Existence of a fractal of iterated function systems containing condensing functions for the degree of nondensifiability

From a fixed point theorists’ view, Hutchinson considered a fractal set as a fixed point problem and applied the Banach contraction principle to prove its existence. In this paper, we present a result about the existence of fractal for a finite iterated condensing function using the degree of nondensifiability.


Introduction
In this paper, fractals are considered throughout the metric fixed point theory view as invariant objects of a given family of maps. More concretely, let F be a nonempty family of self-maps on nonempty set X, in the literature this is called an iterated function system (IFS in short) and define the invariance operator T : P(X) → P(X) ( P(X) denotes family of all subsets) by If H ⊂ T (H) we say that H is a subinvariant set. If (X, d) is a metric space, under an F-fractal we mean a nonempty compact, F-invariant subset of X. In [1], Hutchinson considered a finite set of contractions (ρ n ) N n=1 in a complete metric space (X, d) and in the hyperspace K(X) of all nonempty compact 1. The family Kerμ = {X ∈ M E : μ(X) = 0} is nonempty and Kerμ ⊂ N E . 2. X ⊂ Y ⇒ μ(X) ≤ μ(Y ). 3. μ(X) = μ(X). 4. μ(ConvX) = μ(X).
6. If (X n ) is a sequence of closed sets from M E such that X n+1 ⊂ X n for n = 1, 2, . . . and lim n→∞ μ(X n ) = 0 then the intersection X ∞ = ∩ ∞ n=1 X n is nonempty. Also, μ is called subadditive if μ(X +Y ) ≤ μ(X)+μ(Y ) and μ is homogeneous if μ(λX) = |λ|μ(X) for λ ∈ R. Moreover, μ is said to be sublinear if it is subadditive and homogeneous. If μ(X ∪ Y ) = max{μ(X), μ(Y )} then we say that the m.n.c. μ satisfies the maximum property. The m.n.c. with kernel Kerμ = N E will be called full. The m.n.c μ is said to be regular if it is sublinear, full and satisfies the maximum property.
In the theory of measure of noncompactness, two regular measures of noncompactness are very relevant in this context, the Kuratowski α(X) [13] and the Hausdorff X (X) measure of noncompactness [14] which are defined as  [15]) Suppose that Ω is nonempty, bounded, closed and convex subset of E, μ is a regular m.n.c. on E and T : Ω → Ω a continuous operator. If, for any nonempty subset X of Ω with μ(X) > 0, then T has at least one fixed point in Ω.
A very important fact in Theorem 1.2 is that the m.n.c. μ is regular and, in particular, it satisfies the maximum property.
3. An operator T : Ω → Ω satisfying assumptions of Theorem 1.2, that is, T is continuous and, for any nonempty subset X of Ω with μ(X) > 0, we have μ(T X) < μ(X) is said to be a condensing operator for the m.n.c. μ.
The results of this paper are motivated by the ones given in the recent paper [11]. We next present one of the main results of this paper (Theorem 2 of [11]). Theorem 1.4. Let F be a finite self-maps family of condensing operators for the Hausdorff measure of noncompactness on a nonempty, bounded, closed and convex subset Ω of a real Banach space (E, · ). Then the IFS associated to F has at least one fractal.
Our main purpose in this paper is to prove a similar result as in Theorem 1.4 for a finite family of condensing maps for the degree of nondensifiability (see Sect. 2) instead of the Hausdorff measure of noncompactness.

Background
In this section, we present some basic facts about degree of nondensifiability which will be needed in our further consideration. The details can be found in [16][17][18]. Suppose that (X, d) is a metric space and A is a nonempty bounded subset of X.

Remark 2.2. Notice that if
A is a nonempty and bounded subset of X then we can find an γ-dense curve in A for any γ ≥ diamA. Indeed, we take x 0 ∈ A and the continuous mapping ρ : [0, 1] → X given by ρ(t) = x 0 , for any t ∈ [0, 1]. By Hahn-Mazurkiewicz theorem [19], when A is a connected, compact and locally connected set, we find a continuous mapping ρ : [0, 1] → X, satisfying ρ([0, 1]) = A and, therefore, ρ is an 0-dense curve in A and then γ is called a space-filling curve. Therefore, the γ-dense curves generalize the space-filling curves. Let A to be a nonempty and bounded set, A is said to be densifiable is for every γ > 0 there is an γ-dense curve in A. It is proved (see [17]) that the class of densifiable sets is strictly between the class of Peano Continua (i.e. those sets that are continuous image of I = [0, 1]) and the class of connected and relatively compact sets.
where Γ γ,A denotes the family of γ-dense curves in A.
In [17], it is proved that the degree of nondensifiability is not a measure of noncompactness because it does not satisfy the monotonicity condition with respect to the inclusion. Next, we recollect some properties of the degree of nondensifiability which appear in [16]. (b) For every nonempty bounded subset A of E, the following inequality

holds. (c) Let A and B be nonempty bounded subsets of E, then
We next present some concepts and basic results in the theory of IFSs. More details can be found in [1,11]. Next, we present some lemmas proved in [11] which we will use later.

Lemma 2.6. [11] Let F be a nonempty family of maps on a nonempty set X into itself and H ⊂ X is an F-subinvariant set. Then the limit of H, under Kantorovich iteration, is an F-invariant set.
Lemma 2.7. [11] Let (X, d) be a metric space, F a finite family of continuous maps of X into itself and H ⊂ X is a relatively compact F-invariant set then H is also F-invariant.

Main result
We start this section with the following definition.

Definition 3.1.
Suppose that Ω is a nonempty, bounded, closed and convex and T : Ω → Ω. We say that T is a φ d -condensing operator if T is continuous and such that, for any nonempty and convex subset X of Ω, the following inequality holds.
The following example proves that there exist φ d -condensing operators which are not condensing for the Hausdorff measure of noncompactness. This example appears in Example 3.4 of [20].
Example. Consider the Banach space C([0, 1]) of the real and continuous functions on [0, 1] with the classical supremum norm and let Ω be the bounded, closed and convex subset given by Define the following operator T on Ω as It is easily checked that T maps Ω into itself. Moreover, (in Example 2, pag. 169 of [21]) it is proved that Therefore, T is not a condensing operator for the Hausdorff measure of noncompactness.
On the other hand, in [17] the authors proved that, for any nonempty and convex subset X of Ω with φ d (X) > 0, and, consequently, This proves that T is a φ d -condensing operator.
Next, we present the main result of the paper. F = {f 1 , f 2 , . . . , f n } be a finite family of φ d -condensing self-maps of a nonempty, bounded, closed and convex subset Ω of a real Banach space. Then there exists a convex, compact set P of Ω such that T (P ) ⊂ P .

Theorem 3.2. Let
Proof. Since Ω = ∅, we take x 0 ∈ Ω. Now, we consider the following family It is clear that M = ∅ because Ω ∈ M. Now, we consider the sets We claim that P = Q.
In fact, since for any A ∈ M, x 0 ∈ A, we have x 0 ∈ P and, consequently, P = ∅. Moreover, for any A ∈ M, From this, it follows that T (P ) ⊂ P . Since x 0 ∈ P , P is convex and closed (because it is intersection of convex and closed sets) and T (P ) ⊂ P , we infer For the other inclusion, since Q ⊂ P , we deduce and, since Q is closed, convex and x 0 ∈ Q, we have Q ∈ M. This gives us P ⊂ Q. This proves our claim. Next, we will prove that P is compact. We argue by contradiction.
Since P is bounded, closed, and convex, particularly P is arc-connected. If P is not compact, by Proposition 2.4 (a), φ d (P ) > 0. On the other hand, and, using induction, we get H n ⊂ P, for any n ∈ N.
Therefore, lim H = ∪ n∈N H n ⊂ P . By Lemma 2.6, since H is F-subinvariant set, lim H = ∪ n∈N H n is a F invariant set. Since P is compact, lim H ⊂ P and lim H is a closed contained in a compact, consequently, lim H is compact. Therefore, lim H gives us the desired result. Proof. Applying Lemma 2.7 to the relatively compact set lim H, that was defined in Theorem 3.3, we obtain that lim H is F-invariant. Moreover, in Theorem 3.3 we prove that lim H is a compact subset of Ω, then we obtain the desired result.
In the sequel, we present a Corollary of this result using the following class of functions Q given by Examples of functions belonging to Q are ϕ(t) = ln(1 + t), ϕ(t) = t t + 1 and ϕ(t) = arctan t.
Corollary 3.5. Let F = {f 1 , f 2 , . . . , f n } be a finite family of self-maps of a nonempty, bounded, closed and convex subset Ω of a real Banach space such that, for any nonempty and convex subset X of Ω, we have