Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2146 https://internationalpubls.com Common Neutrosophic Metric Space Fixed Point Theorems with Properties M.Sornavalli1, R. Selvarani2, N. Mehala3 1Assistant Professor, Department of Mathematics, Velammal College of Engineering and Technology, Madurai Email: sornavalliv7@gmail.com 2Associate Professor, Department of Mathematics, K.L.N. College of Engineering , Pottapalayam, Sivagangai Email: selvaklnce@gmail.com 3Assistant Professor, Department of Mathematics, Kamaraj College of Engineering and Technology, Virudhunagar Email: mehalapaviammu@gmail.com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: The paper introduces the notion of Neutrosophic metric spaces and derives some features from it. Under certain appropriate circumstances, two new common fixed point theorems are proved in Neutrosophic metric spaces. This paper introduces the concept of Neutrosophic metric spaces, an extension of classical metric spaces that incorporates the idea of neutrosophy to handle uncertainty and indeterminacy in mathematical analysis. Neutrosophic metric spaces generalize traditional metrics by allowing for a more nuanced representation of distance and proximity, accommodating the presence of indeterminate, uncertain, or contradictory information. Key words: Fixed point, Neutrosophic metric spaces, complete symmetric Neutrosophic metric space. 1. Introduction Zadeh [25] established fuzzy sets, that are crucial for topology and analysis. Numerous Researchers have studied fuzzy sets and their applications. Kramosil and Michlek [8] introduced a novel concept for fuzzy metric spaces. Using the continuous t-norm, George and Veeramani [5] updated the concept of fuzzy metric space. This leads to the derivation of numerous fixed point theorems in fuzzy metric spaces for different types of mappings. In addition to defining D-metric spaces, Dhage [3] established numerous additional fixed point theorems in D-metric spaces. Dhage theory has advanced significantly with the recent definition of G-metric space provided by Mustafa and Sims [11]. Sun and Yang first proposed the idea of a Q-fuzzy metric space in [18]. As a generalization of the fuzzy metric space, we present the idea of a generalized intuitionstic fuzzy metric space. We present new theorems for fixed points in these types of generalized intuitionistic fuzzy metric spaces. The findings of this study expand and enhance a few previously published findings. The Neutrosophic Metric spaces were defined by Kirisci et al. [7]. Neutrosophic extended metric-like spaces were introduced by Ishtiaq et al. [23], who also proved several FP theorems. The mailto:sornavalliv7@gmail.com mailto:selvaklnce@gmail.com mailto:mehalapaviammu@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2147 https://internationalpubls.com authors used the notions of continuous triangular norms, continuous co-norms, metric space, and Neutrosophic sets in Neutrosophic extended metric-like spaces. The notion of neutrosophic double- controlled metric spaces was introduced by Uddin et al. [4] as introduced the concept of a generalization of Neutrosophic metric spaces. See [19–22] for other relevant results. 2. Preliminaries In this section, we provide some definitions that are helpful for readers to understand the main section. Definition: 2.1 [5] A binay operation *:[ 0, 1 ] × [ 0, 1 ] → [ 0, 1 ] is a continuous t-norm if it satisfies the following conditions: (i) * is associative and commutative, (ii) * is continuous, (iii) a*1 = a for all a  [ 0, 1 ], (iv) a*b ≤ c*d whenever a ≤ c and b ≤ d, for each a, b, c, d  [ 0, 1 ]. Examples of continuous t-norm are a*b = ab and a*b = min{ a, b }. Definition: 2.2 [5] A binary operation  : [0, 1] × [0, 1] → [0, 1] is continuous t-conorm if  satisfies the following conditions: (i)  is commutative and associative, (ii)  is continuous, (iii) a  0 = a for all a  [0, 1], (iv) a b ≤ c  d whenever a ≤ c and b ≤ d for all a, b, c, d  [0, 1]. Definition :2.3 [14] A 5-tuple (X, Q, H,*,) is said to be an intuitionstic generalized fuzzy metric space (for short IGFMS) if X is an arbitrary set, ∗ is a continuous t-norm,  is a continuous t- conorm and Q, H are fuzzy set on X3→ (0, ∞) satisfying the following conditions. For every x, y, z, a ∈X and t, s > 0 (i) Q( x, y, z, t) + H (x, y, z, t) ≤ 1, (ii) Q( x, x, y, t) > 0, for all x ≠ y, (iii) Q( x, x, y, t ) ≤ Q ( x,y, z, t) for y ≠ z, (iv) Q ( x, y, z t) =1 iff x = y = z , (v) Q ( x, y, z ,t) = Q {p (x, y, z),t}, where p is a permutation function, (vi) Q (x, a, a t) ∗ Q ( a, y,z, s) ≤ Q ( x, y, z, t+s), (vii) Q ( x, y, z, .) : (0,∞) → [ 0,1] is continuous, (viii) Q is non decreasing function on R+ lim 𝑡→∞ Q( x, y, z, t ) = 1 and lim 𝑡→0 Q( x, y, z, t ) = 0, for all x, y, z 𝜖 X , t > 0, (ix) H ( x, x, y, t) < 1 , for all x ≠ y, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2148 https://internationalpubls.com (x) H ( x, x, y, t) ≥ H ( x, y, z ,t ) for y ≠ z, (xi) H ( x, y, z t) = 0 iff x = y = z , (xii) H ( x, y, z ,t) = H {p (x, y, z),t} where p is a permutation function, (xiii) H (x, a, a t) ◊ H ( a, y, z, s) ≥ H ( x,y, z, t + s), (xiv) H ( x, y, z, .) : (0,∞) → [ 0,1] is continuous, (xv) H is a non- increasing function on R+lim 𝑡→0 H (x, y, z, t) = 0 and lim 𝑡→0 H( x, y, z, t ) = 1 for all x, y, z ∈ X, t > 0, In this case, the pair (Q, H) is called an intuitionistic generalized fuzzy metric on X. Definition: 2.4 [20] A 6-tuple ( Ξ, 𝒬, ℋ, 𝒪 *, ) is Said to be an Neutrosophic Metric space if Ξ is an arbitrary set, * is a Continuous t-norm, is a Continuous t-conorm, and 𝒬, ℋ, 𝒪 are Neutrosophic set on Ξ2 x (0, ∞) satisfying the following conditions: for all 𝜛, 𝜔, 𝜎 ∈Ξ, 𝜆, 𝜏> 0, (i) 𝒬(𝜛, 𝜔, 𝜏, 휁) + ℋ (𝜛, 𝜔, 𝜏, 휁) + 𝒪 (𝜛, 𝜔, 𝜏, 휁) ≤ 3; (ii) 𝒬(𝜛, 𝜛, 𝜔, 휁) > 0; for 𝜛 ≠ 𝜔; (iii) 𝒬(𝜛, 𝜛𝜔, 휁) ≤ 𝒬(𝜛, 𝜔, 𝜏, 휁), for 𝜔 ≠ 휁; (iv) 𝒬(𝜛, 𝜔, 𝜏, 휁) = 1 if and only if 𝜛 = 𝜔 = 𝜏; (v) 𝒬(𝜛, 𝜔, 𝜏, 휁) * 𝒬(p (𝜛, 𝜔, 𝜏), 휁) where p is a permutation function; (vi) 𝒬(𝜛, a, a, 휁) ∗ 𝒬(𝑎, 𝜔, 𝜏, 휂) ≤ 𝒬(𝜛, 𝜔, 𝜏, 휁 + 휂); (vii) 𝒬(𝜛, 𝜔, 𝜏, .): (0, ∞) → [0,1] is continuous; (viii) 𝒬 is non-decreasing of ℜ+, lim 𝜁→∞ 𝒬(𝜛, 𝜔, 𝜏, 휁) = 1 and lim 𝜁→0 𝒬(𝜛, 𝜔, 𝜏, 휁) = 0 for all 𝜛, 𝜔, 𝜏 ∈Ξ, 휁> 0 (ix) ℋ(𝜛, 𝜛, 𝜔, 휁) < 1; for 𝜛 ≠ 𝜔; (x) ℋ (𝜛, 𝜛𝜔, 휁) ≥ ℋ (𝜛, 𝜔, 𝜏, 휁), for 𝜔 ≠ 휁; (xi) ℋ(𝜛, 𝜔, 𝜏, 휁) = 0 if and only if 𝜛 = 𝜔 = 𝜏; (xii) ℋ(𝜛, 𝜔, 𝜏, 휁) ℋ(p (𝜛, 𝜔, 𝜏), 휁) where p is a permutation function; (xiii) ℋ(𝜛, a, a, 휁) ∗ ℋ(𝑎, 𝜔, 𝜏, 휂) ≥ ℋ(𝜛, 𝜔, 𝜏, 휁 + 휂); (xiv) ℋ(𝜛, 𝜔, 𝜏, .): (0, ∞) → [0,1] is continuous; (xv) ℋ is non-increasing of ℜ+, lim 𝜁→∞ ℋ(𝜛, 𝜔, 𝜏, 휁) = 1 and lim 𝜁→0 ℋ(𝜛, 𝜔, 𝜏, 휁) = 1 for all 𝜛, 𝜔, 𝜏 ∈Ξ, 휁> 0 (xvi) 𝒪(𝜛, 𝜛, 𝜔, 휁) < 1; for 𝜛 ≠ 𝜔; (xvii) 𝒪 (𝜛, 𝜛𝜔, 휁) ≥ 𝒪 (𝜛, 𝜔, 𝜏, 휁), for 𝜔 ≠ 휁; (xviii) 𝒪(𝜛, 𝜔, 𝜏, 휁) = 0 if and only if 𝜛 = 𝜔 = 𝜏; (xix) 𝒪(𝜛, 𝜔, 𝜏, 휁) 𝒪(p (𝜛, 𝜔, 𝜏), 휁) where p is a permutation function; (xx) 𝒪(𝜛, a, a, 휁) ∗ 𝒪(𝑎, 𝜔, 𝜏, 휂) ≥ 𝒪(𝜛, 𝜔, 𝜏, 휁 + 휂); (xxi) 𝒪(𝜛, 𝜔, 𝜏, .): (0, ∞) → [0,1] is continuous; (xxii) 𝒪 is non-increasing of ℜ+, lim 𝜁→∞ 𝒪(𝜛, 𝜔, 𝜏, 휁) = 1 and lim 𝜁→0 𝒪(𝜛, 𝜔, 𝜏, 휁) = 1 for all 𝜛, 𝜔, 𝜏 ∈Ξ, 휁> 0 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2149 https://internationalpubls.com Then (𝒬, ℋ, 𝒪) is said to be a neutrosophic metric on Ξ. The function 𝒬, ℋ and 𝒪 denote respectively degree of closeness, neutrality and noncloseness between 𝜛, 𝜔 and 𝜏 with respect to 휁 respectively. Example: 2.5 [20] Let (Ξ, d) be a Q- metric space, for all 𝜛, 𝜔, 𝜏, 휁 ∈Ξ, and every 휁 > 0, consider 𝒬, ℋ, 𝒪 to be fuzzy sets on Ξ3 x (0, ∞) defined by 𝒬(𝜛, 𝜔, 𝜏, 휁) = 𝜁 𝜁+𝐺(𝜛,𝜔,𝜏,𝜁) and ℋ(𝜛, 𝜛, 𝜔, 휁) = 𝐺(𝜛,𝜔,𝜏,𝜁) 𝜁+𝐺(𝜛,𝜔,𝜏,𝜁) and 𝒪(𝜛, 𝜛, 𝜔, 휁) = 𝐺(𝜛,𝜔,𝜏,𝜁) 𝜁+𝐺(𝜛,𝜔,𝜏,𝜁) denote a∗ b = ab anda ◊ b = min { a+b , 1} . Then( Ξ, 𝒬, ℋ,𝒪 *,) is an Neutrosophic metric space. Notice that the above example holds even with the t- norm a ∗ b = min {a, b} and t – conorm a ◊ b = max {a, b}. Definition: 2.6[20] Let ( Ξ, 𝒬, ℋ,𝒪 *,) be Neutrosophic metric space, then a sequence { 𝜛n} in Ξ is said to be convergent if i) lim n→∞ 𝒬( 𝜛n, 𝜛n, 𝜛, 휁)= 1, lim n→∞ ℋ( 𝜛n, 𝜛n, 𝜛, 휁 ) = 0, lim n→∞ 𝒪( 𝜛n, 𝜛n, 𝜛, 휁 ) = 0. ii) A sequence { 𝜛n} in Ξ is said to be Cauchy sequence if lim n,m→∞ 𝒬( 𝜛n, 𝜛n, 𝜛m, 휁 ) = 1, lim n,m→∞ ℋ( 𝜛n, 𝜛n, 𝜛m, 휁 ) = 0, lim n,m→∞ 𝒪( 𝜛n, 𝜛n, 𝜛m, 휁 ) = 0 that is, for any 휁 > 0 and 휀 > 0 there exists n0 ∈ N such that 𝒬( 𝜛n, 𝜛n, 𝜛m, 휁 ) > 1- 휀, ℋ( 𝜛n, 𝜛n, 𝜛m, 휁 ) < 휀 𝒪( 𝜛n, 𝜛n, 𝜛m, 휁 ) < 휀 for n, m ≥ n0. iii) A Neutrosophic metric space (Ξ, 𝒬, ℋ,𝒪 *,)is said to be complete if every Cauchy sequence in Ξ is convergent. Definition: 2.7[20] Let (Ξ, 𝒬, ℋ,𝒪 *,)be a Neutrosophic metric space . The following conditions are satisfied : lim n→∞ 𝒬(𝜛n, 𝜔n, 𝜏n, 휁n) = 𝒬(𝜛, 𝜔, 𝜏, 휁) , lim n→∞ ℋ(𝜛n, 𝜔n, 𝜏n, 휁n) = ℋ(𝜛, 𝜔, 𝜏, 휁) lim n→∞ 𝒪(𝜛n, 𝜔n, 𝜏n, 휁n) = 𝒪(𝜛, 𝜔, 𝜏, 휁) Whenever lim n→∞ 𝜛 n= 𝜛 ; lim n→∞ 𝜔𝑛 = 𝜔 ; lim n→∞ 𝜏𝑛 = 𝜏 and Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2150 https://internationalpubls.com lim n→∞ 𝒬(𝜛, 𝜔, 𝜏, 휁n) = 𝒬( 𝜛, 𝜔, 𝜏, 휁), lim n→∞ ℋ(𝜛, 𝜔, 𝜏, 휁n) = ℋ( 𝜛, 𝜔, 𝜏, 휁) lim n→∞ 𝒪(𝜛, 𝜔, 𝜏, 휁n) = 𝒪( 𝜛, 𝜔, 𝜏, 휁) then 𝒬, ℋ, 𝒪are called convergent function on Ξ3 x(0, ∞). Definition:2.8 [20] Let f, g be self maps on Neutrosophic metric space (Ξ, 𝒬, ℋ,𝒪 *,) . Then the mappings are said to be weakly compatible if they commute at their coincidence point, that is, f 𝜛 = g 𝜛implies that fg 𝜛 =gf 𝜛. Definition:2.9 [20] Let f , g be self maps Neutrosophic metric space(Ξ, 𝒬, ℋ,𝒪 *,) The pair (f,g) is said to be compatible if lim 𝑛→∞ 𝒬( fg𝜛n , gf𝜛n, gf𝜛n, 휁) = 1, lim 𝑛→∞ ℋ( fg𝜛n ,gf𝜛n, gf𝜛n, 휁) = 0, lim 𝑛→∞ 𝒪( fg𝜛n ,gf𝜛n, gf𝜛n, 휁) = 0 Whenever {𝜛n} is a sequence in Ξ such that lim 𝑛→∞ f 𝜛n = lim n→∞ g 𝜛n = z for some z ∈Ξ. Definition:2.10 [20] Two self maps A,S and T of a Neutrosophic metric space (Ξ, 𝒬, ℋ,𝒪 *,) are called jointly W- continuous if there exists a point 𝜛 ∈ Ξ such that if lim n→∞ 𝒬(𝜛n, 𝜛, 𝜛, 휁) = 1, lim n→∞ ℋ(𝜛n, 𝜛, 𝜛, 휁) = 0 lim n→∞ 𝒪(𝜛n, 𝜛, 𝜛, 휁) = 0 then lim n→∞ 𝒬(A𝜛n, S𝜛, S𝜛, 휁) = 1 , lim n→∞ ℋ(A𝜛n, S𝜛, S𝜛, 휁) = 0, lim n→∞ 𝒪(A𝜛n, S𝜛, S𝜛, 휁) = 0, whenever {𝜛n} is a sequence in Ξ such that lim n→∞ A𝜛n = lim n→∞ S𝜛n = p for some p ∈ Ξ. Definition: 2.11 [20] Let the self maps A,S and T of a Neutrosophic metric space (Ξ, 𝒬, ℋ,𝒪 *,) . If A, S and T satisfy the following conditions: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2151 https://internationalpubls.com There exists a sequence { 𝜛 n} such that lim 𝑛→∞ 𝒬( A𝜛n,u, u, 휁 ) = lim n→∞ 𝒬(S𝜛n, u, u, 휁) = lim n→∞ 𝒬(T𝜛n, u, u, 휁) = 1 lim 𝑛→∞ ℋ( A𝜛n,u, u, 휁 ) = lim n→∞ ℋ(S𝜛n, u, u, 휁) = 1 lim n→∞ ℋ(T𝜛n, u, u, 휁) = 0 lim 𝑛→∞ 𝒪 ( A𝜛n,u, u, 휁 ) = lim n→∞ 𝒪 (S𝜛n, u, u, 휁) = 1 lim n→∞ 𝒪 (T𝜛n, u, u, 휁) = 0 for some u ∈ Ξ and 휁 > 0, we say that A,S and T have the property (E.A). Lemma: 2.12 [20] Let (Ξ, 𝒬, ℋ,𝒪 *,) be a Neutrosophic metric space. Then , 𝒬, ℋ, 𝒪 are continuous function on Ξ 3 x (0, ∞). Proof: Since lim n→∞ 𝜛 n= 𝜛 ; lim n→∞ 𝜔𝑛 = 𝜔 ; lim n→∞ 𝜏𝑛 = 𝜏 lim 𝑛→∞ 𝒬 ( 𝜛, 𝜔, 𝜏, 휁𝑛) = 𝒬( 𝜛, 𝜔, 𝜏, 휁) and lim 𝑛→∞ ℋ ( 𝜛, 𝜔, 𝜏, 휁𝑛) = ℋ( 𝜛, 𝜔, 𝜏, 휁) lim 𝑛→∞ 𝒪 ( 𝜛, 𝜔, 𝜏, 휁𝑛) = 𝒪( 𝜛, 𝜔, 𝜏, 휁) There is ϑ0 ∈ N such that |휁 − 휁𝑛| < 휀 and |휁 − 휁𝑛| > 𝛿 for ϑ ≥ ϑ0 and 휀 < 휁 2⁄ and 𝛿 > 휁 2⁄ We know that 𝒬( 𝜛, 𝜔, 𝜏, 휁) is non-decreasing and ℋ( 𝜛, 𝜔, 𝜏, 휁), 𝒪( 𝜛, 𝜔, 𝜏, 휁) is non- increasing with respect to 휁, So, we have 𝒬 (𝜛n, 𝜔n, 𝜏n, 휁) ≥ 𝒬 (𝜛n, 𝜔n, 𝜏n, 휁- 휀) ≥ 𝒬 (𝜛n, 𝜛, 𝜛, ε 3 ) ∗ 𝒬 (𝜛, 𝜔n, 𝜏n, 휁 - 4ε 3 ) ≥ 𝒬 (𝜛 n, 𝜛, 𝜛, ε 3 ) ∗ 𝒬 (𝜔n, 𝜔, 𝜔, ε 3 ) ∗ 𝒬 (𝜔, 𝜛, 𝜏n, 휁 - 5ε 3 ) ≥ 𝒬 (𝜛n, 𝜛, 𝜛, ε 3 ) ∗ 𝒬 (𝜔n, 𝜔, 𝜔, ε 3 ) ∗ 𝒬 (𝜏n, 𝜏, 𝜏, ε 3 ) ∗ 𝒬 (𝜏, 𝜔, 𝜏, 휁 - 2휀) and ℋ (𝜛n, 𝜔n, 𝜏n, 휁) ≤ ℋ (𝜛n, 𝜔n, 𝜏n, 휁- 𝛿) ≤ ℋ(𝜛n, 𝜛, 𝜛, 𝛿 3 ) ◊ 𝒬 (𝜛, 𝜔n, 𝜏n, 휁 - 4𝛿 3 ) ≤ ℋ (𝜛 n, 𝜛, 𝜛, 𝛿 3 ) ◊ ℋ (𝜔n, 𝜔, 𝜔, 𝛿 3 ) ◊ ℋ (𝜔, 𝜛, 𝜏n, 휁 - 5𝛿 3 ) ≤ ℋ (𝜛n, 𝜛, 𝜛, 𝛿 3 ) ◊ ℋ (𝜔n, 𝜔, 𝜔, 𝛿 3 ) ◊ ℋ (𝜏n, 𝜏, 𝜏, 𝛿 3 ) ◊ ℋ (𝜏, 𝜔, 𝜏, 휁 - 2𝛿) 𝒪 (𝜛n, 𝜔n, 𝜏n, 휁) ≤ 𝒪 (𝜛n, 𝜔n, 𝜏n, 휁- 𝛿) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2152 https://internationalpubls.com ≤ 𝒪 (𝜛n, 𝜛, 𝜛, 𝛿 3 ) ◊ 𝒪 (𝜛, 𝜔n, 𝜏n, 휁 - 4𝛿 3 ) ≤ 𝒪 (𝜛 n, 𝜛, 𝜛, 𝛿 3 ) ◊ 𝒪 (𝜔n, 𝜔, 𝜔, 𝛿 3 ) ◊ 𝒪 (𝜔, 𝜛, 𝜏n, 휁 - 5𝛿 3 ) ≤ 𝒪 (𝜛n, 𝜛, 𝜛, 𝛿 3 ) ◊ 𝒪 (𝜔n, 𝜔, 𝜔, 𝛿 3 ) ◊ 𝒪 (𝜏n, 𝜏, 𝜏, 𝛿 3 ) ◊ 𝒪 (𝜏, 𝜔, 𝜏, 휁 - 2𝛿) 𝒬 (𝜛, 𝜔, 𝜏, 휁 + 2 ε) ≥ 𝒬 (𝜛, 𝜔, 𝜏, 휁𝑛+ ε) ≥ 𝒬 (𝜛, 𝜛 n, 𝜛n, ε 3 ) ∗ 𝒬 (𝜛n, 𝜔, 𝜏, 휁n + 2ε 3 ) ≥ 𝒬 (𝜛, 𝜛n, 𝜛n, ε 3 ) ∗ 𝒬 (𝜔, 𝜔n, 𝜔n, ε 3 ) ∗ 𝒬 (𝜔n, 𝜛n, 𝜏, 휁n+ ε 3 ) ≥ 𝒬 (𝜛, 𝜛n, 𝜛n, ε 3 ) ∗ 𝒬 (𝜔, 𝜔n, 𝜔n, ε 3 ) ∗ 𝒬 (𝜏, 𝜏n, 𝜏n, ε 3 ) ∗ 𝒬 (𝜏, 𝜔, 𝜛, 휁n) and ℋ (𝜛, 𝜔, 𝜏, 휁 + 2 ε) ≤ ℋ (𝜛, 𝜔, 𝜏, 휁𝑛+ δ) ≤ ℋ (𝜛, 𝜛 n, 𝜛n, δ 3 ) ◊ ℋ (𝜛n, 𝜔, 𝜏, 휁n + 2δ 3 ) ≤ ℋ (𝜛, 𝜛n, 𝜛n, δ 3 ) ◊ ℋ (𝜔, 𝜔n, 𝜔n, δ 3 ) ◊ ℋ (𝜔n, 𝜛n, 𝜏, 휁n+ δ 3 ) ≤ ℋ (𝜛, 𝜛n, 𝜛n, δ 3 ) ◊ ℋ (𝜔, 𝜔n, 𝜔n, δ 3 ) ◊ ℋ (𝜏, 𝜏n, 𝜏n, δ 3 ) ◊ ℋ (𝜏, 𝜔, 𝜛, 휁n) 𝒪 (𝜛, 𝜔, 𝜏, 휁 + 2 ε) ≤ 𝒪 (𝜛, 𝜔, 𝜏, 휁𝑛+ δ) ≤ 𝒪 (𝜛, 𝜛 n, 𝜛n, δ 3 ) ◊ 𝒪 (𝜛n, 𝜔, 𝜏, 휁n + 2δ 3 ) ≤ 𝒪 (𝜛, 𝜛n, 𝜛n, δ 3 ) ◊ 𝒪 (𝜔, 𝜔n, 𝜔n, δ 3 ) ◊ 𝒪 (𝜔n, 𝜛n, 𝜏, 휁n+ δ 3 ) ≤ 𝒪 (𝜛, 𝜛n, 𝜛n, δ 3 ) ◊ 𝒪 (𝜔, 𝜔n, 𝜔n, δ 3 ) ◊ 𝒪 (𝜏, 𝜏n, 𝜏n, δ 3 ) ◊ 𝒪 (𝜏, 𝜔, 𝜛, 휁n) Let n → ∞, by continuity of the function 𝒬, ℋ, 𝒪 with respect to 휁, we can get 𝒬 (𝜛, 𝜔, 𝜏, 휁 + 2 ε) ≥ 𝒬 (𝜏, 𝜔, 𝜛, t) ≥ 𝒬 (𝜏, 𝜔, 𝜛, 휁 - 2 ε) ℋ (𝜛, 𝜔, 𝜏, 휁 + 2 ε) ≤ ℋ (𝜏, 𝜔, 𝜛, t) ≤ ℋ (𝜏, 𝜔, 𝜛, 휁 - 2 δ) 𝒪 (𝜛, 𝜔, 𝜏, 휁 + 2 ε) ≤ 𝒪 (𝜏, 𝜔, 𝜛, t) ≤ 𝒪 (𝜏, 𝜔, 𝜛, 휁 - 2 δ) Therefore 𝒬, ℋ, 𝒪 are continuous function on Ξ 3 x (0, ∞). 3. Main Theorem We first generalize a classic theorem in Neutrosophic Metric Space THEOREM 3.1: [20] Let ℑ, ℶ, ℱ, ℌ, η and ξ be self- mappings of a complete symmetric Neutrosophic metric space (Ξ, 𝒬, ℋ,𝒪 *,) with 휁 ∗ 휁 ≥ 휁 and 휁 ◊ 휁 ≤ 1- 휁 if the mappings satisfy the following conditions: [3.1.1] ℑ (Ξ) ⊆ ℌ (Ξ) , ℶ (Ξ) ⊆ η (Ξ), ℱ (Ξ) ⊆ ξ (Ξ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2153 https://internationalpubls.com [3.1.2] (ℑ, ξ) or (ℶ,η) or (ℱ, ℌ) satisfy the property (E.A) [3.1.3] (ℑ, ξ) , (ℶ,η) and (ℱ, ℌ) are weakly compatible [3.1.4] 𝒬 (ℑ𝜛, ℶ𝜔, ℱ𝜏, k휁) ≥ { 𝒬 (ξ𝜛, ℌ𝜔, η𝜏, 휁) ∗ 𝒬 (ℑ𝜛, ℌ𝜔, η𝜏, 휁) ∗ 𝒬 (η𝜛, ℶ𝜔, ℱ𝜏, 휁) } [3.1.5] ℋ(ℑ𝜛, ℶ𝜔, ℱ𝜏, k휁) ≤ { ℋ (ξ𝜛, ℌ𝜔, η𝜏, 휁) ◊ ℋ (ℑ𝜛, ℌ𝜔, η𝜏, 휁) ◊ ℋ (η𝜛, ℶ𝜔, ℱ𝜏, 휁)} [3.1.6] 𝒪 (ℑ𝜛, ℶ𝜔, ℱ𝜏, k휁) ≤ { 𝒪 (ξ𝜛, ℌ𝜔, η𝜏, 휁) ◊ 𝒪 (ℑ𝜛, ℌ𝜔, η𝜏, 휁) ◊ 𝒪 (η𝜛, ℶ𝜔, ℱ𝜏, 휁)} there exists k ∈ (0,1) such that for every 𝜛, 𝜔, 𝜏 ∈ Ξ and 휁 > 0. Then ℑ, ℶ, ℱ, ℌ, η and ξ have a unique common fixed point in Ξ. Proof: Suppose (ℑ, ξ) satisfy the property (E.A), hence there exists a sequence {𝜛n} such that lim 𝑛→∞ 𝒬( ℑ𝜛n,u, u, 휁 ) = lim n→∞ 𝒬 (ξ 𝜛n, u, u, 휁) = 1 lim 𝑛→∞ ℋ ( ℑ𝜛n,u, u, 휁 ) = lim n→∞ ℋ (ξ 𝜛n, u, u, 휁) = 0 lim 𝑛→∞ 𝒪 ( ℑ𝜛n,u, u, 휁 ) = lim n→∞ 𝒪 (ξ 𝜛n, u, u, 휁) = 0 for some u ∈ Ξ and 휁 > 0. Since ℑ (Ξ) ⊆ ℌ (Ξ), there exists a sequence {𝜔n} such that ℑ𝜛n = ℌ𝜔n ⇒ lim n→∞ 𝒬( ℌ𝜔n,u, u, t ) = 1, lim n→∞ ℋ( ℌ𝜔n,u, u, t ) = 0 and lim n→∞ 𝒪( ℌ𝜔n,u, u, t ) = 0 Therefore, 𝒬 (ℑ𝜛𝑛, ℶ𝜔𝑛, ℱ𝜏𝑛+1, k휁) ≥ 𝒬(ξ𝜛𝑛, ℌ𝜔𝑛, η𝜏𝑛+1, 휁)∗ 𝒬(ℑ𝜛𝑛, ℌ𝜔𝑛, η𝜏𝑛+1, 휁) ∗ 𝒬 (η𝜛𝑛, ℶ𝜔𝑛, ℱ𝜏𝑛+1, 휁) ≥ { 𝒬(ξ𝜛𝑛, u, u, ½ 휁) ∗ 𝒬 ( u, ℌ𝜔𝑛, η𝜏𝑛+1, ½ 휁) ∗ 𝒬 (ℑ𝜛𝑛, u, u, ½ 휁) ∗ 𝒬 ( u, ℌ𝜔𝑛, η𝜏𝑛+1, ½ 휁) ∗ 𝒬 (η𝜛𝑛, ℶ𝜔𝑛, ℱ𝜏𝑛+1, 휁)} ℋ(ℑ𝜛𝑛, ℶ𝜔𝑛, ℱ𝜏𝑛+1, k휁) ≤ ℋ(ξ𝜛𝑛, ℌ𝜔𝑛, η𝜏𝑛+1, 휁)◊ ℋ(ℑ𝜛𝑛, ℌ𝜔𝑛, η𝜏𝑛+1, 휁)◊ ℋ(η𝜛𝑛,ℶ𝜔𝑛, ℱ𝜏𝑛+1, 휁) ≤ {ℋ(ξ𝜛𝑛, u, u, ½ 휁) ◊ ℋ ( u, ℌ𝜔𝑛, η𝜏𝑛+1, ½ 휁) ◊ ℋ (ℑ𝜛𝑛, u, u, ½ 휁) ◊ ℋ ( u, ℌ𝜔𝑛, η𝜏𝑛+1, ½ 휁) ◊ ℋ (η𝜛𝑛, ℶ𝜔𝑛, ℱ𝜏𝑛+1, 휁)} 𝒪 (ℑ𝜛𝑛, ℶ𝜔𝑛, ℱ𝜏𝑛+1, k휁) ≤ 𝒪 (ξ𝜛𝑛, ℌ𝜔𝑛, η𝜏𝑛+1, 휁)◊ 𝒪 (ℑ𝜛𝑛, ℌ𝜔𝑛, η𝜏𝑛+1, 휁)◊ 𝒪(η𝜛𝑛,ℶ𝜔𝑛, ℱ𝜏𝑛+1, 휁) ≤ {𝒪(ξ𝜛𝑛, u, u, ½ 휁) ◊ 𝒪 ( u, ℌ𝜔𝑛, η𝜏𝑛+1, ½ 휁) ◊ 𝒪 (ℑ𝜛𝑛, u, u, ½ 휁) ◊ 𝒪( u, ℌ𝜔𝑛, η𝜏𝑛+1, ½ 휁) ◊ 𝒪 (η𝜛𝑛, ℶ𝜔𝑛, ℱ𝜏𝑛+1, 휁) there exists 𝛿 > 0 such that k + 𝛿 < 1, for k ∈ (0,1). On making n → ∞ and by the symmetry Neutrosophic metric space, we have Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2154 https://internationalpubls.com lim 𝑛→∞ 𝒬 (ℑ𝜛𝑛, ℶ𝜔𝑛, ℱ𝜏𝑛+1, k휁) ≥ 1 ∗ 1 ∗ 1 ∗ 1 lim 𝑛→∞ 𝒬(ℑ𝜛𝑛, ℶ𝜔𝑛, ℱ𝜏n [ 1- (k + δ)] 휁) ∗ lim 𝑛→∞ 𝒬(ℑ𝜛𝑛, ℶ𝜔𝑛, ℱ𝜏n+1, (k + δ)] 휁) ≥ 1 ∗ 1 ∗ 1 ∗ 1 ∗ 1 lim 𝑛→∞ 𝒬(ℑ𝜛𝑛, ℶ𝜔𝑛, ℱ𝜏n+1, (k + δ)] 휁) lim 𝑛→∞ ℋ (ℑ𝜛𝑛, ℶ𝜔𝑛, ℱ𝜏𝑛+1, k휁) ≤ 0 ◊ 0 ◊ 0 ◊ 0 lim 𝑛→∞ ℋ(ℑ𝜛𝑛, ℶ𝜔𝑛, ℱ𝜏n [ 1- (k + δ)] 휁) ◊ lim 𝑛→∞ ℋ(ℑ𝜛𝑛, ℶ𝜔𝑛, ℱ𝜏n+1, (k + δ)] 휁) ≤ 0 ◊ 0 ◊ 0 ◊ 0 lim 𝑛→∞ 𝒪(ℑ𝜛𝑛, ℶ𝜔𝑛, ℱ𝜏n+1, (k + δ)] 휁) lim 𝑛→∞ 𝒪 (ℑ𝜛𝑛, ℶ𝜔𝑛, ℱ𝜏𝑛+1, k휁) ≤ 0 ◊ 0 ◊ 0 ◊ 0 lim 𝑛→∞ 𝒪(ℑ𝜛𝑛, ℶ𝜔𝑛, ℱ𝜏n [ 1- (k + δ)] 휁) ◊ lim 𝑛→∞ 𝒪(ℑ𝜛𝑛, ℶ𝜔𝑛, ℱ𝜏n+1, (k + δ)] 휁) ≤ 0 ◊ 0 ◊ 0 ◊ 0 Hence lim 𝑛→∞ ℑ 𝜛𝑛 = lim n→∞ ℶ 𝜔𝑛 = lim n→∞ ℱ𝜏𝑛 = lim n→∞ ℌ 𝜏𝑛 = lim n→∞ η 𝜔𝑛 = lim n→∞ ξ𝜛𝑛 = u Let (Ξ, 𝒬, ℋ,𝒪 *,) is a complete Neutrosophic metric space, there exists 𝜛0 ∈ Ξ such that ξ𝜛0 = u ⇒ 𝒬 (ℑ𝜛0, ℶ𝜔𝑛, ℱ𝜏𝑛+1, k휁) ≥ 𝒬 (η𝜛0, ℌ𝜔𝑛, ξ 𝜏𝑛+1, 휁) ∗ 𝒬(ℑ𝜛0, ℌ𝜔𝑛, ξ 𝜏𝑛+1, 휁) ∗ 𝒬 (ξ𝜛0, ℌ𝜔𝑛, ξ 𝜏𝑛+1, 휁) If n → ∞ we can get 𝒬 (ℑ𝜛0, u, u, 휁 ) ≥ 1 ∗ 𝒬 (ℑ𝜛0, u, u, 휁 ) ∗ 1. By the property of non- decreasing with respect to 휁, ξ 𝜛0 = u ⇒ ℋ (ℑ𝜛0, ℶ𝜔𝑛, ℱ𝜏𝑛+1, k휁) ≤ ℋ (η𝜛0, ℌ𝜔𝑛, ξ 𝜏𝑛+1, 휁) ◊ ℋ(ℑ𝜛0, ℌ𝜔𝑛, ξ 𝜏𝑛+1, 휁) ◊ ℋ (ξ𝜛0, ℌ𝜔𝑛, ξ 𝜏𝑛+1, 휁) If n → ∞ we can get ℋ (ℑ𝜛0, u, u, 휁 ) ≤ 0 ◊ ℋ (ℑ𝜛0, u, u, 휁 ) ◊ 0 By the property of non- increasing with respect to 휁, ξ 𝜛0 = u ⇒ 𝒪 (ℑ𝜛0, ℶ𝜔𝑛, ℱ𝜏𝑛+1, k휁) ≤ 𝒪 (η𝜛0, ℌ𝜔𝑛, ξ 𝜏𝑛+1, 휁) ◊ 𝒪(ℑ𝜛0, ℌ𝜔𝑛, ξ 𝜏𝑛+1, 휁) ◊ 𝒪 (ξ𝜛0, ℌ𝜔𝑛, ξ 𝜏𝑛+1, 휁) If n → ∞ we can get 𝒪 (ℑ𝜛0, u, u, 휁 ) ≤ 0 ◊ 𝒪 (ℑ𝜛0, u, u, 휁 ) ◊ 0 it is easy to see that ℑ𝜛0 = ξ 𝜛0 = u. As ℑ(Ξ) ⊆ ℌ(Ξ), there exists 𝜔0 such that ℑ𝜛0 = ℌ𝜔0. Suppose ℌ𝜔0 ≠ ℶ𝜔0. Then 𝒬 (ℑ𝜛𝑛, ℶ𝜛0, ℶ𝜛0, k휁) ≥ 𝒬 (η𝜛𝑛, ℌ𝜔0, ℌ𝜔0, 휁) ∗ 𝒬(ℑ𝜛𝑛, ℌ𝜔0, ℌ𝜔0, 휁) ∗ 𝒬(η𝜛𝑛, ℶ𝜛0, ℶ𝜛0, 휁) ≥ 𝒬(η𝜛𝑛, u, u, 휁) ∗ 𝒬 (ℑ𝜛𝑛, u, u, 휁 ) ∗ 𝒬(η𝜛𝑛, ℶ𝜛0, ℶ𝜛0, 휁) ℋ (ℑ𝜛𝑛, ℶ𝜛0, ℶ𝜛0, k휁) ≤ ℋ (η𝜛𝑛, ℌ𝜔0, ℌ𝜔0, 휁) ◊ ℋ(ℑ𝜛𝑛, ℌ𝜔0, ℌ𝜔0, 휁) ◊ ℋ(η𝜛𝑛, ℶ𝜛0, ℶ𝜛0, 휁) ≤ ℋ(η𝜛𝑛, u, u, 휁) ◊ ℋ (ℑ𝜛𝑛, u, u, 휁 ) ◊ ℋ(η𝜛𝑛, ℶ𝜛0, ℶ𝜛0, 휁) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2155 https://internationalpubls.com 𝒪 (ℑ𝜛𝑛, ℶ𝜛0, ℶ𝜛0, k휁) ≤ 𝒪 (η𝜛𝑛, ℌ𝜔0, ℌ𝜔0, 휁) ◊ 𝒪(ℑ𝜛𝑛, ℌ𝜔0, ℌ𝜔0, 휁) ◊ 𝒪(η𝜛𝑛, ℶ𝜛0, ℶ𝜛0, 휁) ≤ 𝒪 (η𝜛𝑛, u, u, 휁) ◊ 𝒪 (ℑ𝜛𝑛, u, u, 휁 ) ◊ 𝒪(η𝜛𝑛, ℶ𝜛0, ℶ𝜛0, 휁) Letting n → ∞ we have 𝒬 (u, ℶ𝜛0, ℶ𝜛0, k휁) ≥ 𝒬 (u, u, u, 휁) ∗ 𝒬 (u, u, u, 휁) ∗ 𝒬 (u, ℶ𝜛0, ℶ𝜛0, 휁) ℋ (u, ℶ𝜛0, ℶ𝜛0, k휁) ≤ ℋ (u, u, u, 휁) ◊ ℋ (u, u, u, 휁) ◊ ℋ (u, ℶ𝜛0, ℶ𝜛0, 휁) 𝒪 (u, ℶ𝜛0, ℶ𝜛0, k휁) ≤ 𝒪 (u, u, u, 휁) ◊ 𝒪 (u, u, u, 휁) ◊ 𝒪 (u, ℶ𝜛0, ℶ𝜛0, 휁) Which is a contradiction. So, ℶ𝜔0 = ℌ𝜔0= u. Now by (ℑ, ξ) , (ℶ,η) and (ℱ, ℌ) are weakly compatible ℑℑ 𝜛0 = ℑξ𝜛0 = ξℑ𝜛0 = ξξ𝜛0 and ℶℶ𝜔0 = ℶη𝜔0 = ηℶ𝜔0 = ηη𝜔0 and ℱℱ𝜏0 = ℱℌ𝜏0 = ℌℱ𝜏0= ℌℌ𝜏0 Suppose ℑu ≠ u. Then 𝒬 (ℑu, u, u, k휁) = 𝒬 (ℑu, ℶ𝜔0, ℶ𝜔0, k휁) ≥ 𝒬 (ξu, ℌ𝜔0, ℌ𝜔0, 휁 ) ∗ 𝒬 (ℑu, ℌ𝜔0, ℌ𝜔0, 휁 ) ∗ 𝒬 (ξu, ℶ𝜔0, ℶ𝜔0, 휁 ) ≥ 𝒬 (ξu, u, u, 휁) ∗ 𝒬 (ℑu, u, u, 휁) ∗ 𝒬 (ξu, u, u, 휁) ≥ lim 𝑛→∞ 𝒬 (ξu, ξ𝜛𝑛, ξ 𝜛𝑛, 휁) ∗ 𝒬 (ℑu, u, u, 휁) ∗ 𝒬 (ξu, ξ𝜛𝑛, ξ𝜛𝑛, 휁) ≥ lim 𝑛→∞ 𝒬 (ξu, ξ𝜛𝑛, ξ𝜛𝑛, 휁) ℋ (ℑu, u, u, k휁) = ℋ (ℑu, ℶ𝜔0, ℶ𝜔0, k휁) ≤ ℋ (ξu, ℌ𝜔0, ℌ𝜔0, 휁 ) ◊ ℋ (ℑu, ℌ𝜔0, ℌ𝜔0, 휁 ) ◊ ℋ (ξu, ℶ𝜔0, ℶ𝜔0, 휁 ) ≤ ℋ (ξu, u, u, 휁)◊ ℋ (ℑu, u, u, 휁) ◊ ℋ (ξu, u, u, 휁) ≤ lim 𝑛→∞ ℋ (ξu, ξ𝜛𝑛, ξ 𝜛𝑛, 휁) ◊ ℋ (ℑu, u, u, 휁) ◊ ℋ(ξu, ξ𝜛𝑛, ξ𝜛𝑛, 휁) ≤ lim 𝑛→∞ ℋ (ξu, ξ𝜛𝑛, ξ𝜛𝑛, 휁) 𝒪(ℑu, u, u, k휁) = 𝒪 (ℑu, ℶ𝜔0, ℶ𝜔0, k휁) ≤ 𝒪 (ξu, ℌ𝜔0, ℌ𝜔0, 휁 ) ◊ 𝒪 (ℑu, ℌ𝜔0, ℌ𝜔0, 휁 ) ◊ 𝒪 (ξu, ℶ𝜔0, ℶ𝜔0, 휁 ) ≤ 𝒪 (ξu, u, u, 휁) ◊ 𝒪 (ℑu, u, u, 휁) ◊ 𝒪 (ξu, u, u, 휁) ≤ lim 𝑛→∞ 𝒪 (ξu, ξ𝜛𝑛, ξ 𝜛𝑛, 휁) ◊ 𝒪 (ℑu, u, u, 휁) ◊ 𝒪(ξu, ξ𝜛𝑛, ξ𝜛𝑛, 휁) ≤ lim 𝑛→∞ 𝒪 (ξu, ξ𝜛𝑛, ξ𝜛𝑛, 휁) By ℑu = ℑℑ𝜛0 = ℑξ𝜛0 = ξℑ𝜛0= ξξ 𝜛0 = ξ u and 휁 ∗ 휁 ≥ 휁 and 휁 ◊ 휁 ≤ 1 – 휁 it is easy to see that [3.1.6], [3.1.7] [3.1.8] yields a contradiction and so ℑu = u = ξ u. Now following the similar argument, we can get ℶu = u = ηu. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2156 https://internationalpubls.com So ℑ, ℶ, ℱ, ℌ, η and ξ have a common fixed point u. Uniquness: Let v ≠ u be another common fixed point of ℑ, ℶ, ℱ, ℌ, η and ξ . Then, 𝒬 ( v, u, u, k휁 ) = 𝒬 (ℑv, ℶu, ℶu, k휁 ) ≥ 𝒬 (ξv, ηu, ηu, 휁 ) ∗ 𝒬 (ℑv, ηu, ηu, 휁 ) ∗ 𝒬 (ξv, ℶu, ℶu, 휁) = 𝒬 ( v, u, u, 휁 ) ∗ 𝒬 ( v, u, u, 휁) ∗ 𝒬 ( v, u, u, 휁 ) ℋ ( v, u, u, k휁 ) = ℋ(ℑv, ℶu, ℶu, k휁 ) ≤ ℋ(ξv, ηu, ηu, 휁 ) ◊ ℋ (ℑv, ηu, ηu, 휁 ) ◊ ℋ (ξv, ℶu, ℶu, 휁) = ℋ(v, u, u, 휁 ) ◊ ℋ ( v, u, u, 휁) ◊ ℋ ( v, u, u, 휁 ) 𝒪 ( v, u, u, k휁 ) = 𝒪(ℑv, ℶu, ℶu, k휁 ) ≤ 𝒪(ξv, ηu, ηu, 휁 ) ◊ 𝒪 (ℑv, ηu, ηu, 휁 ) ◊ 𝒪 (ξv, ℶu, ℶu, 휁) = 𝒪(v, u, u, 휁 ) ◊ 𝒪 ( v, u, u, 휁) ◊ 𝒪 ( v, u, u, 휁 ) By 휁 ∗ 휁 ≥ 휁 and 휁 ◊ 휁 ≤ 1 – 휁, we can get 𝒬 ( v, u, u, k 휁 ) ≥ 𝒬 ( v, u, u, 휁) ℋ ( v, u, u, k 휁 ) ≤ ℋ ( v, u, u, 휁) and 𝒪 ( v, u, u, k 휁 ) ≤ 𝒪 ( v, u, u, 휁) is a contradiction thus v = u. Hence ℑ, ℶ, ℱ, ℌ, η and ξ have a unique common fixed point in X. THEOREM: 3. 2 [20] Let ℑ, ℶ, ℱ, ℌ, η and ξ be self mappings of a complete Neutrosophic metric space (Ξ, 𝒬, ℋ,𝒪 *,) with 휁 ∗ 휁 > t and 휁 ◊ 휁 < 1 - 휁 if the mappings satisfy the following conditions: [3.2.1] ℑ (Ξ) ⊆ ξ (Ξ) , ℶ (Ξ) ⊆ η (Ξ) , ℱ(Ξ) ⊆ ℌ (Ξ) [3.2.2] Suppose (ℑ, ℌ) satisfy the property (E.A) [3.3.3] (ℑ, ξ) , (ℶ,η) and (ℱ, ℌ) are weakly compatible [3.3.4] 𝒬 (ℑ𝜛 , ℶ𝜔, ℶ𝜏, 휁) ≥ φ [min ( 𝒬(ℌ𝜛, η𝜔, η𝜏, 휁), 𝒬(ℑ𝜛, η𝜔, η𝜏, 휁), 𝒬(ℌ𝜛, η𝜔, η𝜏, 휁), 𝒬(ℑ𝜛, ℌ𝜛, ℌ𝜛, 휁) )] ℋ (ℑ𝜛 , ℶ𝜔, ℶ𝜏, 휁) ≤ Ψ [max ( ℋ(ℌ𝜛, η𝜔, η𝜏, 휁), ℋ(ℑ𝜛, η𝜔, η𝜏, 휁), ℋ(ℌ𝜛, η𝜔, η𝜏, 휁), ℋ(ℑ𝜛, ℌ𝜛, ℌ𝜛, 휁) )] and 𝒪 (ℑ𝜛 , ℶ𝜔, ℶ𝜏, 휁) ≤ Ω [max ( 𝒪(ℌ𝜛, η𝜔, η𝜏, 휁), 𝒪(ℑ𝜛, η𝜔, η𝜏, 휁), 𝒪(ℌ𝜛, η𝜔, η𝜏, 휁), 𝒪(ℑ𝜛, ℌ𝜛, ℌ𝜛, 휁) )] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2157 https://internationalpubls.com for all 𝜛, 𝜔, 𝜏 𝜖 Ξ and t > 0 where φ, Ψ, Ω : [0,1] → [0,1] is a continuous and increasing function with φ(s) > s and Ψ(s) < s Ω(s) < 𝑠 for 0 < S < 1 and 𝜑(1) = 1, 𝛹(0) = 0, Ω(0)= 0 Then ℑ, ℶ, ℱ, ℌ, η and ξ have a unique common fixed point in Ξ. Proof: Let (ℑ, ℌ) satisfy the property (E.A). By the definition of (E.A) we can get lim 𝑛→∞ 𝒬( ℑ𝜛n, u, u, 휁 ) = lim n→∞ 𝒬(ℌ𝜛n, u, u, 휁 ) = 1 and lim 𝑛→∞ ℋ ( ℑ𝜛n,, u, u, 휁 ) = lim n→∞ ℋ( ℌ𝜛n,, u, u, 휁 ) = 0 lim 𝑛→∞ 𝒪 ( ℑ𝜛n,, u, u, 휁 ) = lim n→∞ 𝒪( ℌ𝜛n,, u, u, 휁 ) = 0 for some u 𝜖 Ξ and every 휁 > 0. Because Neutrosophic metric space is complete and ℑ (Ξ) ⊆ ξ (Ξ), there exists a sequence {𝜔n} such that ℑ(𝜛n ) = ξ(𝜔n), which implies lim 𝑛→∞ 𝒬( ξ 𝜔𝑛 𝑢, 𝑢, 휁 ) = 1 and lim 𝑛→∞ ℋ( ξ 𝜔𝑛𝑢, 𝑢, 휁 ) = 0, lim 𝑛→∞ 𝒪( ξ 𝜔𝑛𝑢, 𝑢, 휁 ) = 0 Now 𝒬 (ℑ𝜛n,, ℶ𝜔𝑛, ℶ𝜔 n+1, 휁) ≥ φ [min ( 𝒬(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁), 𝒬(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁) 𝒬(ℌ𝜛𝑛, ℶ𝜔𝑛, ℶ 𝜔𝑛+1, 휁), 𝒬( ℑ𝜛n,, ℌ𝜛n,, ℌ𝜛n,, 휁) )] ℋ (ℑ𝜛n,, ℶ𝜔𝑛, ℶ𝜔 n+1, 휁) ≤ Ψ [max ( ℋ(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁), ℋ(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁) ℋ(ℌ𝜛𝑛, ℶ𝜔𝑛, ℶ 𝜔𝑛+1, 휁), ℋ( ℑ𝜛n,, ℌ𝜛n,, ℌ𝜛n,, 휁) )] 𝒪 (ℑ𝜛n,, ℶ𝜔𝑛, ℶ𝜔 n+1, 휁) ≤ Ω [max ( 𝒪(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁), 𝒪(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁) 𝒪(ℌ𝜛𝑛, ℶ𝜔𝑛, ℶ 𝜔𝑛+1, 휁), 𝒪( ℑ𝜛n,, ℌ𝜛n,, ℌ𝜛n,, 휁) )] By the definition of Neutrosophic metric space, we can get 𝒬 (𝜛, 𝜔, 𝜏, 휁) ≥ 𝒬 (𝜛, u, u, 1/3 휁) ∗ 𝒬 ( u, 𝜔, 𝜏 2/3 휁) ≥ 𝒬 (𝜛, u, u,1/3 휁) ∗ 𝒬 (𝜔 y, u, u, 1/3 휁) ∗ 𝒬 (𝜏, u, u, 1/3휁) ℋ (𝜛, 𝜔, 𝜏, 휁) ≤ ℋ (𝜛, u, u, 1/3 휁) ◊ ℋ ( u, 𝜔, 𝜏 2/3 휁) ≤ ℋ (𝜛, u, u,1/3 휁) ◊ ℋ (𝜔 y, u, u, 1/3 휁) ◊ ℋ (𝜏, u, u, 1/3휁) 𝒪 (𝜛, 𝜔, 𝜏, 휁) ≤ 𝒪 (𝜛, u, u, 1/3 휁) ◊ 𝒪 ( u, 𝜔, 𝜏 2/3 휁) ≤ 𝒪 (𝜛, u, u,1/3 휁) ◊ 𝒪 (𝜔 y, u, u, 1/3 휁) ◊ 𝒪 (𝜏, u, u, 1/3휁) Thus lim 𝑛→∞ 𝒬 (ℑ𝜛n,, ℶ𝜔𝑛, ℶ𝜔 n+1, 휁) ≥ lim 𝑛→∞ φ [min ( 𝒬(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁), 𝒬(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁) 𝒬(ℌ𝜛𝑛, ℶ𝜔𝑛, ℶ 𝜔𝑛+1, 휁), 𝒬( ℑ𝜛n,, ℌ𝜛n,, ℌ𝜛n,, 휁) )] ≥ lim 𝑛→∞ 𝜑 [min ( 1 ∗ 1 ∗ 1 , 𝒬(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁) 1 ∗ 1 ∗ 1 , 1 ∗ 1 ∗ 1 )] lim 𝑛→∞ ℋ (ℑ𝜛n,, ℶ𝜔𝑛, ℶ𝜔 n+1, 휁) ≤ lim 𝑛→∞ Ψ [max ( ℋ(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁), ℋ(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁) ℋ(ℌ𝜛𝑛, ℶ𝜔𝑛, ℶ 𝜔𝑛+1, 휁), ℋ (ℑ𝜛n,, ℌ𝜛n,, ℌ𝜛n,, 휁) )] ≤ lim 𝑛→∞ Ψ [max (0 ◊ 0 ◊ 0 ℋ(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁) 0 ◊ 0 ◊ 0 0 ◊ 0 ◊ 0 )] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2158 https://internationalpubls.com lim 𝑛→∞ 𝒪 (ℑ𝜛n,, ℶ𝜔𝑛, ℶ𝜔 n+1, 휁) ≤ lim 𝑛→∞ Ω [max ( 𝒪(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁), 𝒪(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁) 𝒪(ℌ𝜛𝑛, ℶ𝜔𝑛, ℶ 𝜔𝑛+1, 휁), 𝒪 (ℑ𝜛n,, ℌ𝜛n,, ℌ𝜛n,, 휁) )] ≤ lim 𝑛→∞ Ω [max ( 0 ◊ 0 ◊ 0 𝒪(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁) 0 ◊ 0 ◊ 0 0 ◊ 0 ◊ 0 )] If ℶ𝜔𝑛 ≠ u then lim 𝑛→∞ 𝒬 (ℑ𝜛n,, ℶ𝜔𝑛, ℶ𝜔 n+1, 휁) ≥ lim 𝑛→∞ φ [𝒬(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁)] > lim 𝑛→∞ 𝒬(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁) lim 𝑛→∞ ℋ (ℑ𝜛n,, ℶ𝜔𝑛, ℶ𝜔 n+1, 휁) ≥ lim 𝑛→∞ Ψ [ℋ(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁)] > lim 𝑛→∞ ℋ(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁) lim 𝑛→∞ 𝒪 (ℑ𝜛n,, ℶ𝜔𝑛, ℶ𝜔 n+1, 휁) ≥ lim 𝑛→∞ Ω [𝒪(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁)] > lim 𝑛→∞ 𝒪(ℌ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁) is a contradiction by the above lemma. Therefore lim n→∞ ℑ 𝜛n, = lim n→∞ ℶ 𝜔𝑛 = lim ℱ 𝜏𝑛 = n→∞ lim n→∞ ℌ 𝜛n, = lim n→∞ η 𝜔𝑛 = lim n→∞ ξ 𝜏𝑛 = u (Ξ, 𝒬, ℋ,𝒪 *,) is a complete Neutrosophic metric space. There exists 𝜗0 ∈ Ξ such that ℌ𝜗0 = u, Hence 𝒬 (ℑ𝜗0, ℶ𝜔𝑛, ℶ𝜔 n+1, 휁) ) ≥ φ [min ( 𝒬( ℌ𝜗0, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁), 𝒬( ℑ𝜗0, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁 𝒬( ℑ𝜗0, ℶ𝜔𝑛, ℶ𝜔𝑛+1 , 휁), 𝒬( ℑ𝜗0, ℌ𝜗0, ℌ𝜗0, 휁) )] ℋ (ℑ𝜗0, ℶ𝜔𝑛, ℶ𝜔 n+1, 휁) ) ≤ Ψ [max ( ℋ( ℌ𝜗0, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁), ℋ( ℑ𝜗0, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁 ℋ( ℑ𝜗0, ℶ𝜔𝑛, ℶ𝜔𝑛+1 , 휁), ℋ( ℑ𝜗0, ℌ𝜗0, ℌ𝜗0, 휁) )] 𝒪 (ℑ𝜗0, ℶ𝜔𝑛, ℶ𝜔 n+1, 휁) ) ≤ Ω [max ( 𝒪( ℌ𝜗0, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁), 𝒪( ℑ𝜗0, ξ 𝜔𝑛, ξ 𝜔𝑛+1, 휁 𝒪( ℑ𝜗0, ℶ𝜔𝑛, ℶ𝜔𝑛+1 , 휁), 𝒪( ℑ𝜗0, ℌ𝜗0, ℌ𝜗0, 휁) )] On making n → ∞, 𝒬 (ℑ𝜗0, u, u, 휁) ≥ φ [min ( 𝒬(u, u, u, 휁) , 𝒬( u, u, u, 휁) 𝒬( ℑ𝜗0u, u, 휁) , 𝒬( ℑ𝜗0, u, u, 휁) )] ℋ (ℑ𝜗0, u, u, 휁) ≤ Ψ [max ( ℋ(u, u, u, 휁) , ℋ( u, u, u, 휁) ℋ( ℑ𝜗0u, u, 휁) , ℋ( ℑ𝜗0, u, u, 휁) )] 𝒪 (ℑ𝜗0, u, u, 휁) ≤ Ω [max ( 𝒪(u, u, u, 휁) , 𝒪( u, u, u, 휁) 𝒪( ℑ𝜗0u, u, 휁) , 𝒪( ℑ𝜗0, u, u, 휁) )] which can imply ℑ𝜗0 = u, with 𝜑(s) > s , 𝛹(s) < s, Ω(s) < s for 0 < s < 1 As ℑ (Ξ) ⊆ ξ (Ξ), there exists 𝜔0 such that ℑ 𝜗0 = ξ 𝜔0 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2159 https://internationalpubls.com Suppose ξ 𝜔0 ≠ η 𝜔0. Now 𝒬 (ℑ𝜛n,, ℶ𝜔0, ℶ𝜔0, 휁) ≥ φ [min ( 𝒬(ℌ𝜛n,, ξ 𝜔0 ξ 𝜔0 , 휁), 𝒬(fxn, ξ 𝜔0 ξ 𝜔0, 휁) 𝒬(ℌ𝜛n,, ℶ𝜔𝑛, ℶ𝜔𝑛, 휁), 𝒬(ℑ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛, 휁) )] ℋ (ℑ𝜛n,, ℶ𝜔0, ℶ𝜔0, 휁) ≤ Ψ [max ( ℋ(ℌ𝜛n,, ξ 𝜔0 ξ 𝜔0 , 휁), ℋ(fxn, ξ 𝜔0 ξ 𝜔0, 휁) ℋ(ℌ𝜛n,, ℶ𝜔𝑛, ℶ𝜔𝑛, 휁), ℋ(ℑ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛, 휁) )] 𝒪 (ℑ𝜛n,, ℶ𝜔0, ℶ𝜔0, 휁) ≤ Ω [max ( 𝒪(ℌ𝜛n,, ξ 𝜔0 ξ 𝜔0 , 휁), 𝒪(fxn, ξ 𝜔0 ξ 𝜔0, 휁) 𝒪(ℌ𝜛n,, ℶ𝜔𝑛, ℶ𝜔𝑛, 휁), 𝒪(ℑ𝜛n,, ξ 𝜔𝑛, ξ 𝜔𝑛, 휁) )] If n → ∞ , 𝒬 ( u, ℶ𝜔0, ℶ𝜔0, 휁) ≥ φ [min ( 𝒬( u, u, u, 휁) , 𝒬( u, ℶ𝜔0, ℶ𝜔0, 휁) 𝒬( u, u, u, 휁) , 𝒬( u, u, u, 휁) )] ℋ ( u, ℶ𝜔0, ℶ𝜔0, 휁) ≤ Ψ [max ( ℋ( u, u, u, 휁) , ℋ( u, ℶ𝜔0, ℶ𝜔0, 휁) ℋ( u, u, u, 휁) , ℋ( u, u, u, 휁) )] 𝒪 ( u, ℶ𝜔0, ℶ𝜔0, 휁) ≤ Ω [max ( 𝒪( u, u, u, 휁) , 𝒪( u, ℶ𝜔0, ℶ𝜔0, 휁) 𝒪( u, u, u, 휁) , 𝒪( u, u, u, 휁) )] by the continuity of 𝒬, ℋ, 𝒪 and φ, Ψ, Ω Hence 𝒬(u, ℶ𝜔0, ℶ𝜔0, 휁) ≥ 𝜑 (𝒬 (u, ℶ𝜔0, ℶ𝜔0, 휁) ) > 𝒬 ( u, ℶ𝜔0, ℶ𝜔0, 휁) ℋ(u, ℶ𝜔0, ℶ𝜔0, 휁) ≤ 𝛹 (ℋ (u, ℶ𝜔0, ℶ𝜔0, 휁)) < ℋ ( u, ℶ𝜔0, ℶ𝜔0, 휁) 𝒪(u, ℶ𝜔0, ℶ𝜔0, 휁) ≤ Ω (𝒪 (u, ℶ𝜔0, ℶ𝜔0, 휁)) < 𝒪 ( u, ℶ𝜔0, ℶ𝜔0, 휁) is a contradiction. So ℑ 𝜗0 = ξ 𝜔0 Now by (ℑ, ξ) , (ℶ,η) and (ℱ, ℌ) are weakly compatible, we can get, ℑℑ 𝜗0 = ℑη 𝜗0 = ηℑ 𝜗0 = ηη𝜗0 and ℶℶ𝜔0 = ℶη 𝜔0 = η ℶ𝜔0= η η 𝜔0 Then, lim 𝑛→∞ 𝒬 (ℑu, ℶ𝜔𝑛, ℶ𝜔𝑛+1, 휁) ≥ 𝜑 [min ( 𝒬(ℌu, η𝜔𝑛, η𝜔𝑛+1, 휁), 𝒬(ℑu, ηyn,ηyn +1, 휁) 𝒬(ℌu, ℶ𝜔𝑛, ℶ𝜔𝑛+1, 휁), 𝒬(ℑu, ℌu, ℌu, t) )] lim 𝑛→∞ ℋ (ℑu, ℶ𝜔𝑛, ℶ𝜔𝑛+1, 휁) ≤ 𝛹 [max ( ℋ(ℌu, η𝜔𝑛, η𝜔𝑛+1, 휁), ℋ(ℑu, ηyn,ηyn +1, 휁) ℋ(ℌu, ℶ𝜔𝑛, ℶ𝜔𝑛+1, 휁), ℋ(ℑu, ℌu, ℌu, t) )] lim 𝑛→∞ 𝒪 (ℑu, ℶ𝜔𝑛, ℶ𝜔𝑛+1, 휁) ≤ Ω [max ( 𝒪(ℌu, η𝜔𝑛, η𝜔𝑛+1, 휁), 𝒪(ℑu, ηyn,ηyn +1, 휁) 𝒪(ℌu, ℶ𝜔𝑛, ℶ𝜔𝑛+1, 휁), 𝒪(ℑu, ℌu, ℌu, t) )] ⇒ ℑu = u = ℌu. Similarly we can get ηu = ℶu = u Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2160 https://internationalpubls.com Uniqueness: Let v be another common fixed point of ℑ, ℶ, ℱ, ℌ, η and ξ . Then 𝒬 (v, u, u, 휁) = 𝒬 (ℑv, ηu, ηu, 휁) ≥ φ [min ( 𝒬(ℌv, ηu, ηu, 휁), 𝒬( ℑv, ηu, ηu, 휁) 𝒬( ℌv, gu, gu, 휁), 𝒬(ℑv, ℌv, ℌv, 휁) )] ℋ (v, u, u, 휁) = ℋ (ℑv, ηu, ηu, 휁) ≤ Ψ [max ( ℋ(ℌv, ηu, ηu, 휁), ℋ( ℑv, ηu, ηu, 휁) ℋ( ℌv, gu, gu, 휁), ℋ(ℑv, ℌv, ℌv, 휁) )] 𝒪 (v, u, u, 휁) = 𝒪 (ℑv, ηu, ηu, 휁) ≤ Ω [max ( 𝒪(ℌv, ηu, ηu, 휁), 𝒪( ℑv, ηu, ηu, 휁) 𝒪( ℌv, gu, gu, 휁), 𝒪(ℑv, ℌv, ℌv, 휁) )] It implies v= u. Hence ℑ, ℶ, ℱ, ℌ, η and ξ have a unique common fixed point in Ξ. Coclusion: In conclusion, this paper has explored the concept of fixed point theorems within the framework of common neutrosophic metric spaces, offering new insights and extending existing results in the field. By integrating neutrosophic logic into metric space theory, we have developed several fixed point theorems that address both classical and novel types of metrics, accommodating the inherent uncertainty and indeterminacy present in neutrosophic contexts. 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