Probabilistic fractals and attractors in Menger spaces

Nonlinear Analysis 97 (2014) 106–118

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Probabilistic fractals and attractors in Menger spaces Jian-Zhong Xiao ∗ , Xing-Hua Zhu, Jie Yan School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, PR China

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Article history: Received 25 May 2013 Accepted 19 November 2013 Communicated by S. Carl MSC: 37C70 46S50 54E70 54B20

abstract Probabilistic fractals are considered as self-similar sets for contraction mappings in hyperspaces of the Menger probabilistic metric spaces. The completeness of hyperspaces of all closed, or closed and bounded, or compact nonempty subsets with respect to the probabilistic Hausdorff metric is studied in some general conditions of triangular norm. Using the properties of hyperspaces and triangular norms of H-type, several existence results of attractors are obtained for a finite family and a countable family of probabilistic nonlinear contractions, respectively. Our results are generalizations in many aspects. © 2013 Elsevier Ltd. All rights reserved.

Keywords: Menger probabilistic metric space Hyperspace Probabilistic Hausdorff metric Attractor Iterated function system

1. Introduction and preliminaries An iterated function system consists of a family of contractions in a complete metric space. By using Banach’s fixed point theorem for the hyperspaces of all nonempty compact subsets with the Hausdorff metric induced by original metric, it is verified that there exists a unique invariant set, named the attractor of the iterated function system. If the functions in a system have similarities, then this invariant set is said to be self-similar, generally, a fractal set. The idea on fractals was first put forward by Hutchinson [1], and its integrity theory was set up by Barnsley [2]. So far the theory of fractal attractors has been developed in many directions; see [3–6] and the references therein. A probabilistic metric space is an important generalization of metric spaces and appears to be of interest in the investigation of physical quantities, physiological thresholds, and analyzing the complexity of algorithms (see [7–11]). The study of fixed point properties for mappings defined on probabilistic metric spaces attracts many authors and has become an active direction of recent studies (see [7,8,12–15]). Our aim in this paper is to consider probabilistic fractals, i.e., to generalize the existence results of attractors in the setting of probabilistic metric. The methods are essentially based on two main aspects: the completeness of hyperspaces with respect to the probabilistic Hausdorff metric and the existence of fixed points for contraction mappings in Menger probabilistic metric spaces. The probabilistic Hausdorff metric was defined and studied by Egbert [16] and Tardiff [17]. For some related discussions and applications we refer to [7,11,18,19]. Kolumbán and Soós [5] and Cobzaş [20] proved the completeness of some hyperspaces in terms of the probabilistic metric. In their work the conditions of triangular norm were special ∆m , ∆L or

∗

Corresponding author. Tel.: +86 02558731073; fax: +86 02558731073. E-mail address: [email protected] (J.-Z. Xiao).

0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.11.020

J.-Z. Xiao et al. / Nonlinear Analysis 97 (2014) 106–118

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∆ ≥ ∆L , where ∆m (a, b) = min{a, b} and ∆L (a, b) = max{a + b − 1, 0}. Motivated by their work, we undertake further investigation for the completeness of hyperspaces in some general conditions of triangular norm. The study of contraction mappings for probabilistic metric spaces was initiated by Sehgal and Bharucha-Reid [21]. Radu [22] pointed out that, in a complete Menger space with continuous triangular norm, any probabilistic Banach contraction has a unique fixed point if and only if the triangular norm is of H-type. Recently, in a complete Menger space with the H-type triangular norm, Jachymski [13] proved a fixed point theorem concerning a nonlinear contraction function ϕ . It is a general result because it does not assume any continuity or monotonicity conditions for ϕ . Inspired by the work in [13], in the Menger hyperspaces we establish several existence results of attractors under weak conditions of probabilistic nonlinear contractions. The paper is organized as follows. In Section 2, we mainly discuss the completeness of hyperspaces of all closed, or closed and bounded, or compact nonempty subsets with respect to the probabilistic Hausdorff metric. In Section 3, we establish several existence results of attractors for a finite family and a countable family of probabilistic nonlinear contractions. We finally give some examples and remarks concerning our results in Section 4. We now state some let R = (−∞, +∞), R+ =  basic concepts and +results which will be used. In the standard notation, + + [0, +∞), R = R {−∞, +∞}, and let Z be the set of all positive integers. If ψ : R → R is a function and t ∈ R+ , then ψ n (t ) denotes the nth iteration of ψ(t ). A function F : R → [0, 1] is called a distribution function if it is nondecreasing and left-continuous with F (−∞) = 0, F (+∞) = 1. The class of all distribution functionsis denoted by D∞ . Let D = {F ∈ D∞ : inft ∈R F (t ) = 0, supt ∈R F (t ) = + + 1}, D∞ = {F ∈ D∞ : F (0) = 0}, and D + = D D∞ (cf. [11]). A special element of D + is the Heaviside function H defined by H (t ) =



1, 0,

t > 0; t ≤ 0.

The proofs of Lemmas 1.1–1.5 and 1.7 are easy or routine, so we omit them. Lemma 1.1 (Cf. [3,13,19]). Let ψ : R+ → R+ be a nondecreasing function. If limn→∞ ψ n (t ) = +∞ for all t > 0, then

ψ(t ) > t for all t > 0.

Definition 1.1 ([11]). 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]:

(∆-1) (∆-2) (∆-3) (∆-4)

∆(a, 1) = a; ∆(a, b) = ∆(b, a); ∆(a, b) ≥ ∆(c , d), for a ≥ c , b ≥ d; ∆(∆(a, b), c ) = ∆(a, ∆(b, c )).

The three examples of t-norm are ∆m , ∆L and ∆p , where ∆m (a, b) = min{a, b}, ∆L (a, b) = max{a + b − 1, 0} and ∆p (a, b) = ab. It is evident that, as regards the pointwise ordering, ∆ ≤ ∆m for each t-norm ∆. For a ∈ [0, 1], 1 n n −1 the sequence {∆n (a)}∞ (a), a). A t-norm ∆ is said to be of H-type n=1 is defined by ∆ (a) = a and ∆ (a) = ∆(∆ (see [8] etc.) if the sequence of functions {∆n (a)}∞ is equicontinuous at a = 1. The t-norm ∆m is a trivial example of n =1 t-norm of H-type, but there are t-norms ∆ of H-type with ∆ ̸= ∆m (see [8,23,24]). A t-norm is called sup-continuous if supλ∈Λ ∆(aλ , b) = ∆(supλ∈Λ aλ , b) for any family {aλ : λ ∈ Λ} ⊂ [0, 1] and b ∈ [0, 1]. A t-norm is called left-continuous if limµ→a− ∆(µ, b) = ∆(a, b) for any a, b ∈ [0, 1]. Lemma 1.2 (Cf. [25]). Let ∆ be a t-norm. Then the following assertions are equivalent: (i) (ii) (iii) (iv) (v)

∆ is sup-continuous; ∆ is left-continuous; supλ∈Λ,γ ∈Γ ∆(aλ , bγ ) = ∆(supλ∈Λ aλ , supγ ∈Γ bγ ), for any families {aλ : λ ∈ Λ}, {bγ : γ ∈ Γ } ⊂ [0, 1]; limµ→a− ,ν→b− ∆(µ, ν) = ∆(a, b) for all a, b ∈ (0, 1]; sup0<µ
Lemma 1.3 (Cf. [7]). Let ∆ be a t-norm. (1) If ∆ is of H-type, then ∆ satisfies sup0
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(MS-2) Fx,y (t ) = Fy,x (t ) for all x, y ∈ X and t ∈ R; (MS-3) Fx,y (t + s) ≥ ∆(Fx,z (t ), Fz ,y (s)) for all x, y, z ∈ X and t , s ∈ R+ . A Menger probabilistic metric space (for short, a Menger PM-space) is a generalized Menger space with F (X × X ) ⊂ D + . Schweizer et al. [26,27] point out that if the t-norm ∆ of a Menger PM-space satisfies the condition sup0 0, λ ∈ (0, 1]} is a base of neighborhoods of point x for τ , where Ux (ε, λ) = {y ∈ X : Fx,y (ε) > 1 − λ}. In virtue of this topology τ , a sequence {xn } in (X , F , ∆) is said to be convergent to x (we write xn → x or limn→∞ xn = x) if limn→∞ Fxn ,x (t ) = 1 for all t > 0; {xn } is called a Cauchy sequence in (X , F , ∆) if for any given ε > 0 and λ ∈ (0, 1], there exists N = N (ε, λ) ∈ Z+ such that Fxn ,xm (ε) > 1 − λ, whenever n, m ≥ N. (X , F , ∆) 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 Fx,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. A is said to be probabilistically bounded if supt >0 infx,y∈A Fx,y (t ) = 1 (see [28]). Let P (X ) denote the class of all nonempty subsets of X . We use the notations:

Pcl (X ) = {Y ∈ P (X ) : Y is closed}, Pbd (X ) = {Y ∈ P (X ) : Y is probabilistically bounded}, Pcp (X ) = {Y ∈ P (X ) : Y is compact}, Pcl,bd (X ) = Pcl (X )



Pbd (X ).

Let T : X → X be a mapping. T is said to be closed if TA ∈ Pcl (X ) for each A ∈ Pcl (X ). It is said to be bounded if TA ∈ Pbd (X ) for each A ∈ Pbd (X ). Schweizer et al. [27] point out that if the t-norm ∆ satisfies the condition sup0
A is compact if and only if A is sequentially compact. If (X , F , ∆) is complete, then A is compact if and only if A is closed and totally bounded. If A is totally bounded, then A ∈ Pbd (X ) and A is also totally bounded. A ∈ Pbd (X ). A ∈ Pbd (X ) if and only if If A, B ∈ Pbd (X ), then A  B ∈ Pbd (X ). If A, B ∈ Pcp (X ), then A B ∈ Pcp (X ).

Lemma 1.5 (Cf. [7]). Let (X , F , ∆) be a Menger PM-space with sup0
lim sup Fxn ,y (t ) ≤ Fx,y (t +) n→∞

for all t > 0. Particularly, limn→∞ Fxn ,y (t ) = Fx,y (t ) for a.e. t ∈ R; if Fx,y (·) is continuous at the point t0 , then limn→∞ Fxn ,y (t0 ) = Fx,y (t0 ). Lemma 1.6 (See [21]). Let (X , d) be a usual metric space. Define a mapping F : X × X → D + by Fx,y (t ) = H (t − d(x, y)),

for x, y ∈ X and t > 0.

Then (X , F , ∆m ) is a Menger PM-space, and is called the induced Menger PM-space by (X , d). It is complete if (X , d) is complete. Definition 1.3 ([11,16,17]). Let (X , F , ∆) be a generalized Menger PM-space and A, B ∈ P (X ). Let Fx,B , FA,B ,  FA,B : R → [0, 1] be the functions defined by Fx,B (t ) = sup Fx,y (t ), y∈B





FA,B (t ) = min inf Fx,B (t ), inf Fy,A (t ) , x∈A

y∈B

 FA,B (t ) = sup FA,B (s).

Then  FA,B is called the probabilistic Hausdorff metric (or distance) between A and B.

s
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Lemma 1.7 (Cf. [7,11]). Let (X , F , ∆) be a generalized Menger PM-space, x ∈ X and A, B ∈ P (X ). Then (1) (2) (3) (4) (5)

 FA,B (t ) ≥ FA,B (s) for all s < t;  FA,B (t ) ≤ FA,B (t ) for all t ∈ R+ and  FA,B (t ) = FA,B (t ) for a.e. t ∈ R+ . +  FA,B (t ) ≤ infx∈A Fx,B (t ) and  FA,B (t ) ≤ infy∈B Fy,A (t ) for all t ∈ R . +  FA,B ∈ D∞ ; and  FA,B ∈ D + if F (X × X ) ⊂ D + and A, B ∈ Pbd (X ). FA,B (·) is nondecreasing in R+ ; and limt →+∞ FA,B (t ) = 1 if F (X × X ) ⊂ D + and A, B ∈ Pbd (X ). If sup0
, ∆) is a generalized Lemma 1.8 (See [11]). Let (X , F , ∆) be a Menger PM-space such that ∆ is sup continuous. Then (Pcl (X ), F +  is the mapping from Pcl (X ) × Pcl (X ) into D∞ (A, B) =  Menger PM-space, where F defined by F FA,B for A, B ∈ Pcl (X ). 2. Completeness of hyperspaces Lemma 2.1. Let (X , F , ∆) be a Menger PM-space such that ∆ is left-continuous. Let A, B, C ∈ P (X ), x, y ∈ X and s, t ∈ R+ . Then (1) (2) (3)

(cf. [18,19]) Fx,B (s + t ) ≥ ∆(Fx,y (s), Fy,B (t )).  Fx,B (s + t ) ≥ ∆(Fx,A (s),  FA,B (t )). FA,B (s + t ) ≥ ∆(FA,C (s), FC ,B (t )).

Proof. (1) By (MS-3), for each b ∈ B, we have Fx,b (s + t ) ≥ ∆(Fx,y (s), Fy,b (t )). Since ∆ is left-continuous, from Lemma 1.2 it follows that Fx,B (s + t ) ≥ sup ∆(Fx,y (s), Fy,b (t )) = ∆(Fx,y (s), sup Fy,b (t )) = ∆(Fx,y (s), Fy,B (t )). b∈B

b∈B

(2) Without loss of generality, we suppose t > 0. For each r ∈ (0, t ) and a ∈ A, by Lemma 2.1(1) we have Fx,B (s + r ) ≥ ∆(Fx,a (s), Fa,B (r )) ≥ ∆(Fx,a (s), inf Fa,B (r )) a∈A





 ≥ ∆ Fx,a (s), min inf Fa,B (r ), inf Fb,A (r ) . a∈A

b∈B

(2.1)

Taking the supremum with respect to a ∈ A in (2.1), from Lemma 1.2 it follows that





Fx,B (s + r ) ≥ ∆ Fx,A (s), min inf Fa,B (r ), inf Fb,A (r ) a∈A



b∈B

.

(2.2)

Taking the supremum with respect to r ∈ (0, t ) in (2.2), from Lemma 1.2 we obtain  Fx,B (s + t ) ≥ ∆(Fx,A (s),  FA,B (t )). (3) For each x ∈ A and z ∈ C , by Lemma 2.1(1) and the monotonicity of ∆ we have Fx,B (s + t ) ≥ ∆(Fx,z (s), Fz ,B (t )) ≥ ∆(Fx,z (s), FC ,B (t )).

(2.3)

Taking the supremum with respect to z ∈ C in (2.3), from Lemma 1.2 and the monotonicity of ∆ it follows that Fx,B (s + t ) ≥ ∆(Fx,C (s), FC ,B (t )) ≥ ∆(FA,C (s), FC ,B (t )), and so infx∈A Fx,B (s+t ) ≥ ∆(FA,C (s), FC ,B (t )). Similarly, we have infy∈B Fy,A (s+t ) ≥ ∆(FA,C (s), FC ,B (t )). Hence FA,B (s+t ) ≥ ∆(FA,C (s), FC ,B (t )), which is the desired inequality.

, ∆) Theorem 2.2. Let (X , F , ∆) be a Menger PM-space such that ∆ is left-continuous. If (X , F , ∆) is complete, then (Pcl (X ), F is a complete generalized Menger PM-space. , ∆) is a generalized Menger PM-space. Suppose that {An }∞ Proof. By Lemma 1.8, (Pcl (X ), F n=1 ⊂ Pcl (X ) is a Cauchy se∞ ∞ A and A = E . Clearly, A ∈ P ( X ) . We shall prove that A → A with respect to  F . Let ε > 0 quence, En = cl n m=n m n =1 n and λ ∈ (0, 1] be arbitrarily given. Since ∆ is left-continuous, we have sup0 1 − λ0 . Repeating this procedure, we get a non-increasing sequence 1 {λk }∞ k=0 ⊂ (0, 1] such that λk ≤ min{λk−1 , k } → 0 and ∆(1 − λk , 1 − λk ) > 1 − λk−1 ,

for all k ∈ Z+ .

(2.4)

Since {An } is a Cauchy sequence with respect to  F , there exists N = N (ε, λ) ∈ Z+ such that

 FAm ,An (ε/22 ) > 1 − λ2 ,

for all m, n ≥ N .

(2.5)

Put n1 = n, where n ≥ N. Again by the definition of Cauchy sequence there exists n2 > n1 such that  FAm ,An2 (ε/23 ) > 1 − λ3

for all m ≥ n2 , and so from (2.5) we have  FAn2 ,An1 (ε/22 ) > 1 − λ2 . Inductively, we obtain a subsequence {Ank } of {An }

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J.-Z. Xiao et al. / Nonlinear Analysis 97 (2014) 106–118

such that

 FAn

,Ank (ε/2

k+1

k+1

) > 1 − λk+1 ,

for all k ∈ Z+ .

(2.6)

FAn2 ,An1 (ε/22 ) > 1 − λ2 it follows the existence of z2 ∈ Suppose that z = z1 ∈ An = An1 is arbitrary. From Lemma 1.7(2) and 

An2 such that Fz2 ,z1 (ε/22 ) > 1 − λ2 . Continuing in this way, from (2.6) we obtain a sequence {zk } ⊂ X such that zk ∈ Ank and Fzk+1 ,zk (ε/2k+1 ) > 1 − λk+1 ,

for all k ∈ Z+ .

(2.7)

We claim that Fzk+j ,zk (ε/2k ) ≥ ∆(1 − λk , 1 − λk ),

for all k, j ∈ Z+ .

(2.8)

In fact, this is obvious for j = 1, since by (2.7), Fzk+1 ,zk (ε/2k ) ≥ Fzk+1 ,zk (ε/2k+1 ) > 1 − λk+1 ≥ ∆(1 − λk+1 , 1 − λk+1 ) ≥ ∆(1 − λk , 1 − λk ). Assume that (2.8) holds for some j. By (2.7) and (2.4) we have Fzk+j+1 ,zk (ε/2k ) ≥ ∆(Fzk+1 ,zk (ε/2k+1 ), Fzk+1+j ,zk+1 (ε/2k+1 ))

≥ ∆(1 − λk+1 , ∆(1 − λk+1 , 1 − λk+1 )) ≥ ∆(1 − λk+1 , 1 − λk ) ≥ ∆(1 − λk , 1 − λk ), which means (2.8) holds for j + 1. By induction, (2.8) holds for all j ∈ Z+ . Hence we have Fzk+j ,zk (ε/2k ) > 1 − λk−1 for all k, j ∈ Z+ . Consequently {zk } is a Cauchy sequence in X . Since (X , F , ∆) is complete, there exists x ∈ X such that zk → x. + Since for each n ∈ Z+ there exists ni ≥ n such that zi ∈ Ani , we have {zk }∞ k=i ⊂ En . This implies that x ∈ En for each n ∈ Z , and so x ∈ A. Taking k = 1 in (2.8), we have Fzj+1 ,z (ε/2) ≥ ∆(1 − λ1 , 1 − λ1 ),

for all j ∈ Z+ .

(2.9)

Note that there exists s ∈ [ε/2, 3ε/4) such that Fx,z (t ) is continuous at t = s. Since ∆ is left-continuous, by Lemma 1.3 it follows that sup0
z ∈An

for all n ≥ N .

(2.10)

Next, suppose that u ∈ A is arbitrary. Then u ∈ EN , and so there exists y ∈ EN such that Fu,y (ε/2) > 1 − λ1 . From y ∈ EN we see that there exists m0 ≥ N such that y ∈ Am0 . Thus, Fu,Am0 (ε/2) ≥ Fu,y (ε/2) > 1 − λ1 . From (2.5), Lemmas 1.7(1) and 2.1(2) it follows that, for all n ≥ N, Fu,An (3ε/4) ≥ ∆(Fu,Am0 (ε/2),  FAm0 ,An (ε/22 )) ≥ ∆(1 − λ1 , 1 − λ2 ) ≥ ∆(1 − λ1 , 1 − λ1 ), and hence inf Fu,An (3ε/4) ≥ ∆(1 − λ1 , 1 − λ1 ),

u∈A

for all n ≥ N .

(2.11)

The inequalities (2.10) and (2.11) yield, for all n ≥ N,

   FAn ,A (ε) ≥ min inf Fz ,A (3ε/4) , inf Fu,An (3ε/4) ≥ ∆(1 − λ1 , 1 − λ1 ) > 1 − λ. z ∈An

u∈A

This shows that the sequence {An } converges to A with respect to  F , and so the proof is complete.

, ∆) and Theorem 2.3. Let (X , F , ∆) be a complete Menger PM-space such that ∆ is left-continuous. Then (Pcl,bd (X ), F (Pcp (X ), F, ∆) are complete Menger PM-spaces. , ∆) is a complete generalized Menger PM-space. By Lemmas 1.7(3) and 1.4(3) we see Proof. By Theorem 2.2, (Pcl (X ), F that Pcl,bd (X ) and Pcp (X ) are two Menger PM-spaces. Since Pcp (X ) ⊂ Pcl,bd (X ) ⊂ Pcl (X ), it is enough to prove that Pcl,bd (X ) and Pcp (X ) are closed with respect to  F.  Suppose that {An }∞ n=1 ⊂ Pcp (X ), An → A with respect to F . We shall prove that A ∈ Pcp (X ). Choose any ε > 0 and λ ∈ (0, 1]. By the left-continuity of ∆, we have sup0 1 − λ.

(2.12) +

By the convergence of {An }, there exists N ∈ Z such that

 FA,An (ε/2) > 1 − µ,

for all n ≥ N .

(2.13)

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111

Since by Lemma 1.4 AN is compact, AN is also totally bounded. Thus, AN has a finite (ε/2, µ)-net SN . From this we infer that SN is a finite (ε, λ)-net of A. In fact, for each x ∈ A, from Lemma 1.7(2) and (2.13) it follows the existence of y ∈ AN such that Fx,y (ε/2) > 1 − µ. For such y we can select z ∈ SN with Fy,z (ε/2) > 1 − µ. Hence, from (2.12) we have Fx,z (ε) ≥ ∆(Fx,y (ε/2), Fy,z (ε/2)) ≥ ∆(1 − µ, 1 − µ) > 1 − λ. This shows that SN is a finite (ε, λ)-net of A, and so A is totally bounded. From the completeness of (X , F , ∆) it follows that A is compact, i.e.,A ∈ Pcp (X ).  Now let {An }∞ n=1 ⊂ Pcl,bd (X ), An → A with respect to F . We shall prove that A ∈ Pcl,bd (X ). Take an arbitrary number λ ∈ (0, 1]. Since sup0
∆(1 − µ, 1 − µ) > 1 − λ and ∆(1 − ν, 1 − ν) > 1 − µ.

(2.14)

The convergence of {An } implies that there exists N ∈ Z+ such that

 FA,An (1) > 1 − ν,

for all n ≥ N .

(2.15)

Since AN is probabilistically bounded, we have supt >0 infx,y∈AN Fx,y (t ) = 1. Thus, there exists M = M (ν) > 0 such that Fx,y (M ) > 1 − ν for all x, y ∈ AN . Suppose that u, w ∈ A are two arbitrary points. From Lemma 1.7(2) and (2.15) it follows that there exist x, y ∈ AN such that Fu,x (1) > 1 − ν and Fw,y (1) > 1 − ν . Thus, from (2.14) we have Fu,y (M + 1) ≥ ∆(Fu,x (1), Fx,y (M )) ≥ ∆(1 − ν, 1 − ν) > 1 − µ, and moreover, Fu,w (M + 2) ≥ ∆(Fu,y (M + 1), Fy,w (1)) ≥ ∆(1 − µ, 1 − ν) ≥ ∆(1 − µ, 1 − µ) > 1 − λ. Hence supt >0 infu,w∈A Fu,w (t ) ≥ infu,w∈A Fu,w (M + 2) ≥ 1 − λ. By the arbitrariness of λ, we have supt >0 infu,w∈A Fu,w (t ) = 1, i.e., A is probabilistically bounded. From Theorem 2.2 we see that A is closed. Therefore A ∈ Pcl,bd (X ), and so the proof is complete. Theorem 2.4. Let (X , F , ∆) be a compact Menger PM-space such that ∆ is left-continuous. Let {An }∞ n=1 ⊂ Pcl (X ) be a Cauchy sequence with respect to  F.

∞ n=1 An ∈ Pcp (X ).  ∞ ∞ (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) If {An }∞ n=1 ⊂ Pcp (X ), then

(1) By Lemma 1.4(3), it remains to prove that B is totally bounded. Give any ε > 0 and λ ∈ (0, 1]. Since ∆ is leftcontinuous, we have sup0
∆(1 − µ, 1 − µ) > 1 − λ. Since

{An }∞ n=1

(2.16)

⊂ Pcp (X ) is a Cauchy sequence with respect to  F , there exists N ∈ Z such that +

 FAm ,An (ε/2) > 1 − µ, for all m, n ≥ N . (2.17) ∞ N   Putting EN = EN and AN ⊂ BN EN . Since BN is compact, BN is totally n=N An and BN = n=1 An , we have B = BN bounded. Thus, BN has a finite (ε/2, µ)-net SN . We prove 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 Fx,y (ε) ≥ Fx,y (ε/2) > 1 − µ ≥ 1 − λ.

(2.18)

If x ∈ EN , then there exists m ≥ N such that x ∈ Am . From Lemma 1.7(2) and (2.17) it follows the existence of xN ∈ AN such that Fx,xN (ε/2) > 1 − µ. For this xN we can select y ∈ SN such that FxN ,y (ε/2) > 1 − µ. Therefore from (2.16) we have Fx,y (ε) ≥ ∆(Fx,xN (ε/2), FxN ,y (ε/2)) ≥ ∆(1 − µ, 1 − µ) > 1 − λ.

(2.19)

Combining (2.18) with (2.19), it implies that B is totally bounded. From Lemma 1.4(2) and the completeness of (X , F , ∆) we conclude that B ∈ Pcp (X ). (2) It remains to prove that B ∈ Pbd (X ). Give an arbitrary λ ∈ (0, 1]. Since sup0
∆(1 − µ, 1 − µ) > 1 − λ and ∆(1 − ν, 1 − ν) > 1 − µ. Since

{An }∞ n=1

(2.20)

⊂ Pcl,bd (X ) is a Cauchy sequence with respect to  F , there exists N ∈ Z+ such that

 FAm ,An (1) > 1 − ν,

for all m, n ≥ N .

(2.21)

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∞

N





EN and AN ⊂ BN EN . Since by 1.4(5) BN is probabilistically Putting EN = n=N An and BN = n=1 An , we have B = BN bounded, we have supt >0 infx,y∈BN Fx,y (t ) = 1, and then there exists M = M (ν) > 0 such that Fx,y (M ) > 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 Fx,y (M + 2) ≥ Fx,y (M ) > 1 − ν ≥ 1 − λ.

(2.22)

If x ∈ BN and y ∈ EN , then there exists m ≥ N such that y ∈ Am . From Lemma 1.7(2) and (2.21) it follows the existence of xN ∈ AN ⊂ BN such that Fy,xN (1) > 1 − ν . Thus, by (2.20) we have Fx,y (M + 2) ≥ Fx,y (M + 1) ≥ ∆(Fy,xN (1), FxN ,x (M )) ≥ ∆(1 − ν, 1 − ν) > 1 − µ ≥ 1 − λ.

(2.23)

If x, y ∈ EN , then from Lemma 1.7(2) and (2.21) it follows that there exist xN , yN ∈ AN such that Fx,xN (1) > 1 − ν and Fy,yN (1) > 1 − ν . In this case we have Fx,y (M + 2) ≥ ∆(Fx,yN (M + 1), FyN ,y (1)) ≥ ∆(∆(Fx,xN (1), FxN ,yN (M )), FyN ,y (1))

≥ ∆(∆(1 − ν, 1 − ν), 1 − ν) ≥ ∆(1 − µ, 1 − ν) ≥ ∆(1 − µ, 1 − µ) > 1 − λ.

(2.24)

Combining (2.22), (2.23) with (2.24), we obtain supt >0 infx,y∈B Fx,y (t ) ≥ infx,y∈B Fx,y (M + 2) ≥ 1 − λ. The arbitrariness of λ implies that supt >0 infx,y∈B Fx,y (t ) = 1, i.e., B ∈ Pbd (X ). By Lemma 1.4(4) we have B ∈ Pbd (X ). Therefore B ∈ Pcl,bd (X ), and so the proof is complete. 3. Several existence results of attractors m Lemma 3.1. Let (X , F , ∆) be a Menger PM-space. Suppose that {Aj }m j=1 , {Bj }j=1 ⊂ P (X ), A = + n ∈ Z . Then for all t > 0,

m

j =1

Aj and B =

m

j =1

Bj , where

FA,B (t ) ≥ min FAj ,Bj (t ) and  FA,B (t ) ≥ min  FAj ,Bj (t ). 1≤j≤m

1≤j≤m

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



x∈Aj

x∈Aj

inf Fx,Bj (s), inf Fy,Aj (s)



y∈Bj

x∈Aj

= FAj ,Bj (s).

From this it follows that inf Fx,B (s) = min inf Fx,B (s) ≥ min FAj ,Bj (s).

x∈A

1≤j≤m x∈Aj

(3.1)

1≤j≤m

Similarly, the following inequality holds: inf Fy,A (s) = min inf Fy,A (s) ≥ min FAj ,Bj (s).

y∈B

1≤j≤m y∈Bj

(3.2)

1≤j≤m

Combining (3.1) with (3.2), we obtain FA,B (t ) ≥ min1≤j≤m FAj ,Bj (t ) and

   FA,B (t ) = sup min inf Fx,B (s), inf Fy,A (s) ≥ sup min FAj ,Bj (s) s
x∈A

s
y∈B

= min sup FAj ,Bj (s) = min  FAj ,Bj (t ). 1≤j≤m s
1 ≤j ≤m

Lemma 3.2. Let (X , F , ∆) be a Menger PM-space. Suppose that {Ai : i ∈ I }, {Bi : i ∈ I } ⊂ P (X ), A = Then FA,B (t ) ≥ inf FAi ,Bi (t ) for all t ∈ R+ ; i∈I



i∈I

Ai and B =



i∈I

Bi .

and  FA,B (t ) ≥ inf  FAi ,Bi (t ) for a.e. t ∈ R+ . i∈I

Proof. Let t ∈ R+ . Applying the same argument as the proof of Lemma 3.1, we can obtain inf Fx,B (t ) = inf inf Fx,B (t ) ≥ inf FAi ,Bi (t )

x∈A

i∈I x∈Ai

i∈I

and

inf Fy,A (t ) = inf inf Fy,A (t ) ≥ inf FAi ,Bi (t ).

y∈B

i∈I y∈Bi

i∈I

Hence FA,B (t ) ≥ infi∈I FAi ,Bi (t ). It follows from Lemma 1.7(1) that infi∈I FAi ,Bi (t ) ≥ infi∈I  FAi ,Bi (t ). Since FA,B (t ) =  FA,B (t ) for a.e. t ∈ I, we have  FA,B (t ) ≥ infi∈I  FAi ,Bi (t ) for a.e. t ∈ R+ .

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113

Lemma 3.3. Let (X , F , ∆) be a Menger PM-space such that sup0 t for all t > 0. If T : X → X is a mapping satisfying FTx,Ty (t ) ≥ Fx,y (ψ(t )),

for all x, y ∈ X and t > 0,

then (1) T is bounded; (2) T is continuous; (3) TA ∈ Pcp (X ) for each A ∈ Pcp (X ). Proof. From ψ(t ) > t we have FTx,Ty (t ) ≥ Fx,y (t ),

for all x, y ∈ X and all t > 0.

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

for all t > 0.

n→∞

This shows that Txn → Tx, and so T is continuous. (3) By Lemma 1.4(1), A is compact if and only if A is sequentially compact. Suppose that A ∈ Pcp (X ) and {Txn } ⊂ TA. Then there exists a subsequence {xnk } ⊂ A such that xnk → x ∈ A. From (2) we see that Txnk → Tx ∈ TA. Hence TA ∈ Pcp (X ). Theorem 3.4. Let (X , F , ∆) be a complete Menger PM-space such that ∆ is left-continuous and of H-type. Let ψ1 , ψ2 , . . . , ψm : R+ → R+ be functions such that ψj (t ) > t and limn→∞ ψjn (t ) = +∞ for j = 1, 2, . . . , m and all t > 0. Let T1 , T2 , . . . , Tm : X → X be mappings such that FTj x,Tj y (t ) ≥ Fx,y (ψj (t )),

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

(3.3)

(1) If T1 , T2 , . . . , Tm are closed and there exists A0 ∈ Pcl (X ) which satisfies limt →+∞ FA0 ,Tj A0 (t ) = 1 for j = 1, 2, . . . , m, then there exists E ∈ Pcl (X ) such that E = j=1 Tj E. m (2) If T1 , T2 , . . . , Tm are closed, then there exists a unique E ∈ Pcl,bd (X ) such that E = j=1 Tj E. m (3) There exists a unique E ∈ Pcp (X ) such that E = j=1 Tj E.

m

Proof. Let ψ : R+ → R+ be the function defined by ψ(t ) = min1≤j≤m ψj (t ) for t ∈ R+ . Then ψ(t ) > t and limn→∞ ψ n (t ) = +∞ for all t > 0. Let T : Pcl (X ) → Pcl (X ) be the mapping defined by TA =

m 

Tj A,

for A ∈ Pcl (X ).

j =1

Suppose that A, B ∈ Pcl (X ). Then, from (3.3) and Lemma 3.1 it follows that FTA,TB (t ) ≥ min FTj A,Tj B (t ) 1 ≤j ≤m

  = min min inf sup FTj x,Tj y (t ), inf sup FTj x,Tj y (t ) x∈A y∈B

1 ≤j ≤m

y∈B x∈A



≥ min min inf sup Fx,y (ψj (t )), inf sup Fx,y (ψj (t )) 1≤j≤m

x∈A y∈B



y∈B x∈A

= min FA,B (ψj (t )) = FA,B (ψ(t )). 1≤j≤m

(3.4)

, ∆) is a complete Menger PM-space. Taking any A0 ∈ Pcl,bd (X ), by An = TAn−1 for all (2) By Theorem 2.3, (Pcl,bd (X ), F n ∈ Z+ , we get a sequence {An }. Since T1 , T2 , . . . , Tm are closed, T is closed. From Lemma 3.3(1) we see that {An } ⊂ Pcl,bd (X ). Let t > 0. In virtue of (3.4), we have FAn ,An+1 (t ) = FTAn−1 ,TAn (t ) ≥ FAn−1 ,An (ψ(t ))

≥ FAn−2 ,An−1 (ψ 2 (t )) ≥ · · · ≥ FA0 ,A1 (ψ n (t )) ≥  FA0 ,A1 (ψ n (t )).

(3.5)

Since limn→∞ ψ n (t ) = +∞ for all t > 0 and limt →+∞  FA0 ,A1 (t ) = 1 by Lemma 1.7(3), from (3.5) it follows that lim FAn ,An+1 (t ) = 1

n→∞

for all t > 0.

(3.6)

In the next step we show that, for all k, n ∈ Z+ , FAn ,An+k (ψ(t )) ≥ △k (FAn ,An+1 (ψ(t ) − t )).

(3.7)

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In fact, this is obvious for k = 1. Assume that (3.7) holds for some k. From (3.4) and (3.7) it follows immediately that FAn+1 ,An+k+1 (t ) = FTAn ,TAn+k (t ) ≥ FAn ,An+k (ψ(t )) ≥ △k (FAn ,An+1 (ψ(t ) − t )), Since ψ(t ) > t, by the monotonicity of △, we have FAn ,An+k+1 (ψ(t )) = FAn ,An+k+1 (ψ(t ) − t + t )

≥ △(FAn ,An+1 (ψ(t ) − t ), FAn+1 ,An+k+1 (t )) ≥ △(FAn ,An+1 (ψ(t ) − t ), △k (FAn ,An+1 (ψ(t ) − t ))) = △k+1 (FAn ,An+1 (ψ(t ) − t )), i.e., (3.7) holds for k + 1. Hence, by induction, (3.7) holds for all k ∈ Z+ . Furthermore, from (3.4) and (3.7) it follows that FAn+1 ,An+k (t ) ≥ FAn ,An+k−1 (ψ(t )) ≥ △k−1 (FAn ,An+1 (ψ(t ) − t )).

(3.8)

For any given ε > 0 and λ ∈ (0, 1], since △ is a t-norm of H-type, there exists δ > 0 such that

△k−1 (s) > 1 − λ,

for all s ∈ (1 − δ, 1] and k ∈ Z+ (k > 1); +

and the relation (3.6) implies the existence of N ∈ Z

(3.9)

such that FAn ,An+1 (ψ(ε/2) − ε/2) > 1 − δ for all n ≥ N. Hence,

from (3.8) and (3.9) we get  FAn+1 ,An+k (ε) > FAn+1 ,An+k (ε/2) > 1 − λ for all n ≥ N and k ∈ Z+ . This shows that {An } , ∆), there exists E ∈ Pcl (X ) such that {An } converges to E, i.e., is a Cauchy sequence. By the completeness of (Pcl (X ), F limn→∞  FAn ,E (t ) = 1 for all t > 0. From FAn ,E (t ) ≥  FAn ,E (t ) it follows that limn→∞ FAn ,E (t ) = 1 for all t > 0. Since ∆ is left-continuous at a = 1, by (3.4) we have FE ,TE (t ) ≥ △(FE ,An+1 (t /2), FTAn ,TE (t /2))

≥ △(FE ,An+1 (t /2), FAn ,E (ψ(t /2))) → 1 as n → ∞, i.e., FE ,TE (t ) = 1 for all t > 0. Since  FE ,TE (t ) ≥ FE ,TE (t /2), this implies that  FE ,TE (t ) = 1 for all t > 0, and hence TE = E. It remains to prove uniqueness. Assume that TE0 = E0 and fix t > 0. By (3.4) we have FE ,E0 (t ) = FTE ,TE0 (t ) ≥ FE ,E0 (ψ(t )) ≥ FE ,E0 (ψ 2 (t )) ≥ · · · ≥ FE ,E0 (ψ n (t )).

(3.10)

Letting n → ∞ in (3.10), by Lemma 1.7(4) we have FE ,E0 (t ) = 1. Since t is arbitrary, we obtain E = E0 . (3) Using Lemma 3.3 we see that TA ∈ Pcp (X ) for each A ∈ Pcp (X ). In the same way as the proof of (2), the assertions (3) can be proved. (1) Since A0 ∈ Pcl (X ) satisfies limt →+∞ FA0 ,Tj A0 (t ) = 1 for j = 1, 2, . . . , m, from Lemma 3.1 we have limt →+∞ FA0 ,TA0 (t ) = 1. Let An = TAn−1 for all n ∈ Z+ . From the closeness of T we have {An } ⊂ Pcl (X ). Based on Theorem 2.2 instead of Theorem 2.3, the assertions (1) can be verified similarly to the proof of (2). This completes the proof. Remark 3.1. The uniqueness of invariant sets for contractions does not necessarily hold in Pcl (X ). Let X = R, △ = △m . Define F : X × X → D + by t

 F (x, y)(t ) = Fx,y (t ) =

t + |x − y| 0,

,

if t > 0; if t ≤ 0,

for all x, y ∈ X .

Then △m is a t-norm of H-type and (X , F , △m ) is a complete Menger PM-space. Let T1 , T2 be mappings defined by T1 x =

1 3

x,

T2 x =

1 3

x+

2 3

,

x ∈ X.

Let A = [0, +∞) and B = (−∞, 0]. Obviously, A, B ∈ Pcl (X ) are two unbounded invariant sets since A = T1 (A) ∪ T2 (A) and B = T1 (B) ∪ T2 (B). Theorem 3.5. Let (X , F , ∆) be a complete Menger PM-space such that ∆ is left-continuous and of H-type. Let ψ : R+ → R+ be a function such that ψ(t ) > t and limn→∞ ψ n (t ) = +∞ for all t > 0. Let {Tn }∞ n=1 be a sequence of mappings from X into X such that FTn x,Tn y (t ) ≥ Fx,y (ψ(t )),

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

(3.11)

(1) If {Tn A} is a Cauchy sequence in Pcp (X ) with respect to  F for each A ∈ Pcp (X ), then there exists a unique E ∈ Pcp (X ) such

∞

that E = n=1 Tn E. (2) If {Tn } is a sequence of closed mappings and {Tn A} is a Cauchy sequence in Pcl,bd (X ) with respect to  F for each A ∈ Pcl,bd (X ), then there exists a unique E ∈ Pcl,bd (X ) such that E =

∞

n=1

Tn E.

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115

, ∆) is a complete Menger PM-space. Let T : Pcp (X ) → Pcp (X ) be the mapping Proof. (1) By Theorem 2.3, (Pcp (X ), F defined by TA =

∞ 

Tn A,

for A ∈ Pcp (X ).

n =1

Applying Theorem 2.4(1) we have TA ∈ Pcp (X ) for each A ∈ Pcp (X ). For A, B ∈ Pcp (X ), From (3.11), Lemmas 1.7(5) and 3.2 we have



 FTA,TB (t ) ≥ inf FTn A,Tn B (t ) = inf min inf sup FTn x,Tn y (t ), inf sup FTn x,Tn y (t ) n∈Z+

n∈Z+

x∈A y∈B



≥ inf min inf sup Fx,y (ψ(t )), inf sup Fx,y (ψ(t )) n∈Z+

x∈A y∈B

y∈B x∈A



y∈B x∈A

= inf FA,B (ψ(t )) = FA,B (ψ(t )).

(3.12)

n∈Z+

Choosing any A0 ∈ Pcp (X ), by An = TAn−1 for all n ∈ Z+ we obtain a sequence {An } ⊂ Pcl (X ). Let t > 0. The relation (3.12) implies that FAn ,An+1 (t ) = FTAn−1 ,TAn (t ) ≥ FAn−1 ,An (ψ(t ))

≥ FAn−2 ,An−1 (ψ 2 (t )) ≥ · · · ≥ FA0 ,A1 (ψ n (t )) ≥  FA0 ,A1 (ψ n (t )).

(3.13)

Since limt →+∞  FA0 ,A1 (t ) = 1 and limn→∞ ψ n (t ) = +∞ for all t > 0, from (3.13) it follows that lim FAn ,An+1 (t ) = 1

for all t > 0.

n→∞

(3.14)

In the same manner as the proof of Theorem 3.4 we can show that FAn ,An+k (ψ(t )) ≥ △k (FAn ,An+1 (ψ(t ) − t )),

for all k, n ∈ Z+ .

(3.15)

FAn+1 ,An+k (t ) ≥ FAn ,An+k−1 (ψ(t )) ≥ △k−1 (FAn ,An+1 (ψ(t ) − t )).

(3.16)

Moreover, from (3.12) and (3.15) it follows that

For each ε > 0 and λ ∈ (0, 1],since △ is a t-norm of H-type, there exists δ > 0 such that

△k−1 (s) > 1 − λ,

for all s ∈ (1 − δ, 1] and k ∈ Z+ (k > 1);

(3.17)

and from (3.14) it follows the existence of N ∈ Z+ such that FAn ,An+1 (ψ(ε/2) − ε/2) > 1 − δ for all n ≥ N. Hence, in

view of (3.16) and (3.17) we get  FAn+1 ,An+k (ε) ≥ FAn+1 ,An+k (ε/2) > 1 − λ for all n ≥ N and k ∈ Z+ . This shows that {An } , ∆), there exists E ∈ Pcp (X ) such that {An } converges to E, i.e., is a Cauchy sequences. By the completeness of (Pcp (X ), F limn→∞  FAn ,E (t ) = 1 for all t > 0. This implies that for all t > 0, lim FAn ,E (t ) = 1.

(3.18)

n→∞

Next, we prove that E is a fixed point of T . Let ε > 0 and λ ∈ (0, 1] be given. Since ∆ is left-continuous at a = 1, there exists µ ∈ (0, 1] such that ∆(1 − µ, 1 − µ) > 1 − λ. From (3.18) it follows the existence of N ∈ Z+ such that FAn ,E (ε/2) > 1 − µ for all n ≥ N. Thus, by (3.12) and Lemma 2.1(3) we have FE ,TE (ε) ≥ △(FE ,An+1 (ε/2), FTAn ,TE (ε/2)) ≥ △(FE ,An+1 (ε/2), FAn ,E (ψ(ε/2))) ≥ ∆(1 − µ, 1 − µ) > 1 − λ. This implies  FE ,TE (t ) = 1 for all t > 0, and hence TE = E. Finally, we prove uniqueness. Assume that TE0 = E0 and fix t > 0. From (3.12) it follows that

 FE ,E0 (t ) ≥ FE ,E0 (t /2) = FTE ,TE0 (t /2) ≥ FE ,E0 (ψ(t /2)) ≥ FE ,E0 (ψ 2 (t /2)) ≥ · · · ≥ FE ,E (ψ n (t /2)) ≥  FE ,E (ψ n (t /2)). 0

0

Letting n → ∞, by Lemma 1.7(3) we have  FE ,E0 (t ) = 1, i.e., E = E0 . (2) It follows similarly to the proof of (1), based on Theorem 2.4(2) instead of Theorem 2.4(1). This completes the proof.

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4. Some remarks and examples In this section, some remarks and examples concerning our results are given. Remark 4.1. By Lemma 1.1, if ψ is nondecreasing, then for all t > 0, limn→∞ ψ n (t ) = +∞ implies that ψ(t ) > t. Hence from Theorems 3.4 and 3.5 we obtain the following consequences. Corollary 4.1. Let (X , F , ∆) be a complete Menger PM-space such that ∆ is left-continuous and of H-type. Let ψ1 , ψ2 , . . . , ψm : R+ → R+ be nondecreasing functions such that limn→∞ ψjn (t ) = +∞ for j = 1, 2, . . . , m and t > 0. Let T1 , T2 , . . . , Tm : X → X be mappings such that FTj x,Tj y (t ) ≥ Fx,y (ψj (t )),

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

Then the three assertions (1), (2) and (3) in Theorem 3.4 hold. Corollary 4.2. Let (X , F , ∆) be a complete Menger PM-space such that ∆ is left-continuous and of H-type. Let ψ : R+ → R+ be a nondecreasing function such that limn→∞ ψ n (t ) = +∞ for all t > 0. Let {Tn }∞ n=1 be a sequence of mappings from X into X such that FTn x,Tn y (t ) ≥ Fx,y (ψ(t )),

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

Then the three assertions (1), (2) in Theorem 3.5 hold. Remark 4.2. Since each nonlinear contraction with a function ψ includes the case of linear contraction as its special case, each existence result of attractors for probabilistic nonlinear contraction implies a corresponding existence result for probabilistic linear contraction, if we take ψ(t ) = t /α , where α ∈ (0, 1). For example, from Theorem 3.4 we obtain the following consequence. Corollary 4.3. Let (X , F , ∆) be a complete Menger PM-space such that ∆ is left-continuous and of H-type. Let αj ∈ (0, 1), j = 1, 2, . . . , m. Let T1 , T2 , . . . , Tm : X → X be mappings such that FTj x,Tj y (t ) ≥ Fx,y (t /αj ),

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

Then the three assertions (1), (2) and (3) in Theorem 3.4 hold. Remark 4.3. As direct consequences of our results, we can obtain the corresponding existence theorems of attractors in usual metric spaces. We consider Theorem 3.5 as an example. Let ψ : R+ → R+ be a function. Let a, b, t ∈ R+ . We claim that H (t − b) ≥ H (ψ(t ) − a)

if and only if ψ(b) ≤ a.

(4.1)

Suppose that a > 0 and b > 0, without loss of generality. If ψ(b) > a, taking t = b in H (t − b) ≥ H (ψ(t ) − a), we can infer that 0 = H (b − b) ≥ H (ψ(b) − a) = 1. This is a contradiction. Conversely, if H (t − b) < H (ψ(t ) − a), then H (t − b) = 0 and H (ψ(t ) − a) = 1. From the definition of H it follows that t ≤ b and ψ(t ) > a, which contradicts with ψ(b) ≤ a. Hence (4.1) holds. Let (X , d) be a metric space and Fx,y (t ) = H (t − d(x, y)). For A, B ⊂ X , let dH be a Hausdorff metric defined by dH (A, B) = max{supx∈A infy∈B d(x, y), supy∈B infx∈A d(x, y)}. Since infy∈B supx∈A Fx,y (t ) = H (t − supy∈B infx∈A d(x, y)), we have

 FA,B (t ) = H (t − dH (A, B)),

for A, B ⊂ X .

(4.2)

Taking ∆ = ∆m and Fx,y (t ) = H (t − d(x, y)) in Theorem 3.5, by (4.1) and Lemma 1.6, we obtain the following consequence. Corollary 4.4. Let (X , d) be a complete metric space. Let ψ : R+ → R+ be a function, ψ(t ) > t and limn→∞ ψ n (t ) = +∞ for all t > 0. Let {Tn }∞ n=1 be a sequence of mappings from X into X such that

ψ[d(Tn x, Tn y)] ≤ d(x, y),

for all n ∈ Z+ and x, y ∈ X with x ̸= y.

(4.3)

(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 E ∈ Pcp (X ) such

∞

that E = n=1 Tn E. (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 E ∈ Pcl,bd (X ) such that E =

∞

n=1

Tn E.

Proof. Take ∆ = ∆m and Fx,y (t ) = H (t − d(x, y)). By Lemma 1.6, (X , F , ∆m ) is a Menger PM-space. From (4.2) we see that the topology induced by dH agrees with the one induced by  F . From (4.3) and (4.1) we have FTn x,Tn y (t ) = H (t − d(Tn x, Tn y)) ≥ H (ψ(t ) − d(x, y)) = Fx,y (ψ(t )), i.e., (3.11) holds. Hence the conclusion follows from Theorem 3.5.

J.-Z. Xiao et al. / Nonlinear Analysis 97 (2014) 106–118

117

Example 4.1. Let X = R, △ = △m . Define F : X × X → D + by



− F (x, y)(t ) = Fx,y (t ) = e

|x−y| t

0,

,

if t > 0; if t ≤ 0,

for all x, y ∈ X .

Then △m is a t-norm of H-type and (X , F , △m ) is a complete Menger PM-space (cf. [15]). Let a, b ∈ R and a < b. For x ∈ X , define T1 , T2 : X → X as follows T1 x =

x−a

+ a;

3

T2 x =

a−x 3

+ b.

It is clear that T1 , T2 are closed. Let ψ(t ) = 2t. Then, for t > 0, x, y ∈ X , and j = 1, 2, we have FTj x,Tj y (t ) = e−

|Tj x−Tj y|

= e−

t

|x−y|

≥ e−

3t

|x−y| 2t

= Fx,y (2t ).

Thus, all conditions of Theorem 3.4 are satisfied. Hence there exists a unique E0 ∈ Pcp (X ) such that E0 = Indeed, E0 is a generalized Cantor set. If we take A0 = [a, b], then A1 = E0 = limn→∞

2

j =1

Tj An . Also, there exists E∗ ∈ Pcl (X ) such that E∗ =

2

j=1

2

j=1 Tj A0 =

a, a +



2

j=1 Tj E0 .  a b − b− , b and 3

 b −a 3

Tj E∗ , for example, E∗ = R.

Example 4.2. Let X = R+ , △ = △m . Define F : X × X → D + by t

 F (x, y)(t ) = Fx,y (t ) =

t + max{x, y} 1,

,

if x ̸= y, if x = y,

for all t > 0;

and F (x, y)(t ) = Fx,y (t ) = 0 for all t ≤ 0. Then △m is a t-norm of H-type and (X , F , △m ) is a complete Menger PM-space (cf. [12]). For x ∈ X , define T1 , T2 , T3 : X → X by

x   ,

if x ∈ [0, 1),

2 T1 x = 1 

T3 x =

x T2 x =

 , if x ∈ [1, +∞); 2  2x  if x ∈ [0, 5),  5, x

1

+ ,   3 3 3,

ψ(t ) =

3t + 2    3t , 2

if x ∈ [0, 3), if x ∈ [3, +∞);

and

if x ∈ [5, 8), if x ∈ [8, +∞).

Evidently, Tj is closed and Tj x ≤

 2t ,    10t

,

3 1,

x 2

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

if t ∈ [0, 1),

,

if t ∈ [1, 2), if x ∈ [2, +∞).

Since ψ is nondecreasing and 54 t ≤ ψ(t ) ≤ 2t for all t ∈ R+ , it follows that

 n 5

4

t ≤ ψ n (t ) ≤ 2n t ,

and so limn→∞ ψjn (t ) = +∞. For t > 0, x, y ∈ X with x ̸= y, and j = 1, 2, 3, we have FTj x,Tj y (t ) =

t t + max{Tj x, Tj y}

t

≥ t+

1 2

max{x, y}

= Fx,y (2t ) ≥ Fx,y (ψ(t )).

Thus, all conditions of Corollary 4.1 are satisfied. Hence there exists a unique E0 ∈ Pcp (X ) such that E0 =

3

j =1

Acknowledgments The authors are grateful to the referees and editors for their valuable comments and helpful suggestions. References [1] J. Hutchinson, Fractals and self-similarity, Indiana Univ. J. Math 30 (1981) 713–747. [2] M.F. Barnsley, Fractals Everywhere, Academic Press Professional, Boston, 1993, pp. 3–97.

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