Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1795 https://internationalpubls.com Exploring Quadruple Fixed Points in Fuzzy Metric Spaces for Occasionally Weakly Compatible Mappings 1Sandhya Shukla, 2Priyanka Nigam* 1University Institute of Technology, RGPV Bhopal, 462033, Madhya Pradesh, India 2Presidency University, Bengaluru, 560089, Karnataka, India. Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: In this research paper we prove some quadruple fixed point theorems for occasionally weakly compatible mappings in fuzzy metric space. Keywords: Occasionally weakly compatible mappings, quadruple fixed point, fuzzy metric space. 2000 Mathematics Subject Classification: 47H10; 54H25. 1. Introduction 1 Introduction Zadeh [14] introduced the concept of fuzzy sets, while Kramosil and Michalek [11] developed the idea of fuzzy metric spaces. Later, George and Veermani [5] refined this concept by proposing a new framework for fuzzy metric spaces using continuous t-norms. Numerous researchers have since established common fixed point theorems for mappings under various commutativity conditions. The study of fixed point theorems involving four self-maps initially relied on the assumption of commutativity. Sessa [13] relaxed this assumption by introducing the notion of pairwise weakly commuting maps. Jungck extended this further to pairwise compatible [6] and pairwise weakly compatible mappings [7]. Subsequently, Jungck and Rhoades [8] introduced the concept of occasionally weakly compatible (owc) mappings. The research in works [1], [3], [9], and [10] on quadruple fixed points is truly noteworthy. In this paper we introduce some quadruple fixed point theorems for occasionally weakly compatible mappings in fuzzy metric space. 2 Preliminary Notes Definition 2.1 A fuzzy set A in X is a function with domain X and values in [0,1]. Definition 2.2 A binary operation ∗∶ [0,1] [0,1]→ [0,1] is a continuous t-norm if ∗ is satisfying conditions: (i) ∗ is an commutative and associative; (ii) ∗ is continuous; (iii) 𝑎 ∗ 1 = 𝑎 for all a [0,1]; Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1796 https://internationalpubls.com (iv) 𝑎 ∗ 𝑏𝑐 ∗ 𝑑 whenever ca  𝑎𝑛𝑑 db  and 𝑎, 𝑏, 𝑐, 𝑑[0,1]. Definitions 2.3 A 3-tuple (𝑋, 𝑀,∗) is said to be a fuzzy metric space if X is an arbitrary set ,∗ is a continuous 𝑡 − 𝑛𝑜𝑟𝑚 and M is a fuzzy set on ( ) ,02X satisfying the following conditions, for all 𝑥, 𝑦, 𝑧  𝑋, 𝑠, 𝑡 > 0, (𝑖) 0),,( tyxM ; (𝑖𝑖) 1),,( =tyxM 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 yx = ; (𝑖𝑖𝑖) ),,( tyxM = ),,( txyM ; (𝑖𝑣) ),,( tyxM ∗ ),,( szyM ),,( stzxM + ; (𝑣) ]1,0(),0(:),,( →yxM is continuous. Then M is called a 𝑓𝑢𝑧𝑧𝑦 𝑚𝑒𝑡𝑟𝑖𝑐 on X. Then ),,( tyxM denotes the degree of nearness between x and y with respect to t. Example 2.4 Let ),( dX be a metric space. Denote 𝑎 ∗ 𝑏 = 𝑎𝑏 for all  1,0, ba and let dM be fuzzy sets on ( ) ,02X defined as follows: ),( yxdt t M d + = . Then (𝑋, 𝑀𝑑,∗) is a fuzzy metric space. Lemma 2.5 Let (X, M,*) be a fuzzy metric space. If there exists )1,0(q such that M(x, y, qt)  M(x ,y ,t) for all x, y  X and t>0, then x = y. Definition 2.6 Let X be a non-empty set. An element (𝑥, 𝑦, 𝑧, 𝑡) ∈ 𝑋 × 𝑋 × 𝑋 × 𝑋 is called a quadruple fixed point of a given mapping 𝑓: 𝑋 × 𝑋 × 𝑋 × 𝑋 → 𝑋 if 𝑥 = 𝑓(𝑥, 𝑦, 𝑧, 𝑡), 𝑦 = 𝑓(𝑦, 𝑧, 𝑡, 𝑥), 𝑧 = 𝑓(𝑧, 𝑡, 𝑥, 𝑦), 𝑡 = 𝑓(𝑡, 𝑥, 𝑦, 𝑧). Definition 2.7 An element (𝑥, 𝑦, 𝑧, 𝑡) ∈ 𝑋 × 𝑋 × 𝑋 × 𝑋 is called a quadruple coincidence point of a mapping 𝑓: 𝑋 × 𝑋 × 𝑋 × 𝑋 → 𝑋 and 𝑔: 𝑋 → 𝑋 if 𝑔𝑥 = 𝑓(𝑥, 𝑦, 𝑧, 𝑡), 𝑔𝑦 = 𝑓(𝑦, 𝑧, 𝑡, 𝑥), 𝑔𝑧 = 𝑓(𝑧, 𝑡, 𝑥, 𝑦), 𝑔𝑡 = 𝑓(𝑡, 𝑥, 𝑦, 𝑧) in this case (𝑔𝑥, 𝑔𝑦, 𝑔𝑧, 𝑔𝑡) is called a quadruple point of coincidence. Definition 2.8 The mappings 𝑓: 𝑋 × 𝑋 × 𝑋 × 𝑋 → 𝑋 and 𝑔: 𝑋 → 𝑋 of a set X are occasionally weakly compatible (𝑜𝑤𝑐) iff there is a point (𝑥, 𝑦, 𝑧, 𝑡) ∈ 𝑋 × 𝑋 × 𝑋 × 𝑋 which is a coincidence point of f and g at which f and g commute i.e. (𝑓, 𝑔) are occasionally weakly compatible maps iff 𝑓(𝑥, 𝑦, 𝑧, 𝑡) = 𝑔(𝑥), 𝑓(𝑦, 𝑧, 𝑡, 𝑥) = 𝑔(𝑦), 𝑓(𝑧, 𝑡, 𝑥, 𝑦) = 𝑔(𝑧), 𝑓(𝑡, 𝑥, 𝑦, 𝑧) = 𝑔(𝑡) implies 𝑔𝑓(𝑥, 𝑦, 𝑧, 𝑡) = 𝑓(𝑔𝑥, 𝑔𝑦, 𝑔𝑧, 𝑔𝑡), 𝑔𝑓(𝑦, 𝑧, 𝑡, 𝑥) = 𝑓(𝑔𝑦, 𝑔𝑧, 𝑔𝑡, 𝑔𝑥), 𝑔𝑓(𝑧, 𝑡, 𝑥, 𝑦) = 𝑓(𝑔𝑧, 𝑔𝑡, 𝑔𝑥, 𝑔𝑦), 𝑔𝑓(𝑡, 𝑥, 𝑦, 𝑧) = 𝑓(𝑔𝑡, 𝑔𝑥, 𝑔𝑦, 𝑔𝑧) for (𝑥, 𝑦, 𝑧, 𝑡) ∈ 𝑋 × 𝑋 × 𝑋 × 𝑋. Example 2.10.1 Let (X, ℱ,∗) be a fuzzy metric space, where X = [0,1] with 𝑎 ∗ 𝑏 = min{𝑎, 𝑏} and Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1797 https://internationalpubls.com M(x, y, t) = { t t + |x − y| , if t > 0; 0, if t = 0. Let f: X × X × X → X & 𝑔: X → X be defined by f(x, y, z, w) = 2x + 2y + 2z + w 2 g(x) = { x, if 0 ≤ x < 1; 7 2 , if x ≥ 1. Here, (0,0,0,0) and (1,1,1,1) are two coincidence points of f and g. That is f(0,0,0,0) = 0 = g(0), f(1,1,1,1) = 1 = g(1) but gf(0,0,0,0) = 0 = f(g0, g0, g0, g0), gf(1,1,1,1) ≠ f(g1, g1, g1, g1). Thus f and g are owc but not weakly compatible. Coincidence points verification: 𝑓(0,0,0,0) = 0.0, 𝑔(0) = 0, 𝑔𝑓(0,0,0,0) = 0.0, 𝑓(𝑔0, 𝑔0, 𝑔0, 𝑔0) = 0.0 𝑓(1,1,1,1) = 3.5, 𝑔(1) = 3.5, 𝑔𝑓(1,1,1,1) = 3.5, 𝑓(𝑔1, 𝑔1, 𝑔1, 𝑔1) = 12.25 f and g are occasionally weakly compatible (owc) but not weakly compatible. 3 Main Results Theorem: 3.1 Let (𝑋, 𝑀,  ) be a fuzzy metric space with 𝑡 ∗ 𝑡 = 𝑡 for all 𝑡 ∈ [0,1]. Let 𝐴, 𝐵: 𝑋 × 𝑋 × 𝑋 × 𝑋 → 𝑋 and 𝑆, 𝑇: 𝑋 → 𝑋 be four self-mappings satisfying the following conditions: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1798 https://internationalpubls.com (i) 𝑀(𝐴(𝑥, 𝑦, 𝑧, 𝑝), 𝐵(𝑢, 𝑣, 𝑤, 𝑟), 𝑞𝑡) ≥ 𝑚𝑖𝑛 { 𝑀(𝑆𝑥, 𝑇𝑢, 𝑡), 𝑀(𝐴(𝑥, 𝑦, 𝑧, 𝑝), 𝑆𝑥, 𝑡), 𝑀(𝐵(𝑢, 𝑣, 𝑤, 𝑟), 𝑇𝑢, 𝑡), 𝑀(𝑆𝑥, 𝐵(𝑢, 𝑣, 𝑤, 𝑟), 𝑡), 𝑀(𝐴(𝑥, 𝑦, 𝑧, 𝑝), 𝑇𝑢, 𝑡) } for all 𝑥, 𝑦, 𝑧, 𝑝, 𝑢, 𝑣, 𝑤, 𝑟 ∈ 𝑋 (ii) 𝑦 = 𝐵(𝑥, 𝑦, 𝑧, 𝑝) Moreover if the pairs (𝐴, 𝑆) and (𝐵, 𝑇) are owc, then there exists a unique point 𝑥 in 𝑋 such that 𝐴(𝑥, 𝑥, 𝑥, 𝑥) = 𝑇(𝑥) = 𝐵(𝑥, 𝑥, 𝑥, 𝑥) = 𝑆(𝑥) = 𝑥. Proof: Since the pairs (A,S) and (B,T) are owc so there are points 𝑎, 𝑏, 𝑐, 𝑑, 𝑎′, 𝑏′, 𝑐′, 𝑑′ in X such that 𝐴(𝑎, 𝑏, 𝑐, 𝑑) = 𝑆𝑎, 𝐴(𝑏, 𝑐, 𝑑, 𝑎) = 𝑆𝑏, 𝐴(𝑐, 𝑑, 𝑎, 𝑏) = 𝑆𝑐, 𝐴(𝑑, 𝑎, 𝑏, 𝑐) = 𝑆𝑑 and 𝐵(𝑎′, 𝑏′, 𝑐′, 𝑑′ ) = 𝑇𝑎′ , 𝐵(𝑏′, 𝑐′, 𝑑′, 𝑎′ ) = 𝑇𝑏′, 𝐵(𝑐′, 𝑑′, 𝑎′ , 𝑏′) = 𝑇𝑐′, 𝐵(𝑑′, 𝑎′ , 𝑏′, 𝑐′) = 𝑇𝑑′ 𝑆𝑥 = 𝑆𝐴(𝑎, 𝑏, 𝑐, 𝑑) = 𝐴(𝑆𝑎, 𝑆𝑏, 𝑆𝑐, 𝑆𝑑) = 𝐴(𝑥, 𝑦, 𝑧, 𝑝) 𝑆𝑦 = 𝑆𝐴(𝑏, 𝑐, 𝑑, 𝑎) = 𝐴(𝑆𝑏, 𝑆𝑐, 𝑆𝑑, 𝑆𝑎) = 𝐴(𝑦, 𝑧, 𝑝, 𝑥) 𝑆𝑧 = 𝑆𝐴(𝑐, 𝑑, 𝑎, 𝑏) = 𝐴(𝑆𝑐, 𝑆𝑑, 𝑆𝑎, 𝑆𝑏) = 𝐴(𝑧, 𝑝, 𝑥, 𝑦) 𝑆𝑝 = 𝑆𝐴(𝑑, 𝑎, 𝑏, 𝑐) = 𝐴(𝑆𝑑, 𝑆𝑎, 𝑆𝑏, 𝑆𝑐) = 𝐴(𝑝, 𝑥, 𝑦, 𝑧) We claim that 𝑆𝑎 = 𝑇𝑎′ . If not, by inequality (𝑖) we get 𝑀(𝐴(𝑎, 𝑏, 𝑐, 𝑑), 𝐵(𝑎′, 𝑏′, 𝑐′, 𝑑′), 𝑞𝑡) ≥ 𝑚𝑖𝑛 { 𝑀(𝑆𝑎, 𝑇𝑎′ , 𝑡), 𝑀(𝐴(𝑎, 𝑏, 𝑐, 𝑑), 𝑆𝑎, 𝑡), 𝑀(𝐵(𝑎′ , 𝑏′, 𝑐′, 𝑑′), 𝑇𝑎′ , 𝑡), 𝑀(𝑆𝑎, 𝐵(𝑎′, 𝑏′, 𝑐′, 𝑑′), 𝑡), 𝑀(𝐴(𝑎, 𝑏, 𝑐, 𝑑), 𝑇𝑎′, 𝑡) } or 𝑀(𝑆𝑎, 𝑇𝑎′ , 𝑞𝑡) ≥ 𝑚𝑖𝑛{𝑀(𝑆𝑎, 𝑇𝑎′, 𝑡), 𝑀(𝑆𝑎, 𝑆𝑎, 𝑡), 𝑀(𝑇𝑎′, 𝑇𝑎′, 𝑡), 𝑀(𝑆𝑎, 𝑇𝑎′ , 𝑡)𝑀(𝑆𝑎, 𝑇𝑎′, 𝑡)} = 𝑚𝑖𝑛{𝑀(𝑆𝑎, 𝑇𝑎′, 𝑡), 1,1, 𝑀(𝑆𝑎, 𝑇𝑎′, 𝑡), 𝑀(𝑆𝑎, 𝑇𝑎′ , 𝑡)} = 𝑀(𝑆𝑎, 𝑇𝑎′ , 𝑡) ⇒ 𝑆𝑎 = 𝑇𝑎′ Therefore 𝐴(𝑎, 𝑏, 𝑐, 𝑑) = 𝑇𝑎′ = 𝑆𝑎 = 𝐵(𝑎′ , 𝑏′, 𝑐′, 𝑑′ ) Similarly 𝐴(𝑏, 𝑐, 𝑑, 𝑎) = 𝑇𝑏′ = 𝑆𝑏 = 𝐵(𝑏′, 𝑐′, 𝑑′, 𝑎′ ) 𝐴(𝑐, 𝑑, 𝑎, 𝑏) = 𝑇𝑐′ = 𝑆𝑐 = 𝐵(𝑐′, 𝑑′, 𝑎′, 𝑏′) 𝐴(𝑑, 𝑎, 𝑏, 𝑐) = 𝑇𝑑′ = 𝑆𝑑 = 𝐵(𝑑′, 𝑎′ , 𝑏′, 𝑐′) Thus the pairs (𝐴, 𝑆) and (𝐵, 𝑇) have common coincidence points. Let 𝐴(𝑎, 𝑏, 𝑐, 𝑑) = 𝑇𝑎′ = 𝑆𝑎 = 𝐵(𝑎′, 𝑏′, 𝑐′, 𝑑′ ) = 𝑥 and 𝐴(𝑏, 𝑐, 𝑑, 𝑎) = 𝑇𝑏′ = 𝑆𝑏 = 𝐵(𝑏′, 𝑐′, 𝑑′, 𝑎′ ) = 𝑦 𝐴(𝑐, 𝑑, 𝑎, 𝑏) = 𝑇𝑐′ = 𝑆𝑐 = 𝐵(𝑐′, 𝑑′, 𝑎′ , 𝑏′) = 𝑧 𝐴(𝑑, 𝑎, 𝑏, 𝑐) = 𝑇𝑑′ = 𝑆𝑑 = 𝐵(𝑑′, 𝑎′ , 𝑏′, 𝑐′) = 𝑝 Since (𝐴, 𝑆) and (𝐵, 𝑇) are owc Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1799 https://internationalpubls.com 𝑆𝑥 = 𝑆𝐴(𝑎, 𝑏, 𝑐, 𝑑) = 𝐴(𝑆𝑎, 𝑆𝑏, 𝑆𝑐, 𝑆𝑑) = 𝐴(𝑥, 𝑦, 𝑧, 𝑝) 𝑆𝑦 = 𝑆𝐴(𝑏, 𝑐, 𝑑, 𝑎) = 𝐴(𝑆𝑏, 𝑆𝑐, 𝑆𝑑, 𝑆𝑎) = 𝐴(𝑦, 𝑧, 𝑝, 𝑥) 𝑆𝑧 = 𝑆𝐴(𝑐, 𝑑, 𝑎, 𝑏) = 𝐴(𝑆𝑐, 𝑆𝑑, 𝑆𝑎, 𝑆𝑏) = 𝐴(𝑧, 𝑝, 𝑥, 𝑦) 𝑆𝑝 = 𝑆𝐴(𝑑, 𝑎, 𝑏, 𝑐) = 𝐴(𝑆𝑑, 𝑆𝑎, 𝑆𝑏, 𝑆𝑐) = 𝐴(𝑝, 𝑥, 𝑦, 𝑧) Also 𝑇𝑥 = 𝑇𝐵(𝑎′, 𝑏′, 𝑐′, 𝑑′) = 𝐵(𝑇𝑎′ , 𝑇𝑏′, 𝑇𝑐′, 𝑇𝑑′) = 𝐵(𝑥, 𝑦, 𝑧, 𝑝) and 𝑇𝑦 = 𝑇𝐵(𝑏′, 𝑐′, 𝑑′, 𝑎′) = 𝐵(𝑇𝑏′, 𝑇𝑐′, 𝑇𝑑′, 𝑇𝑎′) = 𝐵(𝑦, 𝑧, 𝑝, 𝑥) 𝑇𝑧 = 𝑇𝐵(𝑐′, 𝑑′, 𝑎′, 𝑏′) = 𝐵(𝑇𝑐′, 𝑇𝑑′, 𝑇𝑎′, 𝑇𝑏′) = 𝐵(𝑧, 𝑝, 𝑥, 𝑦) 𝑇𝑝 = 𝑇𝐵(𝑑′, 𝑎′ , 𝑏′, 𝑐′) = 𝐵(𝑇𝑑′, 𝑇𝑎′ , 𝑇𝑏′, 𝑇𝑐′) = 𝐵(𝑝, 𝑥, 𝑦, 𝑧) Next we show that 𝑥 = 𝑦, for this putting 𝑥 = 𝑎 , 𝑦 = 𝑏 , 𝑧 = 𝑐, 𝑝 = 𝑑, 𝑢 = 𝑏′ , 𝑣 = 𝑐′, 𝑤 = 𝑑′, 𝑟 = 𝑎′ in (i), 𝑀(𝑥, 𝑦, 𝑞𝑡) = 𝑀(𝐴(𝑎, 𝑏, 𝑐, 𝑑), 𝐵(𝑏′, 𝑐′, 𝑑′, 𝑎′), 𝑞𝑡) ≥ 𝑚𝑖𝑛 { 𝑀(𝑆𝑎, 𝑇𝑏′, 𝑡), 𝑀(𝐴(𝑎, 𝑏, 𝑐, 𝑑), 𝑆𝑎, 𝑡), 𝑀(𝐵(𝑏′, 𝑐′, 𝑑′, 𝑎′), 𝑇𝑏′, 𝑡), 𝑀(𝑆𝑎, 𝐵(𝑏′, 𝑐′, 𝑑′, 𝑎′), 𝑡), 𝑀(𝐴(𝑎, 𝑏, 𝑐, 𝑑), 𝑇𝑏′, 𝑡) } = 𝑚𝑖𝑛{𝑀(𝑥, 𝑦, 𝑡), 𝑀(𝑥, 𝑥, 𝑡), 𝑀(𝑦, 𝑦, 𝑡), 𝑀(𝑥, 𝑦, 𝑡), 𝑀(𝑥, 𝑦, 𝑡)} = 𝑀(𝑥, 𝑦, 𝑡) ⟹ 𝑥 = 𝑦 Next we show that 𝑥 = 𝑧, for this putting 𝑥 = 𝑎 , 𝑦 = 𝑏 , 𝑧 = 𝑐, 𝑝 = 𝑑, 𝑢 = 𝑐′ , 𝑣 = 𝑑′, 𝑤 = 𝑎′ , 𝑟 = 𝑏′ in (i), 𝑀(𝑥, 𝑧, 𝑞𝑡) = 𝑀(𝐴(𝑎, 𝑏, 𝑐, 𝑑), 𝐵(𝑐′, 𝑑′, 𝑎′ , 𝑏′), 𝑞𝑡) ≥ 𝑚𝑖𝑛 { 𝑀(𝑆𝑎, 𝑇𝑐′, 𝑡), 𝑀(𝐴(𝑎, 𝑏, 𝑐, 𝑑), 𝑆𝑎, 𝑡), 𝑀(𝐵(𝑐′, 𝑑′, 𝑎′, 𝑏′), 𝑇𝑐′, 𝑡), 𝑀(𝑆𝑎, 𝐵(𝑐′, 𝑑′, 𝑎′, 𝑏′), 𝑡), 𝑀(𝐴(𝑎, 𝑏, 𝑐, 𝑑), 𝑇𝑐′, 𝑡) } = 𝑚𝑖𝑛{𝑀(𝑥, 𝑧, 𝑡), 𝑀(𝑥, 𝑥, 𝑡), 𝑀(𝑧, 𝑧, 𝑡), 𝑀(𝑥, 𝑧, 𝑡), 𝑀(𝑥, 𝑧, 𝑡)} = 𝑀(𝑥, 𝑧, 𝑡) ⟹ 𝑥 = 𝑧 Next we show that 𝑥 = 𝑝, for this putting 𝑥 = 𝑎 , 𝑦 = 𝑏 , 𝑧 = 𝑐, 𝑝 = 𝑑, 𝑢 = 𝑑′ , 𝑣 = 𝑐′, 𝑤 = 𝑎′ , 𝑟 = 𝑏′ in (i), 𝑀(𝑥, 𝑧, 𝑞𝑡) = 𝑀(𝐴(𝑎, 𝑏, 𝑐, 𝑑), 𝐵(𝑑′, 𝑐′, 𝑎′ , 𝑏′), 𝑞𝑡) ≥ 𝑚𝑖𝑛 { 𝑀(𝑆𝑎, 𝑇𝑑′, 𝑡), 𝑀(𝐴(𝑎, 𝑏, 𝑐, 𝑑), 𝑆𝑎, 𝑡), 𝑀(𝐵(𝑑′, 𝑐′, 𝑎′, 𝑏′), 𝑇𝑑′, 𝑡), 𝑀(𝑆𝑎, 𝐵(𝑑′, 𝑐′, 𝑎′ , 𝑏′), 𝑡), 𝑀(𝐴(𝑎, 𝑏, 𝑐, 𝑑), 𝑇𝑑′, 𝑡) } = 𝑚𝑖𝑛{𝑀(𝑥, 𝑝, 𝑡), 𝑀(𝑥, 𝑥, 𝑡), 𝑀(𝑝, 𝑝, 𝑡), 𝑀(𝑥, 𝑝, 𝑡), 𝑀(𝑥, 𝑝, 𝑡)} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1800 https://internationalpubls.com = 𝑀(𝑥, 𝑝, 𝑡) ⟹ 𝑥 = 𝑝 ⇒ 𝑥 = 𝑦 = 𝑧 = 𝑝 Now we prove that 𝑆𝑥 = 𝑇𝑥 Putting 𝑢 = 𝑦, 𝑣 = 𝑧, 𝑤 = 𝑡, 𝑟 = 𝑥 𝑀(𝑆𝑥, 𝑇𝑥, 𝑞𝑡) = 𝑀(𝐴(𝑥, 𝑦, 𝑧, 𝑝), 𝐵(𝑦, 𝑧, 𝑡, 𝑥), 𝑞𝑡) ≥ 𝑚𝑖𝑛 { 𝑀(𝑆𝑥, 𝑇𝑦, 𝑡), 𝑀(𝐴(𝑥, 𝑦, 𝑧, 𝑝), 𝑆𝑥, 𝑡), 𝑀(𝐵(𝑦, 𝑧, 𝑡, 𝑥), 𝑇𝑦, 𝑡), 𝑀(𝑆𝑥, 𝐵(𝑦, 𝑧, 𝑡, 𝑥), 𝑡), 𝑀(𝐴(𝑥, 𝑦, 𝑧, 𝑝), 𝑇𝑦, 𝑡) } = 𝑚𝑖𝑛{𝑀(𝑆𝑥, 𝑇𝑦, 𝑡), 𝑀(𝑆𝑥, 𝑆𝑥, 𝑡), 𝑀(𝑇𝑦, 𝑇𝑦, 𝑡), 𝑀(𝑆𝑥, 𝑇𝑦, 𝑡), 𝑀(𝑆𝑥, 𝑇𝑦, 𝑡)} = 𝑚𝑖𝑛 {𝑀(𝑆𝑥, 𝑇𝑦, 𝑡), 1,1, 𝑀(𝑆𝑥, 𝑇𝑦, 𝑡), 𝑀(𝑆𝑥, 𝑇𝑦, 𝑡)} = 𝑀(𝑆𝑥, 𝑇𝑦, 𝑡) ⟹ 𝑆𝑥 = 𝑇𝑦 ⇒ 𝑆𝑥 = 𝑇𝑥 Also by condition (ii) we have, 𝑥 = 𝐵(𝑥, 𝑥, 𝑥, 𝑥) Thus 𝐴(𝑥, 𝑥) = 𝑇(𝑥) = 𝐵(𝑥, 𝑥) = 𝑆(𝑥) = 𝑥. Example 3.1.1 Let 𝑋 = [0,1] with the metric 𝑑 defined by 𝑑(𝑥, 𝑦) = |𝑥 − 𝑦| and for each 𝑡 ∈ [0,1], define M(x, y, t) = { t t + |x − y| , if t > 0; 0, if t = 0 for all 𝑥, 𝑦 ∈ 𝑋. Clearly (X, ℱ,∗) be a fuzzy metric space, with 𝑎 ∗ 𝑏 = min{𝑎, 𝑏}. Let 𝑆, 𝑇: 𝑋 → 𝑋 and 𝐴, 𝐵: 𝑋 × 𝑋 × 𝑋 × 𝑋 → 𝑋 defined by A(x, y, z, w) = 2x+y+2z+w 2 S(X) = { x, if 0 ≤ x < 1; 7 2 , if x ≥ 1. B(x, y, z, w) = y T(X) = { x, if 0 ≤ x < 1; 5, if x ≥ 1. Clearly all the conditions of the above theorem are satisfied. Also 𝑆𝐴(0,0,0,0) = 𝐴(𝑆0, 𝑆0, 𝑆0, 𝑆0) and 𝑇𝐵(0,0,0,0) = 𝐵(𝑇0, 𝑇0, 𝑇0, 𝑇0) So, (A, S) and (B, T) are owc maps and (0, 0, 0, 0) is the common quadruple fixed point of A, B, S and T. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1801 https://internationalpubls.com All conditions of the theorem are satisfied. Common quadruple fixed point is (0, 0, 0, 0). Theorem: 3.2 Let (𝑋, 𝑀,  ) be a fuzzy metric space with 𝑡 ∗ 𝑡 = 𝑡 for all 𝑡 ∈ [0,1]. Let 𝐴, 𝐵: 𝑋 × 𝑋 × 𝑋 × 𝑋 → 𝑋 and 𝑆, 𝑇: 𝑋 → 𝑋 be four self-mappings satisfying the following conditions: (i) 𝑀(𝐴(𝑥, 𝑦, 𝑧, 𝑝), 𝐵(𝑢, 𝑣, 𝑤, 𝑟), 𝑞𝑡) ≥ { 𝑀(𝐵(𝑢,𝑣,𝑤,𝑟),𝑆𝑥,𝑡).𝑀(𝐵(𝑢,𝑣,𝑤,𝑟),𝑇𝑢,𝑡)+𝑀(𝑆𝑥,𝐵(𝑢,𝑣,𝑤,𝑟),𝑡). 𝑀(𝐵(𝑢,𝑣,𝑤,𝑟),𝑇𝑢,𝑡) 2 } for all 𝑥, 𝑦, 𝑧, 𝑝, 𝑢, 𝑣, 𝑤, 𝑟 ∈ 𝑋 (ii) 𝑦 = 𝐵(𝑥, 𝑦, 𝑧, 𝑝) Moreover if the pairs (𝐴, 𝑆) and (𝐵, 𝑇) are owc, then there exists a unique point 𝑥 in 𝑋 such that 𝐴(𝑥, 𝑥, 𝑥, 𝑥) = 𝑇(𝑥) = 𝐵(𝑥, 𝑥, 𝑥, 𝑥) = 𝑆(𝑥) = 𝑥. Theorem: 3.3 Let (𝑋, 𝑀,  ) be a fuzzy metric space with 𝑡 ∗ 𝑡 = 𝑡 for all 𝑡 ∈ [0,1]. Let 𝐴, 𝐵: 𝑋 × 𝑋 × 𝑋 × 𝑋 → 𝑋 and 𝑆, 𝑇: 𝑋 → 𝑋 be four self-mappings satisfying the following conditions: (i) 𝑀(𝐴(𝑥, 𝑦, 𝑧, 𝑝), 𝐵(𝑢, 𝑣, 𝑤, 𝑟), 𝑞𝑡) ≥ 𝑚𝑖𝑛 {𝑀(𝑆𝑥, 𝑇𝑢, 𝑡), ( 1+𝑀(𝐴(𝑥,𝑦,𝑧,𝑝),𝑆𝑥,𝑡) 1+𝑀(𝐵(𝑢,𝑣,𝑤,𝑟),𝑇𝑢,𝑡) ) , 𝑀(𝐴(𝑥,𝑦,𝑧,𝑝),𝑆𝑥,𝑡) 𝑀(𝐵(𝑢,𝑣,𝑤,𝑟),𝑇𝑢,𝑡) } for all 𝑥, 𝑦, 𝑧, 𝑝, 𝑢, 𝑣, 𝑤, 𝑟 ∈ 𝑋 (ii) 𝑦 = 𝐵(𝑥, 𝑦, 𝑧, 𝑝) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 1802 https://internationalpubls.com Moreover if the pairs (𝐴, 𝑆) and (𝐵, 𝑇) are owc, then there exists a unique point 𝑥 in 𝑋 such that 𝐴(𝑥, 𝑥, 𝑥, 𝑥) = 𝑇(𝑥) = 𝐵(𝑥, 𝑥, 𝑥, 𝑥) = 𝑆(𝑥) = 𝑥. References [1] H. Aydi, E. Karapınar, and İ. Savaş Yüce, "Quadruple fixed point theorems in partially ordered metric spaces depending on another function," ISRN Applied Mathematics, vol. 2012, Article ID 539125, 16 pages, 2012. [2] T. G. Bhaskar and V. 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