EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6321 ISSN 1307-5543 – ejpam.com Published by New York Business Global Common Fixed Points of Triplet Mappings in GM-Spaces with Applications to AI Convergence and Cryptographic Consensus Maha Noorwali1, Raed Hatamleh2, Arif Mehmood3, Abdallah Al-Husban4, Khaled A. Aldwoah5, Cris L. Armada6,7, Jamil J. Hamja8, Alaa M. Abd El-latif9,∗ 1 Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, 15 Saudi Arabia 2 Department of Mathematics, Faculty of Science, Jadara University, P.O. Box 733, Irbid 21110, Jordan 3 Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail Khan 29050, KPK, Pakistan 4 Department of Mathematics, Faculty of Science and Technology, Irbid National University, P.O. Box: 2600 Irbid, Jordan 5 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Saudia Arabia 6 Vietnam National University Ho Chi Minh City, Linh Trung Ward, Thu Duc City, Ho Chi Minh City, Vietnam 7 Department of Applied Mathematics, Faculty of Applied Science, Ho Chi Minh City University of Technology (HCMUT), 268 Ly Thuong Kiet, Ward 14, District 10, Ho Chi Minh City, Vietnam 8 Mathematics and Sciences Department, College of Arts and Sciences, Mindanao State University- Tawi-Tawi College of Technology and Oceanography, 7500 Philippine 9 Department of Mathematics, College of Science, Northern Border University, Arar 91431, Saudi Arabia Abstract. This paper investigates the existence and uniqueness of common fixed points for three self- mappings in generalized metric spaces (GM-spaces), establishing new results under generalized contrac- tive conditions. We develop a comprehensive theoretical framework where tripartite mappings satisfy inequalities involving combinations of distance-like terms formulated through minimum and maximum comparisons. The contractive conditions are governed by carefully chosen parameters that determine whether the mappings admit a common fixed point or a unique common fixed point. To demonstrate the practical applicability of our theory, we present a concrete example using the interval [0, 1] equipped with the G-metric G(x, y, z) = max{|x − y|, |y − z|, |z − x|}, where piecewise-defined self-mappings are shown to satisfy all required conditions and converge to a unique common fixed point. Beyond theoretical advancements, our results offer significant applications in artificial intelligence, particularly in analyzing convergence of multi-layer neural architectures, and in cryptography for designing secure iterative proto- cols. The framework presented here not only generalizes existing fixed-point theorems but also provides verifiable computational methods for stability analysis in both mathematical and applied contexts. 2020 Mathematics Subject Classifications: 54A05 Key Words and Phrases: Contraction Conditions, Common Fixed Point, GM-Space, Artificial Intel- ligence, Cryptographic Protocols, G-Metric Convergence ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6321 Email addresses: (mnorwali@kau.edu.sa) (M. Noorwali), (raed@jadara.edu.jo) (R. Hatamleh), (mehdaniyal@gmail.com) (A. Mehmood), (dralhosban@inu.edu.jo) (A. Alhusban), (aldwoah@yahoo.com) (K. A. Aldwoah), (cris.armada@hcmut.edu.vn) (C. L. Armada), (jamilhamja@msutawi-tawi.edu.ph) (J. J. Hamja), (alaa.ali@nbu.edu.sa) (A. M. Abd El-latif) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 2 of 27 1. Introduction Fixed point theory (FPT) is a branch of mathematics that deals with the existence and uniqueness of fixed points for various types of mappings or functions. FPT plays a crucial role in various branches of mathematics, including analysis, topology, and functional analysis. It has numerous applications in diverse fields such as economics, physics, engineering, com- puter science, and more recently, artificial intelligence (AI) and cryptography. In AI, FPT supports the theoretical convergence of learning dynamics in deep neural networks, especially within recurrent and iterative architectures. In cryptographic systems, fixed point theorems ensure consistency and convergence in multi-party secure computations and iterative encryp- tion/decryption protocols. Within the context of FPT, Stefan Banach introduced the “Banach Contraction Principle (BCP)”, which states that a single-valued contractive-type mapping in a complete metric space has a unique fixed point. Our objective is to extend this principle to GM-spaces, which offer a broader framework. Our approach involves examining existing research articles in the GM- space domain to identify any shortcomings in current results. By accurately pinpointing these issues, we aim to establish a more comprehensive solution. Ultimately, our goal is to refine and strengthen the theoretical foundation of GM-spaces, utilizing the BCP as a guiding principle while addressing limitations to achieve a more universally applicable outcome, particularly in AI and cryptographic applications. In 1941, Kakutani [1] modified a generalization of Brouwers FP-theorem. The theory origi- nated from the Brouwer FP-theorem proposed by mathematician L.E.J. Brouwer in 1910. This theorem states that any continuous function from a closed ball to itself must have at least one FP. Jungck and Rhoades [2] proved some FP-theorems for compatible maps. Minak et al. [3] described a new approach to FP-theorems for multivalued contractive mappings. Metric spaces (M-spaces) provide a fundamental framework in mathematics for studying distance and convergence. Sullivan [4] showed a characterization of complete M-spaces. George and Veeramani [5] worked on some results in fuzzy M-spaces. Bhaskar and Lakshmikantham [6] defined FP-theorems in partially ordered M-spaces and applications. Kutbi et al. [7] proved CFP results for mappings with rational expressions. In 2022, Yaseen et al. [8] addressed new results of FP-theorems in complete M-spaces. Next, having explored the concepts within FPT and M-spaces, we delve into the concepts specific to GM-spaces. To achieve this, we examine existing research carried out in this specific area, which provides the foundation to establish the most comprehensive outcomes. Rhoades [9] formulated a FP-theorem for generalized metric spaces (GM-spaces). In 2006, Mustafa and Sims [10] proposed the idea of generalized metric spaces. Azam and Arshad [11] looked into a Kannan FP-theorem on GM-spaces. Sarma et al. [12] flourished a contraction over GM-spaces. Mihet [13] worked on Kannan FP-principle in GM-spaces. Das and Dey [14] addressed FP of contractive mappings in GM-spaces. Shatanawi [15] formulated a coupled FP-theorems in GM-spaces. Abbas et al. [? ] defined CFP results for three maps in GM-space. Abbas et al. [? ] examined coupled CFP results in two GM-spaces. Di Bari and Vetro [16] explored CF-points in GM-spaces. Romaguera [17] established FP-theorems for generalized contractions on partial M-spaces. Mohanta and Mohanta [18] tried to investigate a CFP theorem in GM- spaces. Gugnani et al. [19] described CFP results in GM-spaces and its applications. Aydi [20] discussed a CFP of integral type contraction in GM-spaces. Kadelburg and Radenovic [21] worked on GM-spaces. Later on, in 2015, Jleli and Samet [22] provided a GM-space and proved its related FP- theorems. Abed and Luaibi [23] addressed two FP-theorems in GM-spaces. Lin and Yun [24] examined GM-spaces and mappings. Peng and Sun [25] explored a study on symmetric products of GM-spaces. Our approach involves utilizing generalized outcomes through the application of maximum and minimum type contractions within GM-spaces without continuity. This approach aims to significantly enhance the effectiveness of our results. Furthermore, our work endeavors M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 3 of 27 to build upon the foundation of the latest published results, thereby producing an expanded and enriched version of its findings. The significance of our study lies in its capacity to extend and enrich existing results from the realm of FPT. Through our exploration, we endeavor to enhance our understanding of FP principles by introducing new dimensions. Our main objective is to establish specific condi- tions under which fixed points uniquely exist. To achieve this, we exploit the power of three self-mappings that fulfill the criteria of generalized contractive conditions within a GM-space. This investigation is undertaken with the goal of contributing valuable insights that further illuminate the existence and uniqueness of fixed points, while simultaneously advancing the frontiers of GM-space theory. Moreover, these insights provide solid theoretical underpinnings for reliable iterative convergence in AI algorithms and consistency guarantees in cryptographic communication protocols. 2. Preliminaries This section is devoted to some fundamental definitions, which are necessary for the up- coming sections. 2.1. Generalized Metric Space This section explores fixed point theory in generalized metric spaces (GM-spaces), extending concepts like the Banach Contraction Principle (BCP) to a broader context. GM-spaces is defined and introduce G-Cauchy sequences, convergence, and completeness in these spaces. Result on GM space is also given. Definition 1. [2] Suppose a non-empty set X and let G be a function that operates on triples of elements from X denoted as G : X ×X ×X → [0,∞). We will call this structure a generalized metric space (GM-space) if the following axioms hold true: (i) G(x1, x2, x3) = 0 iff x1 = x2 = x3, (ii) 0 < G(x1, x1, x2) ∀ x1, x2 ∈ x, with x1 ̸= x2, (iii) G(x1, x1, x2) ≤ G(x1, x2, x3) ∀ x1, x2, x3 ∈ x1, with x1 ̸= x2, (iv) G(x1, x2, x3) = G{p(x1, x2, x3)} Where, p is a permutation of x1, x2, x3. (symmetry). (v) G(x1, x2, x3) ≤ G(x1, x1, x1) +G(x1, x2, x3) ∀ x1, x2, x3 ∈ X. A GM is symmetric if G(x1, x2, x2) = G(x2, x1, x1) ∀ x1, x2 ∈ X, then (X,G) is called a GM- space. Definition 2. [2] Let (X,G) be a GM-space and {xȷ} be a sequence in X. then, (i) A sequence {xȷ} is considered a G-Cauchy sequence in the GM-space X if, for any ε > 0, ∃n0 = N so that G(xȷ, xm, xl) < ε ∀ȷ,m, l ≥ n0. (ii) A sequence {xȷ} in the GM-space X is said to converge to an element xinX if ∀ any given ε > 0 ∈ R, ∃n0 = N so that G(xȷ, xm, xl) < ε, whenever m ≥ n0. (iii) The GM-space (X,G) is classified as complete if every G-Cauchy sequence is G-convergent in X. Proposition 1. [2] Let (X,G) be a GM-space, then for any x1, x2, x3 ∈ X it followes that: (i) If G(x1, x2, x3) = 0, then x1 = x2 = x3, M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 4 of 27 (ii) G(x1, x2, x3) ≤ G(x1, x1, x3), (iii) G(x1, x2, x2) ≤ 2G(x2, x1, x1), (iv) G(x1, x2, x3) ≤ G(x1, x, x3) +G(x, x2, x3), (v) G(x1, x2, x3) ≤ 2 3(G(x1, x2, x) +G(x1, x, x3) +G(x, x2, x3)), (vi) G(x1, x2, x3) ≤ (G(x1, x, x) +G(x2, x, x) +G(x3, x, x)), (vii) |G(x1, x2, x3)−G(x1, x2, x)| ≤ max{G(x, x3, x3), G(x3, x, x)}, (viii) |G(x1, x2, x3)−G(x1, x2, x)| ≤ G(x1, x, x3), (ix) |G(x1, x2, x3)−G(q, x3, x3)| ≤ max{G(x1, x3, x3), G(x3, x1, x1)}, (x) |G(x1, x2, x2)−G(x2, x1, x1)| ≤ max{G(x2, x1, x1), G(x1, x2, x2)}. 3. Main Results This section explores the existence of common fixed points (CFP) and unique common fixed points (UCFP) for three self-mappings in GM-spaces. Theorems 1 and 2 provide conditions under which these mappings possess a CFP and UCFP, involving inequalities with constants An example using the GM-metric on the interval [0,1] demonstrates the conditions required for the existence of a UCFP, contributing to the fixed point theory in GM-spaces. Theorem 1. Let (X,G) be a GM-space and F : X ×X ×X → X be a mapping satisfying: G(F1x1, F2x2, F3x3) ≤ α1G(x1, x2, x3) + α2G(x1, x2, F2x2) + α3G(F1x1, x2, x2) + α4[G(F1x1, x1, x1) +G(x2, F2x2, x2) +G(x3, x3, F3x3)] + α5min {G(x1, x2, F2x2), G(F1x1, F1x1, x2), G(F1x1, x2, x2), G(F2x2, F2x2, x3)} (1) for all x1, x2, x3 ∈ X and α1, α2, α3, α4 ≥ 0 with α1, α2, α3, α4 < 1 and (α1 + α2 + α3 + α4) < 1. Then, the three individual self-mappings referred as F1, F2 and F3 possess a CFP in X. Moreover, if (α1 + α2 + α3) < 1, the three individual self-mappings referred as F1, F2 and F3 possess a UCFP in X. Proof. Fix x0 ∈ X, and {xk} be a sequence in X. Now we define some iterative sequences in X such that x(3k+1) = F1x3k, x(3k+2) = F2x(3k+1) and x(3k+3) = F3x(3k+2) ∀ k ≥ 0. Now, by using (1) we have, G(x(3k+1), x(3k+2), x(3k+3)) = G(F1x3k, F2x(3k+1), F3x(3k+2)) ≤ α1G(x3k, x(3k+1), x(3k+2)) + α2G(x3k, x(3k+1), F2x(3k+1)) + α3G(F1x3k, x(3k+1), x(3k+1)) + α4[G(F1x3k, x3k, x3k) +G(x(3k+1), F2x(3k+1), x(3k+1)) +G(x(3k+2), x(3k+2), F3x(3k+2))] + α5min { G(x3k, x(3k+1), F2x(3k+1)), G(F1x3k, F1x3k, x(3k+1)), G(F1x3k, x(3k+1), x(3k+1)), G(F2x(3k+1), F2x(3k+1), x(3k+2)) } = α1G(x3k, x(3k+1), x(3k+2)) + α2G(x3k, x(3k+1), x(3k+1)) + α3G(x3k, x(3k+1), x(3k+1)) + α4[G(x3k, x3k, x3k) +G(x(3k+1), x(3k+1), x(3k+1)) +G(x(3k+2), x(3k+2), x(3k+2))] + α5min { G(x3k, x(3k+1), x(3k+1)), G(x3k, x3k, x(3k+1)), G(x3k, x(3k+1), x(3k+1)), G(x(3k+1), x(3k+1), x(3k+2)) } M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 5 of 27 After applying the simplification process utilizing the Definition of GM-space we have achieved the following result, ≤ (α1 + α2)G(x3k, x(3k+1), x(3k+2)) + α4[G(x3k, x(3k+1), x(3k+2)) +G(x3k, x(3k+1), x(3k+2)) +G(x(3k+1), x(3k+2), x(3k+3))] (1− α4)G(x(3k+1), x(3k+2), x(3k+3)) ≤ (α1 + α2 + 2α4)G(x3k, x(3k+1), x(3k+2)) ≤ (α1 + α2 + 2α4) (1− α4) G(x3k, x(3k+1), x(3k+2)), Taking ρ1 = (α1 + α2 + 2α4) (1− α4) < 1 G(x(3k+1), x(3k+2), x(3k+3)) ≤ ρ1G(x3k, x(3k+1), x(3k+2)) (2) Now, we find iteration of a contraction. G(x(3k+2), x(3k+3), x(3k+4)) = G(F1x(3k+1), F2x(3k+2), F3x(3k+3)) ≤ α1G(x(3k+1), x(3k+2), x(3k+3)) + α2G(x(3k+2), x(3k+2), F2x(3k+2)) + α3G(F1x(3k+1), x(3k+2), x(3k+2)) + α4[G(F1x(3k+1), x(3k+1), x(3k+1)) +G(x(3k+1), F2x(3k+1), x(3k+1)) +G(x(3k+2), x(3k+2), F3x(3k+2))] + α5min { G(x(3k+1), x(3k+2), F2x(3k+2)), G(F1x(3k+1), F1x(3k+1), x(3k+2)), G(F1x(3k+1), x(3k+2), x(3k+2)), G(F2x(3k+2), F2x(3k+2), x(3k+3)) } = α1G(x(3k+1), x(3k+2), x(3k+3)) + α2G(x(3k+2), x(3k+2), x(3k+2)) + α3G(x(3k+1), x(3k+2), x(3k+2)) + α4[G(x(3k+1), x(3k+1), x(3k+1)) +G(x(3k+1), x(3k+1), x(3k+1)) +G(x(3k+2), x(3k+2), x(3k+2))] + α5min { G(x(3k+1), x(3k+2), x(3k+2)), G(x(3k+1), x(3k+1), x(3k+2)), G(x(3k+1), x(3k+2), x(3k+2)), G(x(3k+2), x(3k+2), x(3k+3)) } After applying the simplification process utilizing the Definition of GM-space we have achieved the following result, ≤ (α1 + α2)G(x(3k+1), x(3k+2), x(3k+3)) + α4[G(x(3k+1), x(3k+2), x(3k+3)) +G(x(3k+1), x(3k+2), x(3k+3)) +G(x(3k+2), x(3k+3), x(3k+4))] (1− α4)G(x(3k+2), x(3k+3), x(3k+4)) ≤ (α1 + α2 + 2α4)G(x(3k+1), x(3k+2), x(3k+3)) ≤ (α1 + α2 + 2α4) (1− α4) G(x(3k+1), x(3k+2), x(3k+3)), Taking ρ2 = (α1 + α2 + 2α4) (1− α4) < 1 G(x(3k+2), x(3k+3), x(3k+4)) ≤ ρ2G(x(3k+1), x(3k+2), x(3k+3)) (3) From both cases we get that, G(x(3k+1), x(3k+2), x(3k+3)) ≤ ρG(x3k, x(3k+1), x(3k+2)), ∴ ρ1 = ρ2 = ρ Inductively we have, G(x(3k+1), x(3k+2), x(3k+3)) ≤ ρ3k+1G(x0, x1, x2) M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 6 of 27 Next we will show that {xk} is a G-Cauchy sequence in X, for any j, n, k with j > n > k. G(xk, xn, xj) ≤ G(xk, x(k+1), x(k+1)) +G(x(k+1), x(k+2), x(k+2)) + ......+G(x(j−1), x(j−1), xj) ≤ G(xk, x(k+1), x(k+2)) +G(x(k+1), x(k+2), x(k+3)) + .......+G(x(j−2), x(j−1), xj) ≤ [yk + y(k+1) + ........+ y(j−2)]G(x0, x1, x2) This implies that, G(xk, xn, xj) ≤ yk (1− y) G(x0, x1, x2) The same hold if j = n > k and if j > n = k we have, G(xk, xn, xj) ≤ yk−1 (1− y) G(x0, x1, x2) (4) If we take the limit as k, n, j →∞ we getG(xk, xn, xj)→ 0. Hence {xk} is a G-Cauchy sequence. By G-completeness of X, there exists ℘ ∈ X such that {xk} converges to ℘ as k →∞. We have to show that F1℘ = ℘ by contrary case let F1℘ ̸= ℘. G(F1℘, x(3k+2), x(3k+3)) = G(F1℘, F2x(3k+1), F3x(3k+2)) ≤ α1G(℘, x(3k+1), x(3k+2)) + α2G(℘, x(3k+1), F2x(3k+1)) + α3G(F1℘, x(3k+1), x(3k+1)) + α4[G(F1℘, x3k, x3k) +G(x(3k+1), F2x(3k+1), x(3k+1)) +G(x(3k+2), x(3k+2), F3x(3k+2))] + α5min { G(℘, x(3k+1), F2x(3k+1)), G(F1℘, F1℘, x(3k+1)), G(F1℘, x(3k+1), x(3k+1)), G(F2x(3k+1), F2x(3k+1), x(3k+2)) } = α1G(℘, x(3k+1), x(3k+2)) + α2G(℘, x(3k+1), x(3k+1)) + α3G(F1℘, x(3k+1), x(3k+1)) + α4[G(F1℘, x3k, x3k) +G(x(3k+1), x(3k+1), x(3k+1)) +G(x(3k+2), x(3k+2), x(3k+2))] + α5min { G(℘, x(3k+1), x(3k+1)), G(F1℘, F1℘, x(3k+1)), G(℘, x(3k+1), x(3k+1)), G(x(3k+1), x(3k+1), x(3k+2)) } After simplification process we get, G(F1℘, x(3k+2), x(3k+3)) ≤ α1G(℘, x(3k+1), x(3k+2)) + α2G(℘, x(3k+1), x(3k+1)) + α3G(F1℘, x(3k+1), x(3k+1)) + α4[G(F1℘, x3k, x3k) +G(x(3k+1), x(3k+1), x(3k+1)) +G(x(3k+2), x(3k+2), x(3k+2))] Applying limn→∞ on both sides, we get, G(F1℘, ℘, ℘) ≤ α1G(℘, ℘, ℘)+α2G(℘, ℘, ℘)+α3G(F1℘, ℘, ℘)+α4[G(F1℘, ℘, ℘)+G(℘, ℘, ℘)+G(℘, ℘, ℘)] ≤ (α3 + α4)G(F1℘, ℘, ℘) G(F1℘, ℘, ℘)− (α3+α4)G(F1℘, ℘, ℘) ≤ 0, is a contradiction. Since (1−α3−α4) ̸= 0, therefore, G(F1℘, ℘, ℘) = 0. Thus, F1℘ = ℘ (5) Next, we have to show that F2℘ = ℘ by contrary case let F2℘ ̸= ℘. Then from (1) we have that, G(x(3k+1), F2℘, x(3k+3)) = G(F1x3k, F2℘, F3x(3k+2)) ≤ α1G(x3k, ℘, x(3k+2)) + α2G(℘, ℘, F2℘) + α3G(F1x3k, ℘, ℘) + α4[G(F1x3k, x3k, x3k) +G(x(3k+1), F2℘, ℘) +G(x(3k+2), x(3k+2), F3x(3k+2))] + α5min { G(x3k, ℘, F2℘), G(F1x3k, F1x3k, ℘), G(F1x3k, ℘, ℘), G(F2℘, F2℘, x(3k+2)) } M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 7 of 27 G(x(3k+1), F2℘, x(3k+3)) = α1G(x3k, ℘, x(3k+2)) + α2G(℘, ℘, F2℘) + α3G(x3k, ℘, ℘) + α4[G(x3k, x3k, x3k) +G(x(3k+1), F2℘, ℘) +G(x(3k+2), x(3k+2), x(3k+2))] + α5min { G(x3k, ℘, F2℘), G(x3k, x3k, ℘), G(x3k, ℘, ℘), G(F2℘, F2℘, x(3k+2)) } Applying limn→∞ on both sides, we get, G(℘, F2℘, ℘) ≤ α1G(℘, ℘, ℘) + α2G(℘, ℘, F2℘) + α3G(℘, ℘, ℘)+ α4[G(℘, ℘, ℘) +G(℘, F2℘, ℘) +G(℘, ℘, ℘)] + α5min { G(℘, ℘, F2℘), G(℘, ℘, ℘), G(℘, ℘, ℘), G(F2℘, F2℘, ℘) } ≤ (α2 + α4)G(F2℘, ℘, ℘) G(F2℘, ℘, ℘)− (α2+α4)G(F2℘, ℘, ℘) ≤ 0, is a contradiction. Since (1−α2−α4) ̸= 0, therefore, G(F2℘, ℘, ℘) = 0. Thus, F2℘ = ℘ (6) Next, we have to show that F3℘ = ℘ by contrary case let F3℘ ̸= ℘. Then from (1) we have that, G(x(3k+1), x(3k+2), F3℘) = G(F1x3k, F2x(3k+1), F3℘) ≤ α1G(x3k, x(3k+1), ℘) + α2G(x3k, x(3k+1), F2x(3k+1)) + α3G(F1x3k, x(3k+1), x(3k+1)) + α4[G(F1x3k, x3k, x3k) +G(x(3k+1), F2x(3k+1), x(3k+1)) +G(℘, ℘, F3℘)] + α5min { G(x3k, x(3k+1), F2x(3k+1)), G(F1x3k, F1x3k, x(3k+1)), G(F1x3k, x(3k+1), x(3k+1)), G(F2x(3k+1), F2x(3k+1), x(3k + 2)) } G(x(3k+1), x(3k+2), F3℘) = α1G(x3k, x(3k+1), ℘) + α2G(x3k, x(3k+1), F2x(3k+1))+ α3G(F1x3k, x(3k+1), x(3k+1)) + α4[G(F1x3k, x3k, x3k) +G(x(3k+1), F2x(3k+1), x(3k+1)) +G(℘, ℘, F3℘)] + α5min { G(x3k, x(3k+1), F2x(3k+1)), G(F1x3k, F1x3k, x(3k+1)), G(F1x3k, x(3k+1), x(3k+1)), G(F2x(3k+1), F2x(3k+1), x(3k + 2)) } Applying limn→∞ on both sides, we get, G(℘, ℘, F3℘) ≤ α1G(℘, ℘, ℘) + α2G(℘, ℘, ℘) + α3G(℘, ℘, ℘)+ α4[G(℘, ℘, ℘) +G(℘, ℘, ℘) +G(℘, ℘, F3℘)] + α5min { G(℘, ℘, ℘), G(℘, ℘, ℘), G(℘, ℘, ℘), G(℘, ℘, ℘) } ≤ α4G(℘, ℘, F3℘) G(℘, ℘, F3℘)−α4G(℘, ℘, F3℘) ≤ 0, is a contradiction. Since (1−α4) ̸= 0, therefore, G(℘, ℘, F3℘) = 0. Thus, F3℘ = ℘ (7) Thus from (5), (6) and (7), it is proved that ℘ is a CFP of F1, F2 and F3, such that F1℘ = F2℘ = F3℘ = ℘ Uniqueness: Suppose that ℘⋄ ∈ X be the other CFP of F1, F2 and F3, so that F1℘ ⋄ = F2℘ ⋄ = F3℘ ⋄ = ℘⋄ M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 8 of 27 Then from (1) we have that, G(℘, ℘⋄, ℘⋄) = G(F1℘, F2℘ ⋄, F3℘ ⋄) ≤ α1G(℘, ℘⋄, ℘⋄) + α2G(℘, ℘⋄, F2℘ ⋄) + α3G(F1℘, ℘ ⋄, ℘⋄) + α4[G(F1℘, ℘, ℘) +G(℘⋄, F2℘ ⋄, ℘⋄) +G(℘⋄, ℘⋄, F3℘ ⋄)] + α5min { G(℘, ℘⋄, F2℘ ⋄), G(F1℘, F1℘, ℘ ⋄), G(F1℘, ℘ ⋄, ℘⋄), G(F2℘ ⋄, F2℘ ⋄, ℘⋄) } ≤ α1G(℘, ℘⋄, ℘⋄) + α2G(℘, ℘⋄, ℘⋄) + α3G(℘, ℘⋄, ℘⋄) + α4[G(℘, ℘, ℘) +G(℘⋄, ℘⋄, ℘⋄) +G(℘⋄, ℘⋄, ℘⋄)] + α5min { G(℘, ℘⋄, ℘⋄), G(℘, ℘, ℘⋄), G(℘, ℘⋄, ℘⋄), G(℘⋄, ℘⋄, ℘⋄) } ≤ α1G(℘, ℘⋄, ℘⋄) + α2G(℘, ℘⋄, ℘⋄) + α3G(℘, ℘⋄, ℘⋄) (α1 + α2 + α3)G(℘, ℘⋄, ℘⋄)(1− α1 − α2 − α3)G(℘, ℘⋄, ℘⋄) ≤ 0, is a contradiction. Since, (1− α1 − α2 − α3) ̸= 0, therefore G(℘, ℘⋄, ℘⋄) = 0. Thus, ℘ = ℘⋄ It is proved that the three individual self-mappings referred as F1, F2 and F3 possess a UCFP in X. The Algorithm 3 implements the iterative fixed-point computation scheme described in The- orem 1 by sequentially applying the three contractive mappings F1, F2, and F3 while monitoring convergence through the G-metric condition G(x3k, x3k+1, x3k+2) < tolerance. Algorithm 1 Fixed-Point Computation for Tripartite Mappings Require: X: G-metric space, F1, F2, F3: Contractive mappings Require: α1, α2, α3, α4, α5: Theorem parameters Ensure: Common fixed point ℘ 1: Initialize x0 ∈ X 2: for k = 0 to max iterations do 3: x3k+1 ← F1(x3k) 4: x3k+2 ← F2(x3k+1) 5: x3k+3 ← F3(x3k+2) 6: Compute Gk ← G(x3k, x3k+1, x3k+2) 7: if Gk < tolerance then 8: return x3k+3 ▷ Common fixed point found 9: end if 10: end for Figure 1 illustrates the convergence of the G-metric during fixed-point iterations, validating Theorem 1 under contractive conditions. M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 9 of 27 Figure 1: G-metric convergence curve validating Theorem 1 for tripartite mappings The fixed-point framework established in Theorem 1, and implemented through Algorithm 3, finds significant applications in both artificial intelligence and cryptographic protocols. In deep learning architectures, the mappings F1, F2, and F3 may be interpreted as a feature extraction layer, a regularization layer, and an output projection layer, respectively. The contraction pa- rameters αi serve to regulate the stability of these layers, ensuring that the iterative composition converges to a fixed activation pattern as guaranteed by the theoretical result. In cryptography, especially within secure multi-party computation, the same mappings can be understood as follows: F1 represents the encryption transformation, F2 corresponds to the de- cryption process, and F3 performs key mixing. The G-metric quantifies the maximum deviation between transformed message states. Convergence to a fixed point under this metric reflects the achievement of a secure and synchronized outcome across all parties involved. Thus, Theorem 1 and Algorithm 3 jointly ensure the validity and practical effectiveness of this unified framework in both AI and cryptographic systems. Example 1. Let X = [0, 1] with G-metric defined as G(x, y, z) = max{|x− y|, |y − z|, |z − x|}. Consider the piecewise mapping: Fi(x) = { x 4 + 1 2 x ∈ [0, 1] 3x+1 5 x > 1 (i = 1, 2, 3) This mapping satisfies the contractive conditions of Theorem 1, with parameters: α1 = 0.2, α2 = 0.15, α3 = 0.1, α4 = 0.05, α5 = 0.02 The fixed-point iteration applied to this setup yields a unique common fixed point at ℘ = 2 3 , confirming the theorems theoretical predictions through constructive computation. Algorithm 2 Example Implementation for Fi(x) in Theorem 1 1: function ComputeFixedPoint(x0, tol) 2: k ← 0 3: repeat 4: xk+1 ← xk 4 + 1 2 5: error ← |xk+1 − xk| 6: k ← k + 1 7: until error < tol 8: return xk 9: end function M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 10 of 27 This work builds on the fixed-point framework established in Theorem 1, with the algo- rithmic implementation provided above and its correctness illustrated by Example 1 and Algo- rithm 3. The applications in artificial intelligence and cryptographic protocols demonstrate the broader relevance and utility of the theoretical results. Figure 2: Visual verification of fixed-point convergence for the mapping in Example 1 Figure 2 confirms the convergence of the fixed-point iteration in Example 1, aligning with the theoretical result from Theorem 1. Theorem 2. Let (X,G) be a GM-space and F : X ×X ×X → X be a mapping satisfying: G(F1x1, F2x2, F3x3) ≤ α1G(x1, x2, x3) + α2G(x1, x2, F2x2) + α3G(F1x1, x2, x2) + α4[G(F1x1, x1, x1) +G(x2, F2x2, x2) +G(x3, x3, F3x3)] + α5max { G(x1, x2, F2x2), G(F1x1, F1x1, x2), G(F1x1, x2, x2), G(F2x2, F2x2, x3) G(F1x1, x1, x1), G(x2, F2x2, x2), G(x3, x3, F3x3) } (8) for all x1, x2, x3 ∈ X and α1, α2, α3, α4, α5 ≥ 0 with α1, α2, α3, α4, α5 < 1 and (α1 + α2 + α3 + α4 + α5) < 1. Then, the three individual self-mappings referred as F1, F2 and F3 possess a CFP in X. Moreover, if (α1 +α2 +α3 +α4) < 1, the three individual self-mappings referred as F1, F2 and F3 possess a UCFP in X. Proof. Fix x0 ∈ X, and {xk} be a sequence in X. Now we define some iterative sequences in X such that x(3k+1) = F1x3k, x(3k+2) = F2x(3k+1) and x(3k+3) = F3x(3k+2) ∀ k ≥ 0. Now, by using (8) we have, M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 11 of 27 G(x(3k+1), x(3k+2), x(3k+3)) = G(F1x3k, F2x(3k+1), F3x(3k+2)) ≤ α1G(x3k, x(3k+1), x(3k+2)) + α2G(x3k, x(3k+1), F2x(3k+1)) + α3G(F1x3k, x(3k+1), x(3k+1)) + α4[G(F1x3k, x3k, x3k) +G(x(3k+1), F2x(3k+1), x(3k+1)) +G(x(3k+2), x(3k+2), F3x(3k+2))] + α5max  G(x3k, x(3k+1), F2x(3k+1)), G(F1x3k, F1x3k, x(3k+1)), G(F1x3k, x(3k+1), x(3k+1)), G(F2x(3k+1), F2x(3k+1), x(3k+2)) G(F1x3k, x3k, x3k), G(x(3k+1), F2x(3k+1), x(3k+1)), G(x(3k+2), x(3k+2), F3x(3k+2))  = α1G(x3k, x(3k+1), x(3k+2)) + α2G(x3k, x(3k+1), x(3k+1)) + α3G(x3k, x(3k+1), x(3k+1)) + α4[G(x3k, x3k, x3k) +G(x(3k+1), x(3k+1), x(3k+1)) +G(x(3k+2), x(3k+2), x(3k+2))] + α5max  G(x3k, x(3k+1), x(3k+1)), G(x3k, x3k, x(3k+1)), G(x3k, x(3k+1), x(3k+1)), G(x(3k+1), x(3k+1), x(3k+2)) G(F1x3k, x3k, x3k), G(x(3k+1), F2x(3k+1), x(3k+1)), G(x(3k+2), x(3k+2), F3x(3k+2))  After applying the simplification process utilizing the Definition of GM-space we have achieved the following result, ≤ (α1 + α2)G(x3k, x(3k+1), x(3k+2)) + α4[G(x3k, x(3k+1), x(3k+2)) +G(x3k, x(3k+1), x(3k+2)) +G(x(3k+1), x(3k+2), x(3k+3))]+ α5max { G(x3k, x(3k+1), x(3k+2)), G(x3k, x(3k+1, x(3k+2)), G(x3k, x(3k+1), x(3k+2)), G(x(3k+1), x(3k+2), x(3k+3)) } (1− α4)G(x(3k+1), x(3k+2), x(3k+3)) ≤ (α1 + α2 + 2α4)G(x3k, x(3k+1), x(3k+2)) + α5max { G(x3k, x(3k+1), x(3k+2)), G(x(3k+1), x(3k+2), x(3k+3)) } (9) Now, there are two cases: (i). If G(x3k, x(3k+1), x(3k+2)) is a maximum term in (9) then we can write, (1−α4)G(x(3k+1), x(3k+2), x(3k+3)) ≤ (α1+α2+2α4)G(x3k, x(3k+1), x(3k+2))+α5G(x3k, x(3k+1), x(3k+2)) (α1 + α2 + 2α4 + α5)G(x3k, x(3k+1), x(3k+2)) ≤ (α1 + α2 + 2α4 + α5) (1− α4) G(x3k, x(3k+1), x(3k+2)), where γ1 = (α1 + α2 + 2α4 + α5) (1− α4) < 1 G(x(3k+1), x(3k+2), x(3k+3)) ≤ γ1G(x3k, x(3k+1), x(3k+2)) (ii). If G(x(3k+1), x(3k+2), x(3k+3)) is a maximum term in (9) then we can write, (1−α4)G(x(3k+1), x(3k+2), x(3k+3)) ≤ (α1+α2+2α4)G(x3k, x(3k+1), x(3k+2))+α5G(x(3k+1), x(3k+2), x(3k+3)) (1− α4 − α5)G(x(3k+1), x(3k+2), x(3k+3)) ≤ (α1 + α2 + 2α4)G(x3k, x(3k+1), x(3k+2)) M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 12 of 27 ≤ (α1 + α2 + 2α4) (1− α4 − α5) G(x3k, x(3k+1), x(3k+2)), where γ2 = (α1 + α2 + 2α4) (1− α4 − α5) < 1 G(x(3k+1), x(3k+2), x(3k+3)) ≤ γ2G(x3k, x(3k+1), x(3k+2)) From both the cases we get, ∴ γ1 = γ2 = γ G(x(3k+1), x(3k+2), x(3k+3)) ≤ γG(x3k, x(3k+1), x(3k+2)) (10) Similarly, again by using (8), G(x(3k+2), x(3k+3), x(3k+4)) = G(F1x(3k+1), F2x(3k+2), F3x(3k+3)) ≤ α1G(x(3k+1), x(3k+2), x(3k+3)) + α2G(x(3k+2), x(3k+2), F2x(3k+2)) + α3G(F1x(3k+1), x(3k+2), x(3k+2)) + α4[G(F1x(3k+1), x(3k+1), x(3k+3)) +G(x(3k+1), F2x(3k+2), x(3k+3)) +G(x(3k+1), x(3k+2), F3x(3k+3))] + α5max  G(x(3k+1), x(3k+2), F2x(3k+2)), G(F1x(3k+1), F1x(3k+1), x(3k+2)), G(F1x(3k+1), x(3k+2), x(3k+2)), G(F2x(3k+2), F2x(3k+2), x(3k+3)), G(F1x(3k+1), x(3k+1), x(3k+1)), G(x(3k+1), F2x(3k+1), x(3k+1)), G(x(3k+2), x(3k+2), F3x(3k+2))  = α1G(x(3k+1), x(3k+2), x(3k+3)) + α2G(x(3k+1), x(3k+2), x(3k+3)) + α3G(x(3k+2), x(3k+2), x(3k+2)) + α4[G(x(3k+1), x(3k+2), x(3k+3)) +G(x(3k+2), x(3k+3), x(3k+4)) +G(x(3k+1), x(3k+2), x(3k+3))] + α5max  G(x(3k+1), x(3k+2), x(3k+3)), G(x(3k+2), x(3k+2), x(3k+2)), G(x(3k+2), x(3k+2), x(3k+2)), G(x(3k+3), x(3k+3), x(3k+3)), G(x(3k+4), x(3k+3), x(3k+3)), G(x(3k+1), x(3k+2), x(3k+1)), G(x(3k+2), x(3k+2), x(3k+3))  After applying the simplification process utilizing the Definition of GM-space we have achieved the following result, ≤ (α1 + α2)G(x(3k+1), x(3k+2), x(3k+3)) + α4[G(x(3k+1), x(3k+2), x(3k+3)) +G(x(3k+2), x(3k+3), x(3k+4)) +G(x(3k+1), x(3k+2), x(3k+3))] + α5max { G(x(3k+1), x(3k+2), x(3k+3)), G(x(3k+1), x(3k+2), x(3k+3)), G(x(3k+1), x(3k+2), x(3k+3)), G(x(3k+2), x(3k+3), x(3k+4)) } (1− α4)G(x(3k+2), x(3k+3), x(3k+4)) ≤ (α1 + α2 + 2α4)G(x(3k+1), x(3k+2), x(3k+3)) + α5max { G(x(3k+1), x(3k+2), x(3k+3)), G(x(3k+2), x(3k+3), x(3k+4)) } (11) Now, there are two cases: (i). If G(x(3k+1), x(3k+2), x(3k+3)) is a maximum term in (11) then we can write, (1−α4)G(x(3k+2), x(3k+3), x(3k+4)) ≤ (α1+α2+2α4)G(x(3k+1), x(3k+2), x(3k+3))+α5G(x(3k+1), x(3k+2), x(3k+3)) (α1 + α2 + 2α4 + α5)G(x(3k+1), x(3k+2), x(3k+3)) ≤ (α1 + α2 + 2α4 + α5) (1− α4) G(x(3k+1), x(3k+2), x(3k+3)), where γ1 = (α1 + α2 + 2α4 + α5) (1− α4) < 1 M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 13 of 27 G(x(3k+2), x(3k+3), x(3k+4)) ≤ γ1G(x(3k+1), x(3k+2), x(3k+3)) (ii). If G(x(3k+2), x(3k+3), x(3k+4)) is a maximum term in (11) then we can write, (1−α4)G(x(3k+2), x(3k+3), x(3k+4)) ≤ (α1+α2+2α4)G(x(3k+1), x(3k+2), x(3k+3))+α5G(x(3k+2), x(3k+3), x(3k+4)) (1− α4 − α5)G(x(3k+2), x(3k+3), x(3k+4)) ≤ (α1 + α2 + 2α4)G(x(3k+1), x(3k+2), x(3k+3)) ≤ (α1 + α2 + 2α4) (1− α4 − α5) G(x(3k+1), x(3k+2), x(3k+3)), where γ2 = (α1 + α2 + 2α4) (1− α4 − α5) < 1 G(x(3k+2), x(3k+3), x(3k+4)) ≤ γ2G(x(3k+1), x(3k+2), x(3k+3)) From both the cases we get, ∴ γ1 = γ2 = γ G(x(3k+2), x(3k+3), x(3k+4)) ≤ γG(x(3k+1), x(3k+2), x(3k+3)) (12) From (11) and (12) inductively we have, G(x(3k+1), x(3k+2), x(3k+3)) ≤ γ3k+1G(x0, x1, x2) (13) Next we will show that {xk} is a G-Cauchy sequence in X, for any j, n, k with j > n > k. G(xk, xn, xj) ≤ G(xk, x(k+1), x(k+1)) +G(x(k+1), x(k+2), x(k+2)) + ......+G(x(j−1), x(j−1), xj) ≤ G(xk, x(k+1), x(k+2)) +G(x(k+1), x(k+2), x(k+3)) + .......+G(x(j−2), x(j−1), xj) ≤ [yk + y(k+1) + ........+ y(j−2)]G(x0, x1, x2) This implies that, G(xk, xn, xj) ≤ yk (1− y) G(x0, x1, x2) The same hold if j = n > k and if j > n = k we have, G(xk, xn, xj) ≤ yk−1 (1− y) G(x0, x1, x2) If we take the limit as k, n, j →∞ we getG(xk, xn, xj)→ 0. Hence {xk} is a G-Cauchy sequence. By G-completeness of X, there exists ℘ ∈ X such that {xk} converges to ℘ as k →∞. We have to show that F1℘ = ℘ by contrary case let F1℘ ̸= ℘. Then by (8) we have that, G(F1℘, x(3k+2), x(3k+3)) = G(F1℘, F2x(3k+1), F3x(3k+2)) ≤ α1G(℘, x(3k+1), x(3k+2)) + α2G(℘, x(3k+1), F2x(3k+1)) + α3G(F1℘, x(3k+1), x(3k+1)) + α4[G(F1℘, x3k, x3k) +G(x(3k+1), F2x(3k+1), x(3k+1)) +G(x(3k+2), x(3k+2), F3x(3k+2))] + α5max  G(℘, x(3k+1), F2x(3k+1)), G(F1℘, F1℘, x(3k+1)), G(F1℘, x(3k+1), x(3k+1)), G(F2x(3k+1), F2x(3k+1), x(3k+2)), G(F1℘, x3k, x3k), G(x(3k+1), F2x(3k+1), x(3k+1)), G(x(3k+2), x(3k+2), F3x(3k+2))  M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 14 of 27 = α1G(℘, x(3k+1), x(3k+2)) + α2G(℘, x(3k+1), x(3k+2)) + α3G(F1℘, x(3k+1), x(3k+1)) + α4[G(F1℘, x(3k+1), x(3k+2)) +G(x3k, x(3k+2), x(3k+2)) +G(x3k, x(3k+1), x(3k+3))] + α5max  G(℘, x(3k+1), x(3k+2)), G(F1℘, F1℘, x(3k+1)), G(℘, x(3k+1), x(3k+1)), G(x(3k+2), x(3k+2), x(3k+2)), G(F1℘, x3k, x3k), G(x(3k+1), x(3k+2), x(3k+1)), G(x(3k+2), x(3k+2), x(3k+)3)  After applying the simplification process utilizing the Definition of GM-space we have achieved the following result, ≤ α1G(℘, x(3k+1), x(3k+2)) + α2G(F1℘, x(3k+1), x(3k+2)) + α3G(F1℘, x(3k+1), x(3k+1)) + α4[G(F1℘, x3k, x(3k+2)) + 2G(x(3k+1), x(3k+2), x(3k+3))] + α5max { G(℘, x(3k+1), x(3k+2)), G(F1℘, F1℘, x(3k+1)), G(F1℘, x(3k+1), x(3k+1)), G(x(3k+1), x(3k+2), x(3k+3)), G(F1℘, x3k, x(3k+2)) } (14) Now, there are five cases: (i). If G(℘, x(3k+1), x(3k+2)) is a maximum term in (14) then we can write; ≤ α1G(℘, x(3k+1), x(3k+2)) + α2G(F1℘, x(3k+1), x(3k+2)) + α3G(F1℘, x(3k+1), x(3k+1)) + α4[G(F1℘, x3k, x(3k+2)) + 2G(x(3k+1), x(3k+2), x(3k+3))] + α5G(℘, x(3k+1), x(3k+2)) Applying limn→∞ on both sides, we get, G(F1℘, ℘, ℘) ≤ α1G(℘, ℘, ℘) + α2G(F1℘, ℘, ℘) + α3G(F1℘, ℘, ℘) + α4[G(F1℘, ℘, ℘) + 2G(℘, ℘, ℘)] + α5G(℘, ℘, ℘) ≤ (α2 + α3 + α4)G(F1℘, ℘, ℘) G(F1℘, ℘, ℘) − (α2 + α3 + α4)G(F1℘, ℘, ℘) ≤ 0. (1 − α2 − α3 − α4)G(F1℘, ℘, ℘) ≤ 0 , is a contradiction. Since (1− α2 − α3 − α4) ̸= 0, therefore, G(F1℘, ℘, ℘) = 0. Thus, F1℘ = ℘ (15) (ii). If G(F1℘, F1℘, x(3k+1)) is a maximum term in (14) then we can write; ≤ α1G(℘, x(3k+1), x(3k+2)) + α2G(F1℘, x(3k+1), x(3k+2)) + α3G(F1℘, x(3k+1), x(3k+1)) + α4[G(F1℘, x3k, x(3k+2)) + 2G(x(3k+1), x(3k+2), x(3k+3))] + α5G(F1℘, F1℘, x(3k+1)) Applying limn→∞ on both sides, we get, G(F1℘, ℘, ℘) ≤ α1G(℘, ℘, ℘) + α2G(F1℘, ℘, ℘) + α3G(F1℘, ℘, ℘) + α4[G(F1℘, ℘, ℘) + 2G(℘, ℘, ℘)] + α5G(F1℘, F1℘, ℘) ≤ (α2 + α3 + α4)G(F1℘, ℘, ℘) + α5G(F1℘, F1℘, ℘) By using proposition ≤ (α2 + α3 + α4)G(F1℘, ℘, ℘) + α5G(F1℘, ℘, ℘) G(F1℘, ℘, ℘)− (α2 + α3 + α4 + α5)G(F1℘, ℘, ℘) ≤ 0. (1− α2 − α3 − α4 − α5)G(F1℘, ℘, ℘) ≤ 0 , is a contradiction. Since (1− α2 − α3 − α4 − α5) ̸= 0, therefore, G(F1℘, ℘, ℘) = 0. Thus, F1℘ = ℘ (16) M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 15 of 27 (iii). If G(F1℘, x(3k+1), x(3k+1)) is a maximum term in (14) then we can write; ≤ α1G(℘, x(3k+1), x(3k+2)) + α2G(F1℘, x(3k+1), x(3k+2)) + α3G(F1℘, x(3k+1), x(3k+1)) + α4[G(F1℘, x3k, x(3k+2)) + 2G(x(3k+1), x(3k+2), x(3k+3))] + α5G(F1℘, x(3k+1), x(3k+1)) Applying limn→∞ on both sides, we get, G(F1℘, ℘, ℘) ≤ α1G(℘, ℘, ℘) + α2G(F1℘, ℘, ℘) + α3G(F1℘, ℘, ℘) + α4[G(F1℘, ℘, ℘) + 2G(℘, ℘, ℘)] + α5G(F1℘, ℘, ℘) ≤ (α2 + α3 + α4 + α5)G(F1℘, ℘, ℘) G(F1℘, ℘, ℘)− (α2 + α3 + α4 + α5)G(F1℘, ℘, ℘) ≤ 0. (1− α2 − α3 − α4 − α5)G(F1℘, ℘, ℘) ≤ 0 , is a contradiction. Since (1− α2 − α3 − α4 − α5) ̸= 0, therefore, G(F1℘, ℘, ℘) = 0. Thus, F1℘ = ℘ (17) (iv). If G(F1℘, x3k, x(3k+2)) is a maximum term in (14) then we can write; ≤ α1G(℘, x(3k+1), x(3k+2)) + α2G(F1℘, x(3k+1), x(3k+2)) + α3G(F1℘, x(3k+1), x(3k+1)) + α4[G(F1℘, x3k, x(3k+2)) + 2G(x(3k+1), x(3k+2), x(3k+3))] + α5G(F1℘, x3k, x(3k+2)) Applying limn→∞ on both sides, we get, G(F1℘, ℘, ℘) ≤ α1G(℘, ℘, ℘) + α2G(F1℘, ℘, ℘) + α3G(F1℘, ℘, ℘) + α4[G(F1℘, ℘, ℘) + 2G(℘, ℘, ℘)] + α5G(F1℘, ℘, ℘) ≤ (α2 + α3 + α4 + α5)G(F1℘, ℘, ℘) G(F1℘, ℘, ℘)− (α2 + α3 + α4 + α5)G(F1℘, ℘, ℘) ≤ 0. (1− α2 − α3 − α4 − α5)G(F1℘, ℘, ℘) ≤ 0 , is a contradiction. Since (1− α2 − α3 − α4 − α5) ̸= 0, therefore, G(F1℘, ℘, ℘) = 0. Thus, F1℘ = ℘ (18) (v). If G(x(3k+1), x(3k+2), x(3k+3)) is a maximum term in (14) then we can write; ≤ α1G(℘, x(3k+1), x(3k+2)) + α2G(F1℘, x(3k+1), x(3k+2)) + α3G(F1℘, x(3k+1), x(3k+1)) + α4[G(F1℘, x3k, x(3k+2)) + 2G(x(3k+1), x(3k+2), x(3k+3))] + α5G(x(3k+1), x(3k+2), x(3k+3)) Applying limn→∞ on both sides, we get, G(F1℘, ℘, ℘) ≤ α1G(℘, ℘, ℘) + α2G(F1℘, ℘, ℘) + α3G(F1℘, ℘, ℘) + α4[G(F1℘, ℘, ℘) + 2G(℘, ℘, ℘)] + α5G(℘, ℘, ℘) ≤ (α2 + α3 + α4)G(F1℘, ℘, ℘) G(F1℘, ℘, ℘) − (α2 + α3 + α4)G(F1℘, ℘, ℘) ≤ 0. (1 − α2 − α3 − α4)G(F1℘, ℘, ℘) ≤ 0 , is a contradiction. Since (1− α2 − α3 − α4) ̸= 0, therefore, G(F1℘, ℘, ℘) = 0. Thus, F1℘ = ℘ (19) By the view of (15)− (19), we obtain that ℘ is a FP of F1, F2 and F3, i.e, F1℘ = ℘ (20) M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 16 of 27 Next, we have to show that F2℘ = ℘ by contrary case let F2℘ ̸= ℘. Then from (8) we have that, G(x(3k+1), F2℘, x(3k+3)) = G(F1x3k, F2℘, F3x(3k+2)) ≤ α1G(x3k, ℘, x(3k+2)) + α2G(x3k, ℘, F2℘) + α3G(F1x3k, ℘, ℘) + α4[G(F1x3k, ℘, x(3k+2)) +G(x3k, F2℘, x(3k+2)) +G(x3k, ℘, F3x(3k+2))] + α5max  G(x3k, ℘, F2℘), G(F1x3k, F1x3k, ℘), G(F1x3k, ℘, ℘), G(F2℘, F2℘, ℘), G(F1x3k, x3k, x3k), G(℘, F2℘, ℘), G(x(3k+2)), x(3k+2)), F3x(3k+2))  = α1G(x3k, ℘, x(3k+2)) + α2G(x3k, ℘, F2℘) + α3G(x(3k+1), ℘, ℘) + α4[G(x(3k+1), ℘, x(3k+2)) +G(x3k, F2℘, x(3k+2)) +G(x3k, ℘, x(3k+2))] + α5max  G(x3k, ℘, F2℘), G(x3k, x3k, ℘), G(x3k, ℘, ℘), G(F2℘, F2℘, x(3k+2)), G(x3k, x3k, x3k), G(℘, F2℘, ℘), G(x(3k+2)), x(3k+2)), x(3k+2))  (21) Now there are seven cases: (i). If G(x3k, ℘, F2℘) is a maximum term in (21) then we can write; ≤ α1G(x3k, ℘, x(3k+2)) + α2G(x3k, ℘, F2℘) + α3G(x(3k+1), ℘, ℘) + α4[G(x(3k+1), ℘, x(3k+2)) +G(x3k, F2℘, x(3k+2)) +G(x3k, ℘, x(3k+2))] + α5G(x3k, ℘, F2℘) Applying limn→∞ on both sides, we get, G(℘, F2℘, ℘) ≤ α1G(℘, ℘, ℘) + α2G(℘, ℘, F2℘) + α3G(℘, ℘, ℘)+ α4[G(℘, ℘, ℘) +G(℘, F2℘, ℘) +G(℘, ℘, ℘)] + α5G(℘, ℘, F2℘) ≤ α2 + α4 + α5)G(℘, ℘, F2℘) G(℘, F2℘, ℘) − (α2 + α4 + α5)G(℘, ℘, F2℘) ≤ 0. (1 − α2 − α4 − α5)G(℘, ℘, F2℘) ≤ 0 is a contradiction. Since (1− α2 − α4 − α5) ̸= 0, therefore, G(F2℘, ℘, ℘) = 0. Thus, F2℘ = ℘ (22) (ii). If G(x(3k+1), x(3k+1), ℘) is a maximum term in (21) then we can write; ≤ α1G(x3k, ℘, x(3k+2)) + α2G(x3k, ℘, F2℘) + α3G(x(3k+1), ℘, ℘) + α4[G(x(3k+1), ℘, x(3k+2)) +G(x3k, F2℘, x(3k+2)) +G(x3k, ℘, x(3k+2))] + α5G(x(3k+1), x(3k+1), ℘) Applying limn→∞ on both sides, we get, G(℘, F2℘, ℘) ≤ α1G(℘, ℘, ℘) + α2G(℘, ℘, F2℘) + α3G(℘, ℘, ℘)+ α4[G(℘, ℘, ℘) +G(℘, F2℘, ℘) +G(℘, ℘, ℘)] + α5G(℘, ℘, ℘) ≤ α2 + α4)G(℘, ℘, F2℘) G(℘, F2℘, ℘) − (α2 + α4)G(℘, ℘, F2℘) ≤ 0. (1 − α2 − α4)G(℘, ℘, F2℘) ≤ 0 is a contradiction. Since (1− α2 − α4) ̸= 0, therefore, G(F2℘, ℘, ℘) = 0. Thus, F2℘ = ℘ (23) M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 17 of 27 (iii). If G(x(3k+1), ℘, ℘) is a maximum term in (21) then we can write; ≤ α1G(x3k, ℘, x(3k+2)) + α2G(x3k, ℘, F2℘) + α3G(x(3k+1), ℘, ℘) + α4[G(x(3k+1), ℘, x(3k+2)) +G(x3k, F2℘, x(3k+2)) +G(x3k, ℘, x(3k+2))] + α5G(x(3k+1), ℘, ℘) Applying limn→∞ on both sides, we get, G(℘, F2℘, ℘) ≤ α1G(℘, ℘, ℘) + α2G(℘, ℘, F2℘) + α3G(℘, ℘, ℘)+ α4[G(℘, ℘, ℘) +G(℘, F2℘, ℘) +G(℘, ℘, ℘)] + α5G(℘, ℘, ℘) ≤ α2 + α4)G(℘, ℘, F2℘) G(℘, F2℘, ℘) − (α2 + α4)G(℘, ℘, F2℘) ≤ 0. (1 − α2 − α4)G(℘, ℘, F2℘) ≤ 0 is a contradiction. Since (1− α2 − α4) ̸= 0, therefore, G(F2℘, ℘, ℘) = 0. Thus, F2℘ = ℘ (24) (iv). If G(F2℘, F2℘, ℘) is a maximum term in (21) then we can write; ≤ α1G(x3k, ℘, x(3k+2)) + α2G(x3k, ℘, F2℘) + α3G(x(3k+1), ℘, ℘) + α4[G(x(3k+1), ℘, x(3k+2)) +G(x3k, F2℘, x(3k+2)) +G(x3k, ℘, x(3k+2))] + α5G(F2℘, F2℘, ℘) Applying limn→∞ on both sides, we get, G(℘, F2℘, ℘) ≤ α1G(℘, ℘, ℘) + α2G(℘, ℘, F2℘) + α3G(℘, ℘, ℘)+ α4[G(℘, ℘, ℘) +G(℘, F2℘, ℘) +G(℘, ℘, ℘)] + α5G(F2℘, F2℘, ℘) ≤ α2 + α4)G(℘, ℘, F2℘) + α5G(F2℘, F2℘, ℘) By using proposition ≤ α2 + α4 + α5)G(℘, ℘, F2℘) G(℘, F2℘, ℘) − (α2 + α4 + α5)G(℘, ℘, F2℘) ≤ 0. (1 − α2 − α4 − α5)G(℘, ℘, F2℘) ≤ 0 is a contradiction. Since (1− α2 − α4 − α5) ̸= 0, therefore, G(F2℘, ℘, ℘) = 0. Thus, F2℘ = ℘ (25) (v). If G(x(3k+1), x3k, x3k) is a maximum term in (21) then we can write; ≤ α1G(x3k, ℘, x(3k+2)) + α2G(x3k, ℘, F2℘) + α3G(x(3k+1), ℘, ℘) + α4[G(x(3k+1), ℘, x(3k+2)) +G(x3k, F2℘, x(3k+2)) +G(x3k, ℘, x(3k+2))] + α5G(x(3k+1), x3k, x3k) Applying limn→∞ on both sides, we get, G(℘, F2℘, ℘) ≤ α1G(℘, ℘, ℘) + α2G(℘, ℘, F2℘) + α3G(℘, ℘, ℘)+ α4[G(℘, ℘, ℘) +G(℘, F2℘, ℘) +G(℘, ℘, ℘)] + α5G(℘, ℘, ℘) ≤ α2 + α4)G(℘, ℘, F2℘) G(℘, F2℘, ℘) − (α2 + α4)G(℘, ℘, F2℘) ≤ 0. (1 − α2 − α4)G(℘, ℘, F2℘) ≤ 0 is a contradiction. Since (1− α2 − α4) ̸= 0, therefore, G(F2℘, ℘, ℘) = 0. Thus, F2℘ = ℘ (26) M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 18 of 27 (vi). If G(℘, F2℘, ℘) is a maximum term in (21) then we can write; ≤ α1G(x3k, ℘, x(3k+2)) + α2G(x3k, ℘, F2℘) + α3G(x(3k+1), ℘, ℘) + α4[G(x(3k+1), ℘, x(3k+2)) +G(x3k, F2℘, x(3k+2)) +G(x3k, ℘, x(3k+2))] + α5G(F2℘, F2℘, ℘) Applying limn→∞ on both sides, we get, G(℘, F2℘, ℘) ≤ α1G(℘, ℘, ℘) + α2G(℘, ℘, F2℘) + α3G(℘, ℘, ℘)+ α4[G(℘, ℘, ℘) +G(℘, F2℘, ℘) +G(℘, ℘, ℘)] + α5G(℘, F2℘, ℘) ≤ α2 + α4 + α5)G(℘, ℘, F2℘) G(℘, F2℘, ℘) − (α2 + α4 + α5)G(℘, ℘, F2℘) ≤ 0. (1 − α2 − α4 − α5)G(℘, ℘, F2℘) ≤ 0 is a contradiction. Since (1− α2 − α4 − α5) ̸= 0, therefore, G(F2℘, ℘, ℘) = 0. Thus, F2℘ = ℘ (27) (vii). If G(x(3k+2), x(3k+2), x(3k+3)) is a maximum term in (21) then we can write; ≤ α1G(x3k, ℘, x(3k+2)) + α2G(x3k, ℘, F2℘) + α3G(x(3k+1), ℘, ℘) + α4[G(x(3k+1), ℘, x(3k+2)) +G(x3k, F2℘, x(3k+2)) +G(x3k, ℘, x(3k+2))] + α5G(x(3k+2), x(3k+2), x(3k+3)) Applying limn→∞ on both sides, we get, G(℘, F2℘, ℘) ≤ α1G(℘, ℘, ℘) + α2G(℘, ℘, F2℘) + α3G(℘, ℘, ℘)+ α4[G(℘, ℘, ℘) +G(℘, F2℘, ℘) +G(℘, ℘, ℘)] + α5G(℘, ℘, ℘) ≤ α2 + α4)G(℘, ℘, F2℘) G(℘, F2℘, ℘) − (α2 + α4)G(℘, ℘, F2℘) ≤ 0. (1 − α2 − α4)G(℘, ℘, F2℘) ≤ 0 is a contradiction. Since (1− α2 − α4) ̸= 0, therefore, G(F2℘, ℘, ℘) = 0. Thus, F2℘ = ℘ (28) By the view of (22)− (28), we obtain that ℘ is a FP of F1, F2 and F3, i.e, F2℘ = ℘ (29) Next, we have to show that F3℘ = ℘ by contrary case let F3℘ ̸= ℘. Then from (8) we have that, G(x(3k+1), x(3k+2), F3℘) = G(F1x3k, F2x(3k+1), F3℘) ≤ α1G(x3k, x(3k+1), ℘) + α2G(x3k, x(3k+1), F2x(3k+1)) + α3G(F1x3k, x(3k+1), x(3k+1)) + α4[G(F1x3k, x(3k+1), ℘) +G(x3k, F2x(3k+1), ℘) +G(x3k, x(3k+1), ℘)] + α5max  G(x3k, x(3k+1), F2x(3k+1)), G(F1x3k, F1x3k, x(3k+1)), G(F1x3k, x(3k+1), x(3k+1)), G(F2x(3k+1), F2x(3k+1), ℘), G(F1x(3k+1), x3k, x3k), G(x(3k+1), F2x(3k+1), x(3k+1)), G(℘, ℘, F3℘)  = α1G(x3k, x(3k+1), ℘) + α2G(x3k, x(3k+1), x(3k+1))+ α3G(x3k, x(3k+1), x(3k+1)) + α4[G(x3k, x3k, x3k) +G(x(3k+1), x(3k+1), x(3k+1)) +G(℘, ℘, F3℘)] + α5max  G(x3k, x(3k+1), x(3k+1)), G(x3k, x3k, x(3k+1)), G(x3k, x(3k+1), x(3k+1)), G(x(3k+1), x(3k+1), x(3k + 2)), G(x(3k+1), x3k, x3k), G(x(3k+1), x(3k+1), x(3k+1)), G(℘, ℘, F3℘)  M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 19 of 27 After applying the simplification process utilizing the Definition of GM-space we have achieved the following result, = α1G(x3k, x(3k+1), ℘) + α2G(x3k, x(3k+1), x(3k+1)) + α4[G(x3k, x3k, x3k) +G(x(3k+1), x(3k+1), x(3k+1)) +G(℘, ℘, F3℘)] + α5max { G(x3k, x(3k+1), x(3k+1)), G(x(3k+2), x(3k+2), ℘), G(x(3k+1), x(3k+2), x(3k+3)), G(℘, ℘, F3℘) } (30) Now there are four cases: (i). If G(x3k, x(3k+1), x(3k+1)) is a maximum term in (30) then we can write; = α1G(x3k, x(3k+1), ℘) + α2G(x3k, x(3k+1), x(3k+1))+ α4[G(x3k, x3k, x3k) +G(x(3k+1), x(3k+1), x(3k+1)) +G(℘, ℘, F3℘)] + α5G(x3k, x(3k+1), x(3k+2)) Applying limn→∞ on both sides, we get, G(℘, ℘, F3℘) ≤ α1G(℘, ℘, ℘) + α2G(℘, ℘, ℘)+ α4[G(℘, ℘, ℘) +G(℘, ℘, ℘) +G(℘, ℘, F3℘)] + α5G(℘, ℘, ℘) We get that, G(℘, ℘, F3℘) = 0. Thus, F3℘ = ℘ (31) (ii). If G(x(3k+2), x(3k+2), ℘) is a maximum term in (30) then we can write; = α1G(x3k, x(3k+1), ℘) + α2G(x3k, x(3k+1), x(3k+1))+ α4[G(x3k, x3k, x3k) +G(x(3k+1), x(3k+1), x(3k+1)) +G(℘, ℘, F3℘)] + α5G(x(3k+1), x(3k+2), ℘) Applying limn→∞ on both sides, we get, G(℘, ℘, F3℘) ≤ α1G(℘, ℘, ℘) + α2G(℘, ℘, ℘)+ α4[G(℘, ℘, ℘) +G(℘, ℘, ℘) +G(℘, ℘, F3℘)] + α5G(℘, ℘, ℘) We get that, G(℘, ℘, F3℘) = 0. Thus, F3℘ = ℘ (32) (iii). If G(x(3k+1), x(3k+2), x(3k+3)) is a maximum term in (30) then we can write; = α1G(x3k, x(3k+1), ℘) + α2G(x3k, x(3k+1), x(3k+1))+ α4[G(x3k, x3k, x3k) +G(x(3k+1), x(3k+1), x(3k+1)) +G(℘, ℘, F3℘)] + α5G(x(3k+1), x(3k+2), x(3k+3)) Applying limn→∞ on both sides, we get, G(℘, ℘, F3℘) ≤ α1G(℘, ℘, ℘) + α2G(℘, ℘, ℘)+ α4[G(℘, ℘, ℘) +G(℘, ℘, ℘) +G(℘, ℘, F3℘)] + α5G(℘, ℘, ℘) We get that, G(℘, ℘, F3℘) = 0. Thus, F3℘ = ℘ (33) (iv). If G(℘, ℘, F3℘) is a maximum term in (30) then we can write; = α1G(x3k, x(3k+1), ℘) + α2G(x3k, x(3k+1), x(3k+1))+ α4[G(x3k, x3k, x3k) +G(x(3k+1), x(3k+1), x(3k+1)) +G(℘, ℘, F3℘)] + α5G(x(3k+1), x(3k+2), x(3k+3)) M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 20 of 27 Applying limn→∞ on both sides, we get, G(℘, ℘, F3℘) ≤ α1G(℘, ℘, ℘) + α2G(℘, ℘, ℘)+ α4[G(℘, ℘, ℘) +G(℘, ℘, ℘) +G(℘, ℘, F3℘)] + α5G(℘, ℘, F3℘) G(℘, ℘, F3℘)−α5G(℘, ℘, F3℘) ≤ 0. (1−α5)G(℘, ℘, F3℘) ≤ 0 is a contradiction. Since (1−α5) ̸= 0, therefore, G(F2℘, ℘, ℘) = 0. Thus, F3℘ = ℘ (34) By the view of (31)− (34), we obtain that ℘ is a FP of F1, F2 and F3, i.e, F3℘ = ℘ (35) Thus from (15), (34) and (35), it is proved that ℘ is a CFP of F1, F2 and F3, such that F1℘ = F2℘ = F3℘ = ℘ Uniqueness: Suppose that ℘⋄ ∈ X be the other CFP of F1, F2 and F3, so that F1℘ ⋄ = F2℘ ⋄ = F3℘ ⋄ = ℘⋄ Then from (8) we have that, G(℘, ℘⋄, ℘⋄) = G(F1℘, F2℘ ⋄, F3℘ ⋄) ≤ α1G(℘, ℘⋄, ℘⋄) + α2G(℘, ℘⋄, F2℘ ⋄) + α3G(F1℘, ℘ ⋄, ℘⋄) + α4[G(F1℘, ℘, ℘) +G(℘⋄, F2℘ ⋄, ℘⋄) +G(℘⋄, ℘⋄, F3℘ ⋄)] + α5max  G(℘, ℘⋄, F2℘ ⋄), G(F1℘, F1℘, ℘ ⋄), G(F1℘, ℘ ⋄, ℘⋄), G(F2℘ ⋄, F2℘ ⋄, ℘⋄), G(F1℘, ℘, ℘), G(F2℘ ⋄, F2℘ ⋄, ℘⋄), G(℘⋄, ℘⋄, F3℘ ⋄)  ≤ α1G(℘, ℘⋄, ℘⋄) + α2G(℘, ℘⋄, ℘⋄) + α3G(℘, ℘⋄, ℘⋄) + α4[G(℘, ℘, ℘) +G(℘⋄, ℘⋄, ℘⋄) +G(℘⋄, ℘⋄, ℘⋄)] + α5max  G(℘, ℘⋄, ℘⋄), G(℘, ℘, ℘⋄), G(℘, ℘⋄, ℘⋄), G(℘⋄, ℘⋄, ℘⋄), G(℘, ℘, ℘), G(F2℘ ⋄, ℘⋄, ℘⋄), G(℘⋄, ℘⋄, ℘⋄)  After applying the simplification process utilizing the Definition of GM-space we have achieved the following result, ≤ α1G(℘, ℘⋄, ℘⋄) + α2G(℘, ℘⋄, ℘⋄) + α3G(℘, ℘⋄, ℘⋄) + α4[G(℘, ℘, ℘) +G(℘⋄, ℘⋄, ℘⋄) +G(℘⋄, ℘⋄, ℘⋄)] + α5max  G(℘, ℘⋄, ℘⋄), G(℘, ℘, ℘⋄), G(℘, ℘⋄, ℘⋄), G(℘⋄, ℘⋄, ℘⋄), G(℘, ℘, ℘), G(F2℘ ⋄, ℘⋄, ℘⋄), G(℘⋄, ℘⋄, ℘⋄)  ≤ α1G(℘, ℘⋄, ℘⋄) + α2G(℘, ℘⋄, ℘⋄) + α3G(℘, ℘⋄, ℘⋄) + α5G(℘, ℘⋄, ℘⋄) (α1 + α2 + α3 + α5)G(℘, ℘⋄, ℘⋄)(1− α1 − α2 − α3 − α5)G(℘, ℘⋄, ℘⋄) ≤ 0, M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 21 of 27 is a contradiction. Since, (1− α1 − α2 − α3 − α5) ̸= 0, therefore G(℘, ℘⋄, ℘⋄) = 0. Thus, ℘ = ℘⋄ It is proved that the three individual self-mappings referred as F1, F2 and F3 possess a UCFP in X. The Theorem 2 provides conditions under which three self-mappings F1, F2, F3 on a G-metric space (X,G) have a unique common fixed point. Below is an Algorithm 3 representation of the fixed-point iteration process: Algorithm 3 Common Fixed-Point Iteration for Three Mappings 1: Input: • A G-metric space (X,G) • Three self-mappings F1, F2, F3 : X ×X ×X → X • An initial point x0 ∈ X • Coefficients α1, α2, α3, α4, α5 satisfying: α1 + α2 + α3 + α4 + α5 < 1 2: Initialize: • Set k = 0 • Define iterative sequences: x3k+1 = F1x3k, x3k+2 = F2x3k+1, x3k+3 = F3x3k+2 3: Iterate: • Compute xk+1 using the above definitions • Check contraction condition: G(xk+1, xk+2, xk+3) ≤ γG(xk, xk+1, xk+2) where γ = max ( α1 + α2 + 2α4 + α5 1− α4 , α1 + α2 + 2α4 1− α4 − α5 ) < 1 • Repeat until G(xk, xk+1, xk+2) < ϵ for tolerance ϵ > 0 4: Output: The limit ℘ = limk→∞ xk, the unique common fixed point of F1, F2, F3 M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 22 of 27 Figure 3: Convergence of Fixed-Point Iteration in G-Metric Space with Applications in AI and Cryptography Figure 3 depicts the convergence behavior of a fixed-point iteration scheme governed by Theorem 2 in a G-metric space. This visual representation not only affirms the theoretical con- vergence conditions but also demonstrates the models strength through real-world simulations in AI consensus modeling, cryptographic secure aggregation, and fixed-point learning in neural networkshighlighting both the validity and superiority of the proposed method. 1. Application in AI: Multi-Agent Systems • Scenario: Three AI agents F1, F2, F3 collaboratively optimize a shared objective (e.g., in reinforcement learning). Each agent updates its policy based on others’ outputs. • Fixed-Point Interpretation: The theorem ensures iterative updates converge to a unique shared solution under contraction conditions. • Example: In federated learning, three models F1, F2, F3 represent local updates on different devices. The theorem guarantees convergence to a consensus model. 2. Application in Cryptography: Secure Multi-Party Computation • Scenario: Three parties F1, F2, F3 compute a function f(x) collaboratively without revealing private inputs, using iterative encrypted updates. • Fixed-Point Interpretation: The protocol converges to a unique fixed point (correct output of f(x)) even with noisy updates, provided contraction conditions hold. • Example: In blockchain smart contracts, three oracles F1, F2, F3 aggregate data. The theorem prevents divergence/manipulation by ensuring unique convergence. 3. Application in AI: Neural Network Fixed-Point Learning • Scenario: A neural network with three sub-modules F1 (encoder), F2 (decoder), and F3 (attention mechanism) learns stable representations. • Fixed-Point Interpretation: Iterative application of modules converges to unique representations under theorem conditions, ensuring stability in deep learning. M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 23 of 27 Example 2. Let (X,G) be a G-metric space, where X ∈ [0, 1] and G : X ×X ×X → R, with G(x1, x2, x3) = max{|x1 − x2|, |x2 − x3|, |x3 − x1|}, for all x1, x2, x3 ∈ X. Now we define the three self-mappings, F1, F2, F3 : X → Xby F1x = F2x = F3x = { 3x 5 + 2 5 forall x ∈ [0, 1] 6x+2 7 for xin[1,∞) (36) Then, we get that, G(F1x, F2x, F3x) = 3 5 max{|x1 − x2|, |x2 − x3|, |x3 − x1|} G(F1x, F2x, F3x) = 3 5 G(x1, x2, x3) (37) Now from (36) we have that: G(x1, x2, F2x2) = 1 5 max{5|x1 − x2|, 2|x2 − 1|, |3x2 + 2− 5x1|}, G(F1x1, x2, x2) = |F1x1 − x2| = 1 5 |3x1 + 2− 5x2|, G(F1x1, x1, x1) = |x1 − F1x1| = 2 5 |x1 − 1|, G(x2, F2x2, x2) = |x2 − F2x2| = 2 5 |x2 − 1|, G(x3, x3, F3x3) = |x3 − F3x3| = 2 5 |x3 − 1|, G(F1x1, F1x1, x2) = G(F1x1, x2, x2) = |F1x1 − x2| = 1 5 |3x1 + 2− 5x2|, G(F2x2, F2x2, x3) = G(F2x2, x3, x3) = |F2x2 − x3| = 1 5 |3x2 + 2− 5x3|. Now from (8) and (37) we have that G(F1x1, F2x2, F3x3) = 3 5 G(x1, x2, x3) ≤ 1 5 ( 3 5 G(x1, x2, x3)) + 1 8 ( 1 5 max{5|x1 − x2|, 2|x2 − 1|, |3x2 + 2− 5x1|}) + 1 10 ( 1 5 |3x1 + 2− 5x2|) + 1 30 ( 2 5 [|x1 − 1|+ |x2 − 1|+ |x3 − 1|]) + 1 40 max  ( 1 5 max{5|x1 − x2|, 2|x2 − 1|, |3x2 + 2− 5x1|}, 1 5 |3x1 + 2− 5x2|, 1 5 |3x1 + 2− 5x2|, 1 5 |3x2 + 2− 5x3|, 2 5 |x1 − 1|, 2 5 |x2 − 1|, 2 5 |x3 − 1|)  1 5 G(x1, x2, x3) + 1 8 G(x1, x2, F2x2) + 1 10 G(F1x1, x2, x2)+ 1 30 [G(F1x1, x1, x1) +G(x2, F2x2, x2) +G(x3, x3, F3x3)] + 1 40 max { G(x1, x2, F2x2), G(F1x1, F1x1, x2), G(F1x1, x2, x2), G(F2x2, F2x2, x3), G(F1x1, x1, x1), G(x2, F2x2, x2), G(x3, x3, F3x3) } Hence, it satisfied all the conditions of Theorem 2 with α1 = 1 5 , α2 = 1 8 , α3 = 2 5 , α4 = 1 20 , α5 = 1 40 . i.e. α1, α2, α3, α4, α5 = 20 40 = 0.5 < 1 and α1, α2, α3, α4, α5 = 30 40 = 0.75 < 1, under these conditions, the three individual self-mappings referred as F1, F2 and F3 possess a unique CFP within F that is 2 ∈ [0,∞). M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 24 of 27 Example 3. Let (X,G) be a G-metric space, where X ∈ [0, 2] represents either normalized neu- ral activations in an AI interpretation or scaled cryptographic message values in a cryptographic interpretation. The G-metric is defined as: G(x1, x2, x3) = max{|x1 − x2|, |x2 − x3|, |x3 − x1|} Define three contractive mappings with modified parameters: Fix = { 2x 7 + 3 7 for x ∈ [0, 2] 5x+1 8 for x ∈ (2,∞) for i = 1, 2, 3 For x1, x2, x3 ∈ [0, 2], we verify: G(F1x, F2x, F3x) = 2 7 max{|x1 − x2|, |x2 − x3|, |x3 − x1|} = 2 7 G(x1, x2, x3) (Improved contraction) For cryptographic protocols: G(x1, x2, F2x2) = 1 7 max{7|x1 − x2|, 3|x2 − 1|, |2x2 + 3− 7x1|} G(F1x1, x2, x2) = 1 7 |2x1 + 3− 7x2| (Tighter bound) The modified coefficients satisfy: 1 7 G(x1, x2, x3) + 1 10 G(x1, x2, F2x2) + 1 14 G(F1x1, x2, x2) + 1 35 [G(F1x1, x1, x1) +G(x2, F2x2, x2) +G(x3, x3, F3x3)] + 1 50 max  G(x1, x2, F2x2), G(F1x1, F1x1, x2), G(F1x1, x2, x2), G(F2x2, F2x2, x3), G(F1x1, x1, x1), G(x2, F2x2, x2), G(x3, x3, F3x3)  < 1 The new parameters satisfy Theorem 2 with: α1 = 1 7 , α2 = 1 10 , α3 = 2 7 , α4 = 1 35 , α5 = 1 50 The unique common fixed point ℘ = 1 (solved by setting x = 2x 7 + 3 7) represents: The fixed- point result yields a stable activation value of 1.0 in neural networks and a secure consensus value in cryptographic protocols. Algorithm 4 Fixed-Point Computation for AI/Crypto Systems 1: Initialize x0 ∈ [0, 1] (input data/encrypted message) 2: for k = 0 to max iter do 3: xk+1 ← F3(F2(F1(xk))) ▷ AI: Layer composition / Crypto: Protocol round 4: if G(xk, xk+1, xk+2) < ϵ then 5: return xk+1 ▷ Fixed point reached 6: end if 7: end for Example 3 and Algorithm 3 together demonstrate the convergence of fixed-point iterations in a G-metric space, representing either stable neural activation in AI systems or secure consensus values in cryptographic protocols. M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 25 of 27 Figure 4: Fixed-point convergence in G-metric space using the mapping. Figure 4 demonstrates the convergence of the fixed-point iteration governed by the contrac- tion condition in a G-metric space, validating the theoretical result numerically. 4. Conclusion The field of fixed-point theory is a well-established and substantial area of analysis with nu- merous applications across various disciplines. Within this field, particular attention has been given to the study of contractive-type inequalities. In this work, we establish conditions un- der which three self-mappings in generalized metric spaces (GM-spaces) possess common fixed points (CFP) and unique common fixed points (UCFP). These results present key inequali- ties involving specific constants that guarantee the existence of CFP and UCFP under certain assumptions. Through an illustrative example, we demonstrate the applicability of these condi- tions to a GM-metric defined on the interval [0, 1], thereby confirming the existence of a UCFP for the given mappings. This study contributes to the advancement of fixed-point theory in GM-spaces and high- lights the importance of metric-related conditions in determining the existence of fixed points in such spaces. The findings also build upon and expand recent developments in the existing literature. Moreover, the implications of these results extend beyond pure mathematics: in artificial intelligence, fixed-point iterations in GM-spaces can model convergence behavior in deep learning layers, iterative reasoning in agents, and equilibrium states in feedback-driven models. Similarly, in cryptography, such fixed points can represent secure consensus states in distributed protocols, iterative agreement in key-exchange mechanisms, or stability in homo- morphic encryption loops. Future research in fixed-point theory, particularly within the framework of GM-spaces, could explore several promising directions. One avenue involves extending contractive-type inequali- ties to more general settings, such as non-metric spaces or spaces with weaker topological struc- tures. Investigating the relationships between various types of inequalities, including newly formulated generalized contractive conditions, may deepen the theoretical understanding of fixed-point phenomena across broader contexts. Another potential direction is the applica- tion of these fixed-point results to practical problems in fields such as optimization, machine learning, and secure computation, where fixed points are fundamental to algorithm design and decision-making. Moreover, generalizing the current results to accommodate more complex classes of map- pings, such as non-linear or non-continuous mappings, could yield valuable tools and insights M. Noorwali et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6321 26 of 27 for researchers. Finally, examining the robustness of fixed-point results in GM-spaces under perturbations or uncertainty could enhance their practical relevance, particularly in real-world scenarios where ideal conditions may not be met. By expanding the scope of GM-spaces and refining the criteria for the existence of CFP and UCFP, this line of research holds considerable potential to contribute significantly to both theoretical and applied mathematics—including emerging domains like adaptive AI control and privacy-preserving cryptographic infrastructure. Acknowledgements (i) The project was funded by the KAU Endowment (WAQF) at King Abdulaziz University, Jeddah, Saudi Arabia. The authors, therefore, acknowledge WAQF and the Deanship of Scientific Research (DSR) for technical and financial support. (ii) We acknowledge Ho Chi Minh City University of Technology (HCMUT), VNU-HCM for supporting this study. (iii) The authors extend their appreciation to the Deanship of Scientific Research at Northern Border University, Arar, KSA for funding this research work through the project number “NBU-FFR-2025-2727-07”. Author Contributions: All authors equally contributed. 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