈ (0, 1), 1 − M( f (x), f (y), u) ≤ q(1 − M(x, y, u)) < q and thus M( f (x), f (y), u) > 1 − q. Example 3.4. Let X = {0, 1, 2, . . . .} and (for x  y) ⎧ if t ≤ 2−Min(x,y) , ⎨0 M(x, y, t) = 1 − 2−Min(x,y) if 2−Min(x,y) < t ≤ 1, ⎩ 1 if t > 1 and f : X → X , f (r ) = r + 1. Then (X, M, TL ) is a fuzzy metric space [20]. Let x, y, ,  be such that M(x, y, ) > 1 − . (a) If 2−Min(x,y) <  ≤ 1, then 1 − 2−Min(x,y) > 1 − . This implies 1 − 2−Min(x+1,y+1) > 1 − 21 , that is, M( f (x), f (y), ) > 1 − 21 . (b) If  > 1 then M( f (x), f (y), ) = 1, hence again M( f (x), f (y), ) > 1 − 21 . Thus, the mapping f is a fuzzy q contraction of (, )-type. Remark 3.5. As it is shown in [13, Proposition 3.7], every (, k) contraction of (, )-type is a B-contraction. On the other hand, see [20, Example 3.1], the mapping f in the preceding example is not a B-contraction on X. Therefore the mapping in Example 3.4 is a fuzzy q-contraction of (, )-type, but it is not a (, )-contraction. ∞ (1 − Lemma 3.6. Every fuzzy  contraction of (, )-type on a fuzzy metric space (X, M, T ) satisfying limn→∞ Ti=n (i)  ()) = 1 for some  ∈ (0, 1) is (uniformly) continuous.

Proof. Let  > 0 and  ∈ (0, 1) be given. We show that there exists  ∈ (0, 1) such that () < . Indeed, if we had ∞ (1 − (i) ()) ≤ T∞ (1 − ) ≤ (t) ≥  for all t ∈ (0, 1), then (n) (t) ≥ , ∀n ∈ N, ∀t ∈ (0, 1). This would imply Ti=n i=n ∞ (1 − (i) ()) = 1. 1 −  for all n, contradicting limn→∞ Ti=n Now, if  ∈ (0, 1) is such that () < , then M(x, y, ) > 1 −  ⇒ M( f (x), f (y), ) > 1 − , concluding the proof of the lemma. 

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Theorem 3.7. Let (X, M, T ) be a complete fuzzy metric space. Let f : X → X be a fuzzy -contraction of (, )-type. Suppose that there exists x ∈ X such that M(x, f (x), 0+) > 0 and that ∞ lim Ti=n (1 − (i) ()) = 1

n→∞

for all  ∈ (0, 1). Then f has a fixed point. Proof. If M(x, f (x), 0+) = 1 then M(x, f (x), t) = 1 for all t > 0, hence f (x) = x. Hence we can suppose that M(x, f (x), 0+) < 1. Let us denote 1 − M(x, f (x), 0+) by 1 . By choosing a  ∈ (1 , 1), we prove by induction on n that M( f n (x), f n+1 (x), t) ≥ 1 − (n) () for all n ∈ N ∪ {0} and t > 0. Since M(x, f (x), t) ≥ M(x, f (x), 0+) = 1 − 1 > 1 −  for all t > 0, the desired inequality is true for n = 0. If we assume that it is true for n ≥ 0, then M( f n+1 (x), f n+2 (x), t) ≥ 1 − ((n) ()) = 1 − (n+1) () ∀t > 0, that is, it is also true for n + 1. Next, let  > 0 and  ∈ (0, 1) be given. Then, for every m ∈ N, M( f n (x), f n+m (x), ) ≥ T (M( f n (x), f n+1 (x), /m), . . . , M( f n+m−1 (x), f n+m (x), /m)) ∞ (1 − (i) ()), ≥ T (1 − (n) (), . . . , 1 − (n+m−1) ()) ≥ Ti=n concluding that the sequence { f n (x)}n∈N is a Cauchy sequence. If z = limn→∞ f n (x), by the continuity of the mapping f it follows that f (z) = z.  Remark 3.8. A fuzzy -contraction of (, )-type may have more than one fixed point. Indeed, let X = [0, ∞) and M(x, y, t) =

min(x, y) ∀t ∈ (0, ∞), max(x, y)

if x  y and M(x, y, t) = 1 ∀t > 0, if x = y. It is known (see [18]) that (X, M, TP ) is a complete fuzzy metric space. The mapping  0 if x = 0, f : X → X, f (x) = 1 if x > 0 is a (, , )-contraction mapping for every , for M( f (x), f (y), ()) ≤ 1 −  imply that (exactly) one of x and y is 0 and then M(x, y, ) = 0 < 1 − . This mapping has two fixed points: x = 0 and 1. However, if M( f (x), f (y), t) > 0 for all x, y ∈ X and t > 0, then the fixed point is unique. To prove this, we note (see [13, Lemma 3.6]) that ∞ (1 − (i) ()) = 1 ⇒ lim n () = 0. lim Ti=n

n→∞

n→∞

If u and v are fixed points for f and M( f (u), f (v), t) > 0 ∀t > 0, then for given t > 0 there is  ∈ (0, 1) such that M( f (u), f (v), t) > 1 − , implying M(u, v, t) = M( f (u), f (v), t) ≥ 1 − n () ∀n ∈ N. ∞ (1 − (i) ()) = 1, it follows that M(u, v, t) = 1, and thus u = v. As limn→∞ Ti=n

Corollary 3.9 (Tirado [24]). Let (X, M, TL ) be a complete fuzzy metric space. If f is a self-mapping of X with the property that there is q ∈ (0, 1) such that 1 − M( f (x), f (y), t) ≤ q(1 − M(x, y, t)) for all x, y ∈ X and t > 0, then f has a unique fixed point.

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Proof. From Example 3.3 it follows that f is a fuzzy -contraction of (, )-type with () = q. Also, as M( f (x), f (y), t) ≥ 1 − q + q M(x, y, t) ≥ 1 − q > 0 for all x, y ∈ X and t > 0, the condition: “M(x, f (x), 0+) > 0 for ∞ ∞ some x ∈ X ” is satisfied. Next, since i=1 i () = i=1 q i  < ∞ ( ∈ (0, 1)), from Proposition 1.2 it follows that (i) ∞ limn→∞ (TL )i=n (1 −  ()) = 1 for all  ∈ (0, 1). By the preceding theorem, there exists z ∈ X such that f (z) = z. The fixed point is unique, since M( f (x), f (y), t) ≥ 1 − q > 0 for all x, y ∈ X and t > 0.  Corollary 3.10. Let (X, M, T ) be a complete fuzzy metric space under a continuous g-convergent t-norm T and f : X → X be a fuzzy q-contraction of (, )-type. If there exists x ∈ X such that M(x, f (x), 0+) > 0, then f has a fixed point. 4. Conclusion Motivated by a celebrated result of Sherwood [23], we introduced a new class of contractions having fixed points on every complete Menger space (S, F, TL ), by weakening the contractive condition of the so-called (, )-contraction mappings. The investigation could be extended to fuzzy quasi-metric spaces, with possible applications to the study of recurrence equations associated to the analysis of Probabilistic Divide and Conquer Algorithms, as in [24]. Acknowledgement The author thanks the referees for carefully reading the first version of the manuscript and for some useful suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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