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On fuzzy contractive mappings in fuzzy metric spaces
Fuzzy Sets and Systems 158 (2007) 915 – 921 www.elsevier.com/locate/fss
On fuzzy contractive mappings in fuzzy metric spaces Dorel Mihe¸t Faculty of Mathematics and Computer Science, West University of Timi¸soara, Bv. V. Parvan 4, 300223 Timi¸soara, Romania Received 30 June 2006; received in revised form 17 November 2006; accepted 17 November 2006 Available online 6 December 2006
Abstract Some results on fuzzy contractive mappings in [V. Gregori, A. Sapena, On fixed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems 125 (2002) 245–253] are extended to Edelstein fuzzy contractive mappings. Some new results are obtained by modifying the notion of convergence in fuzzy metric spaces. Due to specific axiom (GV-2), in fuzzy metric spaces in the sense of George and Veeramani this convergence is Fréchet. © 2006 Elsevier B.V. All rights reserved. Keywords: Fuzzy metric space; Probabilistic metric space; Convergence in the sense of Fréchet; Fuzzy contractive mapping
0. Introduction The concept of fuzzy metric space, intimately related to that of probabilistic metric space of Menger type [3,7,12,19], has been introduced by Kramosil and Michalek [14]. In [5], by modifying the “separation condition” and strengthening some conditions in the definition of Kramosil and Michalek, George and Veeramani have obtained a special class of fuzzy metric spaces (in [17] these spaces are named as “strong fuzzy metric spaces”). Fixed-point theory in fuzzy metric spaces for different contractive-type mappings is closely related to that in probabilistic metric spaces (refer [1, Chapters VIII, IX], [2, Chapter 3], [11, Chapters 3–5], [4,7,9,13,15,16,18,20,21]). In this paper we deal with fuzzy contractive mappings in the sense of Gregori and Sapena [10]. After collecting some specific concepts, it is shown that the class of fuzzy contractive mappings is contained in the class of Edelstein fuzzy contractive mappings. By improving the results of Gregori and Sapena on fuzzy contractive mappings, some specific properties of these contractions are pointed out. The paper is ended with a Conclusions section, including an open question. 1. Preliminaries A generalized Menger space (or a Menger space in the sense of Schweizer and Sklar) is a triple (X, F, T ) where X is a nonempty set, T is a t-norm and F is a mapping from X × X to + , the class of all nondecreasing, left continuous on (0, ∞) mappings F from [0, ∞) to [0, 1] with F (0) = 0, satisfying the following conditions (F (x, y) is denoted by Fxy ): (PM0) Fpq = 0 ⇔ p = q. E-mail addresses: [email protected], [email protected]. 0165-0114/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2006.11.012
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(PM1) Fpq = Fqp ∀p, q ∈ S. (PM2) Fpr (x + y)T (Fpq (x), Fqr (y)) ∀p, q, r ∈ S, ∀x, y 0. Definition 1.1 (Kramosil and Michalek [14]). A fuzzy metric space (in the sense of Kramosil and Michalek) is a triple (X, M, ∗) where X is a nonempty set, ∗ is a continuous t-norm and M is a fuzzy set on X2 × [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) (M(x, y, t) ∗ M(y, z, s)) ∀x, y, z ∈ X ∀t, s > 0.
In the definition of George and Veeramani [5], M is a fuzzy set on X2 × (0, ∞) and (FM-1), (FM-2), (FM-4) are replaced, respectively, with (GV-1), (GV-2), (GV-4) below ( the axiom (GV-2) is reformulated as in [9, Remark 1]): (GV-1) M(x, y, t) > 0 ∀t > 0. (GV-2) M(x, x, t) = 1
∀x ∈ X ∀t > 0
and
x = y ⇒ M(x, y, t) < 1 ∀t > 0. (GV-4) M(x, y, ·) : (0, ∞) → [0, 1] is continuous ∀x, y ∈ X. We will refer to these fuzzy metric spaces as KM fuzzy metric spaces and GV fuzzy metric spaces, respectively. It is worth noting that, by defining the probabilistic metric by Fxy (t) = M(x, y, t), every KM fuzzy metric space (X, M, ∗) becomes a generalized Menger space (under the continuous t-norm ∗). Definition 1.2 (Schweizer and Sklar [19], Kramosil and Michalek [14]). Let (X, M, ∗) be a KM fuzzy metric space. A sequence (xn )n∈N is called M-convergent (convergent, in short) if lim M(xn , x, t) = 1
n→∞
∀t > 0
for some x ∈ X. A sequence (xn )n∈N is called M-Cauchy [19,5] if for each ∈ (0, 1) and t > 0 there exists n0 ∈ N such that M(xn , xm , t) > 1 − for all m, n n0 and a G-Cauchy sequence [7,10,22] if limn→∞ M(xn , xn+m , t) = 1 for each m ∈ N ∗ and t > 0. (X, M, ∗) is said to be M-complete (G-complete) [10,22] if every M-Cauchy (G-Cauchy) sequence is convergent. Definition 1.3 (Grabiec [7]). Let (X, M, ∗) be a KM fuzzy metric space. A mapping f : X → X is called Edelstein fuzzy contractive if it satisfies M(f (x), f (y), ·) > M(x, y, ·) ∀x, y ∈ X, x = y. We note that M(f (x), f (y), ·) > M(x, y, ·) means M(f (x), f (y), ·)M(x, y, ) and M(f (x), f (y), ·) = M(x, y, ·), that is, the following two conditions are satisfied by an Edelstein fuzzy contractive mapping: (i) M(f (x), f (y), t)M(x, y, t)∀x, y ∈ X, ∀t > 0 and (ii) if x = y, then M(f (x), f (y), t) > M(x, y, t) for some t > 0. For more details we refer the reader to [6–8,22].
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2. Main results We begin this section by recalling the results of Gregori and Sapena [10] on fuzzy contractive mappings. Definition 2.1 (Gregori and Sapena [10]). A fuzzy contractive mapping on a KM fuzzy metric space (X, M, ∗) with the property M(x, y, t) = 0 ∀x, y ∈ X, ∀t > 0 is a self-mapping f of X satisfying 1 1 − 1 , ∀x, y ∈ X ∀t > 0, − 1 k M(x, y, t) M(f (x), f (y), t) where k is a fixed number in (0, 1). A sequence (xn ) in X is said to be fuzzy contractive if 1 1 − 1k −1 M(xn+1 , xn+2 , t) M(xn , xn+1 , t)
∀n ∈ N ∀t > 0.
Theorem 2.1 (Gregori and Sapena [10]). (a) (Theorem 4.4) Let (X, M, ∗) be an M-complete GV fuzzy metric space in which fuzzy contractive sequences are M-Cauchy and f : X → X be a fuzzy contractive mapping. Then f has a unique fixed point. (b) (Theorem 5.2) If (X, M, ∗) is a G-complete KM fuzzy metric space satisfying M(x, y, t) > 0 ∀t > 0 and f : X → X is a fuzzy contractive mapping, then f has a unique fixed point. We note that if f is a fuzzy contractive mapping on a KM fuzzy metric space satisfying M(x, y, t) > 0 ∀t > 0, then f is also an Edelstein fuzzy contractive mapping. Indeed, if M(x, y, t) = 1 then 1/(M(f (x), f (y), t)−1) 0, hence M(f (x), f (y), t) is also 1, while if M(x, y, t)<1 then 1 1 −1 k −1 M(f (x), f (y), t) M(x, y, t) <
1 − 1, M(x, y, t)
hence M(f (x), f (y), t) > M(x, y, t). Therefore, M(f (x), f (y), t)M(x, y, t) ∀x, y ∈ X, ∀t > 0. On the other hand, if x = y then, due to (FM-2), there is t > 0 such that M(x, y, t) < 1. This implies M(f (x), f (y), t) > M(x, y, t), that is, M(f (x), f (y), ·) > M(x, y, ·)
∀x, y ∈ X,
x = y.
The converse is not true, as it is illustrated in the following: Example 2.1. Let us consider one of the most familiar examples of fuzzy metric space, [5, Example 2.11], namely the space (X, M, TP ), where X = N ∗ = {1, 2, . . .}, ⎧x ⎪ if x y ⎨ y M(x, y, t) = y ∀t > 0 ⎪ ⎩ if y x x and TP (ab) = ab. In this space, the mapping f : X → X, f (x) = x + 1 is an Edelstein fuzzy contractive mapping for x < y implies x/y < (x + 1)/(y + 1). However, f is not a fuzzy contractive mapping, for (n + 3)/(n + 2) − 1 k((n + 2)/(n + 1) − 1) ∀n ∈ N ∗ would imply k (n + 1)/(n + 2)∀n ∈ N ∗ , contradicting the inequality k < 1. In [7] it has been proven that every Edelstein fuzzy contractive mapping on a compact KM fuzzy metric space has a unique fixed point. In [7, Theorem 8] Grabiec has actually proved that every Edelstein fuzzy contractive mapping f on a KM fuzzy metric space (X, M, ∗) has a unique fixed point provided there is x ∈ X such that the sequence (f n (x))n∈N has a convergent subsequence.
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It should be noticed that Theorem 2.1 remains true even if the hypothesis “f is a fuzzy contractive mapping” is replaced by “f is an Edelstein fuzzy contractive mapping”. Indeed, under the hypotheses of Theorem 2.1(a) and (b), the sequence (f n (x)) is M-convergent for every x ∈ X. We are going to point out a first specific property of fuzzy contractive mappings: as we will show in the next theorem, Theorem 2.1 still holds even if we work in probabilistic metric spaces (the continuity of T is not required). Theorem 2.2. Let (X, F, T ) be a generalized Menger space under a t-norm T satisfying supa<1 T (a, a) = 1. If Fxy (t) > 0 ∀x, y ∈ X, ∀t > 0 and f : X → X is a fuzzy contractive mapping with the property that there is x ∈ X such that the sequence (f n (x))n∈N has a convergent subsequence, then f has a fixed point. Proof. Since f is an Edelstein fuzzy contractive mapping, the inequality Ff (x)f (y) (t)Fxy (t) holds for every ∀x, y ∈ X and t > 0. Let xn = f n (x), n ∈ N (f 0 (x) = x). We have Fx1 x2 (t) Fx0 x1 (t) ∀t > 0 and then, by induction, Fxn+1 xn+2 (t) Fxn xn+1 (t) ∀t > 0. Therefore, for every t > 0, the sequence (Fxn xn+1 (t))n∈N of numbers in (0, 1] is nondecreasing. Fix a t > 0 and denote l = limn→∞ Fxn xn+1 (t). Then l ∈ (0, 1] and since l = 1 would imply 1 1 1 − 1k − 1 < − 1, l l l it follows that l = 1, i.e. limn→∞ Fxn xn+1 (t) = 1 ∀t > 0. Let now (xnk )k∈N be a convergent subsequence of (xn ) converging to y ∈ X, that is, lim Fxnk y (t) = 1
k→∞
∀t > 0.
We show that limk→∞ xnk +1 = y. Indeed, let > 0 be given. Since supa<1 T (a, a) = 1, it follows that there is > 0 such that Fpq () > 1 − ,
Fqr () > 1 − ⇒ Fpr () > 1 − .
By choosing k0 such that Fxnk xnk +1 () > 1− ∀k k0 , Fxnk y () > 1− ∀k k0 one obtains Fxnk +1 y () > 1− ∀k k0 . This means that limk→∞ xnk +1 = y. On the other hand, since Ff (y)f (xnk ) (t) Fyxnk (t) and Fyxnk (t) → 1 for all t > 0, one gets xnk +1 → f (y). Therefore f (y) = y. Remark 2.1. As in the proof of [7, Theorem 8] the continuity of the fuzzy metric plays an essential role, it is natural to ask whether the above theorem remains true for Edelstein fuzzy contractive mappings. In the following we will show that Theorem 2.1(a) can still be improved. The notion of “point-convergence”, introduced in the next definition, is strongly related to that of “point equivalence” (see [9, Definition 6]). Definition 2.2. Let (X, F, ∗) be a fuzzy metric space. A sequence (xn ) in X is said to be point convergent to x ∈ X (we write xn →p x) if there exists t > 0 such that lim M(xn , x, t) = 1.
n→∞
It is easy to see that, endowed with the point convergence, a GV fuzzy metric space (X, F, ∗) is a space with convergence in the sense of Fréchet, that is, the following properties hold: (1) Every sequence in X has at most one limit. (2) Every constant sequence, xn = x ∀n ∈ N, is convergent and limn→∞ xn = x. (3) Every subsequence of a convergent sequence is also convergent and has the same limit as the whole sequence. We prove only the property (1), which is a consequence of the axiom (GV-2).
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Proof of (1). Let (xn ) be a sequence in X and x, y ∈ X be such that xn →p x and xn →p y. Then, for some t, s > 0, M(x, y, t + s) M(x, xn , t) ∗ M(xn , y, s) →n→∞ 1, that is, M(x, y, t + s) = 1 and hence, due to (GV-2), x = y.
Remark 2.2. It is worth noting that if the point convergence in a fuzzy metric space (X, F, ∗) is Fréchet, then (GV-2) holds (thus, the uniqueness of the limit in the point convergence characterizes, in a sense, a fuzzy metric space in the sense of George and Veeramani). Indeed, let x, y ∈ X, x = y. If M(x, y, t) = 1 for some t > 0, then the sequence (xn )n∈N ⊂ X defined as x, y, x, y, . . . has two distinct limits, for the equality M(x, x, t) = M(y, x, t) = 1 implies xn →p x, while M(x, y, t) = M(y, y, t) = 1 implies xn →p y. In the next example we will see that there exist p-convergent but not convergent sequences. Example 2.2. Let (xn )n∈N ⊂ (0, ∞), xn 1 and X = (xn ) ∪ {1}. Define M(xn , xn , t) = 1 ∀n ∈ N, ∀t > 0, M(1, 1, t) = 1 ∀t > 0, M(xn , xm , t) = min{xn , xm } ∀n, m ∈ N, ∀t > 0 and min{xn , t} if 0 < t < 1, M(xn , 1, t) = xn if t > 1 for all n ∈ N. Then (X, M, TM ), where TM (a, b) = min{a, b}, is a fuzzy metric space (see [9, Example 2]). Since limn→∞ M(xn , 1, 21 ) = 21 , (xn ) is not convergent. Nevertheless it is p-convergent to 1, for limn→∞ M(xn , 1, 2) = 1. Theorem 2.3. Let (X, M, ∗) be a GV fuzzy metric space and f : X → X be a fuzzy contractive mapping. Suppose that, for some x ∈ X, the sequence (xn )n∈N , xn = f n (x) of its iterates has a p-convergent subsequence. Then f has a unique fixed point. Proof. As in the proof of Theorem 2.2 we can prove that limn→∞ M(xn , xn+1 , t) = 1 ∀t > 0. We also know that limk→∞ M(xnk , y, s) = 1 for some y ∈ X and s > 0. We show that limk→∞ M(xnk +1 , y, 2s) = 1. Let > 0 be given. Since T is continuous in (1, 1), there is > 0 such that (1 − ) ∗ (1 − ) > 1 − . By choosing k0 such that M(xnk , xnk +1 , s) > 1 − ∀k k0 and M(xnk , y, s) > 1 − ∀k k0 , we get M(xnk +1, y, 2s) (1 − ) ∗ (1 − ) > 1 − ∀k k0 . This means that limk→∞ M(xnk +1 , y, 2s) = 1 i.e. xnk +1 →p y. On the other hand, from M(f (y), f (xnk ), 2s) M(y, xnk , 2s) and M(y, xnk , 2s) → 1 it follows that limk→∞ M(xnk +1 , f (y), 2s) = 1. Since the p-convergence is Fréchet, we deduce that f (y) = y. The uniqueness of the fixed point can be proved as in [10, Theorem 4.4]. It should be noted that a similar theorem does not hold in KM fuzzy metric spaces. This is illustrated in the following: Example 2.3. Let X be the set N ∗ = {1, 2, . . .}. We define (for p = q) the fuzzy mapping M by ⎧ if t = 0, ⎪ ⎨0 − min{p,q} M(p, q, t) = 1 − 2 if 0 < t 1, ⎪ ⎩ 1 if t > 1. As 1 − 1/2min(p,r) min{1 − 1/2min(r,q) , 1 − 1/2min(p,q) }∀p, q, r ∈ N ∗ , (X, M, TM ) is a KM fuzzy metric space satisfying M(x, y, t) = 0 for all x, y ∈ X and t > 0. The mapping f: N ∗ → N ∗ , f (x) = x + 1 is fuzzy contractive. Indeed, if t > 1 then 1 1 1 − 1 = 0 −1 M(f (p), f (q), t) 2 M(p, q, t)
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for every p, q ∈ N ∗ , while if 0 < t 1 and p < q, then 1 1 − 1 = p+1 M(f (p), f (q), t) 2 −1
1 2p+1 − 2
=
1 2
1 −1 . M(p, q, t)
As limn→∞ M(f n (x), 1, s) = 1 for every x ∈ X and s > 1, it follows that xn →p 1. Nevertheless, 1 is not a fixed point of f. Remark 2.3. We note that in Example 2.2, as well as in Example 2.3, there are essentially no nonconstant convergent sequences. 3. Conclusions Some specific properties of fuzzy contractive mappings of Gregori and Sapena [10] have been pointed out. Particularly, an appropriate Fréchet convergence on GV fuzzy metric spaces, weaker than the M-convergence, has been considered. It will be natural to continue the study of this convergence spaces, by finding some more examples and introducing a similar concept for Cauchy sequence, p-completeness, etc. Also, it would be interesting to compare different types of contraction maps in fuzzy metric spaces. This might answer the following open question raised in Remark 2.1: O.Q. 3.1. Is it true that if (X, F, T ) is a generalized Menger space under a t -norm T satisfying supa<1 T (a, a) = 1 and f is an Edelstein fuzzy contractive mapping with the property that there is x ∈ X such that the sequence (f n (x))n∈N has a convergent subsequence, then f has a fixed point? Acknowledgements During preparation of the manuscript, the author has exchanged ideas with Professor E. Pap, Professor V. Radu, Professor S. Romaguera and Professor P. Veeramani. I would like to take this opportunity to thank them for their kindness. References [1] S.S. Chang, Y.J. Cho, S.M. Kang, Probabilistic Metric Spaces and Nonlinear Operator Theory, Sichuan University Press, Chengdu, 1994. [2] Gh. Constantin, I. Istr˘a¸tescu, Elements of probabilistic analysis with applications, Mathematics and Its Applications, East European Series, vol. 36, Editura Academiei, Kluwer Academic Publishers, Bucure¸sti, Dordrecht, 1989. [3] R.J. Egbert, Products and quotients of probabilistic metric spaces, Pacific J. Math. 24 (3) (1968) 437–455. [4] J.-X. Fang, On fixed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems 46 (1992) 107–113. [5] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64 (1994) 395–399. [6] A. George, P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems 90 (1997) 365–368. [7] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems 27 (1988) 385–389. [8] V. Gregori, S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy Sets and Systems 115 (2000) 485–489. [9] V. Gregori, S. Romaguera, Characterizing completable fuzzy metric spaces, Fuzzy Sets and Systems 144 (3) (2004) 411–420. [10] V. Gregori, A. Sapena, On fixed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems 125 (2002) 245–253. [11] O. Hadži´c, E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, Kluwer Academic Publishers, Dordrecht, 2001. [12] U. Höhle, Probabilistische Metriken auf der Menge der nichtnegativen Verteilungsfunctionen, Aequationes Math. 18 (1978) 345–356. [13] I. Istr˘a¸tescu, On some fixed points theorems in generalized Menger spaces, Boll. Un. Mat. Ital. V. Ser. A 13 (1976) 95–100. [14] O. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika 11 (1975) 336–344. [15] D. Mihe¸t, A Banach contraction theorem in fuzzy metric spaces, Fuzzy Sets and Systems 144 (3) (2004) 431–439. [16] V. Radu, Equicontinuous iterates of t-norms and applications to random normed and fuzzy Menger spaces, in: J. Sousa Ramos, D. Gronau, C. Mira, L. Reich, A.N. Sharkowski (Eds.), Iteration Theory (ECIT 02), Grazer Math. Ber. 33 (2004) 323–350. [17] V. Radu, Some suitable metrics on fuzzy metric spaces, Fixed Point Theory 5 (2) (2004) 323–347.
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[18] A. Razani, A contraction theorem in fuzzy metric spaces, Fixed Point Theory Appl. (3) (2005) 257–265. [19] B. Schweizer, A. Sklar, Probabilistic metric spaces, in: North Holland Series in Probability and Applied Mathematics, New York, Amsterdam, Oxford, 1983 and the new edition in Dover Books on Mathematics, Dover, New York, 2005. [20] V.M. Sehgal, A.T. Bharucha-Reid, Fixed points of contraction mappings on probabilistic metric spaces, Math. Systems Theory 6 (1972) 97–104. [21] N.X. Tan, Generalized probabilistic metric spaces and fixed point theorems, Math. Nachr. 129 (1986) 205–218. [22] R. Vasuki, P. Veeramani, Fixed point theorems and Cauchy sequences in fuzzy metric spaces, Fuzzy Sets and Systems 135 (3) (2003) 409–413.
On fuzzy contractive mappings in fuzzy metric spaces Dorel Mihe¸t Faculty of Mathematics and Computer Science, West University of Timi¸soara, Bv. V. Parvan 4, 300223 Timi¸soara, Romania Received 30 June 2006; received in revised form 17 November 2006; accepted 17 November 2006 Available online 6 December 2006
Abstract Some results on fuzzy contractive mappings in [V. Gregori, A. Sapena, On fixed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems 125 (2002) 245–253] are extended to Edelstein fuzzy contractive mappings. Some new results are obtained by modifying the notion of convergence in fuzzy metric spaces. Due to specific axiom (GV-2), in fuzzy metric spaces in the sense of George and Veeramani this convergence is Fréchet. © 2006 Elsevier B.V. All rights reserved. Keywords: Fuzzy metric space; Probabilistic metric space; Convergence in the sense of Fréchet; Fuzzy contractive mapping
0. Introduction The concept of fuzzy metric space, intimately related to that of probabilistic metric space of Menger type [3,7,12,19], has been introduced by Kramosil and Michalek [14]. In [5], by modifying the “separation condition” and strengthening some conditions in the definition of Kramosil and Michalek, George and Veeramani have obtained a special class of fuzzy metric spaces (in [17] these spaces are named as “strong fuzzy metric spaces”). Fixed-point theory in fuzzy metric spaces for different contractive-type mappings is closely related to that in probabilistic metric spaces (refer [1, Chapters VIII, IX], [2, Chapter 3], [11, Chapters 3–5], [4,7,9,13,15,16,18,20,21]). In this paper we deal with fuzzy contractive mappings in the sense of Gregori and Sapena [10]. After collecting some specific concepts, it is shown that the class of fuzzy contractive mappings is contained in the class of Edelstein fuzzy contractive mappings. By improving the results of Gregori and Sapena on fuzzy contractive mappings, some specific properties of these contractions are pointed out. The paper is ended with a Conclusions section, including an open question. 1. Preliminaries A generalized Menger space (or a Menger space in the sense of Schweizer and Sklar) is a triple (X, F, T ) where X is a nonempty set, T is a t-norm and F is a mapping from X × X to + , the class of all nondecreasing, left continuous on (0, ∞) mappings F from [0, ∞) to [0, 1] with F (0) = 0, satisfying the following conditions (F (x, y) is denoted by Fxy ): (PM0) Fpq = 0 ⇔ p = q. E-mail addresses: [email protected], [email protected]. 0165-0114/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2006.11.012
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(PM1) Fpq = Fqp ∀p, q ∈ S. (PM2) Fpr (x + y)T (Fpq (x), Fqr (y)) ∀p, q, r ∈ S, ∀x, y 0. Definition 1.1 (Kramosil and Michalek [14]). A fuzzy metric space (in the sense of Kramosil and Michalek) is a triple (X, M, ∗) where X is a nonempty set, ∗ is a continuous t-norm and M is a fuzzy set on X2 × [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) (M(x, y, t) ∗ M(y, z, s)) ∀x, y, z ∈ X ∀t, s > 0.
In the definition of George and Veeramani [5], M is a fuzzy set on X2 × (0, ∞) and (FM-1), (FM-2), (FM-4) are replaced, respectively, with (GV-1), (GV-2), (GV-4) below ( the axiom (GV-2) is reformulated as in [9, Remark 1]): (GV-1) M(x, y, t) > 0 ∀t > 0. (GV-2) M(x, x, t) = 1
∀x ∈ X ∀t > 0
and
x = y ⇒ M(x, y, t) < 1 ∀t > 0. (GV-4) M(x, y, ·) : (0, ∞) → [0, 1] is continuous ∀x, y ∈ X. We will refer to these fuzzy metric spaces as KM fuzzy metric spaces and GV fuzzy metric spaces, respectively. It is worth noting that, by defining the probabilistic metric by Fxy (t) = M(x, y, t), every KM fuzzy metric space (X, M, ∗) becomes a generalized Menger space (under the continuous t-norm ∗). Definition 1.2 (Schweizer and Sklar [19], Kramosil and Michalek [14]). Let (X, M, ∗) be a KM fuzzy metric space. A sequence (xn )n∈N is called M-convergent (convergent, in short) if lim M(xn , x, t) = 1
n→∞
∀t > 0
for some x ∈ X. A sequence (xn )n∈N is called M-Cauchy [19,5] if for each ∈ (0, 1) and t > 0 there exists n0 ∈ N such that M(xn , xm , t) > 1 − for all m, n n0 and a G-Cauchy sequence [7,10,22] if limn→∞ M(xn , xn+m , t) = 1 for each m ∈ N ∗ and t > 0. (X, M, ∗) is said to be M-complete (G-complete) [10,22] if every M-Cauchy (G-Cauchy) sequence is convergent. Definition 1.3 (Grabiec [7]). Let (X, M, ∗) be a KM fuzzy metric space. A mapping f : X → X is called Edelstein fuzzy contractive if it satisfies M(f (x), f (y), ·) > M(x, y, ·) ∀x, y ∈ X, x = y. We note that M(f (x), f (y), ·) > M(x, y, ·) means M(f (x), f (y), ·)M(x, y, ) and M(f (x), f (y), ·) = M(x, y, ·), that is, the following two conditions are satisfied by an Edelstein fuzzy contractive mapping: (i) M(f (x), f (y), t)M(x, y, t)∀x, y ∈ X, ∀t > 0 and (ii) if x = y, then M(f (x), f (y), t) > M(x, y, t) for some t > 0. For more details we refer the reader to [6–8,22].
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2. Main results We begin this section by recalling the results of Gregori and Sapena [10] on fuzzy contractive mappings. Definition 2.1 (Gregori and Sapena [10]). A fuzzy contractive mapping on a KM fuzzy metric space (X, M, ∗) with the property M(x, y, t) = 0 ∀x, y ∈ X, ∀t > 0 is a self-mapping f of X satisfying 1 1 − 1 , ∀x, y ∈ X ∀t > 0, − 1 k M(x, y, t) M(f (x), f (y), t) where k is a fixed number in (0, 1). A sequence (xn ) in X is said to be fuzzy contractive if 1 1 − 1k −1 M(xn+1 , xn+2 , t) M(xn , xn+1 , t)
∀n ∈ N ∀t > 0.
Theorem 2.1 (Gregori and Sapena [10]). (a) (Theorem 4.4) Let (X, M, ∗) be an M-complete GV fuzzy metric space in which fuzzy contractive sequences are M-Cauchy and f : X → X be a fuzzy contractive mapping. Then f has a unique fixed point. (b) (Theorem 5.2) If (X, M, ∗) is a G-complete KM fuzzy metric space satisfying M(x, y, t) > 0 ∀t > 0 and f : X → X is a fuzzy contractive mapping, then f has a unique fixed point. We note that if f is a fuzzy contractive mapping on a KM fuzzy metric space satisfying M(x, y, t) > 0 ∀t > 0, then f is also an Edelstein fuzzy contractive mapping. Indeed, if M(x, y, t) = 1 then 1/(M(f (x), f (y), t)−1) 0, hence M(f (x), f (y), t) is also 1, while if M(x, y, t)<1 then 1 1 −1 k −1 M(f (x), f (y), t) M(x, y, t) <
1 − 1, M(x, y, t)
hence M(f (x), f (y), t) > M(x, y, t). Therefore, M(f (x), f (y), t)M(x, y, t) ∀x, y ∈ X, ∀t > 0. On the other hand, if x = y then, due to (FM-2), there is t > 0 such that M(x, y, t) < 1. This implies M(f (x), f (y), t) > M(x, y, t), that is, M(f (x), f (y), ·) > M(x, y, ·)
∀x, y ∈ X,
x = y.
The converse is not true, as it is illustrated in the following: Example 2.1. Let us consider one of the most familiar examples of fuzzy metric space, [5, Example 2.11], namely the space (X, M, TP ), where X = N ∗ = {1, 2, . . .}, ⎧x ⎪ if x y ⎨ y M(x, y, t) = y ∀t > 0 ⎪ ⎩ if y x x and TP (ab) = ab. In this space, the mapping f : X → X, f (x) = x + 1 is an Edelstein fuzzy contractive mapping for x < y implies x/y < (x + 1)/(y + 1). However, f is not a fuzzy contractive mapping, for (n + 3)/(n + 2) − 1 k((n + 2)/(n + 1) − 1) ∀n ∈ N ∗ would imply k (n + 1)/(n + 2)∀n ∈ N ∗ , contradicting the inequality k < 1. In [7] it has been proven that every Edelstein fuzzy contractive mapping on a compact KM fuzzy metric space has a unique fixed point. In [7, Theorem 8] Grabiec has actually proved that every Edelstein fuzzy contractive mapping f on a KM fuzzy metric space (X, M, ∗) has a unique fixed point provided there is x ∈ X such that the sequence (f n (x))n∈N has a convergent subsequence.
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It should be noticed that Theorem 2.1 remains true even if the hypothesis “f is a fuzzy contractive mapping” is replaced by “f is an Edelstein fuzzy contractive mapping”. Indeed, under the hypotheses of Theorem 2.1(a) and (b), the sequence (f n (x)) is M-convergent for every x ∈ X. We are going to point out a first specific property of fuzzy contractive mappings: as we will show in the next theorem, Theorem 2.1 still holds even if we work in probabilistic metric spaces (the continuity of T is not required). Theorem 2.2. Let (X, F, T ) be a generalized Menger space under a t-norm T satisfying supa<1 T (a, a) = 1. If Fxy (t) > 0 ∀x, y ∈ X, ∀t > 0 and f : X → X is a fuzzy contractive mapping with the property that there is x ∈ X such that the sequence (f n (x))n∈N has a convergent subsequence, then f has a fixed point. Proof. Since f is an Edelstein fuzzy contractive mapping, the inequality Ff (x)f (y) (t)Fxy (t) holds for every ∀x, y ∈ X and t > 0. Let xn = f n (x), n ∈ N (f 0 (x) = x). We have Fx1 x2 (t) Fx0 x1 (t) ∀t > 0 and then, by induction, Fxn+1 xn+2 (t) Fxn xn+1 (t) ∀t > 0. Therefore, for every t > 0, the sequence (Fxn xn+1 (t))n∈N of numbers in (0, 1] is nondecreasing. Fix a t > 0 and denote l = limn→∞ Fxn xn+1 (t). Then l ∈ (0, 1] and since l = 1 would imply 1 1 1 − 1k − 1 < − 1, l l l it follows that l = 1, i.e. limn→∞ Fxn xn+1 (t) = 1 ∀t > 0. Let now (xnk )k∈N be a convergent subsequence of (xn ) converging to y ∈ X, that is, lim Fxnk y (t) = 1
k→∞
∀t > 0.
We show that limk→∞ xnk +1 = y. Indeed, let > 0 be given. Since supa<1 T (a, a) = 1, it follows that there is > 0 such that Fpq () > 1 − ,
Fqr () > 1 − ⇒ Fpr () > 1 − .
By choosing k0 such that Fxnk xnk +1 () > 1− ∀k k0 , Fxnk y () > 1− ∀k k0 one obtains Fxnk +1 y () > 1− ∀k k0 . This means that limk→∞ xnk +1 = y. On the other hand, since Ff (y)f (xnk ) (t) Fyxnk (t) and Fyxnk (t) → 1 for all t > 0, one gets xnk +1 → f (y). Therefore f (y) = y. Remark 2.1. As in the proof of [7, Theorem 8] the continuity of the fuzzy metric plays an essential role, it is natural to ask whether the above theorem remains true for Edelstein fuzzy contractive mappings. In the following we will show that Theorem 2.1(a) can still be improved. The notion of “point-convergence”, introduced in the next definition, is strongly related to that of “point equivalence” (see [9, Definition 6]). Definition 2.2. Let (X, F, ∗) be a fuzzy metric space. A sequence (xn ) in X is said to be point convergent to x ∈ X (we write xn →p x) if there exists t > 0 such that lim M(xn , x, t) = 1.
n→∞
It is easy to see that, endowed with the point convergence, a GV fuzzy metric space (X, F, ∗) is a space with convergence in the sense of Fréchet, that is, the following properties hold: (1) Every sequence in X has at most one limit. (2) Every constant sequence, xn = x ∀n ∈ N, is convergent and limn→∞ xn = x. (3) Every subsequence of a convergent sequence is also convergent and has the same limit as the whole sequence. We prove only the property (1), which is a consequence of the axiom (GV-2).
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Proof of (1). Let (xn ) be a sequence in X and x, y ∈ X be such that xn →p x and xn →p y. Then, for some t, s > 0, M(x, y, t + s) M(x, xn , t) ∗ M(xn , y, s) →n→∞ 1, that is, M(x, y, t + s) = 1 and hence, due to (GV-2), x = y.
Remark 2.2. It is worth noting that if the point convergence in a fuzzy metric space (X, F, ∗) is Fréchet, then (GV-2) holds (thus, the uniqueness of the limit in the point convergence characterizes, in a sense, a fuzzy metric space in the sense of George and Veeramani). Indeed, let x, y ∈ X, x = y. If M(x, y, t) = 1 for some t > 0, then the sequence (xn )n∈N ⊂ X defined as x, y, x, y, . . . has two distinct limits, for the equality M(x, x, t) = M(y, x, t) = 1 implies xn →p x, while M(x, y, t) = M(y, y, t) = 1 implies xn →p y. In the next example we will see that there exist p-convergent but not convergent sequences. Example 2.2. Let (xn )n∈N ⊂ (0, ∞), xn 1 and X = (xn ) ∪ {1}. Define M(xn , xn , t) = 1 ∀n ∈ N, ∀t > 0, M(1, 1, t) = 1 ∀t > 0, M(xn , xm , t) = min{xn , xm } ∀n, m ∈ N, ∀t > 0 and min{xn , t} if 0 < t < 1, M(xn , 1, t) = xn if t > 1 for all n ∈ N. Then (X, M, TM ), where TM (a, b) = min{a, b}, is a fuzzy metric space (see [9, Example 2]). Since limn→∞ M(xn , 1, 21 ) = 21 , (xn ) is not convergent. Nevertheless it is p-convergent to 1, for limn→∞ M(xn , 1, 2) = 1. Theorem 2.3. Let (X, M, ∗) be a GV fuzzy metric space and f : X → X be a fuzzy contractive mapping. Suppose that, for some x ∈ X, the sequence (xn )n∈N , xn = f n (x) of its iterates has a p-convergent subsequence. Then f has a unique fixed point. Proof. As in the proof of Theorem 2.2 we can prove that limn→∞ M(xn , xn+1 , t) = 1 ∀t > 0. We also know that limk→∞ M(xnk , y, s) = 1 for some y ∈ X and s > 0. We show that limk→∞ M(xnk +1 , y, 2s) = 1. Let > 0 be given. Since T is continuous in (1, 1), there is > 0 such that (1 − ) ∗ (1 − ) > 1 − . By choosing k0 such that M(xnk , xnk +1 , s) > 1 − ∀k k0 and M(xnk , y, s) > 1 − ∀k k0 , we get M(xnk +1, y, 2s) (1 − ) ∗ (1 − ) > 1 − ∀k k0 . This means that limk→∞ M(xnk +1 , y, 2s) = 1 i.e. xnk +1 →p y. On the other hand, from M(f (y), f (xnk ), 2s) M(y, xnk , 2s) and M(y, xnk , 2s) → 1 it follows that limk→∞ M(xnk +1 , f (y), 2s) = 1. Since the p-convergence is Fréchet, we deduce that f (y) = y. The uniqueness of the fixed point can be proved as in [10, Theorem 4.4]. It should be noted that a similar theorem does not hold in KM fuzzy metric spaces. This is illustrated in the following: Example 2.3. Let X be the set N ∗ = {1, 2, . . .}. We define (for p = q) the fuzzy mapping M by ⎧ if t = 0, ⎪ ⎨0 − min{p,q} M(p, q, t) = 1 − 2 if 0 < t 1, ⎪ ⎩ 1 if t > 1. As 1 − 1/2min(p,r) min{1 − 1/2min(r,q) , 1 − 1/2min(p,q) }∀p, q, r ∈ N ∗ , (X, M, TM ) is a KM fuzzy metric space satisfying M(x, y, t) = 0 for all x, y ∈ X and t > 0. The mapping f: N ∗ → N ∗ , f (x) = x + 1 is fuzzy contractive. Indeed, if t > 1 then 1 1 1 − 1 = 0 −1 M(f (p), f (q), t) 2 M(p, q, t)
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for every p, q ∈ N ∗ , while if 0 < t 1 and p < q, then 1 1 − 1 = p+1 M(f (p), f (q), t) 2 −1
1 2p+1 − 2
=
1 2
1 −1 . M(p, q, t)
As limn→∞ M(f n (x), 1, s) = 1 for every x ∈ X and s > 1, it follows that xn →p 1. Nevertheless, 1 is not a fixed point of f. Remark 2.3. We note that in Example 2.2, as well as in Example 2.3, there are essentially no nonconstant convergent sequences. 3. Conclusions Some specific properties of fuzzy contractive mappings of Gregori and Sapena [10] have been pointed out. Particularly, an appropriate Fréchet convergence on GV fuzzy metric spaces, weaker than the M-convergence, has been considered. It will be natural to continue the study of this convergence spaces, by finding some more examples and introducing a similar concept for Cauchy sequence, p-completeness, etc. Also, it would be interesting to compare different types of contraction maps in fuzzy metric spaces. This might answer the following open question raised in Remark 2.1: O.Q. 3.1. Is it true that if (X, F, T ) is a generalized Menger space under a t -norm T satisfying supa<1 T (a, a) = 1 and f is an Edelstein fuzzy contractive mapping with the property that there is x ∈ X such that the sequence (f n (x))n∈N has a convergent subsequence, then f has a fixed point? Acknowledgements During preparation of the manuscript, the author has exchanged ideas with Professor E. Pap, Professor V. Radu, Professor S. Romaguera and Professor P. Veeramani. I would like to take this opportunity to thank them for their kindness. References [1] S.S. Chang, Y.J. Cho, S.M. Kang, Probabilistic Metric Spaces and Nonlinear Operator Theory, Sichuan University Press, Chengdu, 1994. [2] Gh. Constantin, I. Istr˘a¸tescu, Elements of probabilistic analysis with applications, Mathematics and Its Applications, East European Series, vol. 36, Editura Academiei, Kluwer Academic Publishers, Bucure¸sti, Dordrecht, 1989. [3] R.J. Egbert, Products and quotients of probabilistic metric spaces, Pacific J. Math. 24 (3) (1968) 437–455. [4] J.-X. Fang, On fixed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems 46 (1992) 107–113. [5] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64 (1994) 395–399. [6] A. George, P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems 90 (1997) 365–368. [7] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems 27 (1988) 385–389. [8] V. Gregori, S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy Sets and Systems 115 (2000) 485–489. [9] V. Gregori, S. Romaguera, Characterizing completable fuzzy metric spaces, Fuzzy Sets and Systems 144 (3) (2004) 411–420. [10] V. Gregori, A. Sapena, On fixed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems 125 (2002) 245–253. [11] O. Hadži´c, E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, Kluwer Academic Publishers, Dordrecht, 2001. [12] U. Höhle, Probabilistische Metriken auf der Menge der nichtnegativen Verteilungsfunctionen, Aequationes Math. 18 (1978) 345–356. [13] I. Istr˘a¸tescu, On some fixed points theorems in generalized Menger spaces, Boll. Un. Mat. Ital. V. Ser. A 13 (1976) 95–100. [14] O. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika 11 (1975) 336–344. [15] D. Mihe¸t, A Banach contraction theorem in fuzzy metric spaces, Fuzzy Sets and Systems 144 (3) (2004) 431–439. [16] V. Radu, Equicontinuous iterates of t-norms and applications to random normed and fuzzy Menger spaces, in: J. Sousa Ramos, D. Gronau, C. Mira, L. Reich, A.N. Sharkowski (Eds.), Iteration Theory (ECIT 02), Grazer Math. Ber. 33 (2004) 323–350. [17] V. Radu, Some suitable metrics on fuzzy metric spaces, Fixed Point Theory 5 (2) (2004) 323–347.
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