On a topological fuzzy fixed point theorem and its application to non-ejective fuzzy fractals II

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On a topological fuzzy fixed point theorem and its application to non-ejective fuzzy fractals II Jan Andres ∗,1 , Miroslav Rypka Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic Received 3 October 2017; received in revised form 19 September 2018; accepted 25 September 2018

Abstract This paper deals mainly with new very general fuzzy fixed point theorems in metric spaces and their application to fuzzy fractals. It is a natural continuation of our paper (2018) [1] which significantly generalizes and improves the results by Diamond, Kloeden and Pokrovskii (1997) [2], where no application was given. Moreover, besides the existence, some further important properties of fixed point sets (in particular, fractals) like their weak local stability, called none-ejectivity in the sense of Browder (1965) [3], will be established. © 2018 Elsevier B.V. All rights reserved. Keywords: Fuzzy fixed points; Fuzzy fractals; Absolute retracts; Multivalued maps; Admissible maps; Compact absorbing contractions; Non-ejectivity

1. Introduction The main aim of the present paper is two-fold: (i) to extend and significantly improve the main fuzzy fixed point theorem in [2, Theorem 2.4] developed by Diamond, Kloeden and Pokrovskii in 1997, (ii) to apply one of the obtained theorems to so called (non-ejective) fuzzy fractals. Following Hutchinson [4] and Barnsley [5], fractals (both fuzzy and ordinary) will be considered as fixed points of special induced operators, called accordingly the Hutchinson–Barnsley operators. Ordinary (crisp) fractals can be regarded as a particular case of fuzzy fractals, because on each level set a fuzzy fractal reduces to an ordinary one. * Corresponding author.

E-mail addresses: [email protected] (J. Andres), [email protected] (M. Rypka). 1 The first author was supported by the grant IGA_PrF_2017_019 “Mathematical Models” of the Internal Grant Agency of Palacký University in

Olomouc. https://doi.org/10.1016/j.fss.2018.09.013 0165-0114/© 2018 Elsevier B.V. All rights reserved.

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Thus, the crucial role is played here by the fuzzy fixed point theory which can be understood as a natural extension of the ordinary theory. The standard (ordinary) fixed point theory consists of two main parts: the metric (Banach-like) theory (see e.g. [6]) and the topological (Schauder-like) theory (see e.g. [7]). For the appropriate fuzzy extensions, we can adopt this terminology, namely: the metric fuzzy fixed point theory (in particular, metric fuzzy fractals) and the topological fuzzy fixed point theory (in particular, topological fuzzy fractals). For more details concerning the ordinary fractals, see e.g. [8,9], where the terminology however still differs from the one in [10,11]. As far as we know, the topological fuzzy fixed point theorems like those in [1,2] are (if any further) rare and topological fuzzy fractals were probably considered only by ourselves in [1]. The additional notion of non-ejectivity due to Browder [3] can be roughly characterized as a sort of a weak local stability. Trivial examples of ejective (resp. non-ejective) fixed points of a given map are repelling (resp. attractive) fixed points. A concrete example of a non-ejective fixed point is, for instance, a unique fixed point of the stereographically projected map f (x) = x + 1 onto the circle. The origin 0 of the piece-wise linear function  x , for x ∈ (−∞, 0], g(x) = a ax, for x ∈ [0, +∞), where a > 1, is also non-ejective. For those who are familiar with continuous dynamical systems, a typical example of a non-ejective fixed point is a saddle-point in the theory of planar autonomous ordinary differential equations. If a fixed point is not ejective, then it is simply non-ejective. Let us note that the fixed point theory in induced hyperspaces (i.e. spaces of spaces, where the fractals “live”) can be rather curiously more “powerful” than the one in the original spaces. For instance, if X is a locally compact and locally connected metric space (e.g. a finite union of separated rings in R2 ), then the compact continuous self-maps f : X → X (e.g. a rotation) need not evidently possess a fixed point. On the other hand, the related induced (compact) hypermap in the hyperspace of compact subsets of X, endowed with the Hausdorff metric, admits a fixed point (see e.g. [12]). In the present paper, we will show besides other things that the spaces of fuzzy sets under consideration have the same fixed point property, even for arbitrary supporting metric spaces X, i.e. even without any imposed local compactness and any local connectivity. Our paper is organized as follows. After this introduction, the appropriate preliminaries about fuzzy set terminology, retracts, multivalued maps and non-ejective fixed points will be given. We will recall there the related definitions and prove the auxiliary technical lemmas. In particular, we will show that the spaces of fuzzy sets under our consideration are absolute retracts for arbitrary supporting metric spaces, which significantly generalizes an analogy in [2, Theorem 4.1]. It is an essential step in our investigation. In Section 3, fuzzy fixed points will be considered for a very general class of multivalued maps involving only a certain amount of compactness. These maps, operating in spaces of fuzzy sets, which can represent possibility distributions, when they are subnormalized, will not be assumed to be induced from the basic space. The obtained Theorem 3 below is a far generalization of the main stimulation, namely Theorem 1 below due to Diamond, Kloeden and Pokrovskii [2, Theorem 2.4]. This generalization is done in two directions, namely from a single-valued to a multivalued setting and from a compact supporting space to maps having only a certain amount of compactness (CAC-maps). In Section 4, fuzzy fixed point Theorems 5 and 6 below are formulated for “fuzzified” maps. The “fuzzification” of multivalued maps was developed in our paper [1] and, for the sake of completeness, will be recalled here again. Finally, in Section 5, the obtained fuzzy fixed point theorems will be applied to fuzzy fractals. Some nontrivial illustrative examples were presented already in [1]. Here, only simple illustrations will be supplied, indicating rather the potential of our technique than a delicateness of fuzzy fractals themselves. Concretely, the second illustrative example demonstrates the applicability of Theorem 6 below to a fractal (called by the authors of [10,11] as the shark teeth) which cannot be constructed in the frame of the standard metric theory originated by Hutchinson [4] and Barnsley [5]. 2. Preliminaries At first, let us recall the basic fuzzy terminology. We will follow the notation and some results from [13]. A fuzzy subset of X is a function u : X → I , where I = [0, 1], which is usually assumed (like entirely here) to be up-

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per semicontinuous in the single-valued sense, i.e. for any sequence {xn } of elements of X which tends to x ∗ , lim supn→∞ u(xn ) ≤ u(x ∗ ). Let us note that, despite the same name, this notion does not coincide with the one in the multivalued sense defined below. Denote by D(X) the totality of such fuzzy subsets of X. An α-cut [u]α is defined by [u]α := {x ∈ X| u(x) ≥ α}. Notice that, for an upper semicontinuous function u, the α-cuts are compact, provided X is compact. Otherwise, the α-cuts are closed subsets of X. The support supp(u) of u ∈ D(X) is defined as supp(u) := {x ∈ X| u(x) > 0}, where the bar stands for a closure. The height of the fuzzy set u is defined as hgt(u) := max{α ∈ [0, 1], [u]α = ∅}. For u ∈ D(X), the set end(u) := {(x, a) ∈ X × I | u(x) ≥ a} denotes the endograph of u. The space D(X) will be endowed here with the endograph metric dE , i.e. dE (u, v) := dH (end(u), end(v)), for all u, v ∈ D(X), where dH stands for the Hausdorff metric, i.e.  dH (end(u), end(v)) := max sup ( inf d(z1 , z2 ), sup ( z1 ∈end(u) z2 ∈end(v)

inf

z2 ∈end(v) z1 ∈end(u)

 d(z1 , z2 )) ,

and the metric d in X × I is given by d((x, a), (y, b)) = dI (a, b) + dX (x, y). Let, furthermore, (Dδ (X), dE ), δ ∈ (0, 1], be the class of all u ∈ D(X) such that maxx∈X u(x) ≥ δ. All spaces under our consideration will be also metric. A metric space X is locally continuum-connected if for each neighbourhood V ⊂ U of x such that each point of V is contained in a subcontinuum (i.e. a compact connected subset) of U which contains x. A metric space X is called an absolute neighbourhood retract (ANR) if, for each metric space Y , each closed subset A ⊂ Y and each continuous map f : A → X, there is a continuous extension f : U → X, for some open set U ⊂ Y with U ⊃ A; if one can always choose U = Y , then X is called an absolute retract AR. It is well known that an ANR is AR if and only if it is contractible, i.e. homotopic to a one-point space. The following implications are easily seen for a metric space: AR ⇒ ANR ⇒ locally contractible ⇒ locally path-connected ⇒ locally continuum-connected ⇒ locally connected. The last two implications cannot be reversed in general, but they can be reversed in complete metric spaces. The same is true provided the space is locally compact. Let us note that any retract as well as any homeomorphic image of AR (ANR) remains AR (ANR). By a multivalued map, we understand here the mapping f : X → K(Y ), where K(Y ) is the hyperspace of nonempty compact subsets of Y . A map f : X → K(X) is said to be upper semicontinuous (u.s.c.) if, for every open U ⊂ X, the set {x ∈ X|f (x) ⊂ U } is open in X. Although the following classes of maps, containing only a certain amount of compactness, have been usually associated (see e.g. [14, Chapter I.5]) to admissible maps (in the sense of Górniewicz, cf. Definition 2 below), they can be treated separately as below. Definition 1. (cf. [14, Chapter I.5]) A u.s.c. map φ : X → K(X) is called a compact absorbing contraction (φ ∈ CAC(X)) if there exists an open set U ⊂ X such that: a) φ(U ) ⊂ U , b) the closure φ(U ) of φ(U ) is contained in a compact subset of U , c) for every x ∈ X, there exists a natural number nx such that φ nx (x) ⊂ U .

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We say that φ : X → K(X) is a locally compact map provided, for every x ∈ X, there exists an open neighbourhood V of x such that φ|V : V → K(X) is a compact map, i.e. φ|V (V ) is compact. Obviously, a set Y ⊂ X is relatively compact if its closure Y is compact in X. We let: K(X) = {φ : X → K(X) | φ is u.s.c. and compact}. EC(X) = {φ : X → K(X)|φ is u.s.c., locally compact and there exists a natural number n such that the n−th iteration φ n : X → K(X) of φ is a compact map}. n ASC(X) = {φ : X → K(X)|φ is u.s.c., locally compact, the orbit O(x) = ∪∞ n=1 φ (x) is, for every x ∈ X, relan (x) is nonempty and relatively compact}. tively compact and the core C(φ) = ∩∞ φ n=1 CA(X) = {φ : X → K(X)|φ is u.s.c., locally compact and has a compact attractor, i.e., there exists a compact set A ⊂ X such that, for every open set W ⊂ X containing A and for every point x ∈ X, there is nx such that φ nx (x) ⊂ W }. The following hierarchy holds (see [14, Chapter I.5]): K(X) ⊂ EC(X) ⊂ ASC(X) ⊂ CA(X) ⊂ CAC(X).

(1)

Moreover, each of the above inclusions is proper. Let φ ∈CAC(X) and let U be chosen according to Definition 1. Then φU : U → K(U ), defined by the formula φU (x) = φ(x), for every x ∈ U , is obviously a compact map. These classes play an important role for multivalued, admissible (in the sense of Górniewicz) maps in the fixed point theory. Definition 2. A multivalued map φ : X → K(Y ) is called admissible (in the sense of Górniewicz) if there exists a diagram p

q

X← −− →Y in which p :  → X is a Vietoris map and q :  → Y is a continuous map such that φ(x) = q(p −1 (x)). By a Vietoris map p :  → X, we mean the one such that (i) p is onto and proper, i.e. p −1 (K) is compact for every compact K ⊂ X, (ii) for every x ∈ X, the set p −1 (x) is acyclic (i.e. homologically equivalent to a one-point space), where acyclicity ˇ is understood in the sense of Cech homology functor with compact carriers and coefficients in the field Q of ˇ rationals. In other words, p −1 (x), x ∈ X, is acyclic if all of its reduced Cech homology groups over rationals vanish. Let us note that the class of admissible maps is very rich and contains, for instance, u.s.c. maps with compact, contractible values, u.s.c. maps with compact, convex values, etc. All of them are u.s.c. with compact, connected values. For more details, see e.g. [14, Chapter I.4]. The following Lefschetz-type fixed point statement holds for admissible, CAC-maps. Proposition 1. (cf. e.g. [14, Theorem I.6.20]) Let (X, d) be an AR-space and φ : X → K(X) be an admissible, CACmap (see Definitions 1 and 2). Then φ admits a fixed point, i.e. ∃x ∈ X : x ∈ φ(x). Corollary 1. (cf. e.g. [14, Theorem I.6.21]) Let (X, d) be an AR-space and φ : X → K(X) be any of multivalued admissible maps in the series of proper inclusions (1). Then φ admits a fixed point. Definition 3. Let (X, d) be a metric space and f : X → X be a continuous mapping. Denote by Fix(f ) := {x ∈ X|f (x) = x} the set of fixed points to f . We say that a fixed point x0 ∈ Fix(f) is ejective w.r.t. V ∈ U (x0 ), where U (x0 ) stands for the family of all open neighbourhoods of x0 , if for every x ∈ V \{x0 }, there exists an integer n = n(x) ≥ 1 such that

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f n (x) = f ◦ · · · ◦ f (x) ∈ X\V . (n−1)-times If there exists V ∈ U (x0 ) such that x0 is ejective w.r.t. V , then x0 is called ejective. If x0 ∈ Fix(f ) is not ejective, then it is non-ejective in the sense of Browder. Now, let us consider the structure of hyperspaces and fuzzy spaces. Lemma 1. (cf. [15], [12, Theorem 1]) Let (X, d) be a metric space and (K(X), dH ) be the hyperspace of compact subsets of X endowed with the Hausdorff metric dH . Then K(X) is an ANR-space if and only if X is locally continuum connected. Moreover, if it is so, then K(X) is an AR-space if and only if X is still connected. Lemma 2. (cf. [16, Lemma 6]) Let (X, d) be a compact metric space. Then (D(X), dE ) is a compact absolute retract. Lemma 3. Let (X, d) be a compact metric space. Then (Dδ (X), dE ) is a compact absolute retract, for every δ ∈ (0, 1]. Proof. For any u ∈ D(X), define  u(x) − max u(y) + δ, y∈X (r(u))(x) = u(x),

for max u(y) ≤ δ, y∈X

otherwise.

(2)

Since r is obviously a retraction of D(X) onto Dδ (X), δ ∈ (0, 1], and D(X) is, according to Lemma 2, a compact absolute retract, Dδ (X) must be, for every δ ∈ (0, 1], a compact AR, as claimed. 2 Lemma 4. For any (X, d), the spaces (D(X), dE ) as well as (Dδ (X), dE ), δ ∈ (0, 1], are contractible, and so connected. Proof. Consider the homotopy h(λ, u) : [0, 1] × D(X) → D(X), where h(λ, u(x)) = u(x) + (1 − u(x))(1 − λ), λ ∈ [0, 1]. Observe that h(0, u(x)) ≡ 1 ∈ D(X), while h(1, u(x)) = u(x), i.e. the identity on D(X). Since (Dδ (X), dE ) is a retract of (D(X), dE ), both (D(X), dE ) and (Dδ (X), dE ), δ ∈ (0, 1], are contractible, as claimed. 2 Lemma 5. Let (X, d) be a metric space. Then (D(X), dE ) as well as (Dδ (X), dE ), δ ∈ (0, 1], are absolute retracts. Proof. Let us follow the proof of [16, Lemma 6]. In view of Lemma 4, the spaces (D(X), dE ) and (Dδ (X), dE ), δ ∈ (0, 1], are contractible. The homotopy constructed in [2, Theorem 4.2], namely  (1 − 2λ)u(x) + 2λ max{u(x), u1 (x)}, for λ ∈ [0, 12 ], h(λ, u(x)) = (2λ − 1)u(x) + (2 − 2λ) max{u(x), u1 (x)}, for λ ∈ [ 12 , 1], verifies also the local contractibility of (D(X), dE ) in each u1 ∈ D(X). Thus, (D(X), dE ) is connected and locally continuum-connected, and consequently (K(D(X), dE ), dEH ) is, according to Lemma 1, an absolute retract. Now, consider the retraction r : K(D(X)) → D(X), r(U ) = U , where U ∈ K(D(X)). u∈U

It is well defined, because U ⊂ D(X) is a compact subset of D(X) and (D(X), dE ) can be isometrically embedded into (K(D(X)), dEH ). Therefore, (D(X), dE ) is a retract of the AR-space (K(D(X)), dEH ), i.e. an AR itself. Since (Dδ (X), dE ) is a retract of (D(X), dE ), where the retraction r is defined in (2), the proof is complete. 2

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3. Fuzzy fixed point theorems for maps not induced from the basic space In [2, Theorem 4.2], the following fuzzy fixed point theorem was carried out. Theorem 1. (cf. [2]) Let (X, d) be a compact metric space and D(X) be the totality of fuzzy sets u : X → [0, 1] which are upper semicontinuous (in the single-valued sense). Let, furthermore, Dδ (X), δ ∈ (0, 1], be the class of all u ∈ D(X) such that maxx∈X u(x) ≥ δ. Then the metric spaces (D(X), dE ) and (Dδ (X), dE ), for δ ∈ (0, 1], where dE stands for the endograph metric, each have the fixed point property, i.e. that any continuous mapping f : D(X) → D(X), resp. f : Dδ (X) → Dδ (X), δ ∈ (0, 1], has at least one fixed point. The first generalization of Theorem 1 holds for a large class of multivalued, admissible maps in the sense of Górniewicz (see Definition 2). Theorem 2. Let (X, d) be a compact metric space and φ : D(X) → K(D(X)) (resp. φ : Dδ (X) → K(Dδ (X)), δ ∈ (0, 1]) be a multivalued, admissible map in the sense of Definition 2. Then φ admits a fixed point, i.e. ∃u∗ ∈ D(X) : u∗ ∈ φ(u∗ ) (resp. ∃u∗ ∈ Dδ (X) : u∗ ∈ φ(u∗ )). Proof. According to Lemma 2, (D(X), dE ) is a compact AR (resp., according to Lemma 3, (Dδ (X), dE ) is a compact AR). Since φ becomes obviously compact, Proposition 1 implies immediately the existence of a fixed point of φ in D(X), (resp. in Dδ (X), δ ∈ (0, 1]), as claimed. 2 Theorem 2 can be still improved for not necessarily compact spaces and multivalued, admissible maps having only a certain amount of compactness, so called compact absorbing contractions (CAC-maps, see Definition 1), by showing that (D(X), dE ) as well as (Dδ (X), dE ), δ ∈ (0, 1], are absolute retracts, for arbitrary metric spaces (X, d). In other words, the compactness in Theorem 2 can be avoided by means of Lemma 5. Theorem 3. Let (X, d) be a metric space and φ : D(X) → K(D(X)) (resp. φ : Dδ (X) → K(Dδ (X)), δ ∈ (0, 1])) be a multivalued, admissible CAC-mapping in the sense of Definitions 1 and 2. Then φ admits a fixed point, i.e. ∃u∗ ∈ D(X) : u∗ ∈ φ(u∗ ) (resp. ∃u∗ ∈ Dδ (X) : u∗ ∈ φ(u∗ )). Proof. According to Lemma 5, (D(X), dE ) is an AR, resp. (Dδ (X), dE ) is an AR). Proposition 1 immediately implies the existence of u∗ in D(X) resp. in Dδ (X)), as claimed. 2 Applying Corollary 1 (instead of Proposition 1), Theorem 3 can take, in view of (1), the following, more informative form. Corollary 2. Let (X, d) be a metric space and φ : D(X) → K(D(X)) (resp. φ : Dδ (X) → K(Dδ (X)), δ ∈ (0, 1])) be one of multivalued, admissible maps in the series of proper inclusions (1). Then φ admits a fuzzy fixed point. Remark 1. If φ in Theorems 2 and 3 is not admissible, but just compact (or even containing only a certain amount of compactness) u.s.c. map, then it can be fixed point free. On the other hand, one can prove, by means of the well known Knaster–Tarski theorem, the existence of a fixed point of the induced hyper-mapping φ ∗ : K(D(X)) → K(D(X)) (resp. φ ∗ : K(Dδ (X)) → K(Dδ (X)), δ ∈ (0, 1]), i.e. the existence of a compact, invariant subset in D(X) (resp. in Dδ (X), δ ∈ (0, 1]), for φ. Remark 2. In [17], we have formulated sufficient conditions for non-ejective fixed points of multivalued maps on arbitrary ANR-spaces. These conditions, formulated in terms of the generalized Lefschetz numbers, can be adopted to our Theorems 2 and 3. Nevertheless, their verification would not be an easy task. Moreover, the notion of non-ejectivity in [17] slightly differs from the standard one in the sense of Browder (see Definition 3).

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4. Fuzzy fixed point theorems for fuzzified maps Stimulated by [2, Theorem 4.2], formulated here as Theorem 1, we extended it in [1, Theorem 5] in the following way. Theorem 4. Let (X, d) be a metric space and f : X → K(X) be a compact, u.s.c., multivalued map. Then the fuzzified mapping f ∗ : D(X) → D(X) possesses a nonempty, non-degenerated, contractible set of fixed points in (D(X), dE ). Moreover, all the fixed points of f ∗ in (D(X), dE ) are non-ejective in the sense of Browder. By a (Zadeh-like) fuzzified map f ∗ : D(X) → D(X) of a given multivalued mapping f : X → K(X), we mean here again (for more details, see [1]) f ∗ (u)(x) = sup {u(y)}.

(3)

x∈f (y)

For a compact X, it holds [f ∗ (u)]α = f ∗ ([u]α ), for all α ∈ [0, 1]. Let us note that, for single-valued maps f : X → X, the classical Zadeh’s extension f : D(X) → D(X) reduces (3) into f ∗ (u)(x) = sup {u(y)}. x=f (y)

The generalization of Theorem 4 holds for not necessarily compact inducing maps f , i.e. for multivalued CACmaps. Thus, Theorem 4 can be therefore generalized for CAC-maps as follows. Theorem 5. Let (X, d) be a metric space and f : X → K(X) be a CAC-mapping (see Definition 1 and the series of proper inclusions (1)). Then the fuzzified mapping f ∗ : D(X) → D(X) possesses a non-empty, non-degenerated, contractible set of fixed points in (D(X), dE ) which are non-ejective in the sense of Browder (see Definition 3). Proof. By Definition 1 of CAC-maps, there exists an open U ⊂ X and nonempty, compact K ⊂ U such that f (U ) ⊂ K. Thus, we can restrict ourselves to f |K : K → K(K), and apply Theorem 4 to the fuzzified map (f |K )∗ : D(K) → D(K). 2 We can also extend the assertion of Theorem 3 (resp. its part for δ = 1) to its analogy for the spaces (Dδ (X), dE ), δ ∈ (0, 1). Observe that there is a slight exception for (D1 (X), dE ), where the set of fuzzy fixed points can degenerate into a unique fixed point. Theorem 6. Let (X, d) be a metric space and f : X → K(X) be a CAC-mapping (see Definition 1 and the series of proper inclusions (1)). Then the fuzzified mapping f ∗ : Dδ (X) → Dδ (X), δ ∈ (0, 1), possesses a nonempty, non-degenerated, contractible set of fixed points in (Dδ (X), dE ), δ ∈ (0, 1), which are non-ejective in the sense of Definition 3. The fuzzified mapping f ∗ : D1 (X) → D1 (X), possesses a nonempty, contractible set of fixed points in (D1 (X), dE ) which can, however, degenerate into a unique fuzzy set. Proof. By Definition 1 of CAC-maps, there again exists an open U ⊂ X and a nonempty, compact K ⊂ U such that f (U ) ⊂ K. Thus, we can restrict ourselves to f |K : K → K(K), and proceed quite analogously as in the proof of Theorem 4 (see [1, Theorem 5]), but for the fuzzified map (f |K )∗ : Dδ (K) → Dδ (K), δ ∈ (0, 1]. The contractibility of the set Fix((f |K )∗ ) of fixed points of (f |K )∗ can be verified here by means of the homotopy h(λ, u(x)) : [0, 1] × Fix((f |K )∗ ) → Fix((f |K )∗ ), where h(λ, u(x)) = u(x) + (1 − u(x))(1 − λ), λ ∈ [0, 1], x ∈ K. In this way, the whole conclusion can be verified for δ ∈ (0, 1). On the other hand, for (D1 (K), dE ), apart from a nonempty, contractible set of fuzzy fixed points, their multiplicity cannot be obtained in this way. 2

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Remark 3. Observe that the (single-valued) fuzzified maps f ∗ in Theorems 5 and 6 may be discontinuous. Therefore, even if (D1 (X), dE ) would be a Hilbert cube, the Browder-type theorems (see e.g. [9]) need not imply the existence of a non-ejective fixed point of f ∗ in D1 (X). On the other hand, if f : X → K(X) is a multivalued contraction in a complete metric space X, then the unique fuzzy fixed point is attractive, which is impossible for D(X) as well as Dδ (X) with δ ∈ (0, 1). 5. Applications to fuzzy fractals, or so By fuzzy fractals, we understand here the same objects as in [1,8] but, this time, considered in (Dδ (X), dE ), δ ∈ (0, 1]. More precisely, we can give the following definition. Definition 4. Let (X, d) be a metric space and {fi : X → K(X), i = 1, 2, . . . , m} be a system of multivalued maps. By a fuzzy fractal, we mean a fixed point u∗ ∈ Dδ (X), δ ∈ (0, 1], of the (union) Hutchinson–Barnsley fuzzy operator, namely ∗

f :=

m

fi∗ : Dδ (X) → Dδ (X), δ ∈ (0, 1],

i=1

i.e. ∗

∗

f (u ) :=

m

fi∗ (u∗ ) = u∗ ,

i=1

fi∗ :

where Dδ (X) → Dδ (X)(X), δ ∈ (0, 1], i = 1, 2, . . . , m, are defined in (3). By a non-ejective fuzzy fractal, we mean a fuzzy fractal which is still non-ejective in the sense of Definition 3. Theorem 4 was applied in [1, Theorem 6] to fuzzy fractals as follows. Theorem 7. Let (X, d) be a metric space and {fi : X → K(X); i = 1, 2, . . . , m} be a system of compact, u.s.c., multivalued maps. Then there exists a nonempty, non-degenerated, contractible set of fuzzy fractals in (D(X), dE). Moreover, each of them is non-ejective in the sense of Browder. Remark 4. Notice that, for any finite system of u.s.c., multivalued maps fi : X → K(X), i = 1, 2, . . . , n, the equality

m ∗ m fi = fi∗ i=1

i=1

follows from

m  m  m m ∗ fi (u) = fi ([u]α ) = fi ([u]α ) = [fi∗ (u)]α , i=1

α

i=1

i=1

i=1

for any u ∈ D(X) and α ∈ [0, 1]. Moreover, f ∗ (Dδ (X)) ⊂ Dδ (X), δ ∈ (0, 1], holds for any fuzzified mapping f ∗ , because hgt(f ∗ (u)) = hgt(u), for all u ∈ D(X). Unfortunately, neither Theorem 5 nor Theorem 6 can be applied to fuzzy fractals in their full generality, because a finite union of CAC-maps need not be CAC. In particular, the inducing map ∪m i=1 fi for Hutchinson–Barnsley operator f ∗ does not preserve the union of compact invariant subsets to single CAC-maps fi , i = 1, 2, . . . , m, from a given system. That is why, only the special case of Theorem 6 for compact inducing maps fi , i = 1, . . . , m, will be used here, analogously to Theorem 7 (cf. [1, Theorem 6]). The particular case of Theorem 6 will be so applied to fuzzy fractals, considered as fixed points of the Hutchinson– Barnsley operators in the spaces (Dδ (X), dE ), δ ∈ (0, 1], in analogy to Theorem 7.

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Theorem 8. Let (X, d) be a metric space and {fi : X → K(X); i = 1, 2, . . . , m} be a system of compact, u.s.c., multivalued maps. Then there exists a nonempty, non-degenerated contractible set of fuzzy fractals in (Dδ (X), dE ), δ ∈ (0, 1). Moreover, each of them is non-ejective in the sense of Definition 3. In (D1 (X), dE ), there exists a nonempty, contractible set of fuzzy fractals which can degenerate into a unique fuzzy fractal. Proof. Since the maps fi : X → K(X), i = 1, 2, . . . , m, are compact, there exist compact sets Ki ∈ K(X), i = 1, 2, . . . , m, such that fi (X) ⊂ Ki , i = 1, 2, . . . , m. It follows that the set K = ∪m i=1 Ki is compact, i.e. K ∈ K(X), and such that f (K) = ∪m i=1 fi (K) ⊂ K. Since the finite union of u.s.c. maps is also u.s.c. (see e.g. [14, Proposition I.3.18]), Theorem 6 affirmates, in view of Remark 4, the existence of a set of fuzzy fractals for the system {fi : K → K(K); i = 1, 2, . . . , m}, i.e. the fixed points u∗ ∈ Dδ (K), δ ∈ (0, 1], of f ∗ :=

m

fi∗ : Dδ (K) → Dδ (K), δ ∈ (0, 1],

i=1

with all the required properties. 2 The following simple example illustrates the application of Theorem 8, for δ = 1. Example 1. Consider the fuzzification (see (3)) of the system of multivalued maps {[0, 1]2 , fi : [0, 1]2 → K([0, 1]2 ), i = 1, 2, 3},    1      0 0 x x 2 f1 := + 1 , y y 0 12 2 f2

   1 1 x 3, 2 := y 0

   1 x 2 f3 := y 0 where 

0 1 2

1 0 1 3, 2

    1  x 2 + , y 0

     x 0 + , y 0

     x x   y y  1 1 1 1 , · x, , · y := , , , . 3 2 3 2 3 2 3 2

Without fuzzification, this system generates the so called fat Sierpi´nski triangle. The associated “fuzzified” fat Sierpi´nski’s triangle is depicted in Fig. 1. Since all the maps f1 , f2 , f3 , are contractions, the fuzzified mapping f ∗ = f1∗ ∪ f2∗ ∪ f3∗ in (D1 (X), dE ) is a contraction as well, and this fuzzy fractal must be unique and attractive in (D1 (X), dE ). Theorems 7 and 8 can be modified, provided the metric space (X, d) is complete and some of maps from a given system are multivalued contractions. Theorem 9. Let (X, d) be a complete metric space and {fi : X → K(X); i = 1, 2, . . . , m} be a system of u.s.c. multivalued maps. Furthermore, assume that some of maps fi : X → K(X), i = 1, 2, . . . , m, are multivalued contractions and that the remaining maps are compact. Then there exists a nonempty, non-degenerated, contractible set of nonejective fuzzy fractals in (D(X), dE ) (resp. in (Dδ (X), dE ), δ ∈ (0, 1)). In (D1 (X), dE ), there exists an attractive fuzzy fractal. Proof. Let G : K(X) → K(X) be the map induced by the union of contractions and let K0 ∈ K(X) be a set containing the ranges of the remaining compact maps. Since G is a contraction in the complete metric space (K(X), dH ) (see e.g. [14, Appendix A.3], [12]), it admits a unique fixed point M0 ∈ K(X).

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Fig. 1. “Fuzzified” fat Sierpi´nski’s triangle from Example 1.

It was shown in [12, Proposition 2] that the infinite union K := M0 ∪ K0 ∪

∞

Gn (K0 )

n=1

is compact and that the associated compact hyperspace K(K) ⊂ K(X) is a subinvariant attractor of the Hutchinson– Barnsley operator F : K(X) → K(X), F :=

m

fi ,

i=1

i.e., in particular, F (K) =

m

fi (K) ⊂ K,

i=1

which contains the core of f . Thus, we can restrict ourselves to the u.s.c. (cf. [14, Proposition I.3.18]) map f |K : K → K(X) in order to apply Theorem 4 (resp. Theorem 6). Consequently, the claim follows directly from Definition 4 and Remark 4. The case of (D1 (K), dE ) ⊂ (D1 (X), dE ) coincides with the one in (K(K), dH ) ⊂ (K(X), dH ). 2 Unlike Theorems 8 and 9, the application of Theorem 6 to so called Edelstein-type fractals can be illustrated by means of the following example. Example 2. Following [10] and [11], consider the homothetic copies fm (x) = 2−m f (2m x), m ∈ Z, of the piece-wise linear periodic function    x − m, for x ∈ m, m + 12  , f (x) = m − x, for x ∈ m − 12 , m .

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Fig. 2. Shark teeth from Example 2.

Let us define the operator F : [0, 1]2 → K([0, 1]2 ), F = I0 ∪

∞

Fi ,

(4)

i=1

where I0 = [0, 1] × {0},   1 Fi = (x, fmi (x)) | x ∈ [0, 1] i   and mi = log2 log2 (i + 1) , i ∈ N, the symbol  .  stands for the integer part of a given expression. It was shown in [11] (cf. also [10]) that the hypermap F : K([0, 1]2 ) → K([0, 1]2 ) is a homeomorphic image of the Edelstein contraction in the hyperspace (K(0, 1)2 , dH ) which admits a unique fixed point. This fixed point, called there the shark teeth, is an example of a topological fractal in the sense of [10,11], which cannot be obtained as a homeomorphic image of an attractor of the standard hyperbolic iterated function system of Banach contractions. Observe that, despite a countable number of unions in (4), the multivalued mapping F : [0, 1]2 → K([0, 1]2 ) has a closed graph on the compact set [0, 1]2 × [0, 1]2 , and subsequently it is compact and upper semicontinuous (see e.g. [14, Proposition I.3.16]). Hence, the fuzzification F ∗ : D1 ([0, 1]2 ) → D1 ([0, 1]2 ) of F in (D1 ([0, 1]2 ), dE ), which is an absolute retract (see Lemma 3), implies, according to Theorem 6 and a countable analogy of Remark 4 (which holds because of compactness of [0, 1]2 ), the existence of a fuzzy fixed point in (D1 ([0, 1]2 ), dE ). Moreover, one can show that it is again, up to a homeomorphism, the Edelstein attractor. It is depicted in Fig. 2. Remark 5. In [18], the fuzzy set D1 (X) was treated, but endowed with the supremum metric d∞ . Unlike for D(X) and Dδ (X) with δ ∈ (0, 1), we were able to enjoy there all the consequences of the applied Banach fixed point theorem for fuzzy fractals, determined by the systems of multivalued contractions. In particular, we could visualize fuzzy hyperfractals and calculate their Hausdorff dimensions. On the other hand, no results were obtained there for non-contractive maps. Thus, unlike in D(X) and Dδ (X), δ ∈ (0, 1), in D1 (X) metric fractals, obtained by means of the Banach-like fixed point theorems, can meet with topological fractals, obtained by means of the Lefschetz-type theorems like Proposition 1 and Corollary 1 in Section 2.

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Remark 6. Let us note that our topological fuzzy fixed point theorems, namely Theorems 3–8, hold for an arbitrary supporting metric space (X, d) and three of them (Theorems 3, 5, 6) even for not necessarily compact maps. In this light, they are more general than their analogies in hyperspaces (K(X), dH ) of compact subsets of X, endowed with the Hausdorff metric dH . Remark 7. Let us finally note that the applicability of fuzzy fixed point theorems is, like in the case of ordinary (crisp) fixed point theory, very rich. The topological (i.e. Schauder-like or rather Lefschetz-like) fuzzy fixed point theorems can be certainly nontrivially applied, apart from the fuzzy fractals, to further sorts of fuzzy dynamical systems and fuzzy differential equations and inclusions. References [1] J. Andres, M. Rypka, On a topological fuzzy fixed point theorem and its application to non-ejective fuzzy fractals, Fuzzy Sets Syst. 350 (2018) 95–106. [2] P. Diamond, P. Kloeden, A. Pokrovskii, Absolute retracts and a general fixed point theorem for fuzzy sets, Fuzzy Sets Syst. 86 (3) (1997) 377–380. [3] F.E. Browder, A further generalization of the Schauder fixed point theorem, Duke Math. J. 32 (4) (1965) 575–578. [4] J.E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981) 713–747. [5] M.F. Barnsley, Fractals Everywhere, Academic Press, New York, 2003. [6] W.A. Kirk, B. Sims (Eds.), Handbook of Metric Fixed Point Theory, Kluwer, Dordrecht, 2001. [7] R.F. Brown, M. Furi, L. Górniewicz, B. Jiang (Eds.), Handbook of Topological Fixed Point Theory, Springer, Berlin, 2005. [8] J. Andres, J. Fišer, Metric and topological multivalued fractals, Int. J. Bifurc. Chaos 14 (4) (2004) 1277–1289. [9] J. Andres, L. Górniewicz, Note on nonejective topological fractals on Peano’s continua, Int. J. Bifurc. Chaos 24 (11) (2014) 145148. [10] T. Banakh, M. Nowak, A 1-dimensional Peano continuum which is not an IFS attractor, Proc. Am. Math. Soc. 141 (3) (2013) 931–935. [11] M. Nowak, T. Szarek, The shark teeth is a topological IFS-attractor, Sib. Math. J. 55 (2) (2014) 296–300. [12] J. Andres, M. Väth, Calculation of Lefschetz and Nielsen numbers in hyperspaces for fractals and dynamical systems, Proc. Am. Math. Soc. 135 (2) (2007) 479–487. [13] P. Diamond, P.E. Kloeden, Metric Spaces of Fuzzy Sets: Theory and Applications, World Scientific, Singapore, 1994. [14] J. Andres, L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Kluwer, Dordrecht, 2003. [15] D.W. Curtis, Hyperspaces of noncompact metric spaces, Compos. Math. 40 (2) (1980) 139–152. [16] Z. Yang, X. Zhou, A pair of spaces of upper semi-continuous maps and continuous maps, Topol. Appl. 154 (8) (2007) 1737–1747. [17] J. Andres, L. Górniewicz, Fixed point index and ejective fixed points of compact absorbing contraction multivalued mappings, J. Nonlinear Convex Anal. 16 (6) (2015) 1013–1023. [18] J. Andres, M. Rypka, Fuzzy fractals and hyperfractals, Fuzzy Sets Syst. 300 (2016) 146–154.