Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 200 https://internationalpubls.com Fixed Point Results Under Hausdorff Distance in the Fractal Spaces Rita Pal1, J. Leo Amalraj2, V. Venkata Kumar3, G. Venkat Narayanan4 1Department of Applied Mathematics Bhilai Institute of Technology,Bhilai,India Email ID: ritapal001@gmail.com 2Department of Science and Humanities RMK College of Engineering and Technology Puduvoyal, Thiruvallur District, Tamilnadu, India Email ID: leoamalraj@rmkcet.ac.in 3 Department of Mathematics, Aditya Engineering College, Surampalem, India, Email ID: venkatakumar.v@aec.edu.in 4Department of Mathematics St. Joseph’s College of Engineering, OMR, Chennai District Kanchipuram,Tamil Nadu, India Email ID: gvenkatnarayanan@gmail.com Article History: Received: 14-10-2023 Revised: 28-11-2023 Accepted: 18-12-2023 Abstract: We establish a novel notion of Hausdorff distance and explore some of its topological characteristics by using an extended modular metric. We prove a fixed point theorem on generalized modular fractal space from the concept of iterated function system (IFS) and contraction. Keywords: Fixed points; Hausdorff ; fractal space; contraction ; IFS ; Modular metric spaces. 2000 Mathematics Subject Classification: 47H10, 54H25. 1. Introduction The Metric modular space structure was modified by Chistyakov (Chistyakov, 2008; Cho, Saadati and Sadeghi, 2012), in an insightful way and proposed the Hausdorff topology on it, is extremely well-liked in modern study. Now, using nonempty compact subsets, we investigate the Hausdorff distance for a certain (GMMS). In order to demonstrate an intriguing (FPT) fixed point theorem, on a generalised metric modular space, we apply the iterated function system (IFS) and idea of contraction together (Abdou, 2016; Abdou, 2020; Chistyakov, 2008; Chistyakov, 2010; Cho et al., 2012; Ege, Park and Ansari, 2020). Hutchinson studied iterated function system (IFS) and thought about the idea of fractal theory (Hutchinson, 1981). By Ri (Ri, 2016),Barnsley (Good, 1990), Bisht (Bisht, 2018), and Imdad (Imdad, Alfaqih and Khan, 2018), this topic was generalized. A singular nonempty compact set F and F = ⋃ 𝑄𝑚 𝑖=1 i (F) of the complete GMM space (L,T) for a GMMIFS , then a fractal set F is known as the attractor of the relevant generalized modular metric iterated function system. The associated attractor generalized modular metric iterated function system in this context is referred to as generalized modular metric fractal space. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 201 https://internationalpubls.com 2 Preliminiers Now let's review some ideas and fundamental principles. Here, we let P = [0, 1], P0 = (0, 1), Q = [0, ∞), Q0 = (0, ∞) and a set L ≠ ϕ. Definition 2.1 (Azadifa, Maramaei and Sadeghi, 2013) A function T: L × L × L× Q0 → Q is referred to as generalized metric modular (GMM) on L a no -empty set , if it follows the axioms listed below: (GMM-1) T ρ (l, l, n) ∈ Q0 for all l, n ∈ L and ρ ∈ Q0 with l ≠ n. (GMM-2) T ρ (l, m, n) = 0 , ∀ρ ∈ Q0 if l = m = n. (GMM-3) T ρ (l, l, n) ≤ T ρ (l, m, n) , ∀ρ ∈ Q0 if m ≠ n. (GMM-4) T ρ (l, m, n) = T ρ (l, n, m) = T ρ (n, l, m) and so on. (GMM-5) Tρ +δ (l, m, n) ≤ Tρ (l, v, v) + Tδ (v, m, v) for all ρ, δ∈ Q0. Then, (L, T) is referred to as a generalized modular metric on L . Definition 2.2 (Azadifa et al., 2013) Let us set l0 ∈ L and LT = {m ∈ L; lim ρ →0 T ρ (l0, m, n)=0 for some n ∈ L}. The set LT is known as a modular set. Definition 2.3 (Azadifa et al., 2013) Assume that (L, T) be a generalised modular metric (GMM) space. Then, for l0 ∈ LT and c > 0, the T-ball with radius c and center l0 is BT (l0, c) = {m ∈ LT : T ρ (l0, m, m)< c} , ∀ ρ > 0. Proposition 2.3.1 (Azadifa et al., 2013) Assume that (L, T) be a generalised modular metric (GMM) space. Then for l0 ∈ LT and c > 0 , (i) if Tρ (l0, l, m) < c , ∀ ρ > 0, then l, m ∈ BT (l0, c). (ii) if m ∈ BT (l0, c) : BT (m, δ) ⊆ BT (l0, c) and δ >0 . Definition 2.4 (Azadifa et al., 2013) Assume that (L, T) is a generalized (GMMS) modular metric space. Sequence {ln} ⊆ L and Tl is T- convergent to l if it converges to l of τ (T ρ ) , ∀ n∈ N . Proposition 2.4.1 (Azadifa et al., 2013) Assume that (L, T) is a generalized (GMMS) modular metric space and sequence {ln} ⊆ LT , ∀ n ∈ N. Then the followings are satisfied : (1) Sequence{ln} is T-convergent to l. (2) σ ρ T (ln, l) → 0 when n → ∞, (3) T ρ (ln, ln, l) → 0 when n →∞ for all ρ > 0; (4) T ρ (ln, l, l) → 0 as n →∞ for all ρ > 0; (5) T ρ (lm, ln, l) → 0 when m, n →∞ , ∀ ρ > 0. Definition 2.5 (Azadifa et al., 2013) Assume that (L,T) is a generalized (GMMS) modular metric space. Then sequence {ln} ⊆ LT , is called T- Cauchy sequence if, Nε ∈ N : T ρ (ln, lm, lq)< ε , ∀ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 202 https://internationalpubls.com , n, m, q ≥ Nε and for every ε, ρ > 0. If every T- Cauchy sequence in a GMM-space L is a T- convergent sequence in that space, the space is said to be " T-complete." Proposition 2.5.1 (Azadifa et al., 2013) Assume that (L,T) is a generalized(GMMS) modular metric space and sequence {ln} ⊆ LT for all n∈N. Then the followings are equivalent: (1) Sequence {ln} is a T-Cauchy sequence. (2) We can locate Nε ∈ N: T ρ (ln, lm, lm)< ε, for each ε > 0 , ρ > 0, for every n, m ≥ Nε . (3) Sequence{ln} is a Cauchy sequence. Proposition 2.5.2 (Azadifa et al., 2013) Assume that (L, T) is a (GMM) space. Then for L × L × L × Q0 , T is a continuous function. Let us assume that a GMM-space (L, T) has two (nonempty) subsets, T and W. T ρ (l, T , W) = inf{T ρ (l, t, w): t ∈ T , w ∈ W} for l ∈ L and ρ >0 , Proposition 2.5.3 (Azadifa et al., 2013) Assume that (L, T) is a generalized modular metric space . For each M , N, P ∈ H0(L), the function δ |→ supm∈M Tρ (m, N, P ) is continuous on Q0 Proposition 2.5.4 (Alihajimohammad and Saadati, 2021) Assume that (L,T) is a generalized (GMMS) modular metric space . Suppose sequence {ln}⊆ L : Tφn (ρ ) ( l n, l n+1, l n+1) ≤ T ρ (l 0, l 1, l 1) for all ρ ∈ Q0. Then{ln} is a T-Cauchy sequence. Proposition 2.5.5 (Alihajimohammad and Saadati, 2021) Let (L,T) is a generalized (GMM) modular metric space. If T ρ (l ,m,n) = C for all l,m,n ∈ L and ρ ∈ Q0, then C = 0. Lemma 2.6 Let (L, T) is a (GMM) space. Then, for each l ∈ L, M , N ∈ H0(L) and ρ ∈ Q0, there are m0 ∈ M , n0 ∈ N such that T(l, M , N)= T ρ (l, m0,n0). Proof Let l ∈ L, M , N ∈ H0(L) and ρ > 0. By Proposition 2.5.2. the functions t, u |→ T ρ (l, m, n) are continuous. Thus, by compactness of M and N, ∃ m0 ∈ M , n0 ∈ N : inf T ρ (l, m, n)= T ρ (l, m0, n0) , for all m ∈M, n ∈ N Lemma 2.7 Assume that ( L,T) is a generalized (GMMS) modular metric space .Then, for every M ∈ H0(L), N, P ∈ F0(L) and ρ ∈ Q0 we can find m0 ∈ M such that sup T ρ (M , N, P )= T ρ (m0, N, P ). Proof Put δ = supt∈T T ρ (m, N, P ). Then we get a sequence (mn)n in M : δ – 1 𝑛 < T ρ (mn, N, P ) in which n ∈ N. From M ∈ H0(L), a subsequence (tnk )k of (mn)n and m0 ∈ M : mnk → m0 in (L, T). Select n ∈ N, p∈ P . From the Proposition 2.5.2, we get lim 𝑘 𝑇 ρ (mnk , n, p)= T ρ (m0, n, p). Since, for each k ∈ N, δ - 1 𝑛𝑘 < T ρ (mnk , n, p), we get δ ≤ T ρ (m0, n, p). We conclude δ = T ρ (m0, N, P ). Definition 2.8 (Alihajimohammad and Saadati, 2021) Let (L,T) is a generalized (GMM) modular metric and Q : L → L is called a GMM-φ-contractive mapping if Tφ(ρ) ( Q(l), Q(t), Q(u)) ≤ T ρ (l, m, n) for every l, m, n ∈ S and ρ ∈ Q0. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 203 https://internationalpubls.com Definition 2.9 (Alihajimohammad and Saadati, 2021) A generalized modular metric -φ- contractions {Q1, Q2,..., Qm ,: m ≥ 2} is a finite set on a complete GMM-space (L, T) is known as a generalized modular metric iterated function system . 3 Main Results: Mathematical Theorem Let (L, T) is a generalized (GMM) modular metric space and assume the followings: F0(L) = nonempty subsets of L, G0(L) = nonempty finite subsets and H0(L) = nonempty compact subset of L. Then a function HT on H0(L) × H0(L) × H0(L) × Q0 is defined by HT (M , N, P , ρ )= max{supm∈MTρ(m, N, P ),supn∈ NTρ (M ,n,P ), supp∈P T ρ (M , N,p)} for every M , N, P ∈ H0(L) and ρ ∈ Q0. Lemma 3.1 Let (L, T) is a generalized (GMM) modular metric space l ∈ L, M , N ∈ H0(L), P∈ F0(L), and α, β ∈ Q0. Then Tα+β (l, M , P ) ≤ Tα(l, N, N)+Tβ (ul, M ,P), where nl ∈ N satisfies Tα(l, N, N)= Tα(l,nl,nl). Proof: Using Lemma 2.6, Tα(l, N, N) = Tα(l, nl, nl). For each m ∈ M , p ∈ P , we have Tα+β (l, M , P ) ≤ Tα+β (l, m, p) ≤ Tα(l, nl, nl)+ Tβ(nl, m, p). Then Tα+β (l, M , P) ≤ Tα(l, N, N)+ Tβ (nl, M , P ) Theorem 3.2 Let (L,T) be a generalized modular metric (GMM) space. Then (H0(L), HT) is a generalized (GMMS) modular metric space. Proof : Let M , N, P , W ∈ H0(L) and α, β ∈ Q0. By Lemma 2.7, there exist m0 ∈ M , n0 ∈ N, and p0 ∈ P such that: supm∈M T (m, N, P ) = T (m0, N, P ), supn∈N T(M, n, P) = T(M, n0, P ), and supp∈P T(M , P, p)= T(M, P, p0). Then HT (M , N, P , α) ≥ 0. Moreover,it is clear that M = N = P ⇔ HT (M , N, P, α)= 0 Then from Lemma 3.1 we have Supm∈M Tα+β (m, N, W) ≤ Supm∈M Tα (m, P, P) + Supm∈M Tβ (pm, N, W) since{pm : m ∈ M} ⊆ P, Supm∈M Tβ (Pm, N, W) ≤ Supp∈P Tβ(p, N, W) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 204 https://internationalpubls.com Supm∈M Tα +β (m, N, W) ≤ Supm ∈ M Tα (m, P, P) + Sup p∈P Tβ (p, N, W) In the same way, we obtain sup n∈N Tα+β (M , n, W ) ≤ sup n∈N Tα(n, P, P )+ sup p∈P Tβ (p, M , W ), sup w ∈ W Tα+β (M , N, w) ≤ supw ∈ W Tα(w, P , P )+ supp ∈ P Tβ (p, M , W ). Therefore, it is obvious to conclude that HT (M , N, W , α + β) ≤ HT (M , P , P, α)+ HT (P , N, W , β). α |→ HT (M , N, P , α) is continuous on Q0 , by the Proposition 2.5.3. Then (H0(L), HT ) is a Generalized modular metric space. Theorem 3.3 Let (L,T) is a (GMM) space .A function Q : L → L is given by : Tφ(ρ) (Q( l), Q(m), Q(n) ≤ T ρ (l, m, n), for all ρ ∈ Q0 and l, m, n ∈ L. Then the sequence 𝑄𝑛(𝑙)𝑛=1 ∞ is generalized modular metric complete space . Proof Let {ln: Q n(l)}𝑛=1 +∞ }, {ln} is a sequence that complies with the requirements of proposition 2.5.4 T ρ (l, Q(l), Q(l) ≤ T ρ (l, Q(l), Q(l)) (using the induction) If Tφn(ρ )( Qn(l), Qn+1(l), Qn+1(l)) ≤ T ρ (l, Q(l), Q(l)) then Tφ n+1 (ρ )( Qn+1(l), Qn+2(l), Qn+2(l)) = Tφ(φn (ρ ))(Q( Qn(l), Q( Qn+1(l))) Now we have Q (Qn+1(l)) ≤ Tφn (ρ ) (Qn(l), Qn+1(l), Qn+1(l)) ≤ T ρ (l, Q(l), Q(l)) Therefore, Tφn (ρ )( l n, l n+1, l n+1) ≤ T ρ (l 0, l 1, l 1), Hence Qn( l) 𝑛=1 ∞ is generalized modular metric complete space (GMMCS). Theorem 3.4 Suppose (L,T) is a generalized (GMM) space and map Q ,GMM-φ- contractive mapping for all ρ ∈ Q0 and l, m, n ∈ L : Tφ(ρ)(Q(l), Q(t), Q(u) ≤ T ρ (l, m, n) for every l, m, n ∈ L and ρ ∈ Q0.Then in L, Q posseses a unique fixed point . Proof Now from the above theorem 3.3 , we get {ln: {Qn(l)}}n=1 +∞ }is generalized(GMMCS) modular metric complete space , for each l ∈ L and limn→∞ Qn(l) = x ∈L . Letting l0 = l and ln = Qn(l) for each n ≥ 1, since limn→∞ Qn(l)= x, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 205 https://internationalpubls.com we have lim T ρ (ln, x, x)=0for each ρ ∈ Q0 . On the other hand, we recognize Tφ(ρ)(Q(x), ln+1, ln+1) ≤ T ρ (x, ln,ln) for each n ∈ N and each ρ > 0. Then Tφ(ρ)(Q(x), x, x) = .𝑛→∞ 𝑙𝑖𝑚 𝑇φ(ρ)( Q(x), ln+1, ln+1) ≤ .𝑛→∞ 𝑙𝑖𝑚 T ρ (x, ln, ln)=0,for each ρ > 0. . => x = Q(x), Now , To prove : Uniqueness Let y ∈ Q is another point, and ρ ∈ Q0 T ρ (x, x, y) = T ρ (Q(x), Q(x), Q(y)) ≥ Tφ(ρ) (Q(x), Q(x), Q(y)) . since T ρ (l, m, m) is nonincreasing and φ(ρ ) < ρ , we have Tφ(ρ)( Q(x), Q(x), Q(y)) ≥ T ρ (Q(x), Q(x), Q(y)) = Tρ( x, x, y). Hence T ρ (x, x, y)= C From proposition 2.5.5, we get C = 0. Therefore, x = y. 4. Conclusions We have studied certain topological aspects of the Hausdorff distance on GMM and defined a (GMFS) in the sense of Chistyakov by iterated function system. as anapplication Some concepts of fixed point have been implemented in generalized modular metric space and generalized modular metric fractal space (GMMF-space ). 5. Acknowledgement The Authors are grateful to the knowledgeable referee for his insightful observations and comments, which substantially assisted us in significantly improving the manuscript. * References [1] Abdou, A. A. 2016. Some fixed point theorems in modular metric spaces, J. Nonlinear Sci. Appl 9(6): 4381–4387. [2] Abdou, A. A. 2020. Fixed points of kannan maps in modular metric spaces, AIMS Mathematics 5(6): 6395–6403. [3] Alihajimohammad, A. and Saadati, R. 2021. Generalized modular fractal spaces and fixed point theorems, Advances in Difference Equations 2021(1): 1–10. [4] Azadifa, B., Maramaei, M. and Sadeghi, G. 2013. On the modular g-metric spaces and fixed point theorems., Journal of Nonlinear Sciences & Applications (JNSA) 6(4). [5] Bisht, R. K. 2018. Comment on: A new fixed point theorem in the fractal space, Indagationes Mathematicae 29(2): 819–823. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 206 https://internationalpubls.com [6] Chistyakov, V. V. 2008. Modular metric spaces generated by f-modulars, Folia Math 14(3). [7] Chistyakov, V. V. 2010. Modular metric spaces, i: basic concepts, Nonlinear Analysis: Theory, Methods & Applications 72(1): 1–14. [8] Cho, Y. J., Saadati, R. and Sadeghi, G. 2012. Quasi-contractive mappings in modular metric spaces, Journal of Applied Mathematics 2012. [9] Ege, O., Park, C. and Ansari, A. H. 2020. A different approach to complex valued gb g {b}- metric spaces, Advances in Difference Equations 2020(1): 1–13. [10] Good, I. 1990. Fractals everywhere (michael barnsley). [11] Hutchinson, J. E. 1981. Fractals and self similarity, Indiana University Mathematics Journal 30(5): 713–747. [12] Imdad, M., Alfaqih, W. M. and Khan, I. A. 2018. Weak θ-contractions and some fixed point results with applications to fractal theory, Advances in Difference Equations 2018(1): 1– 18. [13] Ri, S.-i. 2016. A new fixed point theorem in the fractal space, Indagationes Mathematicae 27(1): 85–93.