Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 113 https://internationalpubls.com Some Common Fixed Point Theorems in Neutrosophic Metric Spaces Pandiselvi. M1, Jeyaraman. M2 1Research Scholar, PG and Research Department of Mathematics, Raja Doraisingam Govt. Arts College, Sivagangai, Affiliated to Alagappa University, Karaikudi, Tamil Nadu, India; e-mail mpandiselvi2612@gmail.com, ORCID: orcid.org/0000-0003-0210-8843. 2Associate Professor, PG and Research Department of Mathematics, Raja Doraisingam Govt. Arts College, Sivagangai, Affiliated to Alagappa University, Karaikudi, Tamil Nadu, India; e-mail jeya.math@gmail.com, ORCID: orcid.org/0000-0002-0364-1845. Article History: Received: 09-07-2024 Revised: 23-08-2024 Accepted: 05-09-2024 Abstract: In this article, we construct some fixed point results for pair of self mappings and occasionally weakly compatible mappings on neutrosophic metric spaces. In order to show the strength of these results, some motivating examples are established as well. Keywords: Fuzzy metric, Neutrosophic metric space, Occasionally weakly compatible, Self mapping. 1. Introduction The concept of metric spaces and the Banach contraction principle are the backbone of the field of fixed-point theory. Axiomatic interpretation of metric space attracts thousands of researchers towards spaciousness. So far, there have been many generalizations on metric spaces. This tells us of the beauty, attraction and expansion of the concept of metric spaces. Zadeh [12] established the basis for fuzzy mathematics in 1965. Fixed point theory is considered to be the fascination and active area of research and development of nonlinear analysis. Kramosil and Michalek [6] introduced fuzzy metric spaces in a variety of ways in 1975. With the help of continuous t-norm. George and Veeramani [3] present the concept of fuzzy metric spaces in 1994. Atanassov[1] stirred things up by adding the idea of non- membership grade of fuzzy set theory. Smarandache [9] described the concept of neutrosophic logic and neutrosophic sets in 1998. In this study provides a common fixed point theorem for pair of self mappings and occasionally weakly compatible mapping fulfilling various constraints in the neutrosophic metric space 2. Preliminaries Now, we begin with some basic fundamental aspects, notations and definitions. Definition 2.1.[6] A binary operation ∗∶ [0,1] × [0,1] → [0,1], is named continuous t-norm if it meets the following : (i) ∗ is associative and commutative, (ii) ∗ is continuous, (iii) 𝔨 ∗ 1 = 𝔨 for all 𝔨 ∈ [0,1], (iv) 𝔨 ∗ 𝜍̃ ≤ 𝔷 ∗ 𝔡 whenever 𝔨 ≤ 𝔷 and 𝜍̃ ≤ 𝔡. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 114 https://internationalpubls.com Definition 2.2. [6] A binary operation ⨀ ∶ [0,1] × [0,1] → [0,1], is named continuous t-conorm if it meets the following: (i) ⨀ is associative and commutative, (ii) ⨀ is continuous, (iii) 𝔨 ⨀ 0 = 𝔨 for all 𝔨 ∈ [0,1], (iv) 𝔨 ⨀ 𝜍̃ ≤ 𝔷 ⨀ 𝔡 whenever 𝔨 ≤ 𝔷 and 𝜍̃ ≤ 𝔡. Example 2.3.[2] (i) 𝔯 ∗ 𝔰 = min{ 𝔯, 𝔰} for all 𝔯, 𝔰 ∈ [0,1]. (ii) 𝔯 ∗ 𝔰 = max{𝔯 + 𝔰 − 1,0} for all 𝔯, 𝔰 ∈ [0,1]. Example 2.4.[2] (i) 𝔯 ⨀ 𝔰 = max{𝔯, 𝔰 } for all ∈ [0,1]. (ii) 𝔯 ⨀ 𝔰 = min{𝔯 + 𝔰, 1} for all 𝔯, 𝔰 ∈ [0,1]. Definition 2.5. The 6-tuple (Ξ, ℜ, 𝔖, 𝔗 ∗, ⨀) is called a Neutrosophic Metric Space [NMS] if Ξ is an arbitrary non void set, ∗ is a continuous t-norm, ⨀ is a continuous t-conorm and ℜ, 𝔖, 𝔗 ∶ Ξ × Ξ × (0, ∞) → [0,1] are fuzzy sets, fulfilling the following assertions: For all , 𝜍̃, 𝔷 ∈ Ξ ; 𝜚, 𝜌 ∈ (0, ∞). (1) ℜ(𝔨, 𝜍̃, 𝜚) + 𝔖(𝔨, 𝜍̃, 𝜚) + 𝔗(𝔨, 𝜍̃, 𝜚) ≤ 3, (2) 0 ≤ ℜ(𝔨, 𝜍̃, 𝜚) ≤ 1; 0 ≤ 𝔖(𝔨, 𝜍̃, 𝜚) ≤ 1 and 0 ≤ 𝔗(𝔨, 𝜍̃, 𝜚) ≤ 1, (3) ℜ(𝔨, 𝜍̃, 𝜚) > 0, (4) ℜ(𝔨, 𝜍̃, 𝜚) = 1, for all 𝜚 ∈ (0, ∞) ⇔ 𝔨 = 𝜍̃, (5) ℜ(𝔨, 𝜍̃, 𝜚) = ℜ(𝜍̃, 𝔨, 𝜚), (6) ℜ(𝔨, 𝔷 , 𝜚 + 𝜌) ≥ ℜ(𝔨, 𝜍̃, 𝜚) ∗ ℜ(𝜍̃, 𝔷 , 𝜌), (7) ℜ(𝔨, 𝜍̃, 𝜚): (0, ∞) → [0,1] is continuous, (8) 𝔖(𝔨, 𝜍̃, 𝜚) < 1, (9) 𝔖(𝔨, 𝜍̃, 𝜚) = 0, for all 𝜚 ∈ (0, ∞) ⇔ 𝔨 = 𝜍̃, (10) 𝔖(𝔨, 𝜍̃, 𝜚) = 𝔖(𝜍̃, 𝔨, 𝜚), (11) 𝔖(𝔨, 𝔷 , 𝜚 + 𝜌) ≤ 𝔖(𝔨, 𝜍̃, 𝜚) ⨀ 𝔖(�̃�, 𝔷 , 𝜌), (12) 𝔖(𝔨, 𝜍̃, 𝜚): (0, ∞) → [0,1] is continuous, (13) 𝔗(𝔨, 𝜍̃, 𝜚) < 1, (14) 𝔗(𝔨, 𝜍̃, 𝜚) = 0 for all 𝜚 ∈ (0, ∞) ⇔ 𝔨 = 𝜍̃, (15) 𝔗(𝔨, 𝜍̃, 𝜚) = 𝔗(𝜍̃, 𝔨, 𝜚), (16) 𝔗(𝔨, 𝔷 , 𝜚 + 𝜌) ≤ 𝔗(𝔨, 𝜍̃, 𝜚)⨀𝔗(𝜍̃, 𝔷 , 𝜌), Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 115 https://internationalpubls.com (17) 𝔗(𝔨, 𝜍̃, 𝜚): (0, ∞) → [0,1] is continuous. The triplet (ℜ, 𝔖, 𝔗) is named a NMS. The function ℜ(𝔨, 𝜍̃, 𝜚) , 𝔖(𝔨, 𝜍̃, 𝜚) and 𝔗(𝔨, 𝜍̃, 𝜚) indicates the degree of nearness, non-nearness and neutralness between 𝔨 and 𝜍̃ with respect to 𝜚. Example 2.6: Let Ξ = ℝ and let 𝔯 ∗ 𝔰 = min{𝔯, 𝔰} and 𝔯⨀𝔰 = max{𝔯, 𝔰}, for all 𝔯, 𝔰 ∈ [0,1]. For each 𝜚 > 0, 𝔨, 𝜍̃ ∈ Ξ, we define ℜ(𝔨, 𝜍̃, 𝜚) = 𝑒 − |𝔨−�̃�| 𝜚 , 𝔖(𝔨, 𝜍̃, 𝜚) = (𝑒 |𝔨−�̃�| 𝜚 − 1)𝑒 − |𝔨−�̃�| 𝜚 and 𝔗(𝔨, 𝜍̃, 𝜚) = (𝑒 |𝔨−�̃�| 𝜚 − 1). Then (Ξ, ℜ, 𝔖, 𝔗 ∗, ⨀) is a NMS. Definition 2.7: Let (Ξ, ℜ, 𝔖, 𝔗 ∗, ⨀) be NMS and 𝔏 and 𝔐 are self mappings on Ξ. The self mappins 𝔏 and 𝔐 are named to be commuting if 𝔏𝔐(𝔨) = 𝔐𝔏(𝔨), for all 𝔨 ∈ Ξ. The self maps 𝔏 and 𝔐 are named to be compatible if lim 𝑛→∞ |ℜ(𝔏𝔐𝔨𝑛, 𝔐𝔏𝔨𝑛, 𝜚)| = 1, lim 𝑛→∞ |𝔖(𝔏𝔐𝔨𝑛, 𝔐𝔏𝔨𝑛, 𝜚)| = 0 and lim 𝑛→∞ |𝔗(𝔏𝔐𝔨𝑛, 𝔐𝔏𝔨𝑛, 𝜚)| = 0, 𝜚 > 0. Whenever {𝔨𝑛} is a sequence in Ξ such that lim 𝑛→∞ 𝔏 𝔨𝑛 = lim 𝑛→∞ 𝔐 𝔨𝑛, for some 𝔨 ∈ Ξ. Definition 2.8: Let (Ξ, ℜ, 𝔖, 𝔗 ∗, ⨀) be NMS and 𝔏 and 𝔐 are self mappings on Ξ. The self mappings 𝔏 and 𝔐 are named to be Occasionally Weakly Compatible [OWC] if and only if there is a coincidence point 𝔨 in Ξ of 𝔏 and 𝔐 commute. i.e., 𝔏𝔐𝔨 = 𝔐𝔏𝔨. Lemma 2.9: Let (Ξ, ℜ, 𝔖, 𝔗,∗, ⨀) be a NMS with lim 𝜚→∞ ℜ(𝔨, 𝜍̃, 𝜚) = 1, lim 𝜚→∞ 𝔖(𝔨, 𝜍̃, 𝜚) = 0 and lim 𝜚→∞ 𝔗(𝔨, 𝜍̃, 𝜚) = 0, for all 𝔨, 𝜍̃ ∈ Ξ. If ℜ(𝔨, 𝜍̃, 𝔡𝜚) ≥ ℜ(𝔨, 𝜍̃, 𝜚), 𝔖(𝔨, 𝜍̃, 𝔡𝜚) ≤ 𝔖(𝔨, 𝜍̃, 𝜚) and 𝔗(𝔨, 𝜍̃, 𝔡𝜚) ≤ 𝔗(𝔨, 𝜍̃, 𝜚) for some 𝔡 ∈ (0, 1), for all 𝜚 > 0, then 𝔨 = 𝜍̃. Proof: Suppose there exists 𝔡 ∈ (0, 1), such that ℜ(𝔨, 𝜍̃, 𝔡𝜚) ≥ ℜ(𝔨, 𝜍̃, 𝜚),𝔖(𝔨, 𝜍̃, 𝔡𝜚) ≤ 𝔖(𝔨, 𝜍̃, 𝜚) And 𝔗(𝔨, 𝜍̃, 𝔡𝜚) ≤ 𝔗(𝔨, 𝜍̃, 𝜚), for all 𝔨, 𝜍̃ ∈ Ξ and 𝜚 > 0. So that ℜ(𝔨, 𝜍̃, 𝜚) ≥ ℜ (𝔨, 𝜍̃, 𝜚 𝔡 ),𝔖(𝔨, 𝜍̃, 𝜚) ≤ 𝔖 (𝔨, 𝜍̃, 𝜚 𝔡 ) and 𝔗(𝔨, 𝜍̃, 𝜚) ≤ 𝔗 (𝔨, 𝜍̃, 𝜚 𝔡 ). Repeated application gives, ℜ(𝔨, 𝜍̃, 𝜚) ≥ ℜ (𝔨, 𝜍̃, 𝜚 𝔡𝑛) , 𝔖(𝔨, 𝜍̃, 𝜚) ≤ 𝔖 (𝔨, 𝜍̃, 𝜚 𝔡𝑛) and𝔗(𝔨, 𝜍̃, 𝜚) ≤ 𝔗 (𝔨, 𝜍̃, 𝜚 𝔡𝑛) for some positive integer n. On taking 𝑛 → ∞, reduces to ℜ(𝔨, 𝜍̃, 𝜚) ≥ 1 and 𝔖(𝔨, 𝜍̃, 𝜚) ≤ 0 and 𝔗(𝔨, 𝜍̃, 𝜚) ≤ 0. Thus, we have 𝔨 = 𝜍̃. Lemma 2.10: Let {𝔨𝑛} be a sequence in a NMS, (Ξ, ℜ, 𝔖, 𝔗,∗, ⨀) with lim 𝜚→∞ ℜ(𝔨, 𝜍̃, 𝜚) = 1, lim 𝜚→∞ 𝔖(𝔨, 𝜍̃, 𝜚) = 0 and lim 𝜚→∞ 𝔗(𝔨, 𝜍̃, 𝜚) = 0, for all 𝔨, 𝜍̃ ∈ Ξ. If there exists 𝔡 ∈ (0, 1) such that ℜ(𝔨𝑛+1, 𝔨𝑛+2, 𝔡𝜚) ≥ ℜ(𝔨𝑛, 𝔨𝑛+1, 𝜚),𝔖(𝔨𝑛+1, 𝔨𝑛+2, 𝔡𝜚) ≤ 𝔖(𝔨𝑛, 𝔨𝑛+1, 𝜚) and 𝔗(𝔨𝑛+1, 𝔨𝑛+2, 𝔡𝜚) ≤ 𝔗(𝔨𝑛, 𝔨𝑛+1, 𝜚) for all 𝜚 > 0 and n = 0,1,2.... Then{𝔨𝑛} is a Cauchy sequence in Ξ. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 116 https://internationalpubls.com Proof: For n = 0, we have ℜ(𝔨1, 𝔨2, 𝜚) ≥ ℜ (𝔨0, 𝔨1, 𝜚 𝔡 ), 𝔖(𝔨1, 𝔨2, 𝜚) ≤ 𝔖 (𝔨0, 𝔨1, 𝜚 𝔡 ) and 𝔗(𝔨1, 𝔨2, 𝜚) ≤ 𝔗 (𝔨0, 𝔨1, 𝜚 𝔡 ), for all 𝜚 > 0 and 𝔡 ∈ (0, 1). By induction, ℜ(𝔨𝑛+1, 𝔨𝑛+2, 𝜚) ≥ ℜ (𝔨0, 𝔨1, 𝜚 𝔡𝑛+1) , 𝔖(𝔨𝑛+1, 𝔨𝑛+2, 𝜚) ≤ 𝔖 (𝔨0, 𝔨1, 𝜚 𝔡𝑛+1) and 𝔗(𝔨𝑛+1, 𝔨𝑛+2, 𝜚) ≤ 𝔗 (𝔨0, 𝔨1, 𝜚 𝔡𝑛+1), for all n. Thus for any positive integer 𝔮 and using (6), (11) and (16), we have ℜ(𝔨𝑛, 𝔨𝑛+𝔮, 𝜚) ≥ ℜ (𝔨𝑛, 𝔨𝑛+1, 𝜚 𝔮 ) ∗ … ∗ (𝔮 𝑡𝑖𝑚𝑒𝑠) ∗ … ∗ ℜ (𝔨𝑛+𝔮−1, 𝔨𝑛+𝔮, 𝜚 𝔮 ) ≥ ℜ (𝔨0, 𝔨1, 𝜚 𝔮𝔡𝑛 ) ∗ … ∗ (𝔮 𝑡𝑖𝑚𝑒𝑠) ∗ … ∗ ℜ (𝔨0, 𝔨1, 𝜚 𝔮𝔡𝑛+𝔮−1 ). 𝔖(𝔨𝑛, 𝔨𝑛+𝔮, 𝜚) ≤ 𝔖 (𝔨𝑛, 𝔨𝑛+1, 𝜚 𝔮 ) ⨀ … ⨀(𝔮 𝑡𝑖𝑚𝑒𝑠)⨀ … ⨀𝔖 (𝔨𝑛+𝔮−1, 𝔨𝑛+𝔮, 𝜚 𝔮 ) ≤ 𝔖 (𝔨0, 𝔨1, 𝜚 𝔮𝔡𝑛) ⨀ … ⨀(𝔮 𝑡𝑖𝑚𝑒𝑠)⨀ … ⨀𝔖 (𝔨0, 𝔨1, 𝜚 𝔮𝔡𝑛+𝔮−1 ). 𝔗(𝔨𝑛, 𝔨𝑛+𝔮, 𝜚) ≤ 𝔗 (𝔨𝑛, 𝔨𝑛+1, 𝜚 𝔮 ) ⨀ … ⨀(𝔮 𝑡𝑖𝑚𝑒𝑠)⨀ … ⨀𝔗 (𝔨𝑛+𝔮−1, 𝔨𝑛+𝔮, 𝜚 𝔮 ) ≤ 𝔗 (𝔨0, 𝔨1, 𝜚 𝔮𝔡𝑛) ⨀ … ⨀(𝔮 𝑡𝑖𝑚𝑒𝑠)⨀ … ⨀𝔗 (𝔨0, 𝔨1, 𝜚 𝔮𝔡𝑛+𝔮−1 ). Which on taking 𝑛 → ∞, reduces to lim 𝜚→∞ ℜ(𝔨𝑛, 𝔨𝑛+𝔮, 𝜚) ≥ 1 ∗ 1 ∗ … ∗ 1, lim 𝜚→∞ 𝔖(𝔨𝑛, 𝔨𝑛+𝔮, 𝜚) ≤ 0 ⨀ … ⨀ 0 and lim 𝜚→∞ 𝔗(𝔨𝑛, 𝔨𝑛+𝔮, 𝜚) ≤ 0 ⨀ … ⨀ 0. Since 𝔡 < 1, lim 𝜚→∞ ℜ(𝔨𝑛, 𝔨𝑛+𝔮, 𝜚) ≥ 1, lim 𝜚→∞ 𝔗(𝔨𝑛, 𝔨𝑛+𝔮, 𝜚) ≤ 0 and lim 𝜚→∞ 𝔖(𝔨𝑛, 𝔨𝑛+𝔮, 𝜚) ≤ 0. This necessitates that {𝔨𝑛} is a Cauchy sequence in Ξ 3. Main Results In this section, we present the concept of NMS and prove several FP results. Theorem 3.1: Let (Ξ, ℜ, 𝔖, 𝔗,∗, ⨀) be a NMS with lim 𝜚→∞ ℜ(𝔨, 𝜍̃, 𝜚) = 1 , lim 𝜚→∞ 𝔖(𝔨, 𝜍̃, 𝜚) = 0 and lim 𝜚→∞ 𝔗(𝔨, 𝜍̃, 𝜚) = 0, for all 𝔨, 𝜍̃ ∈ Ξ and 𝜚 > 0 and let 𝔏 and 𝔐 be self mapping on Ξ. If there exist 𝔡 ∈ (0, 1) such that ℜ(𝔏𝔨, 𝔐𝜍̃, 𝔡𝜚) ≥ ℜ(𝔨, 𝜍̃, 𝜚), 𝔖(𝔏𝔨, 𝔐𝜍̃, 𝔡𝜚) ≤ 𝔖(𝔨, 𝜍̃, 𝜚) and 𝔗(𝔏𝔨, 𝔐𝜍̃, 𝔡𝜚) ≤ 𝔗(𝔨, 𝜍̃, 𝜚) for all 𝔨, 𝜍̃, ∈ Ξ, and for all 𝜚 > 0 (3.1.1) Then 𝔏 and 𝔐 have a unique common fixed point in Ξ. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 117 https://internationalpubls.com Proof. Let 𝔨0 ∈ Ξ be an arbitrary point and we define the sequence {𝔨𝑛} by 𝔨2𝑛+1 = 𝔏𝔨2𝑛 and 𝔨2𝑛+2 = 𝔐𝔨2𝑛+1; n = 0,1,2,… . Now, for 𝔡 ∈ (0, 1) and for all 𝜚 > 0, then from (3.1.1) we have ℜ(𝔨2𝑛+1, 𝔨2𝑛+2, 𝔡𝜚) = ℜ(𝔏𝔨2𝑛, 𝔐𝔨2𝑛+1, 𝔡𝜚) ≥ ℜ(𝔨2𝑛, 𝔨2𝑛+1, 𝜚), ℜ(𝔨2𝑛, 𝔨2𝑛+1, 𝔡𝜚) = ℜ(𝔏𝔨2𝑛−1, 𝔐𝔨2𝑛, 𝔡𝜚) ≥ ℜ(𝔨2𝑛−1, 𝔨2𝑛, 𝜚). 𝔖(𝔨2𝑛+1, 𝔨2𝑛+2, 𝔡𝜚) = 𝔖(𝔏𝔨2𝑛, 𝔐𝔨2𝑛+1, 𝔡𝜚) ≤ 𝔖(𝔨2𝑛, 𝔨2𝑛+1, 𝜚), 𝔖(𝔨2𝑛, 𝔨2𝑛+1, 𝔡𝜚) = 𝔖(𝔏𝔨2𝑛−1, 𝔐𝔨2𝑛, 𝔡𝜚)𝔖(𝔨2𝑛−1, 𝔨2𝑛, 𝜚) and 𝔗(𝔨2𝑛+1, 𝔨2𝑛+2, 𝔡𝜚) = 𝔗(𝔏𝔨2𝑛, 𝔐𝔨2𝑛+1, 𝔡𝜚) ≤ 𝔗(𝔨2𝑛, 𝔨2𝑛+1, 𝜚), 𝔗(𝔨2𝑛, 𝔨2𝑛+1, 𝔡𝜚) = 𝔗(𝔏𝔨2𝑛−1, 𝔐𝔨2𝑛, 𝔡𝜚)𝔗(𝔨2𝑛−1, 𝔨2𝑛, 𝜚). In general, we have ℜ(𝔨𝑛+1, 𝔨𝑛+2, 𝔡𝜚) ≥ ℜ(𝔨𝑛, 𝔨𝑛+1, 𝜚), 𝔖(𝔨𝑛+1, 𝔨𝑛+2, 𝔡𝜚) ≤ 𝔖(𝔨𝑛, 𝔨𝑛+1, 𝜚) and 𝔗(𝔨𝑛+1, 𝔨𝑛+2, 𝔡𝜚) ≤ 𝔗(𝔨𝑛, 𝔨𝑛+1, 𝜚) for all 𝜚 > 0 and 𝔡 ∈ (0, 1); n = 0,1,2…. By Lemma (2.10) {𝔨𝑛} be a Cauchy sequence in Ξ. Since Ξ is complete then there exists 𝜗 ∈ Ξ such that 𝔨𝑛 → 𝜗 as 𝑛 → ∞ and {𝔨2𝑛}, {𝔨2𝑛+1} are sub sequences of {𝔨2𝑛} converge to the same point 𝜗 ∈ Ξ, i.e. 𝔨2𝑛 → 𝜗, 𝔨2𝑛+1 → 𝜗 as 𝑛 → ∞. Now from equation (3.1.1) we have, ℜ(𝔏𝜗, 𝜗, 𝔡𝜚) = ℜ (𝔏𝜗, 𝜗, 𝔡𝜚 2 + 𝔡𝜚 2 ) ≥ ℜ (𝔏𝜗, 𝔨2𝑛+2, 𝔡𝜚 2 ) ∗ ℜ (𝔨2𝑛+2, 𝜗, 𝔡𝜚 2 ) = ℜ (𝔏𝜗, 𝔐𝔨2𝑛+1, 𝔡𝜚 2 ) ∗ ℜ (𝔨2𝑛+2, 𝜗, 𝔡𝜚 2 ) ≥ ℜ (𝜗, 𝔨2𝑛+1, 𝜚 2 ) ∗ ℜ (𝔨2𝑛+2, 𝜗, 𝔡𝜚 2 ). 𝔖(𝔏𝜗, 𝜗, 𝔡𝜚) = 𝔖 (𝔏𝜗, 𝜗, 𝔡𝜚 2 + 𝔡𝜚 2 ) ≤ 𝔖 (𝔏𝜗, 𝔨2𝑛+2, 𝔡𝜚 2 ) ⨀𝔖 (𝔨2𝑛+2, 𝜗, 𝔡𝜚 2 ) = 𝔖 (𝔏𝜗, 𝔐𝔨2𝑛+1, 𝔡𝜚 2 ) ⨀𝔖 (𝔨2𝑛+2, 𝜗, 𝔡𝜚 2 ) ≤ 𝔖 (𝜗, 𝔨2𝑛+1, 𝜚 2 ) ⨀𝔖 (𝔨2𝑛+2, 𝜗, 𝔡𝜚 2 ) and 𝔗(𝔏𝜗, 𝜗, 𝔡𝜚) = 𝔗 (𝔏𝜗, 𝜗, 𝔡𝜚 2 + 𝔡𝜚 2 ) ≤ 𝔗 (𝔏𝜗, 𝔨2𝑛+2, 𝔡𝜚 2 ) ⨀𝔗 (𝔨2𝑛+2, 𝜗, 𝔡𝜚 2 ) = 𝔗 (𝔏𝜗, 𝔐𝔨2𝑛+1, 𝔡𝜚 2 ) ⨀𝔗 (𝔨2𝑛+2, 𝜗, 𝔡𝜚 2 ) ≤ 𝔗 (𝜗, 𝔨2𝑛+1, 𝜚 2 ) ⨀𝔗 (𝔨2𝑛+2, 𝜗, 𝔡𝜚 2 ). Taking limit 𝑛 → ∞. ℜ(𝔏𝜗, 𝜗, 𝔡𝜚) ≥ 1 ∗ 1 = 1, 𝔖(𝔏𝜗, 𝜗, 𝔡𝜚) ≤ 0⨀0 = 0, 𝔗(𝔏𝜗, 𝜗, 𝔡𝜚) ≤ 0⨀0 = 0. So 𝔏𝜗 = 𝜗; Again, ℜ(𝜗, 𝔐𝜗, 𝔡𝜚) = ℜ (𝜗, 𝔚𝜗, 𝔡𝜚 2 + 𝔡𝜚 2 ) ≥ ℜ (𝜗, 𝔨2𝑛+1, 𝔡𝜚 2 ) ∗ ℜ (𝔨2𝑛+1, 𝔚𝜗, 𝔡𝜚 2 ) = ℜ (𝜗, 𝔨2𝑛+1, 𝔡𝜚 2 ) ∗ ℜ (𝔏𝔨2𝑛, 𝔚𝜗, 𝔡𝜚 2 ) ≥ ℜ (𝜗, 𝔨2𝑛+1, 𝜚 2 ) ∗ ℜ (𝔨2𝑛, 𝜗, 𝔡𝜚 2 ) and Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 118 https://internationalpubls.com 𝔖(𝜗, 𝔐𝜗, 𝔡𝜚) = 𝔖 (𝜗, 𝔚𝜗, 𝔡𝜚 2 + 𝔡𝜚 2 ) ≤ 𝔖 (𝜗, 𝔨2𝑛+1, 𝔡𝜚 2 ) ⨀𝔖 (𝔨2𝑛+1, 𝔚𝜗, 𝔡𝜚 2 ) = 𝔖 (𝜗, 𝔨2𝑛+1, 𝔡𝜚 2 ) ⨀𝔖 (𝔏𝔨2𝑛, 𝔚𝜗, 𝔡𝜚 2 ) ≤ 𝔖 (𝜗, 𝔨2𝑛+1, 𝜚 2 ) ⨀𝔖 (𝔨2𝑛, 𝜗, 𝔡𝜚 2 ). In addition, 𝔗(𝜗, 𝔐𝜗, 𝔡𝜚) = 𝔗 (𝜗, 𝔚𝜗, 𝔡𝜚 2 + 𝔡𝜚 2 ) ≤ 𝔗 (𝜗, 𝔨2𝑛+1, 𝔡𝜚 2 ) ⨀𝔗 (𝔨2𝑛+1, 𝔚𝜗, 𝔡𝜚 2 ) = 𝔗 (𝜗, 𝔨2𝑛+1, 𝔡𝜚 2 ) ⨀𝔗 (𝔏𝔨2𝑛, 𝔚𝜗, 𝔡𝜚 2 ) ≤ 𝔗 (𝜗, 𝔨2𝑛+1, 𝜚 2 ) ⨀𝔗 (𝔨2𝑛, 𝜗, 𝔡𝜚 2 ). On taking limit 𝑛 → ∞. ℜ(𝜗, 𝔐𝜗, 𝔡𝜚) ≥ 1 ∗ 1 = 1,𝔖(𝜗, 𝔐𝜗, 𝔡𝜚) ≤ 0⨀0 = 0, and 𝔗(𝜗, 𝔐𝜗, 𝔡𝜚) ≤ 0⨀0 = 0. So 𝔚𝜗 = 𝜗, and 𝔏𝜗 = 𝔚𝜗 = 𝜗. Hence 𝜗 is a common fixed point of 𝔏 and 𝔚. For uniqueness, let 𝔰 be any another fixed point of 𝔏 and 𝔚. Now from (3.1.1), ℜ(𝜗, 𝔰, 𝔡𝜚) = ℜ(𝔏𝜗, 𝔚𝔰, 𝔡𝜚) ≥ ℜ(𝜗, 𝔰, 𝜚); 𝔖(𝜗, 𝔰 , 𝔡𝜚) = 𝔖(𝔏𝜗, 𝔚𝔰, 𝔡𝜚) ≤ 𝔖(𝜗, 𝔰, 𝜚)and 𝔗(𝜗, 𝔰, 𝔡𝜚) = 𝔗(𝔏𝜗, 𝔚𝔰, 𝔡𝜚) ≤ 𝔗(𝜗, 𝔰, 𝜚). We know that when (Ξ, ℜ, 𝔖,∗, ⨀) be NMS such that lim 𝜚→∞ ℜ(𝔨, 𝜍̃, 𝜚) = 1, lim 𝜚→∞ 𝔖(𝔨, 𝜍̃, 𝜚) = 0 and lim 𝜚→∞ 𝔗(𝔨, 𝜍̃, 𝜚) = 0, for all 𝔨, 𝜍̃ ∈ Ξ. If ℜ(𝔏𝔨, 𝔏𝜍̃, 𝔡𝜚) ≥ ℜ(𝔨, 𝜍̃, 𝜚), 𝔖(𝔏𝔨, 𝔏𝜍̃, 𝔡𝜚) ≤ 𝔖(𝔨, 𝜍̃, 𝜚) and 𝔗(𝔏𝔨, 𝔏𝜍̃, 𝔡𝜚) ≤ 𝔗(𝔨, 𝜍̃, 𝜚), for some 0 < 𝔡 < 1, for all 𝔨, 𝜍̃, ∈ Ξ, 𝜚 ∈ (0, ∞), then 𝔨 = 𝜍̃. Hence 𝜗 = 𝔰. Example: 3.2: Let Ξ = [0, 1]. Consider the metric 𝑑(𝔨, 𝜍̃) = |𝔨 − 𝜍̃| with ℜ(𝔨, 𝜍̃, 𝜚) = 𝜚 𝜚+𝑑(𝔨,�̃�) , 𝔖(𝔨, 𝜍̃, 𝜚) = 𝑑(𝔨,�̃�) 𝜚+𝑑(𝔨,�̃�) and 𝔗(𝔨, 𝜍̃, 𝜚) = 𝑑(𝔨,�̃�) 𝜚 and the self mappings 𝔏 and 𝔚 on Ξ, defined by 𝔏(𝔨) = 𝔨 4 , 𝔚(𝔨) = 𝔨 2 . The self mappings 𝔏 and 𝔚 satisfies all the conditions that are stated in Theorem (3.1), then 𝔏 and 𝔚 have unique common fixed point at 0. Corollary 3.3: Let (Ξ, ℜ, 𝔖, 𝔗,∗, ⨀) be a NMS with lim 𝜚→∞ ℜ(𝔨, 𝜍̃, 𝜚) = 1, lim 𝜚→∞ 𝔖(𝔨, 𝜍̃, 𝜚) = 0 and lim 𝜚→∞ 𝔗(𝔨, 𝜍̃, 𝜚) = 0, for all 𝔨, 𝜍̃ ∈ Ξ and 𝜚 > 0 and let 𝔏 and 𝔐 be self mapping on Ξ. If there exist 𝔡 ∈ (0, 1) such that ℜ(𝔏𝔨, 𝔏𝜍̃, 𝔡𝜚) ≥ ℜ(𝔨, 𝜍̃, 𝜚), 𝔖(𝔏𝔨, 𝔏𝜍̃, 𝔡𝜚) ≤ 𝔖(𝔨, 𝜍̃, 𝜚) and 𝔗(𝔏𝔨, 𝔏𝜍̃, 𝔡𝜚) ≤ 𝔗(𝔨, 𝜍̃, 𝜚) for all 𝔨, 𝜍̃, ∈ Ξ, and for all 𝜚 > 0, then 𝔏 have a unique fixed point in Ξ. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 119 https://internationalpubls.com Theorem3.4: Let (Ξ, ℜ, 𝔖, 𝔗,∗, ⨀) be a NMS with lim 𝜚→∞ ℜ(𝔨, 𝜍̃, 𝜚) = 1, lim 𝜚→∞ 𝔖(𝔨, 𝜍̃, 𝜚) = 0 and lim 𝜚→∞ 𝔗(𝔨, 𝜍̃, 𝜚) = 0, for all 𝔨, 𝜍̃ ∈ Ξ and 𝔄,̈ �̈�, 𝔏 and 𝔚 be self mappings on Ξ. Let the pairs {𝔄,̈ 𝔏} and {�̈�, 𝔚} be OWC. If there exists 𝔡 ∈ (0, 1) such that ℜ(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≥ min{ℜ(𝔏𝔨, 𝔚𝜍̃, 𝜚), ℜ(𝔏𝔨, 𝔄𝔨̈ , 𝜚), ℜ(�̈�𝜍̃, 𝔚𝜍̃, 𝜚), ℜ(�̈�𝔨, 𝔚𝜍̃, 𝜚), ℜ(�̈�𝜍̃, 𝔏𝔨, 𝜚)}(3.4.1) 𝔖(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≤ max{𝔖(𝔏𝔨, 𝔚𝜍̃, 𝜚), 𝔖(𝔏𝔨, 𝔄𝔨̈ , 𝜚), 𝔖(�̈�𝜍̃, 𝔚𝜍̃, 𝜚), 𝔖(�̈�𝔨, 𝔚𝜍̃, 𝜚), 𝔖(�̈�𝜍̃, 𝔏𝔨, 𝜚)}(3.4.2) 𝔗(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≤ max{𝔗(𝔏𝔨, 𝔚𝜍̃, 𝜚), 𝔗(𝔏𝔨, 𝔄𝔨̈ , 𝜚), 𝔗(�̈�𝜍̃, 𝔚𝜍̃, 𝜚), 𝔗(�̈�𝔨, 𝔚𝜍̃, 𝜚), 𝔗(�̈�𝜍̃, 𝔏𝔨, 𝜚)}(3.4.3) for all 𝔨, 𝜍̃ ∈ Ξ and 𝜚 > 0, then 𝔄,̈ �̈�, 𝔏 and 𝔚 have a unique common fixed point in Ξ. Proof: Since the pairs {𝔄,̈ 𝔏} and {�̈�, 𝔚} be OWC, so there are point 𝔨, 𝜍̃ ∈ Ξ such that �̈�(𝔨) = 𝔏(𝔨) and �̈�(𝜍̃) = 𝔚(𝜍̃). Now, by the given conditions (3.4.1) and (3.4.2) we get ℜ(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≥ min{ℜ(𝔏𝔨, 𝔚𝜍̃, 𝜚), ℜ(𝔏𝔨, 𝔄𝔨̈ , 𝜚), ℜ(�̈�𝜍̃, 𝔚𝜍̃, 𝜚), ℜ(�̈�𝔨, �̈�𝜍̃, 𝜚), ℜ(�̈�𝜍̃, 𝔏𝔨, 𝜚)} = min{ℜ(�̈�𝔨, �̈�𝜍̃, 𝜚), ℜ(�̈�𝔨, 𝔄𝔨̈ , 𝜚), ℜ(�̈�𝜍̃, �̈�𝜍̃, 𝜚), ℜ(�̈�𝔨, �̈�𝜍̃, 𝜚), ℜ(�̈�𝜍̃, �̈�𝔨, 𝜚)} = min{ℜ(�̈�𝔨, �̈�𝜍̃, 𝜚), 1, 1, ℜ(�̈�𝔨, �̈�𝜍̃, 𝜚), ℜ(�̈�𝜍̃, �̈�𝔨, 𝜚)} = ℜ(�̈�𝔨, �̈�𝜍̃, 𝜚). 𝔖(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≤ max{𝔖(𝔏𝔨, 𝔚𝜍̃, 𝜚), 𝔖(𝔏𝔨, 𝔄𝔨̈ , 𝜚), 𝔖(�̈�𝜍̃, 𝔚𝜍̃, 𝜚), 𝔖(�̈�𝔨, �̈�𝜍̃, 𝜚), 𝔖(�̈�𝜍̃, 𝔏𝔨, 𝜚)} = max {𝔖(�̈�𝔨, �̈�𝜍̃, 𝜚), 𝔖(�̈�𝔨, 𝔄𝔨̈ , 𝜚), 𝔖(�̈�𝜍̃, �̈�𝜍̃, 𝜚), 𝔖(�̈�𝔨, �̈�𝜍̃, 𝜚), 𝔖(�̈�𝜍̃, �̈�𝔨, 𝜚)} = max{𝔖(�̈�𝔨, �̈�𝜍̃, 𝜚), 0, 0, 𝔖(�̈�𝔨, �̈�𝜍̃, 𝜚), 𝔖(�̈�𝜍̃, �̈�𝔨, 𝜚)} = 𝔖(�̈�𝔨, �̈�𝜍̃, 𝜚). 𝔗(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≤ max{𝔗(𝔏𝔨, 𝔚𝜍̃, 𝜚), 𝔗(𝔏𝔨, 𝔄𝔨̈ , 𝜚), 𝔗(�̈�𝜍̃, 𝔚𝜍̃, 𝜚), 𝔗(�̈�𝔨, �̈�𝜍̃, 𝜚), 𝔗(�̈�𝜍̃, 𝔏𝔨, 𝜚)} = max {𝔗(�̈�𝔨, �̈�𝜍̃, 𝜚), 𝔗(�̈�𝔨, 𝔄𝔨̈ , 𝜚), 𝔗(�̈�𝜍̃, �̈�𝜍̃, 𝜚), 𝔗(�̈�𝔨, �̈�𝜍̃, 𝜚), 𝔗(�̈�𝜍̃, �̈�𝔨, 𝜚)} = max{𝔗(�̈�𝔨, �̈�𝜍̃, 𝜚), 0, 0, 𝔗(�̈�𝔨, �̈�𝜍̃, 𝜚), 𝔗(�̈�𝜍̃, �̈�𝔨, 𝜚)} = 𝔗(�̈�𝔨, �̈�𝜍̃, 𝜚). In view of Lemma (2.9), we have �̈�𝔨 = �̈�𝜍̃ and therefore �̈�𝔨 = 𝔏𝔨 = �̈�𝜍̃ = 𝔚𝜍̃. (3.4.4) Suppose that the pair {𝔄,̈ 𝔏} have an another coincidence point 𝔴 ∈ Ξ. i.e., �̈�𝔴 = 𝔏𝔴. Now, ℜ(�̈�𝔴, �̈�𝜍̃, 𝔡𝜚) ≥ min{ℜ(𝔏𝔴, 𝔚𝜍̃, 𝜚), ℜ(𝔏𝔴, �̈�𝔴, 𝜚), ℜ(�̈�𝜍̃, 𝔚𝜍̃, 𝜚), ℜ(�̈�𝔴, 𝔚𝜍̃, 𝜚), ℜ(�̈�𝜍̃, 𝔏𝔴, 𝜚)} = min{ℜ(�̈�𝔴, �̈�𝜍̃, 𝜚), ℜ(�̈�𝔴, �̈�𝔴, 𝜚), ℜ(�̈�𝜍̃, �̈�𝜍̃, 𝜚), ℜ(�̈�𝔴, �̈�𝜍̃, 𝜚), ℜ(�̈�𝜍̃, �̈�𝔴, 𝜚)} = min{ℜ(�̈�𝔴, �̈�𝜍̃, 𝜚), 1,1, ℜ(�̈�𝔴, �̈�𝜍̃, 𝜚), ℜ(�̈�𝜍̃, �̈�𝔴, 𝜚)} = ℜ(�̈�𝔴, �̈�𝜍̃, 𝜚). 𝔖(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≤ max{𝔖(𝔏𝔴, 𝔚𝜍̃, 𝜚), 𝔖(𝔏𝔴, �̈�𝔴, 𝜚), 𝔖(�̈�𝜍̃, 𝔚𝜍̃, 𝜚), 𝔖(�̈�𝔴, 𝔚𝜍̃, 𝜚), 𝔖(�̈�𝜍̃, 𝔏𝔴, 𝜚)} = max{𝔖(�̈�𝔴, �̈�𝜍̃, 𝜚), 𝔖(�̈�𝔴, �̈�𝔴, 𝜚), 𝔖(�̈�𝜍̃, �̈�𝜍̃, 𝜚), 𝔖(�̈�𝔴, �̈�𝜍̃, 𝜚), 𝔖(�̈�𝜍̃, �̈�𝔴, 𝜚)} = max{𝔖(�̈�𝔴, �̈�𝜍̃, 𝜚), 0,0, 𝔖(�̈�𝔴, �̈�𝜍̃, 𝜚), 𝔖(�̈�𝜍̃, �̈�𝔴, 𝜚)} = 𝔖(�̈�𝔴, �̈�𝜍̃, 𝜚). 𝔗(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≤ max{𝔗(𝔏𝔴, 𝔚𝜍̃, 𝜚), 𝔗(𝔏𝔴, �̈�𝔴, 𝜚), 𝔗(�̈�𝜍̃, 𝔚𝜍̃, 𝜚), 𝔗(�̈�𝔴, 𝔚𝜍̃, 𝜚), 𝔗(�̈�𝜍̃, 𝔏𝔴, 𝜚)} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 120 https://internationalpubls.com = max{𝔗(�̈�𝔴, �̈�𝜍̃, 𝜚), 𝔗(�̈�𝔴, �̈�𝔴, 𝜚), 𝔗(�̈�𝜍̃, �̈�𝜍̃, 𝜚)𝔗(�̈�𝔴, �̈�𝜍̃, 𝜚), 𝔗(�̈�𝜍̃, �̈�𝔴, 𝜚)} = max{𝔗(�̈�𝔴, �̈�𝜍̃, 𝜚), 0,0, 𝔗(�̈�𝔴, �̈�𝜍̃, 𝜚), 𝔗(�̈�𝜍̃, �̈�𝔴, 𝜚)} = 𝔖(�̈�𝔴, �̈�𝜍̃, 𝜚). Again, in view of Lemma (2.9), we have �̈�𝔨 = �̈�𝜍̃. Therefore, �̈�𝔨 = 𝔏𝔨 = �̈�𝜍̃ = 𝔚𝜍̃. (3.4.5) From (3.4.4) and (3.4.5), �̈�𝔨 = �̈�𝔴 and therefore the pair {𝔄,̈ 𝔏} have a unique coincidence point 𝜁 = �̈�𝔨 = 𝔏𝔨. Thus by Lemma (2.9), 𝔴 is the unique common fixed point of the pair {𝔄,̈ 𝔏}. Similarly, we can show that this pair {�̈�, 𝔚} also have a unique common fixed point. Suppose this is 𝜂 ∈ Ξ. Now, ℜ(𝜁, 𝜂, 𝔡𝜚) = ℜ(�̈�𝜁, �̈�𝜂, 𝔡𝜚) ≥ min{ℜ(𝔏𝜁, 𝔚𝜂, 𝜚), ℜ(𝔏𝜁, �̈�𝜁, 𝜚), ℜ(�̈�𝜂, 𝔚𝜂, 𝜚), ℜ(�̈�𝜁, 𝔚𝜂, 𝜚), ℜ(�̈�𝜂, 𝔏𝜁, 𝜚)} = min{ℜ(𝜁, 𝜂, 𝜚), ℜ(𝜁, 𝜁, 𝜚), ℜ(𝜂, 𝜂, 𝜚), ℜ(𝜁, 𝜂, 𝜚), ℜ(𝜂, 𝜁, 𝜚)} = min{ℜ(𝜁, 𝜂, 𝜚), 1,1, ℜ(𝜁, 𝜂, 𝜚), ℜ(𝜂, 𝜁, 𝜚)}= ℜ(𝜁, 𝜂, 𝜚). 𝔖(𝜁, 𝜂, 𝔡𝜚) = 𝔖(�̈�𝜁, �̈�𝜂, 𝔡𝜚) ≤ max{𝔖(𝔏𝜁, 𝔚𝜂, 𝜚), 𝔖(𝔏𝜁, �̈�𝜁, 𝜚), 𝔖(�̈�𝜂, 𝔚𝜂, 𝜚), 𝔖(�̈�𝜁, 𝔚𝜂, 𝜚), 𝔖(�̈�𝜂, 𝔏𝜁, 𝜚)} = max{𝔖(𝜁, 𝜂, 𝜚), 𝔖(𝜁, 𝜁, 𝜚), 𝔖(𝜂, 𝜂, 𝜚), 𝔖(𝜁, 𝜂, 𝜚), 𝔖(𝜂, 𝜁, 𝜚)} = max{𝔖(𝜁, 𝜂, 𝜚), 0,0, 𝔖(𝜁, 𝜂, 𝜚), 𝔖(𝜂, 𝜁, 𝜚)} = 𝔖(𝜁, 𝜂, 𝜚) and 𝔗(𝜁, 𝜂, 𝔡𝜚) = 𝔗(�̈�𝜁, �̈�𝜂, 𝔡𝜚) ≤ max{𝔗(𝔏𝜁, 𝔚𝜂, 𝜚), 𝔗(𝔏𝜁, �̈�𝜁, 𝜚), 𝔗(�̈�𝜂, 𝔚𝜂, 𝜚), 𝔗(�̈�𝜁, 𝔚𝜂, 𝜚), 𝔗(�̈�𝜂, 𝔏𝜁, 𝜚)} = max{𝔗(𝜁, 𝜂, 𝜚), 𝔗(𝜁, 𝜁, 𝜚), 𝔗(𝜂, 𝜂, 𝜚), 𝔗(𝜁, 𝜂, 𝜚), 𝔗(𝜂, 𝜁, 𝜚)} = max{𝔗(𝜁, 𝜂, 𝜚), 0,0, 𝔗(𝜁, 𝜂, 𝜚), 𝔗(𝜂, 𝜁, 𝜚)} = 𝔗(𝜁, 𝜂, 𝜚). Therefore using Lemma (2.9), we have 𝜁 = 𝜂 consequently, 𝜁 is common fixed point of 𝔄,̈ �̈�, 𝔏 and 𝔚. Now, ℜ(𝜁, 𝜏, 𝔡𝜚) = ℜ(�̈�𝜁, �̈�𝜏, 𝔡𝜚) ≥ min{ℜ(𝔏𝜁, 𝔚𝜏, 𝜚), ℜ(𝔏𝜁, �̈�𝜁, 𝜚), ℜ(�̈�𝜏, 𝔚𝜏, 𝜚), ℜ(�̈�𝜁, 𝔚𝜏, 𝜚), ℜ(�̈�𝜏, 𝔏𝜁, 𝜚)} = min{ℜ(𝜁, 𝜏, 𝜚), ℜ(𝜁, 𝜁, 𝜚), ℜ(𝜏, 𝜏, 𝜚), ℜ(𝜁, 𝜏, 𝜚), ℜ(𝜏, 𝜁, 𝜚)} = min{ℜ(𝜁, 𝜏, 𝜚), 1,1, ℜ(𝜁, 𝜏, 𝜚), ℜ(𝜏, 𝜁, 𝜚)} = ℜ(𝜁, 𝜏, 𝜚). 𝔖(𝜁, 𝜏, 𝔡𝜚) = 𝔖(�̈�𝜁, �̈�𝜏, 𝔡𝜚) ≤ max{𝔖(𝔏𝜁, 𝔚𝜏, 𝜚), 𝔖(𝔏𝜁, �̈�𝜁, 𝜚), 𝔖(�̈�𝜏, 𝔚𝜏, 𝜚), 𝔖(�̈�𝜁, 𝔚𝜏, 𝜚), 𝔖(�̈�𝜏, 𝔏𝜁, 𝜚)} = max{𝔖(𝜁, 𝜏, 𝜚), 𝔖(𝜁, 𝜁, 𝜚), 𝔖(𝜏, 𝜏, 𝜚), 𝔖(𝜁, 𝜏, 𝜚), 𝔖(𝜏, 𝜁, 𝜚)} = max{𝔖(𝜁, 𝜏, 𝜚), 0,0, 𝔖(𝜁, 𝜏, 𝜚), 𝔖(𝜏, 𝜁, 𝜚)} = 𝔖(𝜁, 𝜏, 𝜚) and Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 121 https://internationalpubls.com 𝔗(𝜁, 𝜏, 𝔡𝜚) = 𝔗(�̈�𝜁, �̈�𝜏, 𝔡𝜚) ≤ max{𝔗(𝔏𝜁, 𝔚𝜏, 𝜚), 𝔗(𝔏𝜁, �̈�𝜁, 𝜚), 𝔗(�̈�𝜏, 𝔚𝜏, 𝜚), 𝔗(�̈�𝜁, 𝔚𝜏, 𝜚), 𝔗(�̈�𝜏, 𝔏𝜁, 𝜚)} = max{𝔗(𝜁, 𝜏, 𝜚), 𝔗(𝜁, 𝜁, 𝜚), 𝔗(𝜏, 𝜏, 𝜚), 𝔗(𝜁, 𝜏, 𝜚), 𝔗(𝜏, 𝜁, 𝜚)} = max{𝔗(𝜁, 𝜏, 𝜚), 0,0, 𝔗(𝜁, 𝜏, 𝜚), 𝔗(𝜏, 𝜁, 𝜚)} = 𝔗(𝜁, 𝜏, 𝜚). By Lemma (2.9), we have 𝜁 = 𝜏. Hence 𝔄,̈ �̈�, 𝔏 and 𝔚 have a unique common fixed point. Example 3.5: Let Ξ = ℝ. Consider the metric 𝒹(𝔨, 𝜍̃) = |𝔨| + |𝜍̃|, for all 𝔨 ≠ 𝜍̃ and 𝒹(𝔨, 𝜍̃) = 0, for 𝔨 = 𝜍̃ on Ξ. Let 𝔯 ∗ 𝔰 = min{ 𝔯, 𝔰} and 𝔯⨀𝔰 = max{𝔯, 𝔰 }, for all 𝔯, 𝔰 ∈ [0,1]. For each 𝜚 > 0, 𝔨, 𝜍̃ ∈ Ξ, we define ℜ(𝔨, 𝜍̃, 𝜚) = 𝑒 − |𝔨−�̃�| 𝜚 , 𝔖(𝔨, 𝜍̃, 𝜚) = (𝑒 |𝔨−�̃�| 𝜚 − 1)𝑒 − |𝔨−�̃�| 𝜚 and 𝔗(𝔨, 𝜍̃, 𝜚) = (𝑒 |𝔨−�̃�| 𝜚 − 1). Then (Ξ, ℜ, 𝔖, 𝔗 ∗, ⨀) is a NMS with lim 𝜚→∞ ℜ(𝔨, 𝜍̃, 𝜚) = 1, lim 𝜚→∞ 𝔖(𝔨, 𝜍̃, 𝜚) = 0 and lim 𝜚→∞ 𝔗(𝔨, 𝜍̃, 𝜚) = 0, for all 𝔨, 𝜍̃ ∈ Ξ. Now we define the self maps �̈�, 𝔅,̈ 𝔏 and 𝔚 on Ξ by 𝔄 ̈ (𝔨) = 𝔨 9 , �̈�(𝔨) = 𝔨 12 , 𝔏(𝔨) = 𝔨 2 , 𝔚(𝔨) = 𝔨 4 . Let 𝔡 = 1 3 . For 𝔨 ≠ 𝜍̃, ℜ (�̈�𝔨, �̈�𝜍̃, 𝜚 3 )= 𝑒 −3(|�̈�𝔨|+|�̈��̃�|) 𝜚 =𝑒 −3(| 𝔨 9 |+| 𝔨 12 |) 𝜚 = 𝑒 −(| 𝔨 3 |+| 𝔨 4 |) 𝜚 ≥ 𝑒 −(| 𝔨 2 |+| 𝔨 4 |) 𝜚 = ℜ(𝔏𝔨, 𝔚𝜍̃, 𝜚). 𝔖 (�̈�𝔨, �̈�𝜍̃, 𝜚 3 ) = (𝑒 3(|�̈�𝔨|+|�̈��̃�|) 𝜚 − 1)𝑒 −3(|�̈�𝔨|+|�̈��̃�|) 𝜚 = (𝑒 3(| 𝔨 9 |+| 𝔨 12 |) 𝜚 − 1)𝑒 −3(| 𝔨 9 |+| 𝔨 12 |) 𝜚 = (𝑒 (| 𝔨 3 |+| 𝔨 4 |) 𝜚 − 1)𝑒 −(| 𝔨 3 |+| 𝔨 4 |) 𝜚 ≤ (𝑒 (| 𝔨 2 |+| 𝔨 4 |) 𝜚 − 1)𝑒 −(| 𝔨 2 |+| 𝔨 4 |) 𝜚 = 𝔖(𝔏𝔨, 𝔚𝜍̃, 𝜚). 𝔗 (�̈�𝔨, �̈�𝜍̃, 𝜚 3 ) = (𝑒 3(|�̈�𝔨|+|�̈��̃�|) 𝜚 − 1) = (𝑒 3(| 𝔨 9 |+| 𝔨 12 |) 𝜚 − 1) = (𝑒 (| 𝔨 3 |+| 𝔨 4 |) 𝜚 − 1) ≤ (𝑒 (| 𝔨 2 |+| 𝔨 4 |) 𝜚 − 1) = 𝔗(𝔏𝔨, 𝔚𝜍̃, 𝜚). For 𝔨 = 𝜍̃, ℜ (�̈�𝔨, �̈�𝜍̃, 𝜚 3 ) = 1 = ℜ(𝔏𝔨, 𝔚𝜍̃, 𝜚), 𝔖 (�̈�𝔨, �̈�𝜍̃, 𝜚 3 ) = 0 = 𝔖(𝔏𝔨, 𝔚𝜍̃, 𝜚) and 𝔗 (�̈�𝔨, �̈�𝜍̃, 𝜚 3 ) = 0 = 𝔗(𝔏𝔨, 𝔚𝜍̃, 𝜚). So that for any 𝔨, 𝜍̃ ∈ Ξ, ℜ (�̈�𝔨, �̈�𝜍̃, 𝜚 3 ) ≥ ℜ(𝔏𝔨, 𝔚𝜍̃, 𝜚) = min {ℜ(𝔏𝔨, 𝔚𝜍̃, 𝜚), ℜ(𝔏𝔨, 𝔄𝔨̈ , 𝜚), ℜ(�̈�𝜍̃, 𝔚𝜍̃, 𝜚), ℜ(�̈�𝔨, 𝔚𝜍̃, 𝜚), ℜ(�̈�𝜍̃, 𝔏𝔨, 𝜚)}. 𝔖 (�̈�𝔨, �̈�𝜍̃, 𝜚 3 ) ≤ 𝔖(𝔏𝔨, 𝔚𝜍̃, 𝜚) = max{𝔖(𝔏𝔨, 𝔚𝜍̃, 𝜚), 𝔖(𝔏𝔨, 𝔄𝔨̈ , 𝜚), 𝔖(�̈�𝜍̃, 𝔚𝜍̃, 𝜚), 𝔖(�̈�𝔨, 𝔚𝜍̃, 𝜚), 𝔖(�̈�𝜍̃, 𝔏𝔨, 𝜚)}. 𝔗(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≤ 𝔗(𝔏𝔨, 𝔚𝜍̃, 𝜚) 𝔗(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) = max{𝔗(𝔏𝔨, 𝔚𝜍̃, 𝜚), 𝔗(𝔏𝔨, 𝔄𝔨̈ , 𝜚), 𝔗(�̈�𝜍̃, 𝔚𝜍̃, 𝜚), 𝔗(�̈�𝔨, 𝔚𝜍̃, 𝜚), 𝔗(�̈�𝜍̃, 𝔏𝔨, 𝜚)}. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 122 https://internationalpubls.com Hence, the maps �̈�, 𝔅,̈ 𝔏 and 𝔚 satisfies the condition (3.4.1) of Theorem (3.4) for 𝔡 = 1 3 . Also, the pairs {𝔄,̈ 𝔏} and {�̈�, 𝔚} are obviously OWC. Thus all the condition of Theorem (3.4) are satisfied at 𝔨 = 0 is the unique common fixed point of �̈�, 𝔅,̈ 𝔏 and 𝔚 in Ξ. Theorem 3.6: Let (Ξ, ℜ, 𝔖, 𝔗,∗, ⨀) be a NMS with lim 𝜚→∞ ℜ(𝔨, 𝜍̃, 𝜚) = 1, lim 𝜚→∞ 𝔖(𝔨, 𝜍̃, 𝜚) = 0and lim 𝜚→∞ 𝔗(𝔨, 𝜍̃, 𝜚) = 0, for all 𝔨, 𝜍̃ ∈ Ξ and 𝔄,̈ �̈�, 𝔏 and 𝔚 be self mappings on Ξ. Let the pairs {𝔄,̈ 𝔏} and {�̈�, 𝔚} be OWC. If there exists 𝔡 ∈ (0, 1) such that ℜ(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≥ ℏ(𝑚𝑖𝑛{ℜ(𝔏𝔨, 𝔚𝜍̃, 𝜚), ℜ(𝔏𝔨, �̈�𝔨, 𝜚), ℜ(�̈�𝜍̃, 𝔚𝜍̃, 𝜚), ℜ(�̈�𝔨, 𝔚𝜍̃, 𝜚), ℜ(�̈�𝜍̃, 𝔏𝔨, 𝜚)}) 𝔖(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≤ ℏ(𝑚𝑎𝑥{ℜ(𝔏𝔨, 𝔚𝜍̃, 𝜚), ℜ(𝔏𝔨, �̈�𝔨, 𝜚), ℜ(�̈�𝜍̃, 𝔚𝜍̃, 𝜚), ℜ(�̈�𝔨, 𝔚𝜍̃, 𝜚), ℜ(�̈�𝜍̃, 𝔏𝔨, 𝜚)} and 𝔗(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≤ ℏ(𝑚𝑎𝑥{ℜ(𝔏𝔨, 𝔚𝜍̃, 𝜚), ℜ(𝔏𝔨, �̈�𝔨, 𝜚), ℜ(�̈�𝜍̃, 𝔚𝜍̃, 𝜚), ℜ(�̈�𝔨, 𝔚𝜍̃, 𝜚), ℜ(�̈�𝜍̃, 𝔏𝔨, 𝜚)} for all 𝔨, 𝜍̃ ∈ Ξ and 𝜚 > 0, where ℏ ∶ [0,1] → [0,1] with ℏ(𝔨) > 𝑘 for all 𝔨 ∈ [0,1]. Then 𝔄,̈ �̈�, 𝔏 and 𝔚 have a unique common fixed point in Ξ. Proof: The proof follows from Theorem (3.4). Theorem 3.7. Let (Ξ, ℜ, 𝔖, 𝔗,∗, ⨀) be a NMS with lim 𝜚→∞ ℜ(𝔨, 𝜍̃, 𝜚) = 1, lim 𝜚→∞ 𝔖(𝔨, 𝜍̃, 𝜚) = 0 and lim 𝜚→∞ 𝔗(𝔨, 𝜍̃, 𝜚) = 0, for all 𝔨, 𝜍̃ ∈ Ξ and 𝔄,̈ �̈�, 𝔏 and 𝔚 be self mappings on Ξ. Let the pairs {𝔄,̈ 𝔏} and {�̈�, 𝔚} be OWC. If there exists 𝔡 ∈ (0, 1) such that ℜ(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≥ ℜ(𝔏𝔨, 𝔚𝜍̃, 𝜚) ∗ ℜ(�̈�𝔨, 𝔏𝔨, 𝜚) ∗ ℜ(�̈�𝜍̃, 𝔚𝜍̃, 𝜚) ∗ ℜ(�̈�𝔨, 𝔚𝜍̃, 𝜚) (3.7.1) 𝔖(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≤ 𝔖(𝔏𝔨, 𝔚𝜍̃, 𝜚)⨀𝔖(�̈�𝔨, 𝔏𝔨, 𝜚)⨀𝔖(�̈�𝜍̃, 𝔚𝜍̃, 𝜚)⨀𝔖(�̈�𝔨, 𝔚𝜍̃, 𝜚) (3.7.2) 𝔗(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≤ 𝔗(𝔏𝔨, 𝔚𝜍̃, 𝜚)⨀𝔗(�̈�𝔨, 𝔏𝔨, 𝜚)⨀𝔗(�̈�𝜍̃, 𝔚𝜍̃, 𝜚)⨀𝔗(�̈�𝔨, 𝔚𝜍̃, 𝜚) (3.7.3) for all 𝔨, 𝜍̃ ∈ Ξ and 𝜚 > 0. Then 𝔄,̈ �̈�, 𝔏 and 𝔚 have a unique common fixed point in Ξ. Proof: The pairs {𝔄,̈ 𝔏} and {�̈�, 𝔚} be OWC, so there are point 𝔨, 𝜍̃ ∈ Ξ such that �̈�(𝔨) = 𝔏(𝔨) and �̈�(𝜍̃) = 𝔚(𝜍̃). Now, by the given conditions (3.7.1), (3.7.2) and (3.7.3), we get ℜ(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≥ ℜ(𝔏𝔨, 𝔚𝜍̃, 𝜚) ∗ ℜ(�̈�𝔨, 𝔏𝔨, 𝜚) ∗ ℜ(�̈�𝜍̃, 𝔚𝜍̃, 𝜚) ∗ ℜ(�̈�𝔨, 𝔚𝜍̃, 𝜚) = ℜ(�̈�𝔨, �̈�𝜍̃, 𝜚) ∗ ℜ(�̈�𝔨, �̈�𝔨, 𝜚) ∗ ℜ(�̈�𝔨, �̈�𝔨, 𝜚) ∗ ℜ(�̈�𝔨, �̈�𝜍̃, 𝜚) = ℜ(�̈�𝔨, �̈�𝜍̃, 𝜚) ∗ 1 ∗ 1 ∗ ℜ(�̈�𝔨, �̈�𝜍̃, 𝜚) = ℜ(�̈�𝔨, �̈�𝜍̃, 𝜚). 𝔖(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≤ 𝔖(𝔏𝔨, 𝔚𝜍̃, 𝜚)⨀𝔖(�̈�𝔨, 𝔏𝔨, 𝜚)⨀𝔖(�̈�𝜍̃, 𝔚𝜍̃, 𝜚)⨀𝔖(�̈�𝔨, 𝔚𝜍̃, 𝜚) = 𝔖(�̈�𝔨, �̈�𝜍̃, 𝜚)⨀𝔖(�̈�𝔨, �̈�𝔨, 𝜚)⨀𝔖(�̈�𝔨, �̈�𝔨, 𝜚)⨀𝔖(�̈�𝔨, �̈�𝜍̃, 𝜚) = 𝔖(�̈�𝔨, �̈�𝜍̃, 𝜚)⨀0⨀0⨀𝔖(�̈�𝔨, �̈�𝜍̃, 𝜚) = 𝔖(�̈�𝔨, �̈�𝜍̃, 𝜚) and 𝔗(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≤ 𝔗(𝔏𝔨, 𝔚𝜍̃, 𝜚)⨀𝔗(�̈�𝔨, 𝔏𝔨, 𝜚)⨀𝔗(�̈�𝜍̃, 𝔚𝜍̃, 𝜚)⨀𝔗(�̈�𝔨, 𝔚𝜍̃, 𝜚) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 123 https://internationalpubls.com = 𝔗(�̈�𝔨, �̈�𝜍̃, 𝜚)⨀𝔗(�̈�𝔨, �̈�𝔨, 𝜚)⨀𝔗(�̈�𝔨, �̈�𝔨, 𝜚)⨀𝔗(�̈�𝔨, �̈�𝜍̃, 𝜚) = 𝔗(�̈�𝔨, �̈�𝜍̃, 𝜚)⨀0⨀0⨀𝔗(�̈�𝔨, �̈�𝜍̃, 𝜚) = 𝔗(�̈�𝔨, �̈�𝜍̃, 𝜚). In view of Lemma (2.10), we have �̈�𝔨 = �̈�𝜍̃ and therefore �̈�𝔨 = 𝔏𝔨 = �̈�𝜍̃ = 𝔚𝜍̃. (3.7.4) Suppose the pair {𝔄,̈ 𝔏} have an another coincidence point 𝔴 ∈ Ξ, i.e.,�̈�𝔴 = 𝔏𝔴. ℜ(�̈�𝔴, �̈�𝜍̃, 𝔡𝜚) ≥ ℜ(𝔏𝔴, 𝔚𝜍̃, 𝜚) ∗ ℜ(�̈�𝔴, 𝔏𝔴, 𝜚) ∗ ℜ(�̈�𝜍̃, 𝔚𝜍̃, 𝜚) ∗ ℜ(�̈�𝔴, 𝔚𝜍̃, 𝜚) = ℜ(�̈�𝔴, �̈�𝜍̃, 𝜚) ∗ ℜ(�̈�𝔴, �̈�𝔴, 𝜚) ∗ ℜ(�̈�𝔴, �̈�𝔴, 𝜚) ∗ ℜ(�̈�𝔴, �̈�𝜍̃, 𝜚) = ℜ(�̈�𝔴, �̈�𝜍̃, 𝜚) ∗ 1 ∗ 1 ∗ ℜ(�̈�𝔴, �̈�𝜍̃, 𝜚) = ℜ(�̈�𝔴, �̈�𝜍̃, 𝜚). 𝔖(�̈�𝔴, �̈�𝜍̃, 𝔡𝜚) ≤ 𝔖(𝔏𝔴, 𝔚𝜍̃, 𝜚)⨀𝔖(�̈�𝔴, 𝔏𝔴, 𝜚)⨀𝔖(�̈�𝜍̃, 𝔚𝜍̃, 𝜚)⨀𝔖(�̈�𝔴, 𝔚𝜍̃, 𝜚) = 𝔖(�̈�𝔴, �̈�𝜍̃, 𝜚)⨀𝔖(�̈�𝔴, �̈�𝔴, 𝜚)⨀𝔖(�̈�𝔴, �̈�𝔴, 𝜚)⨀𝔖(�̈�𝔴, �̈�𝜍̃, 𝜚) = 𝔖(�̈�𝔴, �̈�𝜍̃, 𝜚)⨀0⨀0⨀𝔖(�̈�𝔴, �̈�𝜍̃, 𝜚) = 𝔖(�̈�𝔴, �̈�𝜍̃, 𝜚) and 𝔗(�̈�𝔴, �̈�𝜍̃, 𝔡𝜚) ≤ 𝔗(𝔏𝔴, 𝔚𝜍̃, 𝜚)⨀𝔗(�̈�𝔴, 𝔏𝔴, 𝜚)⨀𝔗(�̈�𝜍̃, 𝔚𝜍̃, 𝜚)⨀𝔗(�̈�𝔴, 𝔚𝜍̃, 𝜚) = 𝔗(�̈�𝔴, �̈�𝜍̃, 𝜚)⨀𝔗(�̈�𝔴, �̈�𝔴, 𝜚)⨀𝔗(�̈�𝔴, �̈�𝔴, 𝜚)⨀𝔗(�̈�𝔴, �̈�𝜍̃, 𝜚) = 𝔗(�̈�𝔴, �̈�𝜍̃, 𝜚)⨀0⨀0⨀𝔗(�̈�𝔴, �̈�𝜍̃, 𝜚) = 𝔗(�̈�𝔴, �̈�𝜍̃, 𝜚). By lemma (2.10), �̈�𝔴 = �̈�𝜍̃ and consequently �̈�𝔴 = 𝔏𝔴 = �̈�𝜍̃ = 𝔚𝜍̃. (3.7.5) From (3.7.4) and (3.7.5) �̈�𝔨 = �̈�𝔴 and therefore the pair {𝔄,̈ 𝔏} have a unique point of coincidence 𝜁 = �̈�𝔨 = 𝔏𝔨. 𝜁 is the unique common fixed point of {𝔄,̈ 𝔏}. Similarly, we can show that there is unique common fixed point 𝜂 ∈ Ξ of {�̈�, 𝔚}. Now, ℜ(𝜁, 𝜂, 𝔡𝜚) = ℜ(�̈�𝜁, �̈�𝜂, 𝔡𝜚) ≥ ℜ(𝔏𝜁, 𝔚𝜂, 𝜚) ∗ ℜ(�̈�𝜁, 𝔏𝜁, 𝜚) ∗ ℜ(�̈�𝜂, 𝔚𝜂, 𝜚) ∗ ℜ(�̈�𝜁, 𝔚𝜂, 𝜚) = ℜ(�̈�𝜁, �̈�𝜂, 𝜚) ∗ ℜ(�̈�𝜁, �̈�𝜁, 𝜚) ∗ ℜ(�̈�𝜂, �̈�𝜂, 𝜚) ∗ ℜ(�̈�𝜁, �̈�𝜂, 𝜚) = ℜ(�̈�𝜁, 𝔅�̈�, 𝜚) ∗ 1 ∗ 1 ∗ ℜ(�̈�𝜁, �̈�𝜂, 𝜚) = ℜ(�̈�𝜁, 𝔅�̈�, 𝜚) = ℜ(𝜁, 𝜂, 𝜚). 𝔖(𝜁, 𝜂, 𝔡𝜚) = 𝔖(�̈�𝜁, �̈�𝜂, 𝔡𝜚) ≤ 𝔖(𝔏𝜁, 𝔚𝜂, 𝜚)⨀ 𝔖(�̈�𝜁, 𝔏𝜁, 𝜚)⨀ 𝔖(�̈�𝜂, 𝔚𝜂, 𝜚)⨀𝔖(�̈�𝜁, 𝔚𝜂, 𝜚) = 𝔖(�̈�𝜁, �̈�𝜂, 𝜚)⨀ 𝔖(�̈�𝜁, �̈�𝜁, 𝜚)⨀𝔖(�̈�𝜂, �̈�𝜂, 𝜚)⨀𝔖(�̈�𝜁, �̈�𝜂, 𝜚) = 𝔖(�̈�𝜁, 𝔅�̈�, 𝜚)⨀0⨀0⨀𝔖(�̈�𝜁, �̈�𝜂, 𝜚) = 𝔖(�̈�𝜁, 𝔅�̈�, 𝜚) = 𝔖(𝜁, 𝜂, 𝜚) and 𝔗(𝜁, 𝜂, 𝔡𝜚)= 𝔗(�̈�𝜁, �̈�𝜂, 𝔡𝜚) ≤ 𝔗(𝔏𝜁, 𝔚𝜂, 𝜚)⨀ 𝔗(�̈�𝜁, 𝔏𝜁, 𝜚)⨀ 𝔗(�̈�𝜂, 𝔚𝜂, 𝜚)⨀𝔗(�̈�𝜁, 𝔚𝜂, 𝜚) = 𝔗(�̈�𝜁, �̈�𝜂, 𝜚)⨀ 𝔗(�̈�𝜁, �̈�𝜁, 𝜚)⨀𝔗(�̈�𝜂, �̈�𝜂, 𝜚)⨀𝔗(�̈�𝜁, �̈�𝜂, 𝜚) = 𝔗(�̈�𝜁, 𝔅�̈�, 𝜚)⨀0⨀0⨀𝔗(�̈�𝜁, �̈�𝜂, 𝜚) = 𝔗(�̈�𝜁, 𝔅�̈�, 𝜚) = 𝔗(𝜁, 𝜂, 𝜚). By lemma (2.10), we have 𝜁 = 𝜂 and consequently 𝜁 is common fixed Point of 𝔄,̈ �̈�, 𝔏 and 𝔚. For uniqueness, let 𝜏 is an another common fixed point of 𝔄,̈ �̈�, 𝔏 and 𝔚. Therefore, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 124 https://internationalpubls.com ℜ(𝜁, 𝜏, 𝔡𝜚) = ℜ(�̈�𝜁, �̈�𝜏, 𝔡𝜚) ≥ ℜ(𝔏𝜁, 𝔚𝜏, 𝜚) ∗ ℜ(�̈�𝜁, 𝔏𝜁, 𝜚) ∗ ℜ(�̈�𝜏, 𝔚𝜏, 𝜚) ∗ ℜ(�̈�𝜁, 𝔚𝜏, 𝜚) = ℜ(𝜁, 𝜏, 𝜚) ∗ ℜ(𝜁, 𝜁, 𝜚) ∗ ℜ(𝜏, 𝜏, 𝜚) ∗ ℜ(𝜁, 𝜏, 𝜚) = ℜ(𝜁, 𝜏, 𝜚) ∗ 1 ∗ 1 ∗ ℜ(𝜁, 𝜏, 𝜚) = ℜ(𝜁, 𝜏, 𝜚). 𝔖(𝜁, 𝜏, 𝔡𝜚) = 𝔖(�̈�𝜁, �̈�𝜏, 𝔡𝜚) ≤ 𝔖(𝔏𝜁, 𝔚𝜏, 𝜚)⨀𝔖(�̈�𝜁, 𝔏𝜁, 𝜚)⨀𝔖(�̈�𝜏, 𝔚𝜏, 𝜚)⨀𝔖(�̈�𝜁, 𝔚𝜏, 𝜚) = 𝔖(𝜁, 𝜏, 𝜚)⨀𝔖(𝜁, 𝜁, 𝜚)⨀𝔖(𝜏, 𝜏, 𝜚)⨀𝔖(𝜁, 𝜏, 𝜚) = 𝔖(𝜁, 𝜏, 𝜚)⨀0⨀0⨀𝔖(𝜁, 𝜏, 𝜚) = 𝔖(𝜁, 𝜏, 𝜚) and 𝔗(𝜁, 𝜏, 𝔡𝜚) = 𝔗(�̈�𝜁, �̈�𝜏, 𝔡𝜚) ≤ 𝔗(𝔏𝜁, 𝔚𝜏, 𝜚)⨀ 𝔗(�̈�𝜁, 𝔏𝜁, 𝜚)⨀ 𝔗(�̈�𝜏, 𝔚𝜏, 𝜚)⨀𝔗(�̈�𝜁, 𝔚𝜏, 𝜚) = 𝔗(𝜁, 𝜏, 𝜚)⨀ 𝔗(𝜁, 𝜁, 𝜚)⨀ 𝔗(𝜏, 𝜏, 𝜚)⨀𝔗(𝜁, 𝜏, 𝜚) = 𝔗(𝜁, 𝜏, 𝜚)⨀0⨀0⨀ 𝔗(𝜁, 𝜏, 𝜚) = 𝔗(𝜁, 𝜏, 𝜚). In view of Lemma (2.10), we have 𝜁 = 𝜏. Hence 𝔄,̈ �̈�, 𝔏 and 𝔚 have a unique common fixed point. Example 3.8: Let Ξ = ℝ. Consider the metric 𝒹(𝔨, 𝜍̃) = |𝔨| + |𝜍̃|, for all 𝔨 ≠ 𝜍̃ and 𝒹(𝔨, 𝜍̃) = 0, for 𝔨 = 𝜍.̃ Let 𝔯 ∗ 𝔰 = min{ 𝔯, 𝔰} and 𝔯⨀𝔰 = max{𝔯, 𝔰 }, for all 𝔯, 𝔰 ∈ [0,1]. For each 𝜚 > 0, 𝔨, 𝜍̃ ∈ Ξ, we define ℜ(𝔨, 𝜍̃, 𝜚) = 𝑒 − |𝔨−�̃�| 𝜚 , 𝔖(𝔨, 𝜍̃, 𝜚) = (𝑒 |𝔨−�̃�| 𝜚 − 1)𝑒 − |𝔨−�̃�| 𝜚 and 𝔗(𝔨, 𝜍̃, 𝜚) = (𝑒 |𝔨−�̃�| 𝜚 − 1). Then (Ξ, ℜ, 𝔖, 𝔗 ∗, ⨀) is a NMS with lim 𝜚→∞ ℜ(𝔨, 𝜍̃, 𝜚) = 1, lim 𝜚→∞ 𝔖(𝔨, 𝜍̃, 𝜚) = 0 and lim 𝜚→∞ 𝔗(𝔨, 𝜍̃, 𝜚) = 0, for all 𝔨, 𝜍̃ ∈ Ξ. Now we define the self maps �̈�, 𝔅,̈ 𝔏 and 𝔚 on Ξ by 𝔄 ̈ (𝔨) = 𝔨 10 , �̈�(𝔨) = 𝔨 15 , 𝔏(𝔨) = 𝔨, 𝔚(𝔨) = 𝔨 3 . Let 𝔡 = 1 5 . For 𝔨 ≠ 𝜍̃, ℜ (�̈�𝔨, �̈�𝜍̃, 𝜚 5 )= 𝑒 −5(|�̈�𝔨|+|�̈��̃�|) 𝜚 =𝑒 −5(| 𝔨 10 |+| 𝔨 15 |) 𝜚 = 𝑒 −(| 𝔨 2 |+| 𝔨 3 |) 𝜚 ≥ 𝑒 −(|𝔨|+| 𝔨 3 |) 𝜚 = ℜ(𝔏𝔨, 𝔚𝜍̃, 𝜚), 𝔖 (�̈�𝔨, �̈�𝜍̃, 𝜚 3 )= (𝑒 5(|�̈�𝔨|+|�̈��̃�|) 𝜚 − 1) = (𝑒 5(| 𝔨 9 |+| 𝔨 12 |) 𝜚 − 1)𝑒 −3(| 𝔨 9 |+| 𝔨 12 |) 𝜚 =(𝑒 (| 𝔨 2 |+| 𝔨 3 |) 𝜚 − 1)𝑒 −(| 𝔨 2 |+| 𝔨 3 |) 𝜚 ≤ (𝑒 (|𝔨|+| 𝔨 3 |) 𝜚 1)𝑒 −(|𝔨|+| 𝔨 3 |) 𝜚 = 𝔖(𝔏𝔨, 𝔚𝜍̃, 𝜚) 𝔗 (�̈�𝔨, �̈�𝜍̃, 𝜚 5 ) = (𝑒 5(|�̈�𝔨|+|�̈��̃�|) 𝜚 − 1)= (𝑒 5(| 𝔨 2 |+| 𝔨 3 |) 𝜚 − 1) = (𝑒 (| 𝔨 2 |+| 𝔨 3 |) 𝜚 − 1) ≤ (𝑒 (|𝔨|+| 𝔨 3 |) 𝜚 − 1) = 𝔗(𝔏𝔨, 𝔚𝜍̃, 𝜚). For 𝔨 = 𝜍̃. ℜ (�̈�𝔨, �̈�𝜍̃, 𝜚 5 ) = 1 = ℜ(𝔏𝔨, 𝔚𝜍̃, 𝜚), 𝔖 (�̈�𝔨, �̈�𝜍̃, 𝜚 5 ) = 0 = 𝔖(𝔏𝔨, 𝔚𝜍̃, 𝜚) and 𝔗 (�̈�𝔨, �̈�𝜍̃, 𝜚 5 ) = 0 = 𝔗(𝔏𝔨, 𝔚𝜍̃, 𝜚). So that for any 𝔨, 𝜍̃ ∈ Ξ, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 125 https://internationalpubls.com ℜ (�̈�𝔨, �̈�𝜍̃, 𝜚 5 ) ≥ ℜ(𝔏𝔨, 𝔚𝜍̃, 𝜚) = min {ℜ(𝔏𝔨, 𝔚𝜍̃, 𝜚), ℜ(𝔏𝔨, 𝔄𝔨̈ , 𝜚), ℜ(�̈�𝜍̃, 𝔚𝜍̃, 𝜚), ℜ(�̈�𝔨, 𝔚𝜍̃, 𝜚), ℜ(�̈�𝜍̃, 𝔏𝔨, 𝜚)} 𝔖 (�̈�𝔨, �̈�𝜍̃, 𝜚 5 ) ≤ 𝔖(𝔏𝔨, 𝔚𝜍̃, 𝜚) = max{𝔖(𝔏𝔨, 𝔚𝜍̃, 𝜚), 𝔖(𝔏𝔨, 𝔄𝔨̈ , 𝜚), 𝔖(�̈�𝜍̃, 𝔚𝜍̃, 𝜚), 𝔖(�̈�𝔨, 𝔚𝜍̃, 𝜚), 𝔖(�̈�𝜍̃, 𝔏𝔨, 𝜚)} 𝔗(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≤ 𝔗(𝔏𝔨, 𝔚𝜍̃, 𝜚) = max{𝔗(𝔏𝔨, 𝔚𝜍̃, 𝜚), 𝔗(𝔏𝔨, 𝔄𝔨̈ , 𝜚), 𝔗(�̈�𝜍̃, 𝔚�̃�, 𝜚), 𝔗(�̈�𝔨, 𝔚𝜍̃, 𝜚), 𝔗(�̈�𝜍̃, 𝔏𝔨, 𝜚)}. Hence, the maps �̈�, 𝔅,̈ 𝔏 and 𝔚 satisfies the condition (3.7.1), (3.7.2) and (3.7.3) of Theorem (3.7) for 𝔡 = 1 5 . Also, the pairs {𝔄,̈ 𝔏} and {�̈�, 𝔚} are obviously OWC. Thus all the condition of Theorem (3.6) are satisfied at 𝔨 = 0 is the unique common fixed point of �̈�, 𝔅,̈ 𝔏 and 𝔚 in Ξ. Corollary 3.9: Let (Ξ, ℜ, 𝔖, 𝔗,∗, ⨀) be a NMS with lim 𝜚→∞ ℜ(𝔨, 𝜍̃, 𝜚) = 1, lim 𝜚→∞ 𝔖(𝔨, 𝜍̃, 𝜚) = 0 and lim 𝜚→∞ 𝔗(𝔨, 𝜍̃, 𝜚) = 0, for all 𝔨, 𝜍̃ ∈ Ξ and let �̈� and 𝔏 be self mappings on Ξ. Let 𝔄,̈ 𝔏 be self mappings on Ξ . Let the pair {𝔄,̈ 𝔏 } be OWC. If there exists 𝔡 ∈ (0, 1) such that ℜ(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≥ ℜ(𝔏𝔨, 𝔏𝜍̃, 𝜚) ∗ ℜ(�̈�𝔨, 𝔏𝔨, 𝜚) ∗ ℜ(�̈�𝜍̃, 𝔏𝜍̃, 𝜚) ∗ ℜ(�̈�𝔨, 𝔏𝜍̃, 𝜚), 𝔖(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≤ 𝔖(𝔏𝔨, 𝔏𝜍̃, 𝜚)⨀𝔖(�̈�𝔨, 𝔏𝔨, 𝜚)⨀𝔖(�̈�𝜍̃, 𝔏𝜍̃, 𝜚)⨀𝔖(�̈�𝔨, 𝔏𝜍̃, 𝜚)and 𝔗(�̈�𝔨, �̈�𝜍̃, 𝔡𝜚) ≤ 𝔗(𝔏𝔨, 𝔏𝜍̃, 𝜚)⨀𝔗(�̈�𝔨, 𝔏𝔨, 𝜚)⨀𝔗(�̈�𝜍̃, 𝔏𝜍̃, 𝜚)⨀𝔗(�̈�𝔨, 𝔏𝜍̃, 𝜚) for all 𝔨, 𝜍̃ ∈ Ξ and 𝜚 > 0. Then 𝔄 ̈ and 𝔏 have a unique common fixed point in Ξ. Theorem 3.10: Let (Ξ, ℜ, 𝔖, 𝔗,∗, ⨀) be a NMS with lim 𝜚→∞ ℜ(𝔨, 𝜍̃, 𝜚) = 1, lim 𝜚→∞ 𝔖(𝔨, 𝜍̃, 𝜚) = 0 and lim 𝜚→∞ 𝔗(𝔨, 𝜍̃, 𝜚) = 0, for all 𝔨, 𝜍̃ ∈ Ξ . Let the pair {𝔄,̈ 𝔏 } be OWC. If there exists 𝔡 ∈ (0, 1) such that ℜ(𝔏𝔨, 𝔏𝜍̃, 𝔡𝜚) ≥ �̃�ℜ(�̈�𝔨, 𝔄𝜍̃̈ , 𝜚) + �̃� 𝑚𝑖𝑛{ℜ(�̈�𝔨, �̈�𝜍̃, 𝜚), ℜ(𝔏𝔨, �̈�𝔨, 𝜚), ℜ(𝔏𝜍̃, �̈�𝜍̃, 𝜚)} 𝔖(𝔏𝔨, 𝔏𝜍̃, 𝔡𝜚) ≤ �̃�𝔖(�̈�𝔨, �̈�𝜍̃, 𝜚) + �̃� 𝑚𝑎𝑥{𝔖(�̈�𝔨, �̈�𝜍̃, 𝜚)𝔖(𝔏𝔨, �̈�𝔨, 𝜚), 𝔖(𝔏𝜍̃, �̈�𝜍̃, 𝜚)} and 𝔗(𝔏𝔨, 𝔏𝜍̃, 𝔡𝜚) ≤ �̃�𝔗(�̈�𝔨, �̈�𝜍̃, 𝜚) + �̃� 𝑚𝑎𝑥{𝔗(�̈�𝔨, �̈�𝜍̃, 𝜚), 𝔗(𝔏𝔨, �̈�𝔨, 𝜚), 𝔗(𝔏𝜍̃, �̈�𝜍̃, 𝜚)} (3.10.1) Proof: The pairs are OWC, so there exists 𝔨 ∈ Ξ such that �̈�(𝔨) = 𝔏(𝔨). Suppose that there exists another 𝜍̃ ∈ Ξ for which �̈�(𝜍̃) = 𝔏(𝜍̃). From the condition (3.10.1), ℜ(𝔏𝔨, 𝔏𝜍̃, 𝔡𝜚) ≥ �̃� ℜ(�̈�𝔨, �̈�𝜍̃, 𝜚) + �̃� 𝑚𝑖𝑛{ℜ(�̈�𝔨, �̈�𝜍̃, 𝜚), ℜ(𝔏𝔨, �̈�𝔨, 𝜚), ℜ(𝔏𝜍̃, �̈�𝜍̃, 𝜚)} =�̃� ℜ(𝔏𝔨, 𝔏𝜍̃, 𝜚) + �̃� 𝑚𝑖𝑛{ℜ(𝔏𝔨, 𝔏𝜍̃, 𝜚), ℜ(𝔏𝔨, 𝔏𝔨, 𝜚), ℜ(𝔏𝜍̃, 𝔏𝜍̃, 𝜚)} = �̃� ℜ(𝔏𝔨, 𝔏𝜍̃, 𝜚) + �̃� 𝑚𝑖𝑛{ℜ(𝔏𝔨, 𝔏𝜍̃, 𝜚), 1,1} = �̃� ℜ(𝔏𝔨, 𝔏𝜍̃, 𝜚) + �̃�ℜ(𝔏𝔨, 𝔏𝜍̃, 𝜚) = (�̃� + �̃�)ℜ(𝔏𝔨, 𝔏𝜍̃, 𝜚). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 126 https://internationalpubls.com Since �̃� + �̃� ≥ 1, ℜ(𝔏𝔨, 𝔏𝜍̃, 𝔡𝜚) ≥ ℜ(𝔏𝔨, 𝔏𝜍̃, 𝜚). 𝔖(𝔏𝔨, 𝔏𝜍̃, 𝔡𝜚) ≤ �̃� 𝔖(�̈�𝔨, �̈�𝜍̃, 𝜚) + �̃� 𝑚𝑎𝑥{𝔖(�̈�𝔨, �̈�𝜍̃, 𝜚), 𝔖(𝔏𝔨, �̈�𝔨, 𝜚), 𝔖(𝔏𝜍̃, �̈�𝜍̃, 𝜚)} = �̃� 𝔖(𝔏𝔨, 𝔏𝜍̃, 𝜚) + �̃� 𝑚𝑎𝑥{𝔖(𝔏𝔨, 𝔏𝜍̃, 𝜚), 𝔖(𝔏𝔨, 𝔏𝔨, 𝜚), 𝔖(𝔏𝜍̃, 𝔏𝜍̃, 𝜚)} = �̃� 𝔖(𝔏𝔨, 𝔏𝜍̃, 𝜚) + �̃� 𝑚𝑎𝑥{𝔖(𝔏𝔨, 𝔏𝜍̃, 𝜚), 0,0 } = �̃� 𝔖(𝔏𝔨, 𝔏𝜍̃, 𝜚) + �̃�𝔖(𝔏𝔨, 𝔏𝜍̃, 𝜚) = (�̃� + �̃�)𝔖(𝔏𝔨, 𝔏𝜍̃, 𝜚). Since �̃� + �̃� ≥ 1, 𝔖(𝔏𝔨, 𝔏𝜍̃, 𝔡𝜚) ≤ 𝔖(𝔏𝔨, 𝔏𝜍̃, 𝜚) 𝔗(𝔏𝔨, 𝔏𝜍̃, 𝔡𝜚) ≤ �̃� 𝔗(�̈�𝔨, �̈�𝜍̃, 𝜚) + �̃� 𝑚𝑎𝑥{𝔗(�̈�𝔨, �̈�𝜍̃, 𝜚), 𝔗(𝔏𝔨, �̈�𝔨, 𝜚), 𝔗(𝔏𝜍̃, �̈�𝜍̃, 𝜚)} = �̃� 𝔗(𝔏𝔨, 𝔏𝜍̃, 𝜚) + �̃� 𝑚𝑎𝑥{𝔗(𝔏𝔨, 𝔏𝜍̃, 𝜚), 𝔗(𝔏𝔨, 𝔏𝔨, 𝜚), 𝔗(𝔏𝜍̃, 𝔏𝜍̃, 𝜚)} = �̃� 𝔗(𝔏𝔨, 𝔏𝜍̃, 𝜚) + �̃� 𝑚𝑎𝑥{𝔗(𝔏𝔨, 𝔏𝜍̃, 𝜚), 0,0} = �̃� 𝔗(𝔏𝔨, 𝔏𝜍̃, 𝜚) + �̃�𝔗(𝔏𝔨, 𝔏𝜍̃, 𝜚) = (�̃� + �̃�)𝔗(𝔏𝔨, 𝔏𝜍̃, 𝜚). Since �̃� + �̃� ≥ 1, 𝔗(𝔏𝔨, 𝔏𝜍̃, 𝔡𝜚) ≤ 𝔗(𝔏𝔨, 𝔏𝜍̃, 𝜚). In view of Lemma (2.10), we have 𝔏𝔨 = 𝔏𝜍̃ and consequently �̈�𝔨 = �̈�𝜍̃. Therefore the pair {𝔄,̈ 𝔏 } have a unique point of coincidence 𝜁 = �̈�𝔨 = 𝔏𝜍̃. Thus, �̈� and 𝔏 have a unique common fixed point in Ξ. References [1] K. 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