Approach of the value of a rent when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions Ratio Mathematica Volume 43, 2022 Some Results in Hausdorff Neutrosophic Metric Spaces on Hutchinson-Barnsley Operator Vinchu Balan Shakila* Maduraiveeran Jeyaraman† Abstract The main purpose of this paper is to prove the Neutrosophic contraction properties of the Hutchinson-Barnsley operator on the Neutrosophic hyperspace with respect to the Hausdorff Neutrosophic metrics. Also we discuss about the relationships between the Hausdorff Neutrosophic metrics on the Neutrosophic hyperspaces. Our theorems generalize and extend some recent results related with Hutchinson-Barnsley operator in the metric spaces to the Neutrosophic metric spaces. Keywords: Contraction, Hutchinson-Barnsley Operator, Metric Space, Hausdorff Neutrosophic Metric Spaces, Hyperspace. 2010 AMS subject classification: 03E72, 54E35, 54A40, 46S40‡ *Research Scholar, P.G. and Research Department of Mathematics, Raja Doraisingam Government Arts College, Sivagangai. Affiliated to Alagappa University, Karaikudi, Tamilnadu, India. *Department of Mathematics with CA (SF), Sourashtra College, Madurai, Tamilnadu, India. E-mail: shakilavb.math@gmail.com. † P.G. and Research Department of Mathematics, Raja Doraisingam Government Arts College, Sivagangai. Affiliated to Alagappa University, Karaikudi, Tamilnadu, India. E-mail: jeya.math@gmail.com, ORCID: https://orcid.org/0000-0002-0364-1845. ‡Received on May 16th, 2022. Accepted on June 28th, 2022. Published on June 30th, 2022. doi: 10.23755/rm.v41i0.782. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. mailto:jeya.math@gmail.com https://orcid.org/0000-0002-0364-1845 V. B. Shakila and M. Jeyaraman 1. Introduction The Fractal Analysis was introduced by Mandelbrot in 1975 [8] and popularized by various mathematicians [6], [3], [4]. Sets with non-integral Hausdorff dimension, which exceeds its topological dimension, are called Fractals by Mandelbrot [8]. Hutchinson [6] and Barnsley [3] initiated and developed the Hutchinson-Barnsley theory (HB theory) in order to define and construct the fractal as a compact invariant subset of a complete metric space generated by the Iterated Function System (IFS) of contractions. That is, Hutchinson[6] introduced an operator on hyperspace called as Hutchinson-Barnsley operator (HB operator) to define a fractal set as a unique fixed point by using the Banach Contraction Theorem in the metric spaces. Recently in [5], [15], HB operator properties were analyzed in fuzzy metric spaces. Here we introduce the concepts and properties of HB operator in the intuitionistic fuzzy metric spaces. Atanassov [2] introduced and studied the notion of intuitionistic fuzzy set by generalizing the notion of fuzzy set. Park [10] defined the notion of intuitionistic fuzzy metric space as a generalization of fuzzy metric space. In 1998, Smarandache [12,13] characterized the new concept called neutrosophic logic and neutrosophic set and explored many results in it. In the idea of neutrosophic sets, there is T degree of membership, I degree of indeterminacy and F degree of non-membership. Basset et al. [1] Explored the neutrosophic applications in different fields such as model for sustainable supply chain risk management, resource levelling problem in construction projects, Decision Making. In 2019, Kirisci et al [9] defined NMS as a generalization of IFMS and brings about fixed point theorems in complete NMS. Later Jeyaraman at el., [7, 11] proved Fixed point results in non-Archimedean generalized intuitionistic fuzzy metric spaces. In 2020, Sowndrarajan Jeyaraman and Florentin Smarandache [14] proved some fixed point results for contraction theorems in neutrosophic metric spaces. In this paper, we prove the neutrosophic contraction properties of the HB operator on the neutrosophic hyperspace with respect to the Hausdorff neutrosophic metrics. Also we discuss about the relationships between the Hausdorff neutrosophic metrics on the neutrosophic hyperspaces. Here our theorems generalize and extend some recent results related with Hutchinson-Barnsley operator in the metric spaces. 2.Preliminaries Definition 2.1. [3] Let (Σ, 𝑑) be a metric space and 𝒦0(Σ) be the collection of all non-empty compact subsets of Σ. http://fs.unm.edu/NSS2/index.php/111/article/view/753 Some Results in Hausdorff Neutrosophic Metric Spaces on Hutchinson-Barnsley Operator Define 𝑑(𝜁, 𝑄) ≔ 𝑖𝑛𝑓𝑦∈𝑄 𝑑(𝜁, 𝜂) and 𝑑(𝑃, 𝑄) ∶= 𝑠𝑢𝑝𝑥∈𝑃 𝑑(𝜁, 𝑄) for all 𝜁 ∈ Σ and 𝑃, 𝑄 ∈ 𝒦0(Σ). The Hausdorff metric or Hausdorff distance (𝐻𝑑) is a function (𝐻𝑑) ∶ 𝒦0(Σ)x 𝒦0(Σ) → ℝ defined by 𝐻𝑑(𝑃, 𝑄) = 𝑚𝑎𝑥{𝑑(𝑃, 𝑄), 𝑑(𝑄, 𝑃)}. Then 𝐻𝑑 is a metric on the hyperspace of compact sets 𝒦0(Σ) and hence (𝒦0(Σ), 𝐻𝑑) is called a Hausdorff metric space. Theorem 2.2. [3] If (Σ, 𝑑) is a complete metric space, then (𝒦0(Σ), 𝐻𝑑 ) is also a complete metric space. Definition 2.3. [3] Let (Σ, 𝑑) be a metric space and 𝑓𝑛 ∶ Σ → Σ, 𝑛 = 1,2, , … 𝑁0(𝑁0 ∈ ℕ) be 𝑁0 - contraction mappings with the corresponding contractivity ratios 𝑘𝑛 , 𝑛 = 1,2, … 𝑁0. The system {Σ; 𝑓𝑛, 𝑛 = 1,2, … 𝑁0} is called an Iterated Function System (IFS) or Hyperbolic Iterated Function System with the ratio 𝑘 = 𝑚𝑎𝑥𝑛=1 𝑁0 𝑘𝑛. Then the Hutchinson Barnsley Operator (HBO) of the IFS is a function 𝐹 ∶ 𝒦0(Σ) → 𝒦0(Σ) defined by 𝐹(𝑄) = ⋃ 𝑓𝑛 (𝑄), 𝑁0 𝑛=1 for all 𝑄 ∈ 𝒦0(Σ). Theorem 2.4. [3] Let (Σ, 𝑑) be a metric space. Let {Σ; 𝑓𝑛 , 𝑛 = 1,2, … 𝑁0; 𝑁0 ∈ ℕ } be an IFS. Then, the HBO (F) is a contraction mapping on (𝒦0(Σ), 𝐻𝑑 ). Theorem 2.5. [3] Let (Σ, 𝑑) be a complete metric space and {Σ; 𝑓𝑛 , 𝑛 = 1,2,3 … 𝑁0; 𝑁0 ∈ ℕ } be an IFS. Then, there exists only one compact invariant set 𝑃∞ ∈ 𝒦0(Σ) of the HBO (F) or equivalently, F has a unique fixed point namely 𝑃∞ ∈ 𝒦0(Σ). Definition 2.6. [3] The fixed point 𝑃∞ ∈ 𝒦0(Σ) of the HBO F described in the Theorem (2.5) is called the Attractor (Fractal) of the IFS. Sometimes 𝑃∞ ∈ 𝒦0(Σ) is called as Fractal generated by the IFS and so called as IFS Fractal. Definition 2.7. A binary operation ∗ ∶ [0,1] x [0,1] → [0,1] is a continuous t- norm, if * satisfies the following conditions: (a) * is commutative and associative; (b) * is continuous (c) 𝑎 ∗ 1 = 𝑎 for all 𝑎 ∈ [0,1]; (d) 𝑎 ∗ 𝑏 ≤ 𝑐 ∗ 𝑑 whenever 𝑎 ≤ 𝑐 and 𝑏 ≤ 𝑑, and 𝑎, 𝑏, 𝑐, 𝑑 ∈ [0,1]. V. B. Shakila and M. Jeyaraman Definition 2.8. A binary operation ⋄ ∶ [0,1] x [0,1] → [0,1] is a continuous t- norm, if ⋄ satisfies the following conditions: (a) ⋄ is commutative and associative; (b) ⋄ is continuous (c) 𝑎 ⋄ 0 = 𝑎 for all 𝑎 ∈ [0,1]; (d) 𝑎 ⋄ 𝑏 ≤ 𝑐 ⋄ 𝑑 whenever 𝑎 ≤ 𝑐 and 𝑏 ≤ 𝑑, and 𝑎, 𝑏, 𝑐, 𝑑 ∈ [0,1]. Definition 2.9. A 6-tuple (Σ, Ξ, Θ, Υ,∗,⋄) is said to be an Neutrosophic Metric Space (shortly NMS), if Σ is an arbitrary set, ∗ is a neutrosophic CTN, ⋄ is a neutrosophic CTC and Ξ, Θ 𝑎𝑛𝑑 Υ are neutrosophic on Σ x Σ satisfying the following conditions: For all 𝜁, 𝜂, 𝛿,𝜔 ∈ Σ, 𝜆, 𝜇 ∈ 𝑅+. (i) 0 ≤ Ξ ( 𝜁, 𝜂, 𝜆) ≤ 1; 0 ≤ Θ ( 𝜁, 𝜂, 𝜆) ≤ 1; 0 ≤ Υ ( 𝜁, 𝜂, 𝜆) ≤ 1; (ii) Ξ ( 𝜁, 𝜂, 𝜆) + Θ ( 𝜁, 𝜂, 𝜆) + Υ ( 𝜁, 𝜂, 𝜆) ≤ 3; (iii) Ξ ( 𝜁, 𝜂, 𝜆) = 1 if and only if 𝜁 = 𝜂; (iv) Ξ ( 𝜁, 𝜂, 𝜆) = Ξ ( 𝜂, 𝜁, 𝜆) for 𝜆 > 0 (v) Ξ ( 𝜁, 𝜂, 𝜆)∗ Ξ ( 𝜂, 𝛿, 𝜇) ≤ Ξ ( 𝜁, 𝛿, 𝜆 + 𝜇), for all 𝜆 , 𝜇 > 0; (vi) Ξ ( 𝜁, 𝜂, .) : [0, ∞) → [0 , 1] is neutrosophic continuous ; (vii) lim 𝜆→∞ Ξ ( 𝜁, 𝜂, 𝜆) = 1 for all 𝜆 > 0; (viii) Θ ( 𝜁, 𝜂, 𝜆) = 0 if and only if 𝜁 = 𝜂; (ix) Θ ( 𝜁, 𝜂, 𝜆) = Θ ( 𝜂, 𝜁, 𝜆) for 𝜆 > 0; (x) Θ ( 𝜁, 𝜂, 𝜆) ⋄ Θ ( 𝜂, 𝛿, 𝜇) ≥ Θ ( 𝜁, 𝛿, 𝜆 + 𝜇), for all 𝜆 , 𝜇 > 0; (xi) Θ ( 𝜁, 𝜂, .) : [0, ∞) → [0,1] is neutrosophic continuous; (xii) lim 𝜆→∞ Θ ( 𝜁, 𝜂, 𝜆) = 0 for all 𝜆 > 0; (xiii) Υ ( 𝜁, 𝜂, 𝜆) = 0 if and only if 𝜁 = 𝜂 ; (xiv) Υ ( 𝜁, 𝜂, 𝜆) = Υ ( 𝜂, 𝜁, 𝜆) for 𝜆 > 0; (xv) Υ ( 𝜁, 𝜂, 𝜆) ⋄ Υ ( 𝜂, 𝛿, 𝜇) ≥ Υ ( 𝜁, 𝛿, 𝜆 + 𝜇), for all 𝜆 , 𝜇> 0; (xvi) Υ( 𝜁, 𝜂, .) : [0, ∞) → [0,1] is neutrosophic continuous; (xvii) lim 𝜆→∞ Υ ( 𝜁, 𝜂, 𝜆) = 0 for all 𝜆 > 0; Then, (𝛯, 𝛩, 𝛶) is called an NMS on 𝛴. The functions 𝛯, 𝛩 𝑎𝑛𝑑 𝛶denote degree of closedness, neturalness and non-closedness between 𝜁 𝑎𝑛𝑑 𝜂 with respect to 𝜆 respectively. Example 2.10. Let (𝛴, 𝑑) be a metric space. Let 𝛯𝑑, 𝛩𝑑 and 𝛶𝑑 be the functions defined on 𝛴2 x (0, ∞) by 𝛯𝑑 (𝜁, 𝜂, 𝜆) = 𝜆 𝜆+𝑑(𝜁,𝜂) , 𝛩𝑑(𝜁, 𝜂, 𝜆) = 𝑑(𝜁,𝜂) 𝜆+𝑑(𝜁,𝜂) and 𝛶𝑑 (𝜁, 𝜂, 𝜆) = 𝑑(𝜁,𝜂) 𝜆 , for all 𝜁, 𝜂 ∈ 𝛴 and 𝜆 > 0. Then (𝛴, 𝛯𝑑 , 𝛩𝑑, 𝛶𝑑 ,∗,⋄) is a Some Results in Hausdorff Neutrosophic Metric Spaces on Hutchinson-Barnsley Operator NMS which is called standard NMS, and (𝛯𝑑 , 𝛩𝑑, 𝛶𝑑 ) is called as standard NM induced by the metric d. Definition 2.11. Let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄)be a NMS. The open ball 𝐵(𝜁, 𝑟, 𝜆) with 𝜁 ∈ 𝛴 and radius r, 0 < 𝑟 < 1, with respect to 𝜆 > 0 is defined as 𝐵(𝜁, 𝑟, 𝜆) = {𝜂 ∈ 𝛴 ∶ 𝛯 ( 𝜁, 𝜂, 𝜆) > 1 − 𝑟, 𝛩 ( 𝜁, 𝜂, 𝜆) < 𝑟 𝑎𝑛𝑑 𝛶 ( 𝜁, 𝜂, 𝜆) < 𝑟 }. Define 𝜏(𝛯,𝛩,𝛶) = { 𝑃 ⊂ 𝛴 ∶ 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝜁 ∈ 𝑃, ∃ 𝜆 > 0 𝑎𝑛𝑑 𝑟 ∈ (0,1) 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝐵(𝜁, 𝑟, 𝜆) ⊂ 𝑃 }. Then 𝜏(𝛯,𝛩,𝛶) is a topology on 𝛴 induced by a NFM (𝛯, 𝛩, 𝛶). The topologies induced by the metric and the corresponding standard NM are the same. Proposition 2.12. The metric space (𝛴, 𝑑) is complete if and only if the standard NMS (𝛴, 𝛯𝑑 , 𝛩𝑑, 𝛶𝑑 ,∗,⋄)is complete. Definition 2.13. A neutrosophic fuzzy B-contraction (neutrosophic fuzzy Sehgal contraction) on an NMS (Σ, Ξ, Θ, Υ,∗,⋄) is a self –mapping f on Σ for which Ξ(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≥ Ξ(𝜁, 𝜂, λ), Θ(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≤ Θ(𝜁, 𝜂, λ) and Υ(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≤ Υ(𝜁, 𝜂, λ) for all 𝜁, 𝜂 ∈ Σ and λ > 0, where k is a fixed constant in (0,1). 3. Hausdorff Neutrosophic Metric Spaces Definition 3.1. Let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄)be a NMS and 𝜏(Ξ,Θ,Υ) be the topology induced by the NM (Ξ, Θ, 𝛶).We shall denote by 𝒦0(𝑋), the set of all non-empty compact subsets of (𝛴, 𝜏(Ξ,Θ,𝛶)). Define Ξ(𝜁, 𝑄, λ) ∶= sup 𝜂 ∈ 𝑄 Ξ(𝜁, 𝜂, λ) , Ξ(𝑃, 𝑄, λ) ∶= inf 𝜁 ∈ 𝑃 Ξ(𝜁, 𝑄, λ), 𝛩(𝜁, 𝑄, λ) ∶= inf 𝜂 ∈ 𝑄 𝛩(𝜁, 𝜂, λ) , 𝛩(𝑃, 𝑄, λ) ∶= sup 𝜁 ∈ 𝑃 𝛩 (𝜁, 𝑄, λ) and 𝛶(𝜁, 𝑄, λ) ∶= inf 𝜂 ∈ 𝑄 𝛶(𝜁, 𝜂, λ) , 𝛶(𝑃, 𝑄, λ) ∶= sup 𝜁 ∈ 𝑃 𝛩(𝜁, 𝑄, λ), for all 𝜁 ∈ 𝛴 and 𝑃, 𝑄 ∈ 𝒦0(𝑋). Then, we define the Hausdorff NM (𝐻Ξ, 𝐻𝛩 , 𝐻𝛶 ,∗,⋄) as 𝐻Ξ(𝑃, 𝑄, λ) = 𝑚𝑖𝑛{Ξ(𝑃, 𝑄, λ), Ξ(𝑄, 𝑃, λ)}, V. B. Shakila and M. Jeyaraman 𝐻𝛩 (𝑃, 𝑄, λ) = 𝑚𝑎𝑥{𝛩(𝑃, 𝑄, λ), 𝛩(𝑄, 𝑃, λ)}and 𝐻𝛶 (𝑃, 𝑄, λ) = 𝑚𝑎𝑥{𝛶(𝑃, 𝑄, λ), 𝛶(𝑄, 𝑃, λ)}. Here (𝐻Ξ, 𝐻𝛩 , 𝐻𝛶 ) is a NM on the hyperspace of compact sets, 𝒦0(Σ) and hence (𝒦0(Σ), 𝐻Ξ, 𝐻𝛩 , 𝐻𝛶 ,∗,⋄) is called a Hausdorff NMS. Proposition 3.2. Let (Σ, 𝑑) be a metric space. Then the Hausdorff NM (𝐻Ξ𝑑 , 𝐻𝛩 𝑑, 𝐻𝛶 𝑑) of the standard NM (Ξ𝑑, 𝛩𝑑, 𝛶𝑑 ) coincides with the standard NM (𝐻Ξ𝑑 , 𝐻𝛩 𝑑, 𝐻𝛶 𝑑) of the Hausdorff metric (𝐻𝑑) on 𝒦0(Σ). ie., 𝐻Ξ𝑑 (𝑃, 𝑄, λ) = ΞH 𝑑(𝑃, 𝑄, λ), 𝐻𝛩 𝑑 (𝑃, 𝑄, λ) = 𝛩H𝑑 (𝑃, 𝑄, λ) and 𝐻𝛶 𝑑 (𝑃, 𝑄, λ) = 𝛶H 𝑑(𝑃, 𝑄, λ), for all 𝑃, 𝑄 ∈ 𝒦0(Σ) and λ > 0. Proof: Fix λ > 0 and 𝑃, 𝑄 ∈ 𝒦0(Σ).We recall that sup 𝛽 ∈ 𝑄 Ξ𝑑 (𝛼, 𝛽, λ) = λ λ+ inf 𝛽∈𝑄 𝑑(𝛼,𝛽) , inf 𝛽 ∈ 𝑄 𝛩𝑑(𝛼, 𝛽, λ) = 1 1+ λ inf 𝛽∈𝑄 𝑑(𝛼,𝛽) and inf 𝛽 ∈ 𝑄 𝛶𝑑 (𝛼, 𝛽, λ) = 1 λ inf 𝛽∈𝑄 𝑑(𝛼,𝛽) , for all 𝛼 ∈ 𝑃.It follows that Ξ𝑑 (𝛼, 𝑄, λ) = λ λ+d(𝛼,𝑄) ,𝛩𝑑(𝛼, 𝑄, λ) = 1 1+ λ d(𝛼,𝑄) and 𝛶𝑑 (𝛼, 𝑄, λ) = 1 λ d(𝛼,𝑄) for all 𝛼 ∈ 𝑃. Then inf 𝛼 ∈ 𝑃 Ξ𝑑 (𝛼, 𝑄, λ) = λ λ+ sup 𝛼∈ 𝑃 d(𝛼,𝑄) , sup 𝛼 ∈ 𝑃 𝛩𝑑 (𝛼, 𝑄, λ) = 1 1+ sup 𝛼∈ 𝑃 λ d(𝛼,𝑄) and sup 𝛼 ∈ 𝑃 𝛶𝑑 (𝛼, 𝑄, λ) = 1 sup 𝛼∈𝑃 λ d(𝛼,𝑄) . It follows that Ξ𝑑 (𝑃, 𝑄, λ) = λ λ+d(𝑃,𝑄) , 𝛩𝑑(𝑃, 𝑄, λ) = 1 1+ λ d(𝑃,𝑄) = d(𝑃,𝑄) λ+d(𝑃,𝑄) and 𝛶𝑑 (𝑃, 𝑄, λ) = 1 λ d(𝑃,𝑄) = d(𝑃,𝑄) λ . Similarly, we obtain Some Results in Hausdorff Neutrosophic Metric Spaces on Hutchinson-Barnsley Operator Ξ𝑑 (𝑄, 𝑃, λ) = λ λ+d(𝑄,𝑃) , 𝛩𝑑(𝑄, 𝑃, λ) = d(𝑄,𝑃) λ+d(𝑄,𝑃) and 𝛶𝑑 (𝑄, 𝑃, λ) = d(𝑄,𝑃) λ . Therefore, 𝐻Ξ𝑑 (𝑃, 𝑄, λ) = ΞH𝑑 (𝑃, 𝑄, λ), 𝐻𝛩 𝑑 (𝑃, 𝑄, λ) = 𝛩H𝑑 (𝑃, 𝑄, λ) and 𝐻𝛶 𝑑(𝑃, 𝑄, λ) = 𝛶H𝑑 (𝑃, 𝑄, λ). The proof is complete. Using the Proposition 3.2., we can easily compute distances with respect to the Hausdorff NM (𝐻Ξ𝑑, 𝐻𝛩 𝑑 , 𝐻𝛶 𝑑) of the standard NM (Ξ𝑑, 𝛩𝑑, 𝛶𝑑 ) by computing distances with respect to the Hausdorff metric (𝐻𝑑) implied by the metric d. Here, we illustrate this situation with two examples. Example 3.3. Let (ℝ, 𝑑) be the Euclidean metric space and 𝑃 = [𝛼1, 𝛼2] and 𝑄 = [𝛽1, 𝛽2] be two compact intervals of ℝ. Then 𝑑(𝑃, 𝑄) = |𝛼1 − 𝛽1| and 𝑑(𝑄, 𝑃) = |𝛼2 − 𝛽2|and hence 𝐻𝑑 (𝑃, 𝑄) = 𝑚𝑎𝑥{|𝛼1 − 𝛽1|, |𝛼2 − 𝛽2|}; So, by Proposition (3.2), We have Ξ𝑑 (𝑃, 𝑄, λ) = λ λ+𝑚𝑎𝑥{|𝛼1−𝛽1|,|𝛼2−𝛽2|} , 𝛩𝑑(𝑃, 𝑄, λ) = 𝑚𝑎𝑥{|𝛼1−𝛽1|,|𝛼2−𝛽2|} λ+𝑚𝑎𝑥{|𝛼1−𝛽1|,|𝛼2−𝛽2|} and 𝛶𝑑 (𝑃, 𝑄, λ) = 𝑚𝑎𝑥{|𝛼1−𝛽1|,|𝛼2−𝛽2|} λ , for all λ > 0. Example 3.4. Let (Σ, 𝑑) be the discrete metric space such that |Σ| ≥ 2. Let 𝑃 and 𝑄 be two non-empty finite subsets of Σ, with 𝑃 ≠ 𝑄. Then 𝑑(𝑃, 𝑄) = 1 = 𝑑(𝑄, 𝑃) and hence 𝐻𝑑(𝑃, 𝑄) = 1; so by Proposition 3.2., we have 𝐻Ξ𝑑 (𝑃, 𝑄, λ) = λ λ+1 ,𝐻𝛩 𝑑 = 1 1+λ and 𝐻𝛶 𝑑 = 1 λ , for all λ > 0. Definition 3.5. Let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄)be an NMS and 𝜏(𝛯,𝛩,𝛶) be the topology induced by (Ξ, Θ, 𝛶). We observe that (𝒦0(𝒦0(Σ)), 𝐻𝐻𝛯 , ℋ𝐻𝛩 , ℋ𝐻𝛶 ,∗,⋄) is also an NMS, where 𝒦0(𝒦0(Σ)) is the hyperspace of all non-empty compact subsets of (𝒦0(Σ), 𝐻Ξ, 𝐻𝛩 , 𝐻𝛶 ,∗,⋄) and (ℋ𝐻𝛯 , ℋ𝐻𝛩 , ℋ𝐻𝛶 ) is the Hausdorff NM on 𝒦0(𝒦0(Σ)) implied by the Hausdorff NM (Ξ𝑑 , 𝛩𝑑, 𝛶𝑑 ) on 𝒦0(Σ). That is, for all 𝑃 ∈ 𝒦0(Σ) and 𝔓, 𝔔 ∈ 𝒦0(𝒦0(Σ)), 𝐻𝐻𝛯 (𝔓, 𝔔) = 𝑚𝑖𝑛{𝐻Ξ𝔓, 𝔔, 𝐻Ξ(𝔔, 𝔓)},𝐻𝐻𝛩 (𝔓, 𝔔) = 𝑚𝑎𝑥{𝐻𝛩 (𝔓, 𝔔), 𝐻𝛩 (𝔔, 𝔓)} and𝐻𝐻𝛶 (𝔓, 𝔔) = 𝑚𝑎𝑥{𝐻𝛶 (𝔓, 𝔔), 𝐻𝛶 (𝔔, 𝔓)} where V. B. Shakila and M. Jeyaraman 𝐻Ξ(𝔓, 𝔔) ≔ inf 𝑝 ∈ 𝔓 𝐻Ξ(𝑃, 𝔔), 𝐻Ξ(𝑃, 𝔔) ≔ sup 𝑄 ∈ 𝔔 𝐻Ξ(𝑃, 𝑄), 𝐻𝛩 (𝔓, 𝔔) ≔ sup 𝑝 ∈ 𝔓 𝐻𝛩 (𝑃, 𝔔), 𝐻𝛩 (𝑃, 𝔔) ≔ inf 𝑄 ∈ 𝔔 𝐻𝛩 (𝑃, 𝑄) and 𝐻𝛶 (𝔓, 𝔔) ≔ sup 𝑝 ∈ 𝔓 𝐻𝛶 (𝑃, 𝔔), 𝐻𝛶 (𝑃, 𝔔) ≔ inf 𝑄 ∈ 𝔔 𝐻𝛶 (𝑃, 𝑄). Proposition 3.6. Let (Σ, 𝑑) be a metric space and let (𝒦0(Σ), 𝐻d) and (𝒦0(𝒦0(Σ)), ℋ𝐻𝑑 ) be the corresponding Hausdorff metric spaces. Then, the Hausdorff NM (ℋ𝛯𝐻𝑑 , ℋ𝛩𝐻𝑑 , ℋ𝛶𝐻𝑑 ) of the standard NM(𝛯𝐻𝑑 , 𝛩𝐻𝑑 , 𝛶𝐻𝑑 ) coincides with the standard NM (𝛯ℋ𝐻𝑑 , 𝛩ℋ𝐻𝑑 , 𝛶ℋ𝐻𝑑 ) of the Hausdorff metric (ℋ𝐻𝑑 ) on 𝒦0(𝒦0(Σ)), ie. ℋ𝛯𝐻𝑑 (𝔓, 𝔔, λ) = 𝛯ℋ𝐻𝑑 (𝔓, 𝔔, λ), ℋ𝛩𝐻𝑑 (𝔓, 𝔔, λ) = 𝛩ℋ𝐻𝑑 (𝔓, 𝔔, λ) and ℋ𝛶𝐻𝑑 (𝔓, 𝔔, λ) = 𝛶ℋ𝐻𝑑 (𝔓, 𝔔, λ) for all 𝔓, 𝔔 ∈ 𝒦0(𝒦0(Σ)) and λ > 0. Proof: Proposition 3.2. completes the proof. 4. Neutrosophic Hutchinson-Barnsley Operator In this section, we define the Neutrosophic Iterated Function System (NIFS) and Neutrosophic HB Operator on the NMS. Definition 4.1. Let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄) be an NMS and 𝑓𝑛 : 𝛴 → 𝛴, 𝑛 = 1,2,3 … 𝑁0(𝑁0 ∈ ℕ) be 𝑁0 - neutrosophic B-contractions. Then the system {𝛴; 𝑓𝑛 , 𝑛 = 1,2,3 … 𝑁0} is called a NIFS of neutrosophic B-contractions in the NMS (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄). Definition 4.2. Let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄)be a NMS. Let {𝛴; 𝑓𝑛 , 𝑛 = 1,2,3 … 𝑁0} be an NIFS of neutrosophic B-contractions. Then the Neutrosophic Hutchinson-Barnsley Operator (NHBO) of the NIFS is a function 𝐹 ∶ 𝒦0(Σ) → 𝒦0(Σ) defined by 𝐹(𝑄) = ⋃ 𝑓𝑛 (𝑄) 𝑁0 𝑛=1 , for all 𝑄 ∈ 𝒦0(Σ). Definition 4.3. Let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄)be a complete NMS. Let 𝑓𝑛 : 𝛴 → 𝛴, 𝑛 = 1,2,3 … 𝑁0(𝑁0 ∈ ℕ) be a NIFS of neutrosophic B-contractions and F be the NHBO of the NIFS. We say that the set 𝑃∞ ∈ 𝒦0(Σ) is Neutrosophic Attractor (Neutrosophic Fractal) of the given NIFS, if 𝑃∞ is a unique fixed point of the NHBO F. Such 𝑃∞ ∈ 𝒦0(Σ)is also called as Fractal generated by the NIFS and so called NIFS Fractal of neutrosophic B-contractions. Some Results in Hausdorff Neutrosophic Metric Spaces on Hutchinson-Barnsley Operator Properties of NHBO Now, we prove the interesting results about the neutrosophic B-contraction properties of operators with respect to the Hausdorff neutrosophic metric on 𝒦0(Σ). Theorem 4.4. Let (Σ, 𝑑) be a metric space. Let 𝑓: 𝛴 → 𝛴 be a contraction function on (Σ, 𝑑), with a contractivity ratio k. Then H𝛯𝒅 (𝑓(𝑃), 𝑓(𝑄), λ) ≥ H𝛯𝒅 (P, Q, λ), H𝛩𝒅 (𝑓(𝑃), 𝑓(𝑄), λ) ≤ H𝛩𝒅 (P, Q, λ)and H𝛶𝒅 (𝑓(𝑃), 𝑓(𝑄), λ) ≤ H𝛶𝒅 (P, Q, λ), for all 𝑃, 𝑄 ∈ 𝒦0(Σ) and λ > 0. Proof: Fix λ > 0 and let 𝑃, 𝑄 ∈ 𝒦0(Σ). Since f is contraction on (Σ, 𝑑) with the contractivity ratio 𝑘 ∈ (0,1) and by Theorem 2.4. for the case 𝛩 = 1, we have 𝐻𝑑 (𝑓(𝑃), 𝑓(𝑄)) ≤ 𝑘𝐻𝑑 (𝑃, 𝑄). Since λ > 0 and 𝑘 ∈ (0,1), 𝑘λ 𝑘λ+𝐻𝑑(𝑓(𝑃),𝑓(𝑄)) ≥ 𝑘λ 𝑘λ+k𝐻𝑑(𝑃,𝑄) = λ λ+𝐻𝑑(𝑃,𝑄) , 𝐻𝑑(𝑓(𝑃),𝑓(𝑄)) 𝑘λ+ 𝐻𝑑(𝑓(𝑃),𝑓(𝑄)) ≤ 𝑘𝐻𝑑(𝑃,𝑄) 𝑘λ+𝑘𝐻𝑑(𝑃,𝑄) = 𝐻𝑑(𝑃,𝑄) λ+𝐻𝑑(𝑃,𝑄) and 𝐻𝑑(𝑓(𝑃),𝑓(𝑄)) 𝑘λ ≤ 𝑘𝐻𝑑(𝑃,𝑄) 𝑘λ = 𝐻𝑑(𝑃,𝑄) λ . By using the above inequalities and the Proposition 3.2., we have H𝛯𝒅 (𝑓(𝑃), 𝑓(𝑄), kλ) = 𝛯𝐻𝑑 (𝑓(𝑃), 𝑓(𝑄), kλ) = kλ kλ + 𝐻𝑑 (𝑓(𝑃), 𝑓(𝑄)) ≥ λ λ + 𝐻𝑑 (𝑃, 𝑄) = 𝛯𝐻𝑑 (𝑃, 𝑄, λ) = H𝛯𝒅 (𝑃, 𝑄, λ), H𝛩𝒅 (𝑓(𝑃), 𝑓(𝑄), kλ) = 𝛩𝐻𝑑 (𝑓(𝑃), 𝑓(𝑄), kλ) = 𝑘𝐻𝑑(𝑓(𝑃), 𝑓(𝑄)) kλ + 𝐻𝑑(𝑓(𝑃), 𝑓(𝑄)) ≤ 𝐻𝑑(𝑃, 𝑄) λ + 𝐻𝑑(𝑃, 𝑄) = 𝛩𝐻𝑑 (𝑃, 𝑄, λ) = H𝛩𝒅 (𝑃, 𝑄, λ) and Similarly, H𝛶𝒅 (𝑓(𝑃), 𝑓(𝑄), kλ) = 𝛶𝐻𝑑 (𝑓(𝑃), 𝑓(𝑄), kλ) = 𝑘𝐻𝑑 (𝑓(𝑃), 𝑓(𝑄)) kλ + 𝐻𝑑 (𝑓(𝑃), 𝑓(𝑄)) ≤ 𝐻𝑑 (𝑃, 𝑄) λ + 𝐻𝑑 (𝑃, 𝑄) = 𝛶𝐻𝑑 (𝑃, 𝑄, λ) = H𝛶𝒅 (𝑃, 𝑄, λ). The above theorem 4.4. shows that f is a neutrosophic B-contraction on 𝒦0(Σ) with respect to the Hausdorff neutrosophic metric (H𝛯𝒅 , H𝛩𝒅 , H𝛶𝒅 ) implied by the standard metric (𝛯𝑑 , 𝛩𝑑, 𝛶𝑑 ), if f is contraction on a metric space (Σ, 𝑑). The following theorem is somewhat generalization of the Theorem 4.4. V. B. Shakila and M. Jeyaraman Theorem 4.5. Let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄) be a NMS. Let (𝒦0(Σ), 𝐻𝛯 , 𝐻𝛩 , 𝐻𝛶 ,∗,⋄) be the corresponding Hausdorff NMS. Suppose 𝑓: 𝛴 → 𝛴 be a neutrosophic B-Contraction on (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄). Then for 𝑘 ∈ (0,1), 𝐻𝛯 (𝑓(𝑃), 𝑓(𝑄), 𝑘λ) ≥ 𝐻𝛯 (𝑃, 𝑄. λ), 𝐻𝛩 (𝑓(𝑃), 𝑓(𝑄), 𝑘λ) ≤ 𝐻𝛩 (𝑃, 𝑄. λ) and 𝐻𝛶 (𝑓(𝑃), 𝑓(𝑄), 𝑘λ) ≤ 𝐻𝛶 (𝑃, 𝑄. λ) for all 𝑃, 𝑄 ∈ 𝒦0(Σ) and λ > 0. Proof: Fix λ > 0. Let 𝑃, 𝑄 ∈ 𝒦0(Σ). For given 𝑘 ∈ (0,1), we get 𝛯(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≥ Ξ(𝜁, 𝜂, λ), for all 𝜁, 𝜂 ∈ 𝛴, 𝛯(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≥ Ξ(𝜁, 𝜂, λ), for all 𝜁 ∈ 𝑃 𝑎𝑛𝑑 𝜂 ∈ 𝑄, sup 𝜂 ∈ 𝑄 𝛯 (𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≥ sup 𝑦 ∈ 𝑄Ξ (𝜁, 𝜂, λ), for all 𝜁 ∈ 𝑃, 𝛯(𝑓(𝜁), 𝑓(𝑄), 𝑘λ) ≥ Ξ(𝜁, 𝑄, λ),for all 𝜁 ∈ 𝑃, inf 𝜁 ∈ 𝑃 𝛯(𝑓(𝜁), 𝑓(𝑄), 𝑘λ) ≥ inf 𝜁 ∈ 𝑃 Ξ(𝜁, 𝑄, λ), 𝛯(𝑓(𝑃), 𝑓(𝑄), 𝑘λ) ≥ Ξ(𝑃, 𝑄, λ). Similarly 𝛯(𝑓(𝑄), 𝑓(𝜁), 𝑘λ) ≥ Ξ(𝑄, 𝑃, λ). Hence 𝐻𝛯 (𝑓(𝑃), 𝑓(𝑄), 𝑘λ) ≥ 𝐻𝛯 (𝑃, 𝑄. λ) Now, 𝛩(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≤ 𝛩(𝜁, 𝜂, λ), for all 𝜁, 𝜂 ∈ 𝛴, 𝛩(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≤ 𝛩(𝜁, 𝜂, λ), for all 𝜁 ∈ 𝑃 𝑎𝑛𝑑 𝜂 ∈ 𝑄, inf 𝜂 ∈ 𝑄 𝛩(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≤ inf 𝑦 ∈ 𝑄 𝛩(𝜁, 𝜂, λ), for all 𝜁 ∈ 𝑃, 𝛩(𝑓(𝜁), 𝑓(𝑄), 𝑘λ) ≤ 𝛩(𝜁, 𝑄, λ), for all 𝜁 ∈ 𝑃, sup 𝜁 ∈ 𝑃 𝛩 (𝑓(𝜁), 𝑓(𝑄), 𝑘λ) ≤ sup 𝜁 ∈ 𝑃 𝛩 (𝜁, 𝑄, λ). 𝛩(𝑓(𝑃), 𝑓(𝑄), 𝑘λ) ≤ 𝛩(𝑃, 𝑄, λ). Similarly 𝛩(𝑓(𝑄), 𝑓(𝜁), 𝑘λ) ≤ 𝛩(𝑄, 𝑃, λ). Hence 𝐻𝛩 (𝑓(𝑃), 𝑓(𝑄), 𝑘λ) ≤ 𝐻𝛩 (𝑃, 𝑄. λ) and 𝛶(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≤ 𝛶(𝜁, 𝜂, λ), for all 𝜁, 𝜂 ∈ 𝛴, 𝛶(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≤ 𝛶(𝜁, 𝜂, λ), for all 𝜁 ∈ 𝑃 𝑎𝑛𝑑 𝜂 ∈ 𝑄 inf 𝜂 ∈ 𝑄 𝛶(𝑓(𝜁), 𝑓(𝜂), 𝑘λ) ≤ inf 𝑦 ∈ 𝑄 𝛶(𝜁, 𝜂, λ), for all 𝜁 ∈ 𝑃 𝛶(𝑓(𝜁), 𝑓(𝑄), 𝑘λ) ≤ 𝛶(𝜁, 𝑄, λ), for all 𝜁 ∈ 𝑃, sup 𝜁 ∈ 𝑃 𝛶(𝑓(𝜁), 𝑓(𝑄), 𝑘λ) ≤ sup 𝜁 ∈ 𝑃 𝛶(𝜁, 𝑄, λ). 𝛶(𝑓(𝑃), 𝑓(𝑄), 𝑘λ) ≤ 𝛶(𝑃, 𝑄, λ). Similarly 𝛶(𝑓(𝑄), 𝑓(𝜁), 𝑘λ) ≤ 𝛶(𝑄, 𝑃, λ). Hence 𝐻𝛶 (𝑓(𝑃), 𝑓(𝑄), 𝑘λ) ≤ 𝐻𝛶 (𝑃, 𝑄. λ). This completes the proof. Some Results in Hausdorff Neutrosophic Metric Spaces on Hutchinson-Barnsley Operator The above Theorem 4.5. shows that f is a neutrosophic B-contraction on 𝒦0(Σ) with respect to the Hausdorff neutrosophic metric 𝐻𝛯 , 𝐻𝛩 , 𝐻𝛶, if f is neutrosophic B-contraction on neutrosophic metric space (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄). Lemma 4.6. Let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄)be a NMS. If 𝑄, 𝑅 ⊂ 𝛴 such that 𝑄 ⊂ 𝑅, then 𝛯(𝜁, 𝑄, λ) ≤ 𝛯(𝜁, 𝑅, λ), 𝛩(𝜁, 𝑄, λ) ≥ 𝛩(𝜁, 𝑅, λ) and 𝛶(𝜁, 𝑄, λ) ≥ 𝛶(𝜁, 𝑅, λ) for all 𝜁 ∈ 𝛴 and λ > 0. Proof: Fix λ > 0. Let 𝜁 ∈ 𝛴 and 𝑄, 𝑅 ⊂ 𝛴 such that 𝑄 ⊂ 𝑅. Then, Ξ(𝜁, 𝑄, λ) = sup 𝑞 ∈ 𝑄 Ξ (𝜁, 𝑞, λ) ≤ sup 𝑞 ∈ 𝑅 Ξ (𝜁, 𝑞, λ) = Ξ(𝜁, 𝑅, λ), 𝛩(𝜁, 𝑄, λ) = inf 𝑞 ∈ 𝑄 𝛩(𝜁, 𝑞, λ) ≥ inf 𝑞 ∈ 𝑅 𝛩(𝜁, 𝑞, λ) = 𝛩(𝜁, 𝑅, λ) and 𝛶(𝜁, 𝑄, λ) = inf 𝑞 ∈ 𝑄 𝛶(𝜁, 𝑞, λ) ≥ inf 𝑞 ∈ 𝑅 𝛶(𝜁, 𝑞, λ) = 𝛶(𝜁, 𝑅, λ). Lemma 4.7. Let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄)be a NMS. If 𝑄, 𝑅 ⊂ 𝛴 such that 𝑄 ⊂ 𝑅, then 𝛯(𝑃, 𝑄, λ) ≤ 𝛯(𝑃, 𝑅, λ), 𝛩(𝑃, 𝑄, λ) ≥ 𝛩(𝑃, 𝑅, λ) and 𝛶(𝑃, 𝑄, λ) ≥ 𝛶(𝑃, 𝑅, λ) for all 𝑃 ⊂ 𝛴 and λ > 0. Proof: Fix λ > 0. Let 𝑃, 𝑄, 𝑅 ⊂ 𝛴 such that 𝑄 ⊂ 𝑅. By the lemma 4.6., we have 𝛯(𝑃, 𝑄, λ) = inf 𝑝 ∈ 𝑃 𝛯(𝑝, 𝑄, λ), 𝛯(𝑃, 𝑄, λ) ≤ 𝛯(𝑝, 𝑄, λ), for all 𝑝 ∈ 𝑃 𝛯(𝑃, 𝑄, λ) ≤ 𝛯(𝑝, 𝑅, λ), for all 𝑝 ∈ 𝑃, 𝛯(𝑃, 𝑄, λ) ≤ inf 𝑝 ∈ 𝑃 𝛯(𝑝, 𝑅, λ) , 𝛯(𝑃, 𝑄, λ) ≤ 𝛯(𝑃, 𝑅, λ). Similarly, by the lemma 4.6. 𝛩(𝑃, 𝑄, λ) = sup 𝑝 ∈ 𝑃 𝛩 (𝑝, 𝑄, λ), 𝛩(𝑃, 𝑄, λ) ≥ 𝛩(𝑝, 𝑄, λ) for all 𝑝 ∈ 𝑃, 𝛩(𝑃, 𝑄, λ) ≥ 𝛩(𝑝, 𝑅, λ) for all 𝑝 ∈ 𝑃, 𝛩(𝑃, 𝑄, λ) ≥ sup 𝑝 ∈ 𝑃 𝛩 (𝑝, 𝑅, λ), 𝛩(𝑃, 𝑄, λ) ≥ 𝛩(𝑃, 𝑅, λ) and 𝛶(𝑃, 𝑄, λ) = sup 𝑝 ∈ 𝑃 𝛶 (𝑝, 𝑄, λ), 𝛶(𝑃, 𝑄, λ) ≥ 𝛶(𝑝, 𝑄, λ) for all 𝑝 ∈ 𝑃, 𝛶(𝑃, 𝑄, λ) ≥ 𝛶(𝑝, 𝑅, λ) for all 𝑝 ∈ 𝑃, 𝛶(𝑃, 𝑄, λ) ≥ sup 𝑝 ∈ 𝑃 𝛶 (𝑝, 𝑅, λ) 𝛶(𝑃, 𝑄, λ) ≥ 𝛶(𝑃, 𝑅, λ). Lemma 4.8. Let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄)be a NMS. If 𝑃, 𝑄, 𝑅 ⊂ 𝛴, then V. B. Shakila and M. Jeyaraman 𝛯(𝑃 ∪ 𝑄, 𝑅, λ) = 𝑚𝑖𝑛{𝛯(𝑃, 𝑅, λ), 𝛯(𝑄, 𝑅, λ)}, 𝛩(𝑃 ∪ 𝑄, 𝑅, λ) = 𝑚𝑎𝑥{𝛩(𝑃, 𝑅, λ), 𝛩(𝑄, 𝑅, λ)} and 𝛶(𝑃 ∪ 𝑄, 𝑅, λ) = 𝑚𝑎𝑥{𝛶(𝑃, 𝑅, λ), 𝛶(𝑄, 𝑅, λ)},for all λ > 0. Proof: Fix λ > 0. Let 𝑃, 𝑄, 𝑅 ⊂ 𝛴. Then 𝛯(𝑃 ∪ 𝑄, 𝑅, λ) = inf 𝜁 ∈ 𝑃 ∪ 𝑄 𝛯(𝜁, 𝑅, λ) = 𝑚𝑖𝑛 { inf 𝑝 ∈ 𝑃 𝛯(𝑝, 𝑅, λ), inf 𝑞 ∈ 𝑄 𝛯(𝑞, 𝑅, λ),} = 𝑚𝑖𝑛{𝛯(𝑃, 𝑅, λ), 𝛯(𝑄, 𝑅, λ)}, 𝛩(𝑃 ∪ 𝑄, 𝑅, λ) = sup 𝜁 ∈ 𝑃 ∪ 𝑄 𝛩 (𝜁, 𝑅, λ) = 𝑚𝑎𝑥 { sup 𝑝 ∈ 𝑃 𝛩 (𝑝, 𝑅, λ), sup 𝑞 ∈ 𝑄 𝛩 (𝑞, 𝑅, λ),} = 𝑚𝑎𝑥{𝛩(𝑃, 𝑅, λ), 𝛩(𝑄, 𝑅, λ)} and 𝛶(𝑃 ∪ 𝑄, 𝑅, λ) = sup 𝜁 ∈ 𝑃 ∪ 𝑄 𝛶(𝜁, 𝑅, λ) = 𝑚𝑎𝑥 { sup 𝑝 ∈ 𝑃 𝛶 (𝑝, 𝑅, λ), sup 𝑞 ∈ 𝑄 𝛶(𝑞, 𝑅, λ),} = 𝑚𝑎𝑥{𝛶(𝑃, 𝑅, λ), 𝛶(𝑄, 𝑅, λ)} Lemma 4.9. Let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄) be a NMS. Let (𝒦0(Σ), 𝐻𝛯 , 𝐻𝛩 , 𝐻𝛶 ,∗,⋄) be the corresponding Hausdorff NMS. If 𝑃, 𝑄, 𝑅, 𝑆 ∈ 𝒦0(Σ) then 𝐻𝛯 (𝑃 ∪ 𝑄, 𝑅 ∪ 𝑆, λ) ≥ 𝑚𝑖𝑛{𝐻𝛯 (𝑃, 𝑅, λ), 𝐻𝛯 (𝑄, 𝑆, λ)}, 𝐻𝛩 (𝑃 ∪ 𝑄, 𝑅 ∪ 𝑆, λ) ≤ 𝑚𝑎𝑥{𝐻𝛩 (𝑃, 𝑅, λ), 𝐻𝛩 (𝑄, 𝑆, λ)} and 𝐻𝛶 (𝑃 ∪ 𝑄, 𝑅 ∪ 𝑆, λ) ≤ 𝑚𝑎𝑥{𝐻𝛶 (𝑃, 𝑅, λ), 𝐻𝛶 (𝑄, 𝑆, λ)}, for all λ > 0. Proof: Fix λ > 0. Let 𝑃, 𝑄, 𝑅, 𝑆 ∈ 𝒦0(Σ). By using Lemma 4.7. and Lemma 4.8., we get 𝛯(𝑃 ∪ 𝑄, 𝑅 ∪ 𝑆, λ) = 𝑚𝑖𝑛{𝛯(𝑃, 𝑅 ∪ 𝑆, λ), 𝛯(𝑄, 𝑅 ∪ 𝑆, λ)} ≥ 𝑚𝑖𝑛{𝛯(𝑃, 𝑅, λ), 𝛯(𝑄, 𝑆, λ)} ≥ 𝑚𝑖𝑛{𝐻𝛯 (𝑃, 𝑅, λ), 𝐻𝛯 (𝑄, 𝑆, λ)}. Similarly, 𝛯(𝑅 ∪ 𝑆, 𝑃 ∪ 𝑄, λ) ≥ 𝑚𝑖𝑛{𝐻𝛯 (𝑃, 𝑅, λ), 𝐻𝛯 (𝑄, 𝑆, λ)}. Hence, 𝐻𝛯 (𝑃 ∪ 𝑄, 𝑅 ∪ 𝑆, λ) ≥ 𝑚𝑖𝑛{𝐻𝛯 (𝑃, 𝑅, λ), 𝐻𝛯 (𝑄, 𝑆, λ)}. 𝛩(𝑃 ∪ 𝑄, 𝑅 ∪ 𝑆, λ) = 𝑚𝑎𝑥{𝛩(𝑃, 𝑅 ∪ 𝑆, λ), 𝛩(𝑄, 𝑅 ∪ 𝑆, λ)} ≤ 𝑚𝑎𝑥{𝛩(𝑃, 𝑅, λ), 𝛩(𝑄, 𝑆, λ)} ≤ 𝑚𝑎𝑥{𝐻𝛩 (𝑃, 𝑅, λ), 𝐻𝛩 (𝑄, 𝑆, λ)}. Similarly, 𝛩(𝑅 ∪ 𝑆, 𝑃 ∪ 𝑄, λ) ≤ 𝑚𝑎𝑥{𝐻𝛩 (𝑃, 𝑅, λ), 𝐻𝛩 (𝑄, 𝑆, λ)}. Hence, 𝐻𝛩 (𝑅 ∪ 𝑆, 𝑃 ∪ 𝑄, λ) ≤ 𝑚𝑎𝑥{𝐻𝛩 (𝑃, 𝑅, λ), 𝐻𝛩 (𝑄, 𝑆, λ)} and 𝛶(𝑃 ∪ 𝑄, 𝑅 ∪ 𝑆, λ) = 𝑚𝑎𝑥{𝛶(𝑃, 𝑅 ∪ 𝑆, λ), 𝛶(𝑄, 𝑅 ∪ 𝑆, λ)} ≤ 𝑚𝑎𝑥{𝛶(𝑃, 𝑅, λ), 𝛶(𝑄, 𝑆, λ)} ≤ 𝑚𝑎𝑥{𝐻𝛶 (𝑃, 𝑅, λ), 𝐻𝛶 (𝑄, 𝑆, λ)}. Similarly, 𝛶(𝑅 ∪ 𝑆, 𝑃 ∪ 𝑄, λ) ≤ 𝑚𝑎𝑥{𝐻𝛶 (𝑃, 𝑅, λ), 𝐻𝛶 (𝑄, 𝑆, λ)}. Hence, 𝐻𝛶 (𝑅 ∪ 𝑆, 𝑃 ∪ 𝑄, λ) ≤ 𝑚𝑎𝑥{𝐻𝛶 (𝑃, 𝑅, λ), 𝐻𝛶 (𝑄, 𝑆, λ)}. This completes the proof. Some Results in Hausdorff Neutrosophic Metric Spaces on Hutchinson-Barnsley Operator The following theorem is a generalized version of the Theorem 4.5. Theorem 4.10. Let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄)be a NMS. Let (𝒦0(Σ), 𝐻𝛯 , 𝐻𝛩 , 𝐻𝛶 ,∗,⋄) be the corresponding Hausdorff NMS. Suppose 𝑓𝑛 : 𝛴 → 𝛴, 𝑛 = 1,2, … 𝑁0; 𝑁0 ∈ ℕ, is a neutrosophic B-Contraction on (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄). Then the neutrosophic HBO is also a neutrosophic B-Contraction on (𝒦0(Σ), 𝐻𝛯 , 𝐻𝛩 , 𝐻𝛶 ,∗,⋄). Proof: Fix λ > 0. Let 𝑃, 𝑄 ∈ 𝒦0(Σ). By using the Lemma 4.9. and Theorem 4.5. for a given 𝑘 ∈ (0,1), we get 𝐻𝛯 (𝐹(𝑃), 𝐹(𝑄), 𝑘λ) = 𝐻𝛯 (⋃ 𝑓𝑛 𝑁0 𝑛=1 (𝑃), ⋃ 𝑓𝑛 𝑁0 𝑛=1 (𝑄), 𝑘λ) ≥ 𝑁0 min 𝑛 = 1 𝐻𝛯 (𝑓𝑛(𝑃), 𝑓𝑛 (𝑄), 𝑘λ) ≥ 𝐻𝛯 (𝑃, 𝑄, λ), 𝐻𝛩 (𝐹(𝑃), 𝐹(𝑄), 𝑘λ) = 𝐻𝛩 (⋃ 𝑓𝑛 𝑁0 𝑛=1 (𝑃), ⋃ 𝑓𝑛 𝑁0 𝑛=1 (𝑄), 𝑘λ) ≤ 𝑁0 max 𝑛 = 1 𝐻𝛩 (𝑓𝑛 (𝑃), 𝑓𝑛 (𝑄), 𝑘λ) ≤ 𝐻𝛩 (𝑃, 𝑄, λ) and 𝐻𝛶 (𝐹(𝑃), 𝐹(𝑄), 𝑘λ) = 𝐻𝛶 (⋃ 𝑓𝑛 𝑁0 𝑛=1 (𝑃), ⋃ 𝑓𝑛 𝑁0 𝑛=1 (𝑄), 𝑘λ) ≤ 𝑁0 max 𝑛 = 1 𝐻𝛶 (𝑓𝑛(𝑃), 𝑓𝑛 (𝑄), 𝑘λ) ≤ 𝐻𝛶 (𝑃, 𝑄, λ). This completes the proof. From the above Theorem 4.10., we conclude that the operator F is a neutrosophic B-contraction on 𝒦0(𝛴) with respect to the Hausdorff neutrosophic metric (𝐻𝛯 , 𝐻𝛩 , 𝐻𝛶 ), if 𝑓𝑛 is neutrosophic B-contraction on an neutrosophic metric space(𝛴, 𝛯, 𝛩, 𝛶,∗,⋄) for each 𝑛 ∈ {1,2, … 𝑁0}. 5. Hausdorff Neutrosophic Metrics On 𝓚𝟎(Σ) and 𝓚𝟎(𝓚𝟎(Σ)) Now, we investigate the relationships between the hyperspaces 𝒦0(Σ) and 𝒦0(𝒦0(Σ)) and the Hausdorff neutrosophic metrics 𝐻𝛯 and ℋ𝐻𝛯 . V. B. Shakila and M. Jeyaraman Theorem 5.1. Let (𝛴, 𝛯, 𝛩, 𝛶,∗,⋄) be a NMS. Let (𝒦0(Σ), 𝐻𝛯 , 𝐻𝛩 , 𝐻𝛶 ,∗,⋄) and (𝒦0(𝒦0(Σ)), ℋ𝐻𝛯 , ℋ𝐻𝛩 , ℋ𝐻𝛶 ,∗,⋄) be the corresponding Hausdorff Neutrosophic hyper spaces. Let 𝔓, 𝔔 ∈ 𝒦0(𝒦0(Σ)) be such that {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄, 𝑄 ∈ 𝔔} ∈ 𝒦0(Σ). Then 𝐻𝛯 ({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄 ∶ 𝑄 ∈ 𝔔}, λ) ≥ ℋ𝐻𝛯 (𝔓, 𝔔, λ), 𝐻𝛩 ({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ : 𝑄 ∈ 𝔔}, λ) ≤ ℋ𝐻𝛩 (𝔓, 𝔔, λ) and 𝐻𝛶 ({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄: 𝑄 ∈ 𝔔}, λ) ≤ ℋ𝐻𝛶 (𝔓, 𝔔, λ) for all λ > 0. Proof: Fix λ > 0. Firstly, we note that 𝛯(𝑄, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = inf 𝑞 ∈ 𝑄 𝛯(𝑞, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = inf 𝑞 ∈ 𝑄 sup {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓} 𝛯(𝑞, 𝑝, λ) = inf 𝑞 ∈ 𝑄 sup 𝑃 ∈ 𝔓 sup 𝑝 ∈ 𝑃 𝛯 (𝑞, 𝑝, λ) ≥ sup 𝑃 ∈ 𝔓 inf 𝑞 ∈ 𝑄 sup 𝑝 ∈ 𝑃 𝛯 (𝑞, 𝑝, λ) = sup 𝑃 ∈ 𝔓 𝛯(𝑄, 𝑃, λ). It follows that 𝛯({𝑞 ∈ 𝑄, 𝑄 ∈ 𝔔}, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = inf {𝑞 ∈ 𝑄: 𝑄 ∈ 𝔔} 𝛯(𝑞, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = inf 𝑄 ∈ 𝔔 inf 𝑞 ∈ 𝑄 𝛯(𝑞, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = inf 𝑄 ∈ 𝔔 𝛯(𝑄, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) ≥ inf 𝑄 ∈ 𝔔 sup 𝑃 ∈ 𝔓 𝛯(𝑄, 𝑃, λ). Similarly, 𝛯({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄, 𝑄 ∈ 𝔔}, λ) ≥ inf 𝑃 ∈ 𝔓 sup 𝑄 ∈ 𝔔 𝛯(𝑃, 𝑄, λ). Hence, 𝐻𝛯 ({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄, 𝑄 ∈ 𝔔}, λ) = 𝑚𝑖𝑛 { 𝛯({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄, 𝑄 ∈ 𝔔}, λ), 𝛯({{𝑞 ∈ 𝑄, 𝑄 ∈ 𝔔}, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) } ≥ 𝑚𝑖𝑛 { inf 𝑃 ∈ 𝔓 sup 𝑄 ∈ 𝔔 𝛯(𝑃, 𝑄, λ) , inf 𝑄 ∈ 𝔔 sup 𝑃 ∈ 𝔓 𝛯(𝑄, 𝑃, λ)} ≥ 𝑚𝑖𝑛 { inf 𝑃 ∈ 𝔓 sup 𝑄 ∈ 𝔔 𝐻𝛯 (𝑃, 𝑄, λ) , inf 𝑄 ∈ 𝔔 sup 𝑃 ∈ 𝔓 𝐻𝛯 (𝑄, 𝑃, λ)} = 𝑚𝑖𝑛{𝐻𝛯 (𝔓, 𝔔, λ), 𝐻𝛯 (𝔔, 𝔓, λ)} = ℋ𝐻𝛯 (𝔓, 𝔔, λ). Secondly, we note that 𝛩(𝑄, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = sup 𝑞 ∈ 𝑄 𝛩(𝑞, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) Some Results in Hausdorff Neutrosophic Metric Spaces on Hutchinson-Barnsley Operator = 𝑠𝑢𝑝 𝑞 ∈ 𝑄 inf {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓} 𝛩(𝑞, 𝑝, λ) = sup 𝑞 ∈ 𝑄 inf 𝑃 ∈ 𝔓 inf 𝑝 ∈ 𝑃 𝛩(𝑞, 𝑝, λ) ≤ inf 𝑃 ∈ 𝔓 sup 𝑞 ∈ 𝑄 inf 𝑝 ∈ 𝑃 𝛩(𝑞, 𝑝, λ) = inf 𝑃 ∈ 𝔓 𝛩(𝑄, 𝑃, λ). It follows that 𝛩({𝑞 ∈: 𝑄 ∈ 𝔔}, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = sup {𝑞 ∈ 𝑄: 𝑄 ∈ 𝔔} 𝛩(𝑞, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = sup 𝑄 ∈ 𝔔 sup 𝑞 ∈ 𝑄 𝛩(𝑞, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = sup 𝑄 ∈ 𝔔 𝛩(𝑄, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) ≤ sup 𝑄 ∈ 𝔔 inf 𝑃 ∈ 𝔓 𝛩(𝑄, 𝑃, λ). Similarly, 𝛩({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄: 𝑄 ∈ 𝔔}, λ) ≤ sup 𝑃 ∈ 𝔓 inf 𝑄 ∈ 𝔔 𝛩(𝑃, 𝑄, λ). Hence, 𝐻𝛩 ({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄, 𝑄 ∈ 𝔔}, λ) = 𝑚𝑎𝑥 { 𝛩({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄: 𝑄 ∈ 𝔔}, λ), 𝛩({{𝑞 ∈ 𝑄: 𝑄 ∈ 𝔔}, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) } ≤ 𝑚𝑎𝑥 { sup 𝑃 ∈ 𝔓 𝑖𝑛𝑓 𝑄 ∈ 𝔔 𝛩(𝑃, 𝑄, λ) , sup 𝑄 ∈ 𝔔 𝑖𝑛𝑓 𝑃 ∈ 𝔓 𝛩(𝑄, 𝑃, λ)} ≤ 𝑚𝑎𝑥 { sup 𝑃 ∈ 𝔓 𝑖𝑛𝑓 𝑄 ∈ 𝔔 𝐻𝛩 (𝑃, 𝑄, λ) , sup 𝑄 ∈ 𝔔 𝑖𝑛𝑓 𝑃 ∈ 𝔓 𝐻𝛩 (𝑄, 𝑃, λ)} = 𝑚𝑎𝑥{𝐻𝛩 (𝔓, 𝔔, λ), 𝐻𝛩 (𝔔, 𝔓, λ)} = ℋ𝐻𝛩 (𝔓, 𝔔, λ) and Lastly, we note that 𝛶(𝑄, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = sup 𝑞 ∈ 𝑄 𝛶 (𝑞, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = 𝑠𝑢𝑝 𝑞 ∈ 𝑄 inf {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓} 𝛶(𝑞, 𝑝, λ) = sup 𝑞 ∈ 𝑄 inf 𝑃 ∈ 𝔓 inf 𝑝 ∈ 𝑃 𝛶(𝑞, 𝑝, λ) ≤ inf 𝑃 ∈ 𝔓 sup 𝑞 ∈ 𝑄 inf 𝑝 ∈ 𝑃 𝛶(𝑞, 𝑝, λ) = inf 𝑃 ∈ 𝔓 𝛶(𝑄, 𝑃, λ) It follows that 𝛶({𝑞 ∈: 𝑄 ∈ 𝔔}, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = sup {𝑞 ∈ 𝑄: 𝑄 ∈ 𝔔} 𝛶(𝑞, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = sup 𝑄 ∈ 𝔔 sup 𝑞 ∈ 𝑄 𝛶 (𝑞, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) = sup 𝑄 ∈ 𝔔 𝛶(𝑄, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) ≤ sup 𝑄 ∈ 𝔔 inf 𝑃 ∈ 𝔓 𝛶(𝑄, 𝑃, λ). Similarly, 𝛶({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈: 𝑄 ∈ 𝔔}, λ) ≤ sup 𝑃 ∈ 𝔓 inf 𝑄 ∈ 𝔔 𝛶(𝑃, 𝑄, λ). Hence, 𝐻𝛶 ({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄, 𝑄 ∈ 𝔔}, λ) V. B. Shakila and M. Jeyaraman = 𝑚𝑎𝑥 { 𝛶({𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, {𝑞 ∈ 𝑄: 𝑄 ∈ 𝔔}, λ), 𝛶({{𝑞 ∈ 𝑄: 𝑄 ∈ 𝔔}, {𝑝 ∈ 𝑃: 𝑃 ∈ 𝔓}, λ) } ≤ 𝑚𝑎𝑥 { sup 𝑃 ∈ 𝔓 𝑖𝑛𝑓 𝑄 ∈ 𝔔 𝛶(𝑃, 𝑄, λ) , sup 𝑄 ∈ 𝔔 𝑖𝑛𝑓 𝑃 ∈ 𝔓 𝛶(𝑄, 𝑃, λ)} ≤ 𝑚𝑎𝑥 { sup 𝑃 ∈ 𝔓 𝑖𝑛𝑓 𝑄 ∈ 𝔔 𝐻𝛶 (𝑃, 𝑄, λ) , sup 𝑄 ∈ 𝔔 𝑖𝑛𝑓 𝑃 ∈ 𝔓 𝐻𝛶 (𝑄, 𝑃, λ)} = 𝑚𝑎𝑥{𝐻𝛶 (𝔓, 𝔔, λ), 𝐻𝛶 (𝔔, 𝔓, λ)} = ℋ𝐻𝛶 (𝔓, 𝔔, λ). The proof is complete. 6. Conclusions In this paper, we proved the neutrosophic contraction properties of the Hutchinson-Barnsley operator on the neutrosophic hyperspace with respect to the Hausdorff neutrosophic metrics. Also we discussed about the relationships between the Hausdorff neutrosophic metrics on the neutrosophic hyperspaces. This paper will lead our direction to develop the Hutchinson-Barnsley Theory in the sense of neutrosophic B-contractions in order to define a fractal set in the neutrosophic metric spaces as a unique fixed point of the Neutrosophic HBO. References [1] M. Abdel-Basset, M. 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