Iterated function systems consisting of F-contractions

In this paper we consider a particular case of a contractive self-mapping on a complete metric space, namely the F-contraction introduced by Wardowski (Fixed Point Theory Appl. 87, 2012, doi:10.1186/1687-1812-2012-94), and provide some new properties of it. As an application, we investigate the iterated function systems (IFS) composed of F-contractions extending some fixed point results from the classical Hutchinson-Barnsley theory of IFS consisting of Banach contractions. Some illustrative examples are given. MSC:28A80, 47H10, 54E50.


Introduction
The fixed point theory is, without doubt, one of the most modern and dynamic areas of research, mainly because of its large range of applications. Fractal theory is very much connected to the fixed point theory, mainly because of the role that the Banach-Caccioppoli-Picard Principle plays. It states that if X is a complete metric space, and f is a mapping of X into itself which satisfies d f (x), f(y) ≤ ρd(x, y), for some ρ ∈ [0, 1) and all x, y ∈ X, then f has a fixed point x, and the successive approximations f n (x) converge to x for whatever x ∈ X.
Many generalizations of the notion of contraction operator were given, among them we notice the notion of ϕ-contraction (with ϕ a comparison function as it is later defined in the paper). ϕ-Contractions were studied much in the mathematical literature, among the papers in which they were treated we remark [1][2][3][4].
In case f L < ∞, we say that f is lipschitzian. In case f L < 1, we call f a contraction.
See [9] for the definition of the Monge-Kantorovich norm.

Theorem 3 (Tihonov Theorem). Let X be a locally convex space and
K ⊂ X a non empty compact convex set.
Then ∀f : K → K continuous admits a fixed point.
In (  The Fractal Set The Hutchinson operator for iterated function systems of ϕ-contractions In the sequel, the set Θ = {1, 2, . . . , N}, N ∈ N * , or Θ = N * or (Θ, K, W ) is a field over K.
Let ϕ be a comparison function and (ω θ ) θ∈Θ a family of ϕ-contractions. Let us define F : K(X) → K(X) by We consider K(X) endowed with the Hausdorff-Pompeiu metric d H . Then (K(X), d H ) is a compact metric space.
We observe that the set operator is correctly defined, as θ∈Θ ω θ (A) is a closed subset of the compact metric space (X, d), so θ∈Θ ω θ (A) is a non empty compact set. Theorem 5. F is a ϕ-function (ϕ-contraction). (see [10]) It will be sufficient to prove that We will prove first the inequality From the inequality In an analogous way, As a conclusion, According to the Boyd-Wong theorem, the application F has a unique fixed point A 0 ∈ K(X). (F (A 0 ) = A 0 , the attractor set).
The Fractal Measure I. We consider first the case when  II. Secondly, we consider the case when

Proposition 3. The Markov operator
and ω 1 , ω 2 , . . . , ω n , . . . : X → X are a CIFS of ϕ-contractions (countable iterated function system). Consider a set of probabilities ( and as M(X) is a compact convex subset of a normed space, M has a fixed point, the fractal measure.
Consider the Markov operator (the push-forward measure). Proof.

Theorem 7.
The Markov operator has a fixed point μ 0 (not necessarily unique, though).
Another proof of the existence of the attractor measure (case I.) We suppose that we have: For this N ∈ N, N ≥ 2, we define We have on C the lexicographic metric, given by: and k is the least index n ≥ 1 for which σ n = τ n . If 1 ≤ m ≤ N we can define the translation operators where m * σ := mσ 1 σ 2 · · · σ n · · · .
The operators s 1 , . . . , s N are similitudes with the similarity ratio 1 M , meaning The metric space (C, d C ) is a compact metric space, and so is K(C), d H . We consider As C = N m=1 s m (C), the attractor (the unique fixed point of S) is the total space C.

Then (M(C), d MK ) is a compact metric space.
A natural probability on {1, 2, . . . , N} that is associated to the set of numbers (p 1 , . . . , p N ) is obtained defining on "generators": If we take into account the fact that we can endow C with the product measure Supposing that we have to the IFSp F = (X, ω 1 , . . . , ω N , p 1 , . . . , p N ) we associate the operators: The set of ϕ-contractions is a set of uniformly Meir-Keeler functions (or S is a ϕ-contraction), so there is A 0 ∈ K(X) a unique attractor set, which satisfies the relation If σ ∈ C is a given code, and x ∈ X is an "initial" point, we emphasize our reasoning on the orbit ω σ1 • · · · • ω σ k (x) k≥1 from the complete metric space (X, d).
As we have: we infer that the limit does not depend on the intial point x ∈ X.
This way, it will be correctly defined We obtained this way an applicationφ : C → X.
As a consequence, We'll prove that A is the attractor of the set application S. For this, we should prove that We notice that Reciprocally, if x ∈ A, ∃ σ ∈ C such that x =φ(σ). As σ = σ 1 σ 2 · · · σ n · · · = s σ1 (σ 2 · · · σ k · · · ) = s σ1 (σ ), we have that and we have proved this way the inclusion

So, we have the equality
which implies that A is the attractor set of the IFS of ϕ-contractions (X, ω 1 , . . . , ω N ).
This way, if C is the attractor set for the IFS (C, s 1 , . . . , s N ), then the direct imageφ(C) will be the attractor set for the IFS of ϕ-contractions (X, ω 1 , . . . , ω N ).
As the direct image operator is linear, it means that which implies thatφ * (π) is an attractor measure for the IFS of ϕ-contractions considered. We haven't proved its uniqueness, though. The Markov Operator (a different approach) Let (X, d) be a compact metric space and (ω i ) i=1,N be a family of ϕcontractions.
For μ : B X → [0, 1] a Borel measure and f ∈ C(X), we define We also suppose we have ( We have: We obtain this way the operator in functions Obviously, B S is a linear operator and we have: As we can easily check that B S (1) = 1, we have that B S op = 1.
We also have the following properties: Proof. Let ε > 0. According to the definition of the ϕ-contractions (ϕ-functions), we have: we have that This way, Passing to the supremum with x, y ∈ X and d(x, y) ≤ ε, we obtain We have: [2] (ε) (f ).
This way, by induction, As a consequence, we can prove that the sequence sup X B [n] S (f ) n≥1 is monotonically decreasing and the sequence inf X B [n] S (f ) n≥1 is monotonically increasing.
We obviously have Then, both sequences are also bounded. Let We obviously have c ≥ c.
In fact, we will prove more, namely that c = c not = c. From the inequality As ϕ is a comparison function, ϕ [n] (δ) n≥1 is a monotonically decreasing sequence, convergent to 0 ⇒ is nondecreasing, and right-continuous at 0, and ω 0 (f ) = 0.

Iterated Function Systems Consisting of ϕ-Contractions 2217
Applying this inequality for the iterates Letting n tend to infinity, we have: As a consequence, there exists c : C(X) → R such that We will enumerate the main properties of the functional c.

Proof. (i) Let f, g ∈ C(X).
Then In a similar way, Replacing f with −f and g with −g, we'll have In a similar way, (ii) From this, we infer that c(0) = 0 (taking We have the positive homogeneity (and then the homogeneity as follows): Let a > 0 and f ∈ C(X). We have: But c(−f ) = −c(f ), and as a consequence Let f ∈ C(X). We have the inequality: S (f ), ∀n ≥ 1.
We have: Consider the following known lemma: Lemma 1. Let f n : X → R, n ≥ 1, a sequence of functions and (α n ) n , (β n ) n 2 sequences of real numbers such that  I  I  I  I  I  I  I  I  I  I  I ≤ B [n] , and the limit being unique, we have that From the properties i)-vi) and the Riesz representation theorem, ∃!ν ∈ M(X) such that X fdν = c(f ), ∀f ∈ C(X).
We have:  We also consider We also give We consider We have: which means that the series ∞ i=1 p i X f • ω i dν is absolutely convergent, and the operator is correctly defined. Analogously, We also notice that Let us justify the equality. We have For ε > 0 and f ∈ C(X), defining we have: Using induction, [n] (δ) (f ), and in a similar way as in the finite case, We can prove that the functional c : C(X) → R has the following properties: The uniqueness is similar to the finite case. The general case By now, we have worked in the following situations: We consider in the general case a field (Θ, Σ, W ), (X, d) a compact metric space and ω : X × Θ → X an application (B X ⊗ Σ, B X ) measurable.
We denote ω(x, θ) = ω θ (x). We suppose that all ω θ are ϕ-contractions. The application It means that the application This application is also bounded. Then, we can define Using the abbreviated notation we obtain the correct definition and we can easily check that B C(X) ⊂ C(X).
Similarly as in the preceding situations, Proof. Indeed, ε → ω ε (f ) being nondecreasing, if we translate the uniform continuity of f on X.