Common Attractors for Generalized F -Iterated Function Systems in G -Metric Spaces

: In this paper, we study the generalized F -iterated function system in G -metric space. Several results of common attractors of generalized iterated function systems obtained by using generalized F -Hutchinson operators are also established. We prove that the triplet of F -Hutchinson operators defined for a finite number of general contractive mappings on a complete G -metric space is itself a generalized F -contraction mapping on a space of compact sets. We also present several examples in 2-D and 3-D for our results.

In his 1981 seminal work, Hutchinson [15] established mathematical foundations for iterated function systems (IFSs) and showed that the Hutchinson operator defined on R k has as its fixed point a bounded and closed subset of R k called an attractor of IFS [16,17].Several researchers have obtained useful results for iterated function systems (see [18,19] and references therein).Nazir, Silvestrov, and Abbas [20] established fractals by employing F-Hutchinson maps in the setup of metric space.Recently, Navascués [21] presented the approximation of fixed points and fractal functions by means of different iterative algorithms.Navascués et al. [22] established some useful results of the collage type for Reich mutual contractions in b-metric and strong b-metric spaces.Thangaraj et al. [23] constructed an iterated function system called Controlled Kannan Iterated Function System based on Kannan contraction maps in a controlled metric space and used it to develop a new kind of invariant set, known as a Controlled Kannan Attractor or Controlled Kannan Fractal.Recently, Nazir and Silvestrov [24] investigated a generalized iterated function system based on pair of self-mappings and obtained the common attractors of these maps in complete dislocated metric spaces, established the well-posedness of the attractor problems of rational contraction maps in the framework of dislocated metric spaces, and obtained the generalized collage theorem in dislocated metric spaces.
In this paper, we consider the triplet of generalized F-contractive operators and define generalized F-Hutchinson operators to obtain the common attractors in complete G-metric spaces.The contractive conditions are different from those in [24], and both dislocated metric spaces and G-metric spaces are independent to each other.We construct some new common attractor point results based on a generalized F-iterated function system in G-metric spaces.We define F-Hutchinson operators with a finite number of general F-contractive operators in the complete G-metric space and show that these operators are themselves general F-contractions.It is worth mentioning that we are obtaining these results without using any type of commuting conditions of selfmaps in non-symmetric G-metric space.At the end, we present several nontrivial examples of common attractors as a result of F-Hutchinson operators.
Mustafa and Sims [5] established the following notion of G-metric.

Definition 2 ([25]
).Let {y n } be a sequence in G-metric space (Z, G).Then, (a) {y n } ⊂ Z is G-convergent sequence if, for any ε > 0, there is a point y ∈ Z and a natural number N such that for all n, m ≥ N, G(y, y n , y m ) < ε; {y n } converges to y ∈ Z whenever G(y m , y n , y) → 0 as m, n → ∞ and {y n } is Cauchy whenever G(y m , y n , y l ) → 0 as m, n, l → ∞.

Remark 1 ([28]
).In G-metric space (Z, G), let H G : CB(Z) × CB(Z) × CB(Z) → [0, +∞) be a mapping defined as In G-metric space (Z, G), for P, Q, R, S, U , V ∈ C G (Z), the following are satisfied: Similarly, Hence, Wardowski [29] defined F-contraction maps for fixed point results as follows.Let F : R + → R be a continuous map satisfying the following conditions: (F 3 ) There exists θ ∈ (0, 1) such that lim We denote a set  as a collection of all F-contractions.Definition 4. In G-metric space (Z, G), a self-map h : Z → Z is called an F-contraction on Z if for all u, v, w ∈ Z, there exists F ∈  and τ > 0 such that τ + F(G(hu, hv, hw)) ≤ F(G(u, v, w)) whenever G(hu, hv, hw) > 0.
We discuss F-iterated function systems in G-metric space.First, we define generalized F-contractive operators as a preliminary result.Definition 5.In G-metric space (Z, G), let f, g, h : Z → Z be three self-mappings.A triplet (f, g, h) is called a generalized F-contraction mappings if for all u, v, w ∈ Z, there exists F ∈  and τ > 0 such that τ + F(G(fu, gv, hw)) ≤ F(G(u, v, w)) whenever G(fu, gv, hw) > 0.
Theorem 1.Consider G-metric space (Z, G) and let f, g, h : Z → Z be continuous maps.If the triplet of mappings (f, g, h) is a generalized F-contraction, then (i) the elements in C G (Z) are mapped to elements in C G (Z) under f, g and h; (ii) if for an arbitrary U ∈ C G (Z), the mappings f, h, g : C G (Z) → C G (Z) are defined as then, the triplet (f, g, h) is a generalized F-contraction on (C G (Z), H G ).
Proof.(i) Since f is a continuous and the image of a compact subset under a continuous mapping, f : for all u, v, w ∈ Z such that G(fu, gv, hw) > 0. Now, and hence, Consequently, there exists τ * > 0 such that Thus, the triplet (f, g, h) is a generalized F-contraction mappings on (C G (Z), H G ).
Proposition 2. In G-metric space (Z, G), suppose the mappings f k , g k , h k : Z → Z for k = 1, . . ., q are continuous and satisfy Proof.We give a proof by induction.If q = 1, then, the result is true trivially.For Hence, the result is true for q = 2. Suppose that for q = n, the result holds, that is, , and from Lemma 1 (iii), we have Hence, the result is true for q = n + 1.Thus, the triplet (Υ, Ψ, Φ) is also a generalized F-contraction on C G (Z). where . ., q be continuous maps, where each triplet (f k , g k , h k ) for k = 1, . . ., q is a generalized F-contraction, then {Z; Consequently, a generalized F-iterated function system in G-metric space is a finite collection of generalized F-contractions on Z. Definition 8. Let (Z, G) be a complete G-metric space and U ⊆ Z a non-empty compact set.Then, U is the common attractor of the mappings Υ, Ψ, Φ : where the limit is taken relative to the G-Hausdorff metric.

Main Results
Now, we establish the results of common attractors of generalized F-Hutchinson contraction in G-metric spaces.
Theorem 2. In a complete G-metric space (Z, G), let {Z; (f k , g k , h k ), k = 1, . . ., q} be the generalized F-iterated function system.Define Υ, Ψ, Φ : If the mappings (Υ, Ψ, Φ) are generalized F-Hutchinson contractive operators, then Υ, Ψ and Φ have a unique common attractor U * ∈ C G (Z), that is, Additionally, for any arbitrarily chosen initial set R 0 ∈ C G (Z), the sequence of compact sets converges to the common attractor U * .
We proceed by showing that Υ, Ψ, and Φ have a unique common attractor.
is an attractor of Υ and from the Proof above, U * is a common attractor for Υ, Ψ and Φ.The same is true for k = 3n + 1 or k = 3n + 2. We assume that R k ̸ = R k+1 for all k ∈ N, then by using (G 3 ), we have where Thus from (3), we have Similarly, one can show that Thus, for all k, Thus, Thus, On taking limit as k → ∞, we obtain As lim By the convergence of the series To prove that Υ(U * ) = U * , when assuming the contrary we have where Thus, (4) implies and taking the limit as k → +∞ yields which is a contradiction as τ > 0. Thus, Υ(U * ) = U * .Following the conclusion above, U * is the common attractor of Υ, Ψ, and Φ.
For uniqueness, we consider V as another common attractor of Υ, Ψ and Φ with H G (U * , V, V) > 0.Then, where Thus, (5) ) from which we conclude that H G (U * , V, V) = 0, and thus, U * = V.Hence, U * is a unique common attractor of Υ, Ψ, and Φ. Remark 2. In Theorem 2, take the collection S G (Z), of all singleton subsets of Z, then S G (Z) ⊆ C G (Z). Furthermore, if we take the mappings (f k , g k , h k ) = (f, g, h) for each k, where f = f 1 , g = g 1 and h = h 1 , then the operators (Υ, Ψ, Φ) become Thus, we obtain the following result on common fixed point.
Corollary 1.Let {Z; (f k , g k , h k ), k = 1, 2, . . ., q} be a generalized F-iterated function system in a complete G-metric space (Z, G) and define the maps f, g, h : Z → Z as in Remark 2. If there exists τ > 0 such that for v 1 , v 2 , v 3 ∈ Z having G(fv 1 , gv 2 , hv 3 ) > 0, the following holds Then, f, g, and h have a unique common fixed point u ∈ Z.Additionally, for an arbitrary element u 0 ∈ Z, the sequence {u 0 , fu 0 , gfu 0 , hgfu 0 , fhgfu 0 , • • • } converges to the common fixed point of f, g, and h.
Corollary 2. In a complete G-metric space (Z, G), let {Z; Then, there exists unique U * ∈ C G (Z) that satisfies Additionally, for any arbitrarily chosen initial set R 0 ∈ C G (Z), the sequence of compact sets converges to the common attractor U * .
Proof.From Theorem 2, we obtain that there exists unique U * ∈ C G (Z) that satisfy is also an attractor of Υ m .Fol- lowing the similar steps for those in Proof of Theorem 2, we obtain that * ) is also the common attractor of Υ m , Ψ m and Φ m .By the uniqueness of the common attractor, and G-metric on Z be defined as The maps f 1 , f 2 , g 1 , g 2 , h 1 , and h 2 are continuous and non commutative.Now, we show that for F ∈  and τ > 0, the mappings ; ; ; . Now, by taking F(λ) = ln(λ) for λ > 0, τ = ln( 20  19 ), and for u, v, w Again for u, v, w ∈ Z, we have ; ; ; holds.Thus, all the conditions of Theorem 2 are satisfied, and moreover, for any initial set R 0 ∈ C G (Y), the sequence {R 0 , Υ(R 0 ), ΨΥ(R 0 ), ΦΨΥ(R 0 ), ΥΦΨΥ(R 0 ), • • • } of compact sets is convergent and has a limit, the common attractor of Υ, Ψ, and Φ.
holds.Thus, all the conditions of Theorem 2 are satisfied, and moreover, for any initial set R 0 ∈ C G (Z 3 ), the sequence {R 0 , Υ(R 0 ), ΨΥ(R 0 ), ΦΨΥ(R 0 ), ΥΦΨΥ(R 0 ), • • • } of compact sets is convergent and has a limit, the common attractor of Υ, Ψ, and Φ (see Figure 4).The Figure 4 shows the convergence process of sequence steps at n = 2, 4, 6, and 8 in (a), (b), (c), and (d), respectively.The green points in the figures show the data points of convergence steps and the blue lines show the movements of data points in the different places.Interchanging the order of variables in maps yields a new form of common attractor of Υ, Ψ, and Φ (see Figure 5).The green points in the figures show the data points of convergence steps and the blue lines show the movements of data points in the different places.
a non empty subset of Z}.B(Z) = {W : U is a non empty bounded subset of Z}.CL(Z) = {U : U is a non empty closed subset of Z}.CB(Z) = {U : U is a non empty closed and bounded subset of Z}.

Figure 1
shows the convergence process of sequence steps at n = 2, 4, 6, and 8 in (a), (b), (c), and (d), respectively.The green points in the figures show the data points of convergence steps and the blue lines show the movements of data points in the different places.

Figure 2
shows the convergence process of sequence steps at n = 2, 4, 6, and 8 in (a), (b), (c), and (d), respectively.The green points in the figures show the data points of convergence steps and the blue lines show the movements of data points in the different places.

Figure 2 .
Figure 2. Iteration steps of the convergence to the common attractor of Υ, Ψ, and Φ.If we are interchanging the order of variables in maps, then we obtain a new form of common attractor of Υ, Ψ, and Φ, see for example in Figure3.The green points in the figures show the data points of convergence steps and the blue lines show the movements of data points in the different places.

Figure 4 .
Figure 4. Iteration steps to the convergence of the common attractor of Υ, Ψ, and Φ.

Figure 5 .
Figure 5. Iteration steps to the convergence of the common attractor of Υ, Ψ, and Φ.