Iterated function systems and attractors in the KM fuzzy metric spaces

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Iterated function systems and attractors in the KM fuzzy metric spaces Jian-Zhong Xiao ∗ , Xing-Hua Zhu, Pan-Pan Jin School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, PR China Received 25 May 2013; received in revised form 5 July 2014; accepted 9 July 2014

Abstract Fractal attractors are considered as invariant sets for contraction mappings in hyperspaces of the KM fuzzy metric spaces. Using the properties of Hausdorff fuzzy metrics, several existence results of attractors are obtained for a finite family and a countable family of nonlinear contractions. In particular, unbounded invariant sets are also examined for the iterated function systems in KM fuzzy metric spaces. Our results are generalizations in some aspects and analogous to their counterparts in usual metric spaces. © 2014 Elsevier B.V. All rights reserved.

Keywords: Fuzzy metric space; Hyperspace; Hausdorff fuzzy metric; Attractor; Iterated function system

1. Introduction Fuzzy metric spaces were introduced by Kramosil and Michálek [14], George and Veeramani [6], which could be considered as modifications of the concept of probabilistic metric spaces given by Menger [15]. As important generalizations of classical metrics, these fuzzy metrics can be easily included within fuzzy systems since the value given by them can be directly interpreted as a fuzzy certainty degree of nearness. Recently, they have been applied to color image processing and analysis of algorithms (see [9,21]). The study of fixed point properties for mappings defined on fuzzy metric spaces attracted many authors, and has become an active direction of recent research (see [5,7,8,16,17,25,27] etc.). Since their introduction in [2,13], Iterated Function Systems (IFS) have become a very useful way of constructing fractal objects. So far the theory of fractal attractors or invariant sets has been developed by many researchers, see [1, 3,24] and the references therein. In the classical setting, since the values of Hausdorff metrics between two unbounded sets are sometimes infinite, it is impossible to obtain the existence results of unbounded invariant sets. To the best of our knowledge, there is no work reported on the unbounded invariant sets. However, the values of fuzzy Hausdorff

* Corresponding author. Fax: +86 02558731073.

E-mail address: [email protected] (J.-Z. Xiao). http://dx.doi.org/10.1016/j.fss.2014.07.004 0165-0114/© 2014 Elsevier B.V. All rights reserved.

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metrics are always finite, no matter the boundedness of sets. We would like to carry on the discussion of invariant sets in fuzzy setting. Our aim in this paper is to consider fuzzy fractals, i.e., to generalize the existence results of attractors in the KM fuzzy metric spaces. Some properties of the Hausdorff fuzzy metric for a given fuzzy metric space were investigated by several authors (see [11,19,20]). Using these properties we prove the completeness of hyperspaces of all closed and bounded subsets with respect to the Hausdorff fuzzy metric. Based on the completeness of hyperspaces we establish several existence results of attractors for a finite family and a countable family of fuzzy nonlinear contractions. In particular, unbounded invariant sets are also examined for the IFS in the KM fuzzy metric spaces. Our results are generalizations in some aspects and analogous to their counterpart in usual metric spaces (see Section 5). The structure of the paper is as follows. In Section 2 we give some basic concepts and auxiliary results which will be needed. In Section 3 we mainly discuss the boundedness of subsets in the KM fuzzy metric spaces and the completeness of some hyperspaces in fuzzy setting. In Section 4 we establish several existence results of attractors for a finite family and a countable family of fuzzy nonlinear contractions. Finally some examples and remarks concerning our results are given in Section 5. 2. Preliminaries We now state some basic concepts and results which will be used. In the sequel the letters R, R+ and Z+ will denote the sets of real numbers, nonnegative real numbers and positive integer numbers, respectively. Definition 2.1. (See Schweizer and Sklar [22], Hadži´c and Pap [12].) A function ∗ : [0, 1] × [0, 1] → [0, 1] is called a triangular norm (for short, a t-norm) if the following conditions are satisfied for any a, b, c, d ∈ [0, 1]: a ∗ b = b ∗ a; (a ∗ b) ∗ c = a ∗ (b ∗ c); a ∗ 1 = a; a ∗ b ≥ c ∗ d if a ≥ c and b ≥ d. A t-norm ∗ is said to be continuous if a ∗ b 1 is continuous at each point (a, b) ∈ [0, 1] × [0, 1]. For a ∈ [0, 1], the sequence {∗n a}∞ n=1 is defined by ∗ a = a and ∞ n n−1 n ∗ a = (∗ a) ∗ a. A t-norm ∗ is said to be of H-type if the sequence of functions {∗ a}n=1 is equicontinuous at a = 1. The t-norm ∗m defined by a ∗m b = min{a, b} is a trivial example of t-norm of H-type, but there are t-norms ∗ of H-type with ∗ = ∗m (see [12,18,26]). Definition 2.2. (See Kramosil and Michálek [14], Mihe¸t [17].) A triplet (X, M, ∗) is called a fuzzy metric space in the sense of Kramosil and Michálek (shortly, KM fuzzy metric space) if X is an arbitrary non-empty set, ∗ is a continuous t-norm and M is a fuzzy set on X × X × R+ satisfying the following conditions: (KM-1) (KM-2) (KM-3) (KM-4) (KM-5)

M(x, y, 0) = 0 for all x, y ∈ X; M(x, y, t) = 1 for all t > 0 if and only if x = y; M(x, y, t) = M(y, x, t), for all x, y ∈ X and t > 0; M(x, y, ·) : R+ → [0, 1] is left continuous; M(x, y, t) ∗ M(y, z, s) ≤ M(x, z, t + s), for all x, y, z ∈ X and t, s > 0.

From (KM-5) and (KM-2) we deduce that if t > s > 0 then M(x, y, t) = M(x, y, s + t − s) ≥ M(x, y, s) ∗ M(y, y, t − s) = M(x, y, s). Hence, if (X, M, ∗) is a KM fuzzy metric space then M(x, y, ·) is nondecreasing in R+ for all x, y ∈ X (see [7]). In Definition 2.2, if M is a fuzzy set on X × X × (0, +∞), and (KM-1), (KM-2), (KM-4) are replaced with the following (GV-1), (GV-2), (GV-4), respectively, then (X, M, ∗) is called a fuzzy metric space in the sense of George and Veeramani [6] (shortly, GV fuzzy metric space): (GV-1) M(x, y, t) > 0 for all t > 0 and x, y ∈ X; (GV-2) M(x, y, t) = 1 for all t > 0 if and only if x = y; (GV-4) M(x, y, ·) : (0, +∞) → [0, 1] is continuous.

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It is easy to see that, a GV fuzzy metric space can be considered a KM fuzzy metric space if we extend M by defining M(x, y, 0) = 0 for all x, y ∈ X. On the other hand, by defining the probabilistic metric Fx,y (t) = M(x, y, t), every KM fuzzy metric space (X, M, ∗) becomes a generalized Menger space under the continuous t-norm ∗ (see [16]). It is worth noting that (see [11,14,16,22,23] etc.), every KM fuzzy metric space (X, M, ∗) is a first countable Hausdorff topological space in the (ε, λ)-topology τM , i.e., the family of sets {Ux (ε, λ) : ε > 0, λ ∈ (0, 1]} is a base of neighborhoods of point x for τM , where   Ux (ε, λ) = y ∈ X : M(x, y, ε) > 1 − λ . By means of this topology τM , a sequence {xn } in (X, M, ∗) is said to be convergent to x (we write xn → x or limn→∞ xn = x) if limn→∞ M(xn , x, t) = 1 for all t > 0; {xn } is called a Cauchy sequence in (X, M, ∗) if for any given ε > 0 and λ ∈ (0, 1], there exists N = N (ε, λ) ∈ Z+ such that M(xn , xm , ε) > 1 − λ, whenever n, m ≥ N . (X, M, ∗) is said to be complete, if each Cauchy sequence in X converges to some point in X. Let A ⊂ X and x0 ∈ X. x0 is called a point of closure of A if there exists a sequence {xn } ⊂ A which converges to x0 . A denotes the set of all points of closure of A. A is said to be closed if A = A. A is said to be open if X\A is closed. A is open if and only if for each x ∈ A there exists Ux (ε, λ) such that Ux (ε, λ) ⊂ A. A is said to be compact, if every open cover of A has a finite subcover. A is said to be sequentially compact if every sequence of points of A has a subsequence which converges to a  point of A. Let ε > 0 and λ ∈ (0, 1]. A is said to have a finite (ε, λ)-net if there exists a finite set S ⊂ A such that A ⊂ x∈S Ux (ε, λ), i.e., for each y ∈ A there is x ∈ S such that M(x, y, ε) > 1 − λ. A is said to be totally bounded if for each ε > 0 and λ ∈ (0, 1], A has a finite (ε, λ)-net. The number β(A) = sup inf M(x, y, t) t>0 x,y∈A

is called the defective diameter of a set A. A is said to be fuzzy bounded if β(A) = 1. Lemma 2.1. (See [6].) Let (X, d) be a usual metric space. Define a function Md : X × X × R+ → [0, 1] by Md (x, y, t) =

t , t + d(x, y)

for x, y ∈ X and t ∈ R+ .

Then (X, Md , ∗m ) is a KM fuzzy metric space (and also is a GV fuzzy metric space), and Md is called the standard fuzzy metric induced by d. The topology τMd coincides with the topology on X deduced from d (see [10,11]). Let P(X) be the collection of non-empty subsets of X. By Pcl (X), Pcp (X) and Pbd (X) we will denote the collections of non-empty closed, compact, fuzzy bounded subsets of a KM fuzzy metric space (X, M, ∗), respectively. The notation Pcl,bd (X) means that Pcl,bd (X) = Pcl (X) ∩ Pbd (X). Definition 2.3. Let (X, M, ∗) be a KM fuzzy metric space and T : X → X a mapping. T is said to be closed if T A ∈ Pcl (X) for each A ∈ Pcl (X). T is said to be bounded if T A ∈ Pbd (X) for each A ∈ Pbd (X). T is said to be compact if T A ∈ Pcp (X) for each A ∈ Pbd (X). In the light of Mišˇcenko’s well-known theorem we see that every compact and first countable Hausdorff topological space is also metrizable. It is known that metrizability is a hereditary property. So the following lemma is true, and we omit its proof. Lemma 2.2. (Cf. Gregori and Romaguera [10].) Let (X, M, ∗) be a KM fuzzy metric space and A, B ∈ P(X). (1) (2) (3) (4)

A is compact if and only if A is sequentially compact. If A is compact, then A is totally bounded. If (X, M, ∗) is complete, then A is compact if and only if A is closed and totally bounded. If A, B ∈ Pcp (X), then A ∪ B ∈ Pcp (X).

Definition 2.4. (See Rodríguez-López et al. [19,20] and Egbert [4].) Let (X, M, ∗) be a KM fuzzy metric space and  A, B ∈ P(X). Let M(x, B, ·), MH (A, B, ·), M(A, B, ·) : R+ → [0, 1] be the functions defined by

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M(x, B, t) = sup M(x, y, t);

  MH (A, B, t) = min inf M(x, B, t), inf M(y, A, t) ; x∈A

y∈B

 M(A, B, 0) = 0;

 M(A, B, t) = sup MH (A, B, s)

y∈B

for t > 0.

0
 Then M(A, B, ·) is called the Hausdorff fuzzy metric (or distance) between A and B. Lemma 2.3. (Cf. Rodríguez-López et al. [20], Gutiérrez García et al. [11] and Egbert [4].) Let (X, M, ∗) be a KM fuzzy metric space, x ∈ X and A, B ∈ P(X). Then   (1) M(A, B, t) ≥ MH (A, B, s) for all s < t ; M(A, B, t) ≤ MH (A, B, t) for all t ∈ R+ and  M(A, B, t) = MH (A, B, t) (2) (3) (4) (5) (6)

for a.e. t ∈ R+ .

  M(A, B, t) ≤ infx∈A M(x, B, t) and M(A, B, t) ≤ infy∈B M(y, A, t) for all t ∈ R+ .   M(x, B, ·) = M(x, B, ·), MH (A, B, ·) = MH (A, B, ·) and M(A, B, ·) = M(A, B, ·).  M(x, B, ·), MH (A, B, ·) and M(A, B, ·) are nondecreasing in R+ .  B, ·) are left continuous in R+ . M(x, B, ·) and M(A, If limt→+∞ M(x, y, t) = 1 for all x, y ∈ X, then limt→+∞ M(x, B, t) = 1.

From [19, Theorem 3], [20, Corollary of Theorem 5] and [11, Theorem 2.5] we see that the following lemma is true, and its direct proof is omitted.  ∗) is a KM fuzzy metric space. If (X, M, ∗) Lemma 2.4. Let (X, M, ∗) be a KM fuzzy metric space. Then (Pcl (X), M,  ∗) and (Pcp (X), M,  ∗). is complete, so are (Pcl (X), M, Let ϕ : R+ → R+ be a function and t ∈ R+ . Then ϕ n (t) denotes the nth iteration of ϕ(t) in the sequel. The proof of the following lemma is easy, so we omit it. Lemma 2.5. Let ϕ : R+ → R+ be a nondecreasing function. If limn→∞ ϕ n (t) = 0 for all t > 0, then ϕ(t) < t for all t > 0. 3. Properties of hyperspaces In this section, we mainly discuss the boundedness of subsets in KM fuzzy metric spaces and the completeness of hyperspaces of all closed and bounded subsets with respect to the Hausdorff fuzzy metric. Lemma 3.1. Let (X, M, ∗) be a KM fuzzy metric space, and let x, y ∈ X, {xn } ⊂ X and xn → x. Then lim inf M(xn , y, t) ≥ M(x, y, t) n→∞

and

lim sup M(xn , y, t) ≤ M(x, y, t+) n→∞

for all t > 0. Particularly, limn→∞ M(xn , y, t) = M(x, y, t) for a.e. t ∈ R+ ; if M(x, y, ·) is continuous at the point t0 , then limn→∞ M(xn , y, t0 ) = M(x, y, t0 ). Proof. For each ε ∈ (0, t), from (KM-5) we have M(xn , y, t) ≥ M(xn , x, ε) ∗ M(x, y, t − ε);

(3.1)

M(x, xn , ε) ∗ M(xn , y, t) ≤ M(x, y, t + ε).

(3.2)

Since ∗ is continuous, it follows from (3.1) and (3.2) that lim inf M(xn , y, t) ≥ M(x, y, t − ε) and n→∞

lim sup M(xn , y, t) ≤ M(x, y, t + ε). n→∞

By (KM-4) and the arbitrariness of ε we obtain the desired conclusion. 2 Lemma 3.2. Let (X, M, ∗) be a KM fuzzy metric space. Let A, B, C ∈ P(X), x, y ∈ X and s, t ∈ R+ . Then

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(1) M(x, B, s + t) ≥ M(x, y, s) ∗ M(y, B, t). (2) MH (A, B, s + t) ≥ MH (A, C, s) ∗ MH (C, B, t). Proof. (1) For each b ∈ B, from (KM-5) we have M(x, b, s + t) ≥ M(x, y, s) ∗ M(y, b, t). Thus, by taking the supremum with respect to b ∈ B, the inequality follows from the continuity of ∗. (2) For each x ∈ A and z ∈ C, by Lemma 3.2(1) we have M(x, B, s + t) ≥ M(x, z, s) ∗ M(z, B, t) ≥ M(x, z, s) ∗ MH (C, B, t).

(3.3)

Taking the supremum with respect to z ∈ C in (3.3), it follows that M(x, B, s + t) ≥ M(x, C, s) ∗ MH (C, B, t) ≥ MH (A, C, s) ∗ MH (C, B, t), and so infx∈A M(x, B, s +t) ≥ MH (A, C, s) ∗MH (C, B, t). Similarly, we have infy∈B M(y, A, s +t) ≥ MH (A, C, s) ∗ MH (C, B, t). Hence MH (A, B, s + t) ≥ MH (A, C, s) ∗ MH (C, B, t), which is the desired inequality. 2 Lemma 3.3. Let (X, M, ∗) be a KM fuzzy metric space and A ∈ P(X). (1) If A is totally bounded, then A is totally bounded. (2) If A ∈ Pbd (X), then A ∈ Pbd (X). Proof. (1) For each ε > 0 and λ ∈ (0, 1], by the continuity of ∗ there is μ ∈ (0, λ] such that (1 − μ) ∗ (1 − μ) > 1 − λ.

(3.4)

Since A is also totally bounded, A has a finite (ε/2, μ)-net S. Next we prove that S is a finite (ε, λ)-net of A. In fact, for each x ∈ A, from the definition of A there exists xA ∈ A such that M(x, xA , ε/2) > 1 − μ. For such xA we can select y ∈ S with M(xA , y, ε/2) > 1 − μ. Hence, from (3.4) we have M(x, y, ε) ≥ M(x, xA , ε/2) ∗ M(xA , y, ε/2) ≥ (1 − μ) ∗ (1 − μ) > 1 − λ,  i.e., x ∈ y∈S Uy (ε, λ). This shows that A is totally bounded. (2) Let A ∈ Pbd (X). Since supt>0 infx,y∈A M(x, y, t) = 1, for each λ ∈ (0, 1) there exists Γλ > 0 such that M(x, y, Γλ ) > 1 − λ,

for all x, y ∈ A.

(3.5)

Suppose that x, y ∈ A are arbitrary. Then there exist {xn }, {yn } ⊂ A such that lim M(xn , x, Γλ ) = 1 and

n→∞

lim M(yn , y, Γλ ) = 1.

(3.6)

n→∞

Thus, from (3.5) we have M(x, y, 3Γλ ) ≥ M(x, yn , 2Γλ ) ∗ M(yn , y, Γλ )  ≥ M(x, xn , Γλ ) ∗ M(xn , yn , Γλ ) ∗ M(yn , y, Γλ )  ≥ M(x, xn , Γλ ) ∗ (1 − λ) ∗ M(yn , y, Γλ ).

(3.7)

Letting n → ∞ in (3.7), from (3.6) we have M(x, y, 3Γλ ) ≥ 1 − λ, and so inf M(x, y, 3Γλ ) ≥ 1 − λ.

x,y∈A

By the arbitrariness of λ we obtain supt>0 infx,y∈A M(x, y, t) = 1, i.e., A ∈ Pbd (X).

2

Example 3.1. A compact set is not necessarily fuzzy bounded in the KM fuzzy metric spaces. Let X = R+ , ∗ = ∗m . Define M : X × X × R+ → [0, 1] by M(x, y, 0) = M(y, x, 0) = 0 and

1 , if x = y, M(x, y, t) = 2 for all t > 0. 1, if x = y,

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Then (X, M, ∗m ) is a KM fuzzy metric space. Let a, b ∈ R+ with a = b and A = {a, b}. It is evident that A is compact. But β(A) = sup inf M(x, y, t) = sup M(a, b, t) = t>0 x,y∈A

t>0

1 2

shows that A is not fuzzy bounded. Lemma 3.4. Let (X, M, ∗) be a KM fuzzy metric space and A, B ∈ P(X). Suppose that limt→+∞ M(x, y, t) = 1 for all x, y ∈ X. Then (1) (2) (3) (4)

If A, B ∈ Pbd (X), then A ∪ B ∈ Pbd (X). If A is totally bounded, then A is fuzzy bounded; and Pcp (X) ⊂ Pcl,bd (X).  B, t) = 1. If A, B ∈ Pbd (X), then limt→+∞ MH (A, B, t) = limt→+∞ M(A, If limt→+∞ MH (A, B, t) = 1, then β(A) = β(B).

Proof. (1) For each λ ∈ (0, 1], by the continuity of ∗ there exist μ ∈ (0, λ] and ν ∈ (0, μ] such that (1 − μ) ∗ (1 − μ) > 1 − λ

and (1 − ν) ∗ (1 − ν) > 1 − μ.

(3.8)

Let A, B ∈ Pbd (X), a ∈ A and b ∈ B. Since limt→+∞ M(a, b, t) = 1, supt>0 infx,y∈A M(x, y, t) = 1 and supt>0 infx,y∈B M(x, y, t) = 1, there exists Γ > 0 such that M(a, b, Γ ) > 1 − ν;

M(x, a, Γ ) > 1 − ν,

∀x ∈ A;

M(y, b, Γ ) > 1 − ν,

∀y ∈ B.

(3.9)

Let x, y ∈ A ∪ B be arbitrary. Without loss of generality we suppose x ∈ A and y ∈ B. Then from (3.9) and (3.8) we have  M(x, y, 3Γ ) ≥ M(x, a, Γ ) ∗ M(a, b, Γ ) ∗ M(b, y, Γ )  ≥ (1 − ν) ∗ (1 − ν) ∗ (1 − ν) ≥ (1 − μ) ∗ (1 − μ) > 1 − λ, which shows that supt>0 infx,y∈A∪B M(x, y, t) = 1. Hence A ∪ B ∈ Pbd (X). (2) Let A be a totally bounded set, and a, b ∈ A be arbitrary. For each λ ∈ (0, 1], by the continuity of ∗ there exist μ ∈ (0, λ] and ν ∈ (0, μ] such that (3.8) holds. Since A has a finite (1, ν)-net S, there exist x0 , y0 ∈ S such that M(a, x0 , 1) > 1 − ν

and

M(b, y0 , 1) > 1 − ν.

(3.10)

Since S is a finite set, from the assertion (1) we see that supt>0 infx,y∈S M(x, y, t) = 1. Thus there exists Γ > 0 such that infx,y∈S M(x, y, Γ ) > 1 − ν, which follows that M(x0 , y0 , Γ ) > 1 − ν.

(3.11)

Hence, from (3.10), (3.11) and (3.8) we have  M(a, b, Γ + 2) ≥ M(a, x0 , 1) ∗ M(x0 , y0 , Γ ) ∗ M(y0 , b, 1)  ≥ (1 − ν) ∗ (1 − ν) ∗ (1 − ν) ≥ (1 − μ) ∗ (1 − μ) > 1 − λ. This shows that supt>0 infa,b∈A M(a, b, t) = 1, i.e., A ∈ Pbd (X). For A ∈ Pcp (X), from Lemma 2.2 we see that A is totally bounded, and so A is fuzzy bounded. Hence A ∈ Pcl,bd (X). (3) If A, B ∈ Pbd (X), then from the assertion (1) we see that A ∪ B ∈ Pbd (X). For each λ ∈ (0, 1) there exists Γλ > 0 such that infx,y∈A∪B M(x, y, Γλ ) > 1 − λ. This follows that M(x, y, Γλ ) > 1 − λ,

∀x ∈ A, ∀y ∈ B.

(3.12)

From (3.12) we have inf sup M(x, y, Γλ ) ≥ 1 − λ

y∈B x∈A

and

inf sup M(x, y, Γλ ) ≥ 1 − λ.

x∈A y∈B

(3.13)

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Combining two inequalities in (3.13) we get MH (A, B, Γλ ) ≥ 1 − λ. Since MH (A, B, ·) is nondecreasing, we have MH (A, B, t) ≥ 1 − λ for t > Γλ . This shows that limt→+∞ MH (A, B, t) = 1. From Lemma 2.3 (1) we also have  limt→+∞ M(A, B, t) = 1. (4) Suppose that β(B) = r and a1 , a2 ∈ A are two arbitrary points. Then 0 ≤ r ≤ 1. Without loss of generality we suppose r > 0. For each ε ∈ (0, r), by the continuity of ∗ and 1 ∗ r = r there exist δ ∈ (0, ε] and η ∈ (0, δ] such that (1 − δ) ∗ (r − δ) > r − ε

and

(1 − η) ∗ (r − η) > r − δ.

Since supt>0 infx,y∈B M(x, y, t) = r, there exists Γ > 0 such that M(x, y, Γ ) > r − η,

∀x, y ∈ B.

Since limt→+∞ MH (A, B, t) = 1, there exists Γ0 > 0 such that MH (A, B, Γ0 ) > 1 − η. Thus, there exist b1 , b2 ∈ B such that M(a1 , b1 , Γ0 ) > 1 − η

and M(a2 , b2 , Γ0 ) > 1 − η.

Hence we have  M(a1 , a2 , Γ + 2Γ0 ) ≥ M(a1 , b1 , Γ0 ) ∗ M(b1 , b2 , Γ ) ∗ M(b2 , a2 , Γ0 )  ≥ (1 − η) ∗ (r − η) ∗ (1 − η) ≥ (r − δ) ∗ (1 − δ) > r − ε. This implies that supt>0 infa1 ,a2 ∈A M(a1 , a2 , t) ≥ r, i.e., β(A) ≥ β(B). Applying the same argument we have β(B) ≥ β(A). This completes the proof. 2 Example 3.2. For the fuzzy unbounded sets A and B, it does not necessarily hold lim MH (A, B, t) = 1.

t→+∞

Let X = R, ∗ = ∗m . Define M : X × X × R+ → [0, 1] by M(x, y, t) =

t . t + |x − y|

Then (X, M, ∗m ) is a KM fuzzy metric space. Let A = [0, +∞), B = (−∞, 0] and C = [1, +∞). It is easy to see that MH (A, B, t) ≡ 0 and limt→+∞ MH (A, C, t) = 1. Example 3.3. Let l 1 be the set of all sequences x = {xn } of numbers such that Define M : l 1 × l 1 × R+ → [0, 1] by M(x, y, 0) = 0 and M(x, y, t) =

t+

∞

t

n=1 |xn

− yn |

,

∞

n=1 |xn |

converges and ∗ = ∗m .

for t > 0 and x = {xn }, y = {yn } ∈ l 1 .

Then (l 1 , M, ∗m ) is a KM fuzzy metric space satisfying limt→+∞ M(x, y, t) = 1 for all x, y ∈ l 1 . Let A = {en : n ∈ t Z+ } ⊂ l 1 , where en is a sequence whose nth term is 1 and kth term is 0 when k = n. If k = n, then M(ek , en , t) = t+2 , and hence we have t β(A) = sup inf M(x, y, t) = sup = 1. x,y∈A t + 2 t>0 t>0 Taking / Uen (1/2, 1/2) when k = n. Thus, for any finite set S ⊂ A we have  (ε, λ) = (1/2, 1/2), we see that ek ∈ A ⊂ x∈S Ux (1/2, 1/2), i.e., A does not have a finite (1/2, 1/2)-net. This shows that A is fuzzy bounded, but it is not totally bounded. Hence the converse implication of Lemma 3.4 (2) does not hold. Let B = {nen : n ∈ Z+ } ⊂ l 1 . t If k = n, then M(kek , nen , t) = t+k+n , and hence β(B) = supt>0 infx,y∈B M(x, y, t) = 0. This shows that B is not fuzzy bounded.  ∗) is a complete KM fuzzy Theorem 3.5. Let (X, M, ∗) be a complete KM fuzzy metric space. Then (Pcl,bd (X), M, metric space.

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 ∗) is a complete KM fuzzy metric space. Since Proof. By Lemma 2.4, (Pcl (X), M, Pcl,bd (X) ⊂ Pcl (X),  it is enough to prove that Pcl,bd (X) is closed with respect to M. ∞  Let {An }n=1 ⊂ Pcl,bd (X), An → A with respect to M. We shall prove that A ∈ Pcl,bd (X). Take λ ∈ (0, 1]. By the continuity of ∗ there exist μ ∈ (0, λ] and ν ∈ (0, μ] such that (1 − μ) ∗ (1 − μ) > 1 − λ

and (1 − ν) ∗ (1 − ν) > 1 − μ.

From the convergence of {An } we see that there exists N  M(A, An , 1) > 1 − ν,

∈ Z+

(3.14)

such that

for all n ≥ N.

(3.15)

Since AN is fuzzy bounded, we have supt>0 infx,y∈AN M(x, y, t) = 1. Thus, there exists Γ > 0 such that M(x, y, Γ ) > 1 − ν for all x, y ∈ AN . Suppose that u, w ∈ A are two arbitrary points. From Lemma 2.3 (2) and (3.15) it follows that there exist x0 , y0 ∈ AN such that M(u, x0 , 1) > 1 − ν and M(w, y0 , 1) > 1 − ν. Thus, from (3.14) we have  M(u, w, Γ + 2) ≥ M(u, x0 , 1) ∗ M(x0 , y0 , Γ ) ∗ M(y0 , w, 1)  ≥ (1 − ν) ∗ (1 − ν) ∗ (1 − ν) ≥ (1 − μ) ∗ (1 − μ) > 1 − λ. Hence supt>0 infu,w∈A M(u, w, t) ≥ 1 − λ. By the arbitrariness of λ, we have supt>0 infu,w∈A M(u, w, t) = 1, i.e., A is fuzzy bounded. From Lemma 2.4 we see that A ∈ Pcl (X). Therefore A ∈ Pcl,bd (X), which is the desired conclusion. 2 Theorem 3.6. Let (X, M, ∗) be a complete KM fuzzy metric space. Let {An }∞ n=1 ⊂ Pcl (X) be a Cauchy sequence with  respect to M. ∞ (1) If {An }∞ n=1 An ∈ Pcp (X). n=1 ⊂ Pcp (X), then  ∞ (2) If {An }n=1 ⊂ Pcl,bd (X), then ∞ n=1 An ∈ Pcl,bd (X).  Proof. Suppose that B = ∞ n=1 An . Then we have B ∈ Pcl (X). (1) It remains to prove that B is totally bounded. Give any ε > 0 and λ ∈ (0, 1]. Since ∗ is continuous, there exists μ ∈ (0, λ] such that (1 − μ) ∗ (1 − μ) > 1 − λ.

(3.16)

+  Since {An }∞ n=1 ⊂ Pcp (X) is a Cauchy sequence with respect to M, there exists N ∈ Z such that

 m , An , ε/2) > 1 − μ, for all m, n ≥ N. (3.17) M(A ∞ N Putting EN = n=N An and BN = n=1 An , we have B = BN ∪ EN and AN ⊂ BN ∩ EN . Since BN is compact, BN is totally bounded. Thus, BN has a finite (ε/2, μ)-net SN . We claim that SN is a finite (ε, λ)-net of B. In fact, for each x ∈ B, we have x ∈ BN or x ∈ EN . If x ∈ BN , then there exists y ∈ SN such that M(x, y, ε) ≥ M(x, y, ε/2) > 1 − μ ≥ 1 − λ.

(3.18)

If x ∈ EN , then there exists m ≥ N such that x ∈ Am . From (3.17) and Lemma 2.3 (2) it follows that there exists xN ∈ AN such that M(x, xN , ε/2) > 1 − μ. For this xN we can select y ∈ SN such that M(xN , y, ε/2) > 1 − μ. Therefore from (3.16) we have M(x, y, ε) ≥ M(x, xN , ε/2) ∗ M(xN , y, ε/2) ≥ (1 − μ) ∗ (1 − μ) > 1 − λ.

(3.19)

Combining (3.18) with (3.19), it implies that B is totally bounded. From Lemma 3.3 (1) it follows that B is totally bounded. From Lemma 2.2 and the completeness of (X, M, ∗) we conclude that B is compact, i.e., B ∈ Pcp (X). (2) It remains to prove that B ∈ Pbd (X). Give an arbitrary λ ∈ (0, 1]. Applying the same argument as above, we have μ ∈ (0, λ] and ν ∈ (0, μ] such that

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(1 − μ) ∗ (1 − μ) > 1 − λ

and (1 − ν) ∗ (1 − ν) > 1 − μ. +  Since {An }∞ n=1 ⊂ Pcl,bd (X) is a Cauchy sequence with respect to M, there exists N ∈ Z such that

(3.20)

 m , An , 1) > 1 − ν, for all m, n ≥ N. (3.21) M(A ∞ N Setting EN = n=N An and BN = n=1 An , we have B = BN ∪EN and AN ⊂ BN ∩EN . Since by Lemma 3.4 (1) BN is fuzzy bounded, we have supt>0 infx,y∈BN M(x, y, t) = 1, and then there exists Γ > 0 such that M(x, y, Γ ) > 1 − ν for all x, y ∈ BN . Suppose that x, y ∈ B are two arbitrary points. If x, y ∈ BN , then it is clear that M(x, y, Γ + 2) ≥ M(x, y, Γ ) > 1 − ν ≥ 1 − λ.

(3.22)

If x ∈ BN and y ∈ EN , then there exists m ≥ N such that y ∈ Am . From (3.21) and Lemma 2.3 (2) it follows that there exists xN ∈ AN ⊂ BN such that M(y, xN , 1) > 1 − ν. Thus, by (3.20) we have M(x, y, Γ + 2) ≥ M(x, y, Γ + 1) ≥ M(y, xN , 1) ∗ M(xN , x, Γ ) ≥ (1 − ν) ∗ (1 − ν) > 1 − μ ≥ 1 − λ.

(3.23)

If x, y ∈ EN , then from (3.21) and Lemma 2.3 (2) it follows that there exist xN , yN ∈ AN such that M(x, xN , 1) > 1 − ν and M(y, yN , 1) > 1 − ν. In this case we have  M(x, y, Γ + 2) ≥ M(x, xN , 1) ∗ M(xN , yN , Γ ) ∗ M(yN , y, 1)  ≥ (1 − ν) ∗ (1 − ν) ∗ (1 − ν) ≥ (1 − μ) ∗ (1 − μ) > 1 − λ. (3.24) Combining (3.22), (3.23) with (3.24), we have supt>0 infx,y∈B M(x, y, t) ≥ infx,y∈B M(x, y, Γ + 2) ≥ 1 − λ. The arbitrariness of λ shows that supt>0 infx,y∈B M(x, y, t) = 1, i.e., B ∈ Pbd (X). By Lemma 3.3 (2) we have B ∈ Pbd (X). Therefore B ∈ Pcl,bd (X), which is the desired conclusion. 2 m Lemma 3.7. Let (X, M, ∗) be a KM fuzzy metric space. Suppose that {Aj }m j =1 , {Bj }j =1 ⊂ P(X), A = m B = j =1 Bj , where n ∈ Z+ . Then for all t > 0,

m

j =1 Aj

and

  j , Bj , t). M(A, B, t) ≥ min M(A

MH (A, B, t) ≥ min MH (Aj , Bj , t) and 1≤j ≤m

1≤j ≤m

Proof. Since Bj ⊂ B for each j , we have M(x, B, s) ≥ M(x, Bj , s) for x ∈ X and s > 0. Thus,   inf M(x, B, s) ≥ inf M(x, Bj , s) ≥ min inf M(x, Bj , s), inf M(y, Aj , s) = MH (Aj , Bj , s). x∈Aj

x∈Aj

x∈Aj

y∈Bj

From this it follows that inf M(x, B, s) = min inf M(x, B, s) ≥ min MH (Aj , Bj , s). 1≤j ≤m x∈Aj

x∈A

(3.25)

1≤j ≤m

Similarly, the following inequality holds: inf M(y, A, s) = min inf M(y, A, s) ≥ min MH (Aj , Bj , s). 1≤j ≤m y∈Bj

y∈B

(3.26)

1≤j ≤m

Combining (3.25) with (3.26), we obtain MH (A, B, t) ≥ min1≤j ≤m MH (Aj , Bj , t) and    M(A, B, t) = sup min inf M(x, B, s), inf M(y, A, s) ≥ sup min MH (Aj , Bj , s) 0
= min

x∈A

0
y∈B

 j , Bj , t). sup MH (Aj , Bj , s) = min M(A

1≤j ≤m 0
1≤j ≤m

2

Lemma 3.8.  Let (X, M, ∗) be a KM fuzzy metric space. Suppose that {Ai : i ∈ I }, {Bi : i ∈ I } ⊂ P(X), A = and B = i∈I Bi . Then MH (A, B, t) ≥ inf MH (Ai , Bi , t) i∈I

  i , Bi , t) M(A, B, t) ≥ inf M(A i∈I

for all t ∈ R+

for a.e. t ∈ R+ .

and

 i∈I

Ai

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Proof. Let t ∈ R+ . Applying the same argument as the proof of Lemma 3.7, we can obtain inf M(x, B, t) = inf inf M(x, B, t) ≥ inf MH (Ai , Bi , t)

x∈A

i∈I x∈Ai

i∈I

and

inf M(y, A, t) = inf inf M(y, A, t) ≥ inf MH (Ai , Bi , t).

y∈B

i∈I y∈Bi

i∈I

Hence MH (A, B, t) ≥ infi∈I MH (Ai , Bi , t). It follows from Lemma 2.3 (1) that infi∈I MH (Ai , Bi , t) ≥    i , Bi , t). Since MH (A, B, t) = M(A,  i , Bi , t) for infi∈I M(A B, t) for a.e. t ∈ I , we have M(A, B, t) ≥ infi∈I M(A + a.e. t ∈ R . 2 Lemma 3.9. Let (X, M, ∗) be a KM fuzzy metric space. Let ϕ : R+ → R+ be functions such that ϕ(t) ≤ t for all t > 0. If T : X → X is a mapping satisfying M(T x, T y, ϕ(t)) ≥ M(x, y, t), for all x, y ∈ X and t > 0, Then (1) T is bounded. (2) T is continuous. (3) T A ∈ Pcp (X) for each A ∈ Pcp (X). Proof. From ϕ(t) ≤ t we have M(T x, T y, t) ≥ M(x, y, t),

for all x, y ∈ X and t > 0.

(1) If A ∈ Pbd (X), then supt>0 infx,y∈A M(T x, T y, t) ≥ supt>0 infx,y∈A M(x, y, t) = 1, i.e., T A ∈ Pbd (X). Hence T is bounded. (2) Suppose that xn → x. Then lim inf M(T xn , T x, t) ≥ lim M(xn , x, t) = 1, n→∞

n→∞

for all t > 0.

This shows that T xn → T x, and so T is continuous. (3) It follows from the assertion (2). 2 4. Several existence results of attractors Theorem 4.1. Let (X, M, ∗) be a complete KM fuzzy metric space such that ∗ is of H-type and limt→+∞ M(x, y, t) = 1 for all x, y ∈ X. Let ϕ1 , ϕ2 , · · · , ϕm : R+ → R+ be functions such that ϕj (t) < t and limn→∞ ϕjn (t) = 0 for j = 1, 2, · · · , m and t > 0. Let T1 , T2 , · · · , Tm : X → X be mappings such that

M Tj x, Tj y, ϕj (t) ≥ M(x, y, t), for j = 1, 2, · · · , m; x, y ∈ X and t > 0. (4.1) Then  (1) there exists a unique F0 ∈ Pcp (X) such that F0 = m j =1 Tj F0 .  (2) if T1 , T2 , · · · , Tm are closed, then there exists a unique F0 ∈ Pcl,bd (X) such that F0 = m j =1 Tj F0 . (3) if T1 , T2 , · · · , Tm are closed and there exists A0 ∈ Pcl (X) which satisfies A0 ∈ / Pbd(X) and limt→+∞ MH (A0 , Tj A0 , t) = 1 for j = 1, 2, · · · , m, then there exists an F0 ∈ Pcl (X) such that F0 = m j =1 Tj F0 and F0 ∈ / Pbd (X). Proof. Let ϕ : R+ → R+ be the function defined by ϕ(t) = max1≤j ≤m ϕj (t) for t ∈ R+ . Then ϕj (t) < t and limn→∞ ϕ n (t) = 0 for all t > 0. Let T : Pcl (X) → P(X) be the mapping defined by TA=

m 

Tj A,

for A ∈ Pcl (X).

j =1

Take A, B ∈ Pcl (X). Thus, from (4.1) and Lemma 3.7 it follows that

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MH T A, T B, ϕ(t) ≥ min MH Tj A, Tj B, ϕ(t) ≥ min MH Tj A, Tj B, ϕj (t) 1≤j ≤m 1≤j ≤m 



 = min min inf sup M Tj x, Tj y, ϕj (t) , inf sup M Tj x, Tj y, ϕj (t) 1≤j ≤m

x∈A y∈B

1≤j ≤m

x∈A y∈B

y∈B x∈A

  ≥ min min inf sup M(x, y, t), inf sup M(x, y, t) y∈B x∈A

= MH (A, B, t).

(4.2)

 ∗) is a complete KM fuzzy metric space. Suppose that A0 ∈ Pcl,bd (X) is (2) By Theorem 3.5, (Pcl,bd (X), M, arbitrary and An = T An−1 for all n ∈ Z+ . Since T1 , T2 , · · · , Tm are closed, T is closed. Hence from Lemma 3.9 (1) we obtain a sequence {An } ⊂ Pcl,bd (X). Suppose that t > 0. Using the relation (4.2) we get

MH An , An+1 , ϕ n (t) = MH T An−1 , T An , ϕ n (t) ≥ MH An−1 , An , ϕ n−1 (t)

≥ · · · ≥ MH A1 , A2 , ϕ(t) ≥ MH (A0 , A1 , t). (4.3) Since by Lemma 3.4 (3) lims→∞ MH (A0 , A1 , s) = 1, for each λ ∈ (0, 1] there exists t0 > 0 such that MH (A0 , A1 , t0 ) > 1 − λ. Since ϕ n (t0 ) → 0, there exists N ∈ Z+ such that ϕ n (t0 ) < t for all n ≥ N . From (4.3) and the monotonicity of MH (An , An+1 , ·) it follows that MH (An , An+1 , t) ≥ MH (An , An+1 , ϕ n (t0 )) ≥ MH (A0 , A1 , t0 ) > 1 − λ, and so we have lim MH (An , An+1 , t) = 1

n→∞

for all t > 0.

(4.4)

We claim that, for all k, n ∈ Z+ , 

MH (An , An+k , t) ≥ ∗k MH An , An+1 , t − ϕ(t) .

(4.5)

Indeed, this is obvious for k = 1. Assume that (4.5) holds for some k. It follows from (4.2) and (4.5) that 

MH An+1 , An+k+1 , ϕ(t) ≥ MH (An , An+k , t) ≥ ∗k MH An , An+1 , t − ϕ(t) .

(4.6)

Since ϕ(t) < t, by (4.6) we have

MH (An , An+k+1 , t) = MH An , An+k+1 , t − ϕ(t) + ϕ(t) 

≥ ∗ MH An , An+1 , t − ϕ(t) , MH An+1 , An+k+1 , ϕ(t)



 ≥ ∗ MH An , An+1 , t − ϕ(t) , ∗k MH An , An+1 , t − ϕ(t)

= ∗k+1 MH An , An+1 , t − ϕ(t) , which shows (4.5) holds for k + 1. By induction, (4.5) holds for all k ∈ Z+ . Suppose that ε > 0 and λ ∈ (0, 1] are given. Since ∗ is a t-norm of H-type, there exists δ > 0 such that ∗k (s) > 1 − λ,

for all s ∈ (1 − δ, 1] and k ∈ Z+ .

(4.7)

From (4.4) it follows the existence of N ∈ Z+ such that MH (An , An+1 , ε/2 − ϕ(ε/2)) > 1 − δ for all n ≥ N . Hence,  n , An+k , ε) ≥ MH (An , An+k , ε/2) > 1 − λ for all n ≥ N and k ∈ Z+ . Therefore, from (4.5) and (4.7) we get M(A  ∗), there exists F0 ∈ Pcl,bd (X) such that {An } {An } is a Cauchy sequence. By the completeness of (Pcl,bd (X), M,  n , F0 , t), we have  converges to F0 , i.e., limn→∞ M(An , F0 , t) = 1 for all t > 0. Since MH (An , F0 , t) ≥ M(A lim MH (An , F0 , t) = 1 for all t > 0.

n→∞

Now we show that F0 is a fixed point of T . From (4.2) and the continuity of ∗ it follows that

MH (F0 , T F0 , t) ≥ MH F0 , An+1 , t − ϕ(t) ∗ MH T An , T F0 , ϕ(t)

≥ MH F0 , An+1 , t − ϕ(t) ∗ MH (An , F0 , t) → 1 as n → ∞,  0 , T F0 , t) ≥ MH (F0 , T F0 , t/2), we have M(F  0 , T F0 , t) = 1 for all i.e., MH (F0 , T F0 , t) = 1 for all t > 0. Since M(F t > 0, and hence T F0 = F0 . To prove uniqueness, we assume T E0 = E0 and fix t > 0. Since MH (F0 , E0 , s) → 1 as

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s → +∞ by Lemma 3.4 (3), for each λ ∈ (0, 1] there exists t0 > 0 such that MH (F0 , E0 , t0 ) > 1 − λ. Since ϕ n (t0 ) → 0 as n → ∞, there exists N ∈ Z+ such that ϕ N (t0 ) < t. Hence, by (4.2) we have

MH (F0 , E0 , t) ≥ MH F0 , E0 , ϕ N (t0 ) = MH T F0 , T E0 , ϕ N (t0 ) ≥ MH F0 , E0 , ϕ N−1 (t0 )

≥ · · · ≥ MH F0 , E0 , ϕ(t0 ) ≥ MH (F0 , E0 , t0 ) > 1 − λ, which shows that MH (F0 , E0 , t) = 1 for all t > 0. Since t is arbitrary, we have F0 = E0 . (1) Using Lemma 3.9 (3) we see that T A ∈ Pcp (X) for each A ∈ Pcp (X). Based on Lemma 2.4 instead of Theorem 3.5, the assertions (1) can be verified similarly to the proof of (2). (3) Since A0 ∈ Pcl (X) satisfies limt→+∞ MH (A0 , Tj A0 , t) = 1 for j = 1, 2, · · · , m, from Lemma 3.7 we have limt→+∞ MH (A0 , T A0 , t) = 1. Let An = T An−1 for all n ∈ Z+ . From the closeness of T we have {An } ⊂ Pcl (X). Based on Lemma 2.4 instead of Theorem 3.5, similarly to the proof of (1), we can show that there exists an F0 ∈  / Pbd (X). Pcl (X) such that F0 = m j =1 Tj F0 and limn→∞ MH (An , F0 , t) = 1 for all t > 0. It remains to prove that F0 ∈ / Pbd (X), we can suppose that β(A0 ) = r < 1. From Lemma 3.4 (4) and Since A0 ∈ lim MH (A0 , A1 , t) = lim MH (A0 , T A0 , t) = 1

t→+∞

t→+∞

we see that β(A1 ) = r. By induction we can show that β(An ) = r for each n ∈ Z+ . Assume that β(F0 ) = 1 and 0 < λ < 1 − r. By the continuity of ∗ there exist μ ∈ (0, λ] and ν ∈ (0, μ] such that (1 − ν) ∗ (1 − ν) > 1 − μ

and (1 − μ) ∗ (1 − μ) > 1 − λ.

Since supt>0 infx,y∈F0 M(x, y, t) = 1, there exists Γ > 0 such that M(x, y, Γ ) > 1 − ν,

∀x, y ∈ F0 .

Since limn→∞ MH (An , F0 , t) = 1 for all t > 0, there exists N ∈ Z+ such that MH (AN , F0 , 1) > 1 − ν. Let a, b ∈ AN be two arbitrary points. Then there exist x0 , y0 ∈ F0 such that M(a, x0 , 1) > 1 − ν

and

M(b, y0 , 1) > 1 − ν.

Thus we have  M(a, b, Γ + 2) ≥ M(a, x0 , 1) ∗ M(x0 , y0 , Γ ) ∗ M(y0 , b, 1)  ≥ (1 − ν) ∗ (1 − ν) ∗ (1 − ν) ≥ (1 − μ) ∗ (1 − μ) > 1 − λ. This implies that β(AN ) = supt>0 infa,b∈AN M(a, b, t) ≥ 1 − λ > r, a contradiction to β(AN ) = r. Therefore / Pbd (X). This completes the proof. 2 β(F0 ) < 1, i.e., F0 ∈ From Theorem 4.1 we see that in a complete KM fuzzy metric space with an H-type t-norm, an IFS which consists of m nonlinear contractions {T1 , T2 , · · · , Tm } has a unique compact attractor F0 . Also, if T1 , T2 , · · · , Tm are closed, then the iterative sequence generated by a closed unbounded set A0 converges to an unbounded invariant set F0 . Corollary 4.2. Let (X, M, ∗) be a complete KM fuzzy metric space such that ∗ is of H-type and limt→+∞ M(x, y, t) = 1 for all x, y ∈ X. Let ϕ1 , ϕ2 , · · · , ϕm : R+ → R+ be functions such that ϕj (t) < t and limn→∞ ϕjn (t) = 0 for j = 1, 2, · · · , m and all t > 0. Let T1 , T2 , · · · , Tm : X → X be mappings such  that (4.1) holds. If T1 , T2 , · · · , Tm are closed, then for any A0 ∈ Pcl,bd (X), the sequence {An } generated by An = m j =1 Tj An−1 converges to a compact set with  respect to M. Proof. Suppose that limn→∞ An = F0 . By Theorem 4.1 (2), F0 ∈ Pcl,bd (X) is a  unique set satisfying F0 =  m m T F ; and by Theorem 4.1 (1) there exists unique E ∈ P (X) such that E = j 0 0 cp 0 j =1 j =1 Tj E0 . Since Pcp (X) ⊂ Pcl,bd (X) by Lemma 3.4 (2), we have F0 = E0 . 2 Theorem 4.3. Let (X, M, ∗) be a complete KM fuzzy metric space such that ∗ is of H-type and limt→+∞ M(x, y, t) = 1 for all x, y ∈ X. Let ϕ : R+ → R+ be a function such that ϕ(t) < t and limn→∞ ϕ n (t) = 0 for all t > 0. Let {Tn }∞ n=1 be a sequence of mappings from X into X such that

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M Tn x, Tn y, ϕ(t) ≥ M(x, y, t),

for all n ∈ Z+ and x, y ∈ X and t > 0.

13

(4.8)

 for each A ∈ Pcp (X), then there exists a unique (1) If {Tn A} is a Cauchy sequence in Pcp (X) with respect to M ∞ F0 ∈ Pcp (X) such that F0 = n=1 Tn F0 .  for each (2) If {Tn } is a sequence of closed mappings and {Tn A} is a Cauchy sequence in Pcl,bd (X) with respect to M ∞ A ∈ Pcl,bd (X), then there exists a unique F0 ∈ Pcp (X) such that F0 = n=1 Tn F0 , and for any A0 ∈ Pcl,bd (X), the   sequence {Ak } generated by Ak = ∞ n=1 Tn Ak−1 converges to F0 with respect to M.  ∗) is a complete KM fuzzy metric space. Let T : Pcp (X) → Pcp (X) be the Proof. (1) By Lemma 2.4, (Pcp (X), M, mapping defined by TA=

∞ 

Tn A,

for A ∈ Pcp (X).

n=1

Using Theorem 3.6 (1) we see that T A ∈ Pcp (X) for each A ∈ Pcp (X). For each A, B ∈ Pcp (X), from (4.8) it follows that 





 MH Tn A, Tn B, ϕ(t) = min inf sup M Tn x, Tn y, ϕ(t) , inf sup M Tn x, Tn y, ϕ(t) x∈A y∈B

y∈B x∈A

  ≥ min inf sup M(x, y, t), inf sup M(x, y, t) = MH (A, B, t). x∈A y∈B

y∈B x∈A

(4.9)

Moreover, from (4.9), Lemmas 2.3 (3) and 3.8 we have



MH T A, T B, ϕ(t) ≥ inf MH Tn A, Tn B, ϕ(t) n∈Z+

≥ inf MH (A, B, t) = MH (A, B, t). n∈Z+

(4.10)

Suppose that A0 ∈ Pcp (X) and An = T An−1 for all n ∈ Z+ . Then we have {An } ⊂ Pcp (X). Let t > 0. In view of (4.10), we have

MH An , An+1 , ϕ n (t) = MH T An−1 , T An , ϕ n (t) ≥ MH An−1 , An , ϕ n−1 (t)

≥ · · · ≥ MH A1 , A2 , ϕ(t) ≥ MH (A0 , A1 , t)  0 , A1 , t). (4.11) ≥ M(A  0 , A1 , t) = 1. Note By Lemma 2.3 (4), MH (An , An+1 , ·) is nondecreasing on R+ . By Lemma 3.4 (3), limt→+∞ M(A n that limn→∞ ϕ (t) → 0 as n → ∞. In the same manner as the proof of Theorem 4.1, from (4.11) we can show that lim MH (An , An+1 , t) = 1

n→∞

for all t > 0.

Since ϕ(t) < t, by induction and Lemma 3.2 (2) we have 

MH (An , An+k , t) ≥ ∗k MH An , An+1 , t − ϕ(t) ,

(4.12)

for all k, n ∈ Z+ .

(4.13)

Suppose that ε > 0 and λ ∈ (0, 1] are given. Since ∗ is a t-norm of H-type, there exists δ > 0 such that ∗k (s) > 1 − λ,

for all s ∈ (1 − δ, 1] and k ∈ Z+ .

(4.14)

From (4.12) it follows the existence of N ∈ Z+ such that MH (An , An+1 , ε/2 − ϕ(ε/2)) > 1 − δ for all n ≥ N .  n , An+k , ε) ≥ MH (An , An+k , ε/2) > 1 − λ for all n ≥ N and k ∈ Z+ , which Hence, by (4.13) and (4.14) we get M(A  ∗) implies the existence of F0 ∈ Pcp (X) means that {An } is a Cauchy sequence. The completeness of (Pcp (X), M,  such that {An } converges to F0 , i.e., limn→∞ M(An , F0 , t) = 1 for all t > 0. Applying Lemma 2.3 (1), we also have limn→∞ MH (An , F0 , t) = 1 for all t > 0. Next, we show that E is a fixed point of T . Indeed, for each ε > 0 and λ ∈ (0, 1], from the continuity of ∗ it follows the existence of μ ∈ (0, λ] such that (1 − μ) ∗ (1 − μ) > 1 − λ; and from limn→∞ MH (An , F0 , t) = 1 it follows the existence of N ∈ Z+ such that MH (An , F0 , ε − ϕ(ε)) > 1 − μ for all n ≥ N . Thus, by (4.10) and Lemma 3.2 (2) we have

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 0 , T F0 , 2ε) ≥ MH (F0 , T F0 , ε) ≥ MH F0 , An+1 , ε − ϕ(ε) ∗ MH T An , T F0 , ϕ(ε) M(F

≥ MH F0 , An+1 , ε − ϕ(ε) ∗ MH (An , F0 , ε) ≥ (1 − μ) ∗ (1 − μ) > 1 − λ,  0 , T F0 , t) = 1 for all t > 0, and hence T F0 = F0 . Finally, we prove uniqueness. Assume which implies that M(F  0 , E0 , s) → 1 as s → +∞ by Lemma 3.4 (3), for each λ ∈ (0, 1] there exists T E0 = E0 and fix t > 0. Since M(F  0 , E0 , t0 ) > 1 − λ. Since ϕ n (t0 ) → 0 as n → ∞, there exists N ∈ Z+ such that ϕ N (t0 ) < t. t0 > 0 such that M(F Hence, in virtue of (4.10) we have

MH (F0 , E0 , t) ≥ MH F0 , E0 , ϕ N (t0 ) = MH T F0 , T E0 , ϕ N (t0 ) ≥ MH F0 , E0 , ϕ N−1 (t0 )

 0 , E0 , t0 ) > 1 − λ, ≥ · · · ≥ MH F0 , E0 , ϕ(t0 ) ≥ MH (F0 , E0 , t0 ) ≥ M(F  0 , E0 , t) = 1 for t > 0, i.e., which shows that MH (F0 , E0 , t) = 1 for t > 0. From Lemma 2.3 (1) it follows that M(F F0 = E0 . (2) It can be verified in the same way as the proof of (1) and Corollary 4.2, based on Theorem 3.6 (2) instead of Theorem 3.6 (1). 2 5. Some remarks and examples In this section, some remarks and examples concerning our results are given. Remark 5.1. By Lemma 2.5, if ϕ is nondecreasing, then for all t > 0, limn→∞ ϕ n (t) = 0 implies that ϕ(t) < t. Hence from Theorems 4.1 and 4.3 we obtain the following consequences. Corollary 5.1. Let (X, M, ∗) be a complete KM fuzzy metric space such that ∗ is of H-type and limt→+∞ M(x, y, t) = 1 for all x, y ∈ X. Let ϕ1 , ϕ2 , · · · , ϕm : R+ → R+ be nondecreasing functions limn→∞ ϕjn (t) = 0 for j = 1, 2, · · · , m and all t > 0. Let T1 , T2 , · · · , Tm : X → X be mappings such that

M Tj x, Tj y, ϕj (t) ≥ M(x, y, t), for j = 1, 2, · · · , m; x, y ∈ X and t > 0. Then the three assertions (1), (2) and (3) in Theorem 4.1 hold. Corollary 5.2. Let (X, M, ∗) be a complete KM fuzzy metric space such that ∗ is of H-type and limt→+∞ M(x, y, t) = 1 for all x, y ∈ X. Let ϕ : R+ → R+ be a nondecreasing function limn→∞ ϕ n (t) = 0 for all t > 0. Let {Tn }∞ n=1 be a sequence of mappings from X into X such that

M Tn x, Tn y, ϕ(t) ≥ M(x, y, t), for all n ∈ Z+ and x, y ∈ X and t > 0. Then the two assertions (1), (2) in Theorem 4.3 hold. Remark 5.2. Since each nonlinear contraction with a function ϕ includes the case of linear contraction as its special case, each existence result of fuzzy attractors for nonlinear contraction implies a corresponding existence result for linear contraction, if we take ϕ(t) = αt, where α ∈ (0, 1). For example, from Theorem 4.1 we obtain the following consequence. Corollary 5.3. Let (X, M, ∗) be a complete KM fuzzy metric space such that ∗ is of H-type and limt→+∞ M(x, y, t) = 1 for all x, y ∈ X. Let αj ∈ (0, 1), j = 1, 2, · · · , m. Let T1 , T2 , · · · , Tm : X → X be mappings such that M(Tj x, Tj y, αj t) ≥ M(x, y, t),

for j = 1, 2, · · · , m; x, y ∈ X and t > 0.

Then the three assertions (1), (2) and (3) in Theorem 4.1 hold. Remark 5.3. As direct consequences of our results in Section 4, we can obtain the corresponding existence theorems of attractors in usual metric spaces. We consider Theorem 4.3 as a example. Let (X, d) be a mett . For A, B ⊂ X, let dH be a Hausdorff metric defined by dH (A, B) = ric space and Md (x, y, t) = t+d(x,y) max{supx∈A infy∈B d(x, y), supy∈B infx∈A d(x, y)}. Then we have

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d (A, B, t) = M

t , t + dH (A, B)

Taking ∗ = ∗m and Md (x, y, t) =

15

for A, B ⊂ X.

t t+d(x,y)

in Theorem 4.3, by Lemma 2.1, we obtain the following consequence.

Corollary 5.4. Let (X, d) be a complete metric space. Let ϕ : R+ → R+ be a function ϕ(t) < t and limn→∞ ϕ n (t) = 0 for all t > 0. Let {Tn }∞ n=1 be a sequence of mappings from X into X such that ϕ(t) d(x, y), for all n ∈ Z+ , t > 0 and x, y ∈ X. t (1) If {Tn A} is a Cauchy sequence in Pcp (X) with respect to dH for each A ∈ Pcp (X), then there exists a unique  F0 ∈ Pcp (X) such that F0 = ∞ n=1 Tn F0 . (2) If {Tn } is a sequence of closed mappings and {Tn A} is a Cauchy sequence in Pcl,bd (X) with respect to dH for each  A ∈ Pcl,bd (X), then there exists a unique F0 ∈ Pcp (X) such that F0 = ∞ n=1 Tn F0 , and for any A0 ∈ Pcl,bd (X), the ∞ sequence {Ak } generated by Ak = n=1 Tn Ak−1 converges to F0 with respect to dH . d(Tn x, Tn y) <

Example 5.1. Let X = R, ∗ = ∗m . Define M : X × X × R+ → [0, 1] by

|x−y| e− t , if t > 0; M(x, y, t) = for all x, y ∈ X. 0, if t = 0, Then ∗m is a t-norm of H-type and (X, M, ∗m ) is a complete KM fuzzy metric space (cf. [27]). For x ∈ X, define T1 , T2 : X → X as follows x x T1 x = ; T2 x = 6 − . 3 2 It is clear that T1 , T2 are closed. Let ϕ(t) = t/2. Then, for t > 0, x, y ∈ X, and j = 1, 2, we have   2|T1 x−T1 y| 2|x−y| |x−y| t t M T1 x, T1 y, = e− 3t ≥ e− t = M(x, y, t); = e− 2   2|T2 x−T2 y| |x−y| t t = e− t = M(x, y, t). = e− M T2 x, T2 y, 2  Thus, all conditions of Theorem 4.1 are satisfied. Hence there exists a unique F0 ∈ Pcp (X) such that F0 = 2j =1 Tj F0 .  Indeed, F0 is a generalized Cantor set. If we take A0 = [0, 6], then A1 = 2j =1 Tj A0 = [0, 2] ∪ [3, 6] and F0 =   limn→∞ 2j =1 Tj An . Also, there exists an F∗ ∈ Pcl (X) such that F∗ ∈ / Pcp (X) and F∗ = 2j =1 Tj F∗ , for example, F∗ = X. This illustrates that the uniqueness of invariant sets for IFS does not necessarily hold in Pcl (X). Example 5.2. Let X = R+ , ∗ = ∗m . Define M : X × X × R+ → [0, 1] by M(x, y, 0) = M(y, x, 0) and

t , if x = y, M(x, y, t) = t+max{x,y} for all t > 0. 1, if x = y, Then ∗m is a t-norm of H-type and (X, M, ∗m ) is a complete KM fuzzy metric space. For x ∈ X, define T1 , T2 , T3 : X → X by

x

x , if x ∈ [0, 3), 2 , if x ∈ [0, 1), T1 x = 1 T2 x = 3 and 1, if x ∈ [3, +∞); 2 , if x ∈ [1, +∞); ⎧ 2x if x ∈ [0, 5), ⎪ ⎨ 5, x 1 T3 x = 3 + 3 , if x ∈ [5, 8), ⎪ ⎩ 3, if x ∈ [8, +∞). Evidently, Tj is closed and Tj x ≤

x 2

for x ∈ X and j = 1, 2, 3. Let ϕ : R+ → R+ be defined by

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ϕ(t) =

⎧t , ⎪ ⎨2 ⎪ ⎩

if t ∈ [0, 1),

2t 2+t , 2t 3,

if t ∈ [1, 2), if t ∈ [2, +∞).

Since ϕ is nondecreasing and 12 t ≤ ϕ(t) ≤ 23 t for all t ∈ R+ , it follows that  n  n 2 1 n t ≤ ϕ (t) ≤ t, 2 3 and so limn→∞ ϕjn (t) = 0. For t > 0, x, y ∈ X with x = y, and j = 1, 2, 3, we have

M Tj x, Tj y, ϕ(t) = ≥

ϕ(t) ϕ(t) ≥ ϕ(t) + max{Tj x, Tj y} ϕ(t) + 12 max{x, y} t 2 t 2

+ 12 max{x, y}

= M(x, y, t).

Thus, all conditions of Corollary 5.1 are satisfied. Hence there exists a unique F0 ∈ Pcp (X) such that F0 =

3

j =1 Tj F0 .

Remark 5.4. Let X = R and T : X → X be a mapping defined by 1 T x = x, 3

x ∈ X.

If we define M : X × X × R+ → [0, 1] by M(x, y, 0) = 0 and M(x, y, t) =

t t + |x − y|

for t > 0,

then (X, M, ∗m ) is a complete KM fuzzy metric space. It is evident that M(T x, T y, 12 t) ≥ M(x, y, t), i.e., T is a contraction in (X, M, ∗m ). From this it implies that   1   B, t) for A, B ∈ Pcl (X), M T A, T B, t ≥ M(A, 2  ∗m ). Hence we can obtain some unbounded invariant sets, such as [0, +∞), i.e., T is still a contraction in (Pcl (X), M, (−∞, 0], etc. In the classical setting, we take the metric d on X by d(x, y) = |x − y|,

for x, y ∈ X.

It is evident that d(T x, T y) ≤ 12 d(x, y), i.e., T is a contraction in (X, d). Since for A, B ∈ Pcl (X) it is sometimes possible dH (A, B) = +∞, we do not know whether dH (T A, T B) ≤ 12 dH (A, B) holds on the Hausdorff metric dH . Of course we can take the bounded metric d  on X by   d  (x, y) = min 1, |x − y| ,

for x, y ∈ X.

But it only holds d  (T x, T y) ≤ d  (x, y), and so we do not know whether T is a contraction in Pcl (X) on the Hausdorff  . If we take the bounded metric d ∗ on X by metric dH d ∗ (x, y) =

|x − y| , 1 + |x − y|

for x, y ∈ X.

Then T is not a linear contraction in (X, d ∗ ), and so the discussion on unbounded invariant sets is still difficult. Fuzzy metrics provide a natural tool to deal with unbounded invariant sets. It seems that the fuzzy metrics are superior to the classical ones in a sense.

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6. Conclusion In this paper, we have studied the topological structures of three hyperspaces of the KM fuzzy metric spaces. These hyperspaces are the collections of non-empty closed, compact, fuzzy bounded subsets, respectively; and they are endowed with the Hausdorff fuzzy metrics. By means of the completeness of hyperspaces we have obtained several existence results of attractors for a finite family and a countable family of fuzzy nonlinear contractions. Roughly speaking, in a complete KM fuzzy metric space with an H-type t-norm, an IFS which consists of finite or countable fuzzy nonlinear contractions has a unique compact attractor, and the iterative sequence generated by any compact set or any closed bounded set can converge to this attractor. In particular, the iterative sequence generated by any closed unbounded set can converge to an unbounded invariant set. Fuzzy metrics provide a natural tool to search compact attractors or unbounded invariant sets. In the proof of Theorem 4.1 (3), the new notion of defective diameter (i.e., β(A) for a set A) plays important role. We now only know that it is a description of the unboundedness of sets. Many algebraic and topological properties with respect to it are still not clear. For example, what relations between β(T A) and β(A), for an operator T and a set A? It seems interesting to study further. Acknowledgements The authors are grateful to the referees and the editors for their valuable comments and helpful suggestions. Particularly, some guidance by the referees in study of unbounded invariant sets and statement of motivation lead to the improvement of the paper. 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] [26] [27]

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