Ratio Mathematica Volume 47, 2023 Some fixed point results in fuzzy metric space using intimate mappings Vijayabaskerreddy Bonuga* Srinivas Veladi † Abstract The aim of this research paper is to prove the existence and uniqueness of common fixed point theorems for four self-mappings in fuzzy metric space using the notion of Intimate mappings.We also provide appropriate illustrations to justify the key points mentioned in the main results. Keywords: Fuzzy metric space, Intimate mappings, E.A property, Common E.A property. 2010 AMS subject classification: 54H25, 47H10.‡ * Department of Mathematics, Sreenidhi Institute of Science and Technology, Hyderabad, India; Email: basker.bonuga@gmail.com. † Department of Mathematics, University college of Science, Osmania University Hyderabad, India; Email:srinivasmaths4141@gmail.com. ‡ Received on August 12, 2022. Accepted on January 2, 2023. Published on January 13, 2023. doi: 10.23755/rm.v41i0.823. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 173 Vijayabaskerreddy Bonuga and Srinivas Veladi 1. Introduction L.A Zadeh [1] initiated the new concept, as extension an of classical set namely Fuzzy set. Lateron the notion of fuzzy metric space was introduced by Kramosil and Mechalek in [2]. Further this was altered by George and veeramani [4] in order to obtain Harsdorff topology for the class of fuzzy metric spaces. Thereafter many fixed point theorems came into light under various conditions like ([5],[6],[9],[10],[11],[13],[16],[19]) in fuzzy metric space. Under other conditions, Sahu and others [12] developed the notion of generalized compatible mappings of type (𝒜) called Intimate mappings. These were further extended by Chugh and Madhu Aggarwal [13] which resulted in the formation of some results in Hausdorff uniform spaces. Further some more results can be witnessed like [14] using intimate mappings in complex valued metric space. Apart from this Praveenkumar and others [15] proved some theorems in multiplicative metric space (MMS) using the notion of intimate mappings and subsequently many results came into existence on this platform like ([17],[18]). The concept of non-compatible mappings extended as the E. A property was introduced in metric space by Aamri and Matouwakil [20]. Consequently, the concept of improved E.A property resulted in the formation of common property E.A was introduced by Yicheng liu et al. [21]. The important note of this artice is to extend the notion of intimate mappings in fuzzy metric space using recent concepts like the different forms of E.A properties.In this process we prove three unique common fixed point theorems using these concepts. Cocequently these results stand as generalizations of some of the existing results like [16] [19]. Furthermore, some illustrations are provided to support our findings. 2. Definitions and Preliminaries Definition 2.1 (B.Schweizer and A.Sklar [7]):A binary operation ∗:[0,1] × [0, 1]→[0,1] is said to be continuous triangular norm (i. e continuous 𝓉 − norm) if the following assertions hold: (CT-i) * is continuous;(CT-ii)𝒶 ∗ 𝒷 ≤ c ∗ 𝒹 where 𝒶 ≤ 𝒷, 𝒸 ≤ 𝒹 and 𝒶, 𝒷, 𝒸, 𝒹 ∈ [0,1];(CT-iii)𝒶 ∗ 1 = 𝒶 for 𝒶 ∈ [0,1]; (CT-iv) ∗ is associative and commutative. Definition 2.2 (Kramosil and Mechalek [2]): A triplet (𝕏,𝑀𝐾𝑀,*) is fuzzy metric space (i.e., FMS) if 𝕏 is a arbitrary set, * is continuous 𝓉 − norm and 𝑀𝐾𝑀 is fuzzy set on 𝕏2× (0, ∞) satisfying the following conditions for all 𝓍, 𝓎, 𝓏𝕏 such that 𝓉, 𝓈(0, ∞): (KMFM-i) MKM(𝓍, 𝓎, 0) = 0 (KMFM-ii) MKM(𝓍, 𝓎, 𝓉) = 1 ∀𝓉 > 0 ⟺ 𝓍 = 𝓎 (KMFM-iii) MKM(𝓎, 𝓍, 𝓉) = MKM(𝓍, 𝓎, 𝓉) (KMFM-iv) MKM(𝓍, 𝓏, 𝓉 + 𝓈) ≥ MKM (𝓍, 𝓎, 𝓉) ∗ MKM (𝓎, 𝓏, 𝓈) (KMFM-v) MKM(𝓍, 𝓎, . ): [0.1]→[0,1] left continuous. Example 2.3 (George &Veeramani [4]): Consider(𝕏, 𝒹𝓊) is a metric space and define 𝑀𝐾𝑀(𝓍, 𝓎, 𝓉) = 𝓉 𝓉 + 𝒹𝓊(𝓍, 𝓎) then ( 𝕏, MKM,∗) is FMS where ∀𝓍, 𝓎𝕏, 𝓉 > 0 and ∗ is continuous 𝓉 − norm with 𝒶 ∗ 𝒷 = min {𝒶, 𝒷}. 174 Some fixed point results in fuzzy metric space using intimate mappings In the entire paper, (𝕏, 𝑀𝐾𝑀,∗) is to be assumed FMS with the condition (KFFM-6) : lim 𝓉→∞ MKM(𝓍, 𝓎, 𝓉) = 1 for all 𝓍, 𝓎, 𝕏. Definition 2.4 (Grabiec [3]): Let 〈𝓍𝓃 〉 be sequence in FMS (𝕏, MKM,∗), 〈𝓍𝓃 〉 then converges to a point ℓ ∈ 𝕏 if lim 𝓃⟶∞ MKM(𝓍𝓃, ℓ, 𝓉) = 1, ∀𝓉 > 0. Definition 2.5 (Garbaic [3]): Let 〈𝓍𝓃 〉 be a sequence in FMS (𝕏, MKM,∗), this sequence < 𝓍𝓃 > in 𝕏 is said to be Cauchy sequence in FMS if lim 𝓃→∞ MKM(𝓍𝓃+𝓅, 𝓍𝓃 , 𝓉) = 1, ∀𝓉 > 0 and 𝓅 > 0. Definition 2.6 (Garbiec [3]): If every Cauchy sequence is convergent in (𝕏, MKM,∗) then we say that it is complete. Lemma 2.7 (S.N. Mishra et al [5]): Let(𝕏, MKM,∗) be a FMS if there exists 𝓀 ∈ (0,1) such that MKM(𝓍, 𝓎, 𝓀𝓉) ≥ MKM(𝓍, 𝓎, 𝓉) then 𝓍 = 𝓎. Definition 2.8 ([5],[10]): Let 𝔖 and 𝔗 be two self mappings of a FMS(𝕏, MKM,∗). Then 𝔖 and 𝔗 are (1) compatible if lim 𝓃→∞ M𝐾𝑀(𝔖𝔗𝓍𝓃 , 𝔗𝔖𝓍𝓃 , 𝓉) = 1 whenever a sequence 〈𝓍𝑛 〉 in 𝕏 provided lim 𝓃→∞ 𝔖𝓍𝓃 = lim 𝓃→∞ 𝔗𝓍𝓃 = 𝓉 for some 𝓉𝕏 (2) compatible of type (𝒜) if lim 𝓃→∞ M𝐾𝑀(𝔖𝔗𝓍𝓃 , 𝔗𝔗𝓍𝓃 , 𝓉) = 1 lim 𝓃→∞ M𝐾𝑀 (𝔗𝔖𝓍𝓃 , 𝔖𝔖𝓍𝓃 , 𝓉) = 1 whenever 〈𝑥𝓃 〉 in 𝕏 such that lim n→∞ 𝔖𝓍𝓃 = lim n→∞ 𝔗𝓍𝓃 = 𝓉 for some 𝓉𝕏. Now we discuss some definitions related to intimate mappings in FMS. Definition 2.9: Let 𝔄 𝑎𝑛𝑑 𝔖 be two mappings of a FMS (𝕏, MKM,∗) into itself. Then 𝔄 and 𝔖 are said to be (1). 𝒜-Intimate mappings if α MKM(𝔄𝔖𝓍𝓃, 𝔄𝓍𝓃 , 𝓉) ≥ α MKM(𝔖𝔖𝓍𝓃 , 𝔖𝓍𝓃 , 𝓉) where α = lim 𝓃⟶∞ Sup or lim 𝓃⟶∞ Inf and 〈𝓍𝓃 〉 is a sequence in 𝕏 ∋ lim 𝓃→∞ 𝔄𝓍𝓃 = lim 𝓃→∞ 𝔖S𝓍𝓃 = 𝓉 for some 𝓉𝕏. (2). 𝒮-Intimate mapping if α M𝐾𝑀(𝔖𝔄𝓍𝓃, 𝔖𝓍𝓃 , 𝓉) ≥ α M𝐾𝑀(𝔄𝔄𝓍𝓃, 𝔄𝓍𝓃 , 𝓉) where α = lim 𝓃⟶∞ Sup or lim 𝓃⟶∞ Inf and a sequence 〈𝓍𝑛〉 in 𝕏 ∋ lim 𝓃→∞ 𝔄𝓍𝓃 = lim 𝓃→∞ 𝔖𝓍𝓃 = 𝓉 for some 𝓉𝕏. Proposition 2.10: Let 𝔄 𝑎𝑛𝑑 𝔖 be two self mappings of a FMS (𝕏, 𝑀𝐾𝑀 ,∗). Suppose 𝔄 and 𝔖 are compatible mappings of type (𝒜) then the pair of mappings 𝔄 𝑎𝑛𝑑 𝔖 are 𝒜 − intimate mappings and 𝒮-intimate mappings. Proof:Since 𝔄 𝑎𝑛𝑑 𝔖are compatible of type (𝒜), we have lim 𝓃→∞ M𝐾𝑀 (𝔄𝔖𝓍𝓃, 𝔖𝔖𝑥𝓃, 𝓉) = 1 and lim n→∞ M𝐾𝑀(𝔖𝔄𝓍𝓃, 𝔄𝔄𝑥𝓃, 𝓉) = 1 whenever 〈𝓍𝓃〉 in 𝕏 ∋ lim n→∞ 𝔄 𝓍𝓃 = lim 𝓃→∞ 𝔖𝓍𝓃 = 𝓉 for some 𝓉𝕏. Now M𝐾𝑀 (𝔄𝔖𝓃𝓃, 𝔄𝓍𝓃, (2 − β)𝓉) = M𝐾𝑀 (𝔄𝔖𝓍𝓃, 𝔄𝓍𝓃, (1 + 𝑘1)𝓉) ≥ M𝐾𝑀 (𝔄𝔖𝓍𝓃, 𝔖𝔖𝓍𝓃, 𝑘1𝓉) ∗ M𝐾𝑀 (𝔖𝔖𝓍𝓃, 𝔄𝓍𝓃, 𝓉). 175 Vijayabaskerreddy Bonuga and Srinivas Veladi By taking 𝑘1 = 1 − β and 0 < 𝑘1 < 1 and letting 𝓃 → ∞ and 𝛽 → 1 we obtain M𝐾𝑀(𝔄𝔖𝓍𝓃, 𝔄𝓍𝓃 , 𝓉) ≥ M𝐾𝑀(𝔄𝔖𝓍𝓃 , 𝔖𝔖𝓍𝓃 , 𝑘1𝓉) ∗ M𝐾𝑀(𝔖𝔖𝓍𝓃, 𝔄𝓍𝓃 , 𝓉) = M𝐾𝑀(𝔖𝔖𝓍𝓃, A𝓍𝓃 , 𝓉). By applying limit supremum on both sides, α M𝐾𝑀(𝔄𝔖𝓍𝓃 , 𝔄𝓍𝓃 , 𝓉) ≥ α M𝐾𝑀(𝔖𝔖𝓍𝓃 , 𝔄𝓍𝓃 , 𝓉) this implies 𝔄 𝑎𝑛𝑑 𝔖 are 𝒜-intimate mappings whenever {𝓍𝓃 } is a sequence in 𝕏 such that lim n→∞ 𝔄𝓍𝓃 = lim n→∞ 𝔖𝓍𝓃 = 𝓉 for some 𝓉𝕏. Likewise, we can prove that the pair of these mappings is 𝒮-intimate. Proposition 2.11: Let 𝔄 and 𝔖 be two self mappings on FMS.𝔄 and 𝔖 are 𝒜-intimate mappings and 𝔄t1=𝔖t1=𝑝,𝑝𝕏 then M𝐾𝑀(𝔄p, p, 𝓉) ≥ M𝐾𝑀(𝔖p, p, 𝓉). Proof: Suppose that {𝑥𝓃 } ∈ 𝕏 is a sequence such that 𝔄xn = 𝔖xn → 𝔄t1 = 𝔖t1 = 𝑝 for some 𝑝, 𝓉𝕏. Since the pair of mappings 𝔄 and 𝔖 are 𝒜 − intimate, then we obtain 𝑀𝐾𝑀(𝔄𝑝, 𝑝, 𝓉) = lim 𝓃→∞ M𝐾𝑀(𝔄𝔖𝓍𝓃 , 𝔄𝓍𝓃 , 𝓉) ≥ lim 𝓃→∞ M𝐾𝑀(𝔖𝔖𝓍𝓃 , 𝔖𝓍𝓃 , 𝓉) = 𝑀𝐾𝑀 (𝔖𝑝, 𝑝, 𝓉). Thus M𝐾𝑀 (𝔄p, p, 𝓉) ≥ M𝐾𝑀 (𝔖p, p, 𝓉). Remark 2.12: A pair of mappings 𝔄 and 𝔖 is 𝒜-intimate or 𝒮-intimate but not compatible mapping of type (𝒜). The following example revels the relation between intimate mappings and compatible mappings of type (𝒜). Example 2.13: Suppose 𝕏 = [0,1]. Define two self-mappings 𝔄 and 𝔖 as follows 𝔄(𝓍) = 5 𝓍+5 𝔖(𝓍) = 1 𝓍+1 for every 𝓍 in [0,1]. Consider a sequence 〈𝑥𝑛 〉 = 1 𝓃 𝓃 ∈ ℕ.Then lim 𝓃→∞ 𝔄𝑥𝑛 = lim𝔖𝑥𝑛 𝓃→∞ = 1. Consequently, lim 𝓃→∞ 𝑀(𝔄𝔖𝑥𝑛, 𝔄𝑥𝑛, 𝓉) = 6𝓉 6𝓉+1 and lim 𝓃→∞ 𝑀(𝔖𝔖𝑥𝑛, 𝔖𝑥𝑛 , 𝓉) = 2𝓉 2𝓉+1 . Hence lim 𝑛→∞ 𝑀𝐾𝑀 (𝔄𝔖𝑥𝑛, 𝔄𝑥𝑛 , 𝓉) lim 𝑛→∞ 𝑀𝐾𝑀 (𝔖𝔖𝑥𝑛, 𝔖𝑥𝑛 , 𝓉), for all 𝓉 > 0. Thus, the pair (𝔄, 𝔖) is 𝒜-intimate. On the other hand, the (𝔄, 𝔖) are not compatible of type (𝒜),since lim 𝓃→∞ M𝐾𝑀(𝔄𝔖𝓍𝓃, 𝔖𝔖𝑥𝓃 , 𝓉) = 3𝓉 3𝓉+1 ≠ 1 and lim n→∞ M𝐾𝑀(𝔖𝔄𝓍𝓃, 𝔄𝔄𝑥𝓃 , 𝓉) = 3𝓉 3𝓉+1 ≠ 1. Definition 2.14[20]: Define 𝔄 and 𝔖 as two self maps of FMS (𝕏, 𝑀𝐾𝑀 ,∗) then we say that 𝔄 and 𝔖 satisfy the property E.A if there exists a sequence 〈𝑥𝓃 〉 ∈ 𝕏 such that lim 𝓃→∞ 𝔄𝓍𝓃 = lim 𝓃→∞ 𝔖𝓍𝓃 = 𝓉 for some 𝓉 ∈ 𝕏. Definition 2.15[21]: Suppose 𝔄 , 𝔓 , 𝔅 and 𝔗 are four self maps on FMS (𝕏, MKM,∗) then we say that (𝔄, 𝔓)and (𝔅, 𝔗) satisfy common property E.A whenever two sequences 〈x𝓃 〉 and 〈γ𝓃 〉 in 𝕏 satisfying lim 𝓃→∞ 𝔄 𝓍𝓃 = lim 𝓃→∞ 𝔖𝓍𝓃 = lim 𝓃→∞ 𝔅 γ𝓃 = lim 𝓃→∞ 𝔗γ𝓃 = 𝓉 for some 𝓉 ∈ 𝕏. 176 Some fixed point results in fuzzy metric space using intimate mappings 3. Main results 3.1 Theorem: Let (𝕏, MKM,∗) be a complete fuzzy metric space. Suppose 𝔓,𝔔, 𝔖 and 𝔄 are self maps on 𝕏 satisfying the conditions (𝒞 − 1) 𝔓(𝕏)  𝔖(𝕏) and 𝔔(𝕏)  𝔄(𝕏) (𝒞 − 2) M𝐾𝑀(𝔓𝓍, 𝔔γ, k𝓉) ≥ M𝐾𝑀(𝔄𝓍, 𝔖𝛾, 𝓉) ∗ M𝐾𝑀(𝔓𝓍, 𝔄𝓍, 𝓉) ∗ M𝐾𝑀(𝔔γ, 𝔖𝛾, 𝓉) ∗ M𝐾𝑀(𝔓𝓍, 𝔖𝛾, 𝓉) where 𝑘 ∈ (0,1) and for all 𝓍, γ ∈ 𝕏 (𝒞 − 3) 𝔄( 𝕏 ) is complete (𝒞 − 4) the pair of mappings 𝔄 and 𝔓 𝑖𝑠 𝒜 − intimate and the other pair of mappings also 𝔖 and 𝔔 is 𝒮 − intimate. Then 𝔓, 𝔔,𝔖 and 𝔄 have a unique common fixed point in 𝕏. Proof: Let𝓍0 be any arbitrary point of 𝕏. Since from the condition 𝔓(𝕏)  𝔖(𝕏) of (𝒞 − 1) , there exists a point 𝓍1∈𝕏 such that 𝔓𝓍0=𝔖𝓍1=𝛾0. Now for this 𝓍1 and applying the (𝒞 − 1)[i.e 𝔔(𝕏)  𝔄(𝕏)] ∃𝓍2𝕏 such that 𝔔𝓍1=𝔄𝓍2=𝛾1. Inductively, we establish two real sequences < 𝓍𝓃 > and < γ𝓃 > in 𝕏 ∋ 𝛾2𝑛=𝔓𝓍2𝑛= 𝔖𝓍2𝓃+1 and 𝛾2𝑛+1 = 𝔔𝓍2𝓃+1 = 𝔄𝑥2𝓃+2 for 𝓃 0. By taking 𝓍 = 𝓍2𝓃 , 𝛾 = 𝓍2𝓃+1in the inequality (𝒞 − 2), M𝐾𝑀(𝔓𝓍2n, 𝔔𝓍2𝓃+1, k𝓉) ≥ M𝐾𝑀(𝔄𝓍2n, 𝔖𝓍2𝓃+1, 𝓉) ∗ M𝐾𝑀(𝔓𝓍2n, 𝔄𝓍2𝓃 , 𝓉) ∗ M𝐾𝑀(𝔔𝓍2𝓃+1, 𝔖𝓍2𝓃+1, 𝓉) ∗ M𝐾𝑀(𝔓𝓍2n, 𝔖𝓍2𝓃+1, 𝓉) which implies that an 𝓃→∞ M𝐾𝑀(𝛾2𝑛, 𝛾2𝑛+1, k𝓉) ≥ M𝐾𝑀 (𝛾2𝑛−1, 𝛾2𝑛, 𝓉) ∗ M𝐾𝑀(𝛾2𝑛, 𝛾2𝑛−1, 𝓉) ∗ M𝐾𝑀(𝛾2𝑛+1, 𝛾2𝑛, 𝓉) ∗ M𝐾𝑀 (𝛾2𝑛, 𝛾2𝑛, 𝓉). This yield M𝐾𝑀(𝛾2𝑛, 𝛾2𝑛+1, k𝓉) ≥ M𝐾𝑀(𝛾2𝑛−1, 𝛾2𝑛, 𝓉) ∗ M𝐾𝑀(𝛾2𝑛+1, 𝛾2𝑛, 𝓉) ∗ M𝐾𝑀(𝛾2𝑛, 𝛾2𝑛−1, 𝓉) ∗ 1. Again, by the condition KMFM-3, we get M𝐾𝑀(𝛾2𝑛, 𝛾2𝑛+1, k𝓉) ≥ M𝐾𝑀(𝛾2𝑛−1, 𝛾2𝑛, 𝓉) ∗ M𝐾𝑀(𝛾2𝑛, 𝛾2𝑛+1, 𝓉) which implies (since 𝔞 ∗ 𝔟 = min{𝔞, 𝔟}.) M𝐾𝑀(𝛾2𝑛, 𝛾2𝑛+1, k𝓉) ≥ M𝐾𝑀(𝛾2𝑛−1, 𝛾2𝑛, 𝓉). In general MKM(𝛾𝓃+1, 𝛾𝓃+2, 𝑘𝓉) ≥ 𝑀𝐾𝑀 (𝛾𝓃 , 𝛾𝓃+1, 𝓉) … . . (𝜎 − 1) for all 𝓃 = 1,2,3. . , and 𝓉 > 0. From (𝜎 − 1), 177 Vijayabaskerreddy Bonuga and Srinivas Veladi [M𝐾𝑀(γ𝓃, γ𝓃+1, 𝓉)] ≥ M𝐾𝑀 (γ𝓃−1, γ𝓃 , 𝓉 k ) ≥ M𝐾𝑀 (γ𝓃−2, γ𝓃−1, 𝓉 k2 ) ≥ ⋯ … ≥ M𝐾𝑀 (γ0, γ1, 𝓉 k 𝓃 ) → 1 as 𝓃 → ∞. . . . . . . (𝜎 − 2) For any 𝓉 > 0 and 𝜆𝑀𝐾 ∈ (0,1) we consider ∀ 𝓃 > 𝓃0 ∈ ℕ such that M𝐾𝑀(γ𝓃, γ𝓃+1, 𝓉) > (1 − MK) … (𝜎 − 3). For 𝓂, 𝓃 ∈ ℕ . Suppose 𝓂 ≥ 𝓃, then we have that [MMK(γ𝓃 , γ𝓂 , 𝓉)] ≥ min {MMK (γ𝓃 , γ𝓃+1, 𝓉 𝓂 − 𝓃 ) ∗ MMK (γ𝓃+1, γ𝓃+2, 𝓉 𝓂 − 𝓃 ) ∗. . .. MMK (γ𝓂−1, γ𝓂 , 𝓉 𝓂 − 𝓃 ) ≥ (1 − MK) ∗ (1 − MK) ∗. . . (1 − MK). . (𝓂 − 𝓃) times. This implies MMK(γ𝓂−1, γ𝓂 , 𝓉) ≥ (1 − MK) Therefore < γ𝓃 > is cauchy sequence in FMS. Since (𝕏, 𝑀𝐾𝑀,∗) is complete FMS, so sequence {γ𝓃 } converges to p*X. Further fuzzy cauchy sequence {γ𝓃 } has convergent subsequence {γ2𝓃+1} and {γ2𝓃 }. From the above argument, γ2𝓃+1 = 𝔔𝓍2𝓃+1 = 𝔄𝓍2𝓃+2→p ∗ and γ2𝓃 = 𝔓𝓍2𝓃 = 𝔖𝓍2𝓃+1→p ∗ as 𝓃→∞ … (𝜎 − 4) Now suppose that the range set 𝔄(X) is complete then  a point u𝕏 ∋ 𝔄u=p*..(𝜎 − 5). Now we claim that 𝔓u=p* from the inequality, put 𝓍 = 𝓊 and γ = 𝓍2𝓃+1 we have M𝐾𝑀(𝔓u, 𝔔𝓃2𝓃+1, k𝓉) ≥ M𝐾𝑀(𝔄u, 𝔖𝓍2𝓃+1, 𝓉) ∗ M𝐾𝑀(𝔓u, 𝔄u, 𝓉) ∗ M𝐾𝑀(𝔔𝓍2𝓃+1, 𝔖𝓍2𝓃+1, 𝓉) ∗ M𝐾𝑀(𝔓u, 𝔖S𝓍2𝓃+1, 𝓉). Taking limit as 𝓃→∞ M𝐾𝑀(𝔓u, p ∗, k𝓉) ≥ M𝐾𝑀(p ∗, p ∗, 𝓉) ∗ M𝐾𝑀(𝔓u, p ∗, 𝓉) ∗ M𝐾𝑀(p ∗, p ∗, 𝓉) ∗ M𝐾𝑀(𝔓u, p ∗, 𝓉). This gives 𝔓u=p*. That is 𝔓u=𝔄u=p*…... (𝜎 − 6) Let us prove that Qv=p*. Using the equation ((𝜎 − 6) with contained inequality 𝔓 (𝕏) ⊆ 𝔖 (𝕏), p*=𝔓u  𝔓(𝕏)  𝔖(𝕏) then ∃ a point v𝕏 ∋ 𝔖v=𝔓u=p*…. (𝜎 − 7). Put 𝓍=u and 𝛾 = 𝑣 in (𝒞 − 2) then we obtain M𝐾𝑀(𝔓u, 𝔔v, k𝓉) ≥ M𝐾𝑀 (Au, 𝔖v, 𝓉) ∗ M𝐾𝑀(𝔓u, 𝔄u, 𝓉) ∗ M𝐾𝑀(𝔔v, 𝔖v, 𝓉) ∗ M𝐾𝑀 (𝔓u, 𝔖v, 𝓉). By using(𝜎 − 7) we get M𝐾𝑀(p ∗, 𝔔v, k𝓉) ≥ M𝐾𝑀(p ∗, 𝔖v, 𝓉) ∗ M𝐾𝑀(p ∗, p ∗, 𝓉) ∗ M𝐾𝑀(𝔔v, p ∗, 𝓉) ∗ M𝐾𝑀(p ∗, p ∗, 𝓉) this gives M𝐾𝑀(p ∗, 𝔔v, 𝑘𝓉) ≥ M𝐾𝑀(𝔔v, p ∗, k𝓉). Consequently M𝐾𝑀(p ∗, 𝔔v, k𝓉) ≥ M𝐾𝑀(p ∗, 𝔔v, k𝓉) 178 Some fixed point results in fuzzy metric space using intimate mappings this implies 𝔔v=p*. This shows that 𝔔v=𝔖v=p*…... (𝜎 − 8) Since 𝔓u=𝔄u=p* and (𝔄,𝔓) is 𝒜-intimate we have M𝐾𝑀(𝔄p*, p*,𝓉) ≥ M𝐾𝑀 (𝔓p*, p*, 𝓉)…. (𝜎 − 9). Suppose that 𝔓p*≠ p*. Put 𝓍 = p ∗, γ = v in (𝒞 − 2) then we get, M𝐾𝑀(𝔓p ∗, 𝔔v, k𝓉) ≥ M𝐾𝑀(𝔄p ∗, 𝔖v, 𝓉) ∗ M𝐾𝑀 (𝔓p ∗, 𝔄p ∗, 𝓉) ∗ M𝐾𝑀(𝔔v, 𝔖v, 𝓉) ∗ M𝐾𝑀(𝔓p ∗, 𝔖v, 𝓉). Using (𝜎 − 8) we get, MKM(𝔓p ∗, p ∗, k𝓉) ≥ MKM(𝔄p ∗, p ∗, 𝓉) ∗ MKM(𝔓p ∗, 𝔄p ∗, 𝓉) ∗ MKM(p ∗, p ∗, 𝓉) ∗ MKM(𝔓p ∗, p ∗, 𝓉). By applying (KMFM-iv) we get M𝐾𝑀(𝔓p ∗, p ∗, k𝓉) ≥ M𝐾𝑀 (𝔓p ∗, p ∗, 𝓉) ∗ M𝐾𝑀(𝔓p ∗, p ∗, 𝓉/2) ∗ M𝐾𝑀(p ∗, 𝔄p ∗, 𝓉/2) ∗ M𝐾𝑀(p ∗, p ∗, 𝓉) ∗ M𝐾𝑀 (𝔓p ∗, p ∗, 𝓉). By using (𝜎 − 9) we get M𝐾𝑀(𝔓p ∗, p ∗, k𝓉) ≥ M𝐾𝑀(𝔓p ∗, p ∗, 𝓉/2). This gives 𝔓p*=p* …...(𝜎 − 10). From (𝜎 − 9) and (𝜎 − 10) we write M𝐾𝑀(𝔄p*, p*,𝓉) ≥ 1 this gives 𝔄p*=p*……(𝜎 − 11) using (𝜎 − 10) and (𝜎 − 11) we get 𝔄p*=𝔓p*=p*…... (𝜎 − 12) Also, 𝔔v=𝔖v=p* and using the pair (𝔖, 𝔔) as 𝒮-intimate then we have M𝐾𝑀(𝔖p ∗, p ∗, 𝓉) ≥ M𝐾𝑀(𝔔p ∗, p ∗, k𝓉)…. (𝜎 − 13) Suppose that 𝔔p*≠p*. Put 𝑥 = u and γ = 𝑝 ∗ in the inequality M𝐾𝑀(𝔓u, 𝔔p ∗, k𝓉) ≥ M𝐾𝑀 (𝔄u, 𝔖p ∗, 𝓉) ∗ M𝐾𝑀(𝔓u, 𝔄u, 𝓉) ∗ M𝐾𝑀(𝔔p ∗, 𝔖p ∗, 𝓉) ∗ M𝐾𝑀 (𝔓u, 𝔖p ∗, 𝓉) using (𝜎 − 6) and (KMFM-iv) we get, M𝐾𝑀(p ∗, 𝔔p ∗, k𝓉) ≥ M𝐾𝑀(p ∗, 𝔖p ∗, 𝓉) ∗ M𝐾𝑀 (p ∗, p ∗, 𝓉) ∗ M𝐾𝑀 (𝔓p ∗, p ∗, 𝓉 2 ) ∗ M𝐾𝑀 (p ∗, 𝔖p ∗, 𝓉 2 ) ∗ M𝐾𝑀(p ∗, 𝔖p ∗, 𝓉) on using (𝜎 − 13) we get M𝐾𝑀(p ∗, 𝔔p ∗, k𝓉) ≥ M𝐾𝑀(p ∗, 𝔔p ∗, 𝓉) ∗ M𝐾𝑀 (𝔔p ∗, p ∗, 𝓉 2 ) ∗ M𝐾𝑀(𝔔p ∗, p ∗, 𝓉/2) ∗ M𝐾𝑀(p ∗, 𝔔p ∗, 𝓉). This implies M𝐾𝑀(p ∗, 𝔔p ∗, k𝓉) ≥ M𝐾𝑀(p ∗, 𝔔p ∗, 𝓉/2). 179 Vijayabaskerreddy Bonuga and Srinivas Veladi This gives 𝔔p*=p*…(𝜎 − 14). From (𝜎 − 13) and (𝜎 − 14) we get M𝐾𝑀(𝔖p*, p*,𝓉)≥1 𝔖p*=p*…...(𝜎 − 15) . Using (𝜎 − 14) and (𝜎 − 15) we get 𝔔p*=𝔖p*=p*. …. (𝜎 − 16). Using (𝜎 − 12) and (𝜎 − 16) we conclude that 𝔄p*=𝔓p*=𝔔p*=𝔖p*=p*. Hence the result. We can prove the uniqueness of the fixed point easily. Example 3.1.1: Suppose (𝕏, MKM, *) is a standard FMS with 𝒶 ∗ 𝒶𝒶 ∀𝒶 ∈ [0,1], where 𝔄, 𝔖, 𝔓 and 𝔔:𝕏→𝕏 as 𝔓(𝑥) = 𝔔(x) = { 𝑥 + 0.125 if 0 ≤ 𝑥 < 0.125 0.25 if 0.125 ≤ 𝑥 ≤ 1 𝔄(𝑥) = 𝔖(x) = { 2𝑥 if 0 ≤ 𝑥 < 0.125 0.25 if 0.125 ≤ 𝑥 ≤ 1 𝔓(𝕏) = 𝔔(𝕏) = [0.125,0.25] and 𝔄(𝕏) = 𝔖(𝕏) = [0,0.25] these sets satisfy the condition (𝒞 − 1). Now assume 〈𝓍𝓃 〉 = {0.125 + 1 𝓃 } then lim n→∞ 𝔄𝑥𝓃 = lim 𝓃→∞ 𝔓𝑥𝓃 = 0.25. Also we have, lim 𝓃→∞ 𝔄𝔓𝓍𝓃 = lim 𝓃→∞ 𝔄𝔓(0.125 + 1 𝓃 ) = lim 𝓃→∞ 𝔄(0.25) = 0.125. lim 𝓃→∞ MKM(𝔄𝔓𝓍𝓃 , 𝔄𝓍𝓃 , 𝓉) lim n→∞ MKM(𝔓𝔓𝓍𝓃 , 𝔓𝓍𝓃 , 𝓉), for 𝓉 > 0. Thus, the pair (𝔄,𝔓) is 𝒜-intimate. Further lim 𝓃→∞ MKM(𝔖𝔔𝓍𝓃, 𝔖𝓍𝓃 , 𝓉) lim 𝓃→∞ MKM(𝔔𝔔𝓍𝓃 , 𝔔𝓍𝓃 , 𝓉). Thus, the pair (𝔖,𝔔) is 𝒮-intimate. Moreover, it satisfies the contraction condition of the theorem. Clearly 0.25 is the unique common fixed point for these four mappings. Theorem.3.2: Let (𝕏, MKM,∗)be a fuzzy metric space. Suppose 𝔓,𝔔, 𝔖 and 𝔄 are self maps on 𝕏 satisfies the conditions (𝒞 − 1), (𝒞 − 2), (𝒞 − 3) and (𝒞 − 4) with (𝒞 − 5):(𝔓, 𝔄) or (𝔔, 𝔖) satisfy E.A property then 𝔓, 𝔔,𝔖 and 𝔄 have a unique common fixed point in 𝕏. Proof: Suppose the pair (𝔔, 𝔖) satisfies E.A property then ∃ sequence 〈𝑥𝑛 〉 in 𝕏 such that lim 𝓃⟶∞ 𝔔𝑥𝑛 = lim 𝓃⟶∞ 𝔖𝑥𝑛 = 𝑝 ∗ for some 𝑝 ∗∈ 𝕏. Since 𝔔(𝕏) ⊆ 𝔄(𝕏) then ∃ 〈𝑥𝑛〉 in 𝕏 such that 𝔔𝑥𝓃 = 𝔄𝑦𝓃 . Hence lim 𝓃⟶∞ 𝔄𝛾𝓃 = p ∗. ….(𝜑 − 1). Now we show that lim 𝓃⟶∞ 𝔓𝛾𝓃 = 𝑝 ∗. Put 𝑥 = 𝛾𝑛 and 𝛾 = 𝑥𝑛 we obtain, 180 Some fixed point results in fuzzy metric space using intimate mappings M𝐾𝑀(𝔓𝛾𝓃, 𝔔𝑥𝑛 , k𝓉) ≥ M𝐾𝑀 (𝔄𝛾𝓃 , 𝔖𝑥𝑛 , 𝓉) ∗ M𝐾𝑀(𝔓𝛾𝓃 , 𝔄𝛾𝓃 , 𝓉) ∗ M𝐾𝑀(𝔔𝑥𝑛, 𝔖𝑥𝑛 , 𝓉) ∗ M𝐾𝑀 (𝔓𝛾𝓃 , 𝔖𝑥𝑛 , 𝓉). Letting 𝓃 ⟶ ∞ and using 𝑝𝛾𝓃 ⟶ 𝑝 ∗ we get lim 𝑛→∞ 𝔔𝑥𝑛 = lim 𝑛→∞ 𝔖𝑥𝑛 = lim 𝑛→∞ 𝔄𝛾𝑛 = lim 𝑛→∞ 𝔓𝛾𝑛 = 𝑝 ∗. Suppose that 𝔄(𝕏) is closed subspace of 𝕏, ∃ 𝑢 ∈ 𝕏 such that p*=𝔄𝑢. . . (𝜑 − 2). We show that 𝔄𝑢 = 𝔓𝑢. Put 𝑥 = 𝑢 and 𝛾 = 𝑥𝑛 in (𝒞 − 2) then we get M𝐾𝑀(𝔓u, 𝔔𝑥𝑛 , k𝓉) ≥ M𝐾𝑀(𝔄u, 𝔖𝑥𝑛 , 𝓉) ∗ M𝐾𝑀 (𝔓u, 𝔄u, 𝓉) ∗ M𝐾𝑀(𝔔𝑥𝑛, 𝔖𝑥𝑛 , 𝓉) ∗ M𝐾𝑀(𝔓u, 𝔖𝑥𝑛, 𝓉). This implies 𝔓𝑢 = 𝑝 ∗…(𝜑 − 3). From (𝜑 − 2) and (𝜑 − 2) we get 𝔄𝑢 = 𝔓𝑢 = 𝑝 ∗ ….(𝜑 − 4). And since (𝔄, 𝔓) is 𝒜 − 𝑖ntimate then we get 𝔄𝑝 ∗= 𝔓𝑝 ∗= 𝑝 ∗. . . . . . . (𝜑 − 5). Since 𝔓(𝕏) ⊆ 𝔖(𝕏) then there exists a point 𝑣 ∈ 𝕏 such that 𝔓𝑢 = 𝔖𝑣 = 𝑝 ∗. . . . . . (𝜑 − 6). Now put 𝑥 = 𝑢 and 𝛾 = 𝑣 in (𝒞 − 2) then this gives M𝐾𝑀(𝔓u, 𝔔𝑣, k𝓉) ≥ M𝐾𝑀(𝔄u, 𝔖𝑣, 𝓉) ∗ M𝐾𝑀(𝔓u, 𝔄u, 𝓉) ∗ M𝐾𝑀(𝔔𝑣, 𝔖𝑣, 𝓉) ∗ M𝐾𝑀 (𝔓u, 𝔖𝑣, 𝓉) implies M𝐾𝑀(𝑝 ∗, 𝔔𝑣, k𝓉) ≥ M𝐾𝑀 (p ∗, 𝑝 ∗, 𝓉) ∗ M𝐾𝑀(𝑝 ∗, p ∗, 𝓉) ∗ M𝐾𝑀(𝔔𝑣, p ∗, 𝓉) ∗ M𝐾𝑀 (𝑝 ∗, 𝑝 ∗, 𝓉). This implies 𝔔𝑣 = 𝑝 ∗ therefore 𝔖𝑣 = 𝔔𝑣 = 𝑝 ∗. . . . . . . (𝜑 − 7), and since (𝔖, 𝔔) is 𝒮 − intimate then we get 𝔖𝑝 ∗= 𝔔𝑝 ∗= 𝑝 ∗ …..(𝜑 − 8). Using (𝜑 − 7) (𝜑 − 8) and we conclude that 𝔄𝑝 ∗= 𝔓𝑝 ∗= 𝔔𝑝 ∗= 𝔖𝑝 ∗= 𝑝 ∗. We can prove the uniqueness of the common fixed point easily. Example 3.2.1: Suppose (𝕏, MKM, *) is a standard FMS with 𝒶 ∗ 𝒶𝒶 ∀𝒶 ∈ [1,11), where 𝔄, 𝔖, 𝔓 and 𝔔:𝕏→𝕏 as 𝔓(𝓍) = 𝔔(𝓍) = { 1 if x ∈ {1} ∪ (3,11) 1 + 𝓍 if 1 < 𝓍 ≤ 3 𝔖(𝓍) = { 1 if 𝓍 = 1 6 if 1 < 𝓍 ≤ 3 𝓍 − 2 if 3 < 𝓍 < 11 𝔄(x) = { 1 if 𝓍 = 1 4 if 1 < 𝓍 ≤ 3 3𝓍−1 8 if 3 < 𝓍 < 11 181 Vijayabaskerreddy Bonuga and Srinivas Veladi 𝔓(𝕏) = 𝔔(𝕏) = {1} ∪ (2,4] and 𝔖(𝕏) = {1} ∪ {6} ∪ (1,9) 𝔄(𝕏) = {1} ∪ {4} ∪ (1,4) = [1,4] these sets satisfy the conditions (𝒞 − 2)and (𝒞 − 3). Now assume 〈𝓍𝓃 〉 = {3 + 1 𝓃 } then lim n→∞ 𝔄𝓍𝓃 = lim 𝓃→∞ 𝔓𝓍𝓃 = 1 and this implies (𝔓, 𝔄) satisfies E.A property and also we have, lim n→∞ 𝔄𝔓𝓍𝓃 = lim 𝓃→∞ 𝔓𝔓𝓍𝓃 = 1. This gives lim 𝓃→∞ MKM(𝔄𝔓𝓍𝓃, 𝔄𝓍𝓃 , 𝓉) lim 𝓃→∞ MKM(𝔓𝔓𝓍𝓃, 𝔓𝓍𝓃 , 𝓉) for 𝓉 > 0. Thus, the pair (𝔄,𝔓) is 𝒜-intimate. Since lim n→∞ 𝔖𝓍𝓃 = lim 𝓃→∞ 𝔔𝓍𝓃 = 1 and lim n→∞ 𝔖𝔔𝓍𝓃 = lim 𝓃→∞ 𝔔𝔔𝓍𝓃 = 1 this gives lim 𝓃→∞ MKM(𝔖𝔔𝓍𝓃 , 𝔖𝓍n, 𝓉) lim 𝓃→∞ MKM(𝔔𝔔𝓍𝓃 𝔔𝓍𝓃 , 𝓉). Thus, the pair (𝔖,𝔔) is 𝒮-intimate. Moreover, it satisfies the contraction condition of the theorem. Clearly 1 is the unique common fixed point for these four mappings. Finally, we discuss another theorem. 3.3 Theorem: Let(𝕏, MKM,∗) be a FMS. Suppose 𝔓,𝔔, 𝔖 and 𝔄 are self maps on 𝕏 satisfying the conditions (𝒞 − 2) and (𝒞 − 4) in addition to (𝒞 − 6) 𝔄(𝕏) and 𝔖(𝕏) are closed subsets of 𝕏 (𝒞 − 7)the pairs (𝔓, 𝔄) and (𝔔, 𝔖) share the common property E. A. Then 𝔓, 𝔔,𝔖 and 𝔄 have a unique common fixed point in 𝕏. Proof: In view of the condition (𝒞 − 7)there exists two sequences 〈𝑥𝓃 〉 and 〈γn〉 in 𝕏 such that lim 𝓃⟶∞ 𝔓𝑥𝓃 = lim 𝓃⟶∞ 𝔄𝑥𝓃 = lim 𝓃⟶∞ 𝔔𝛾𝓃 = lim 𝓃⟶∞ 𝔖𝛾𝓃 = 𝑝 ∗ for some 𝑝 ∗∈ 𝕏. From the (𝒞 − 6) we have𝔄(𝕏) is closed subset of 𝕏,consequently lim 𝓃⟶∞ 𝔓𝑥𝓃 = 𝑝 ∗∈ 𝔄(𝕏). This means there exists appoint 𝑢 ∈ 𝕏 such that 𝔄𝑢 = 𝑝 ∗. Now we assert that 𝔓𝑢 = 𝔄𝑢. Put 𝑥 = 𝑢 and 𝛾 = 𝛾𝑛, we get M𝐾𝑀(𝔓u, 𝔔𝑦𝑛 , k𝓉) ≥ M𝐾𝑀 (𝔄u, 𝔖𝑦𝑛, 𝓉) ∗ M𝐾𝑀(𝔓u, 𝔄u, 𝓉) ∗ M𝐾𝑀(𝔔𝑦𝑛, 𝔖𝑦𝑛, 𝓉) ∗ M𝐾𝑀 (𝔓u, 𝔖𝑦𝑛, 𝓉) Which on making 𝓃 → ∞, with 𝔄𝑢 = 𝑝 ∗ reduces to 𝔓𝑢 = 𝑝 ∗. This implies 𝔓𝑢 = 𝔄𝑢 = 𝑝 ∗ which signifies that 𝑢 is coincident point of the pair (𝔓, 𝔄). On the other hand, 𝔖(𝕏) is closed subset of 𝕏 therefore lim 𝓃⟶∞ 𝔖𝛾𝓃 = 𝑝 ∗ ∈ 𝔖(𝕏) and hence we can find a point 𝑤 ∈ 𝕏 ∋ 𝔖𝑤 = 𝑝 ∗. Now we show that 𝔖𝑤 = 𝔔𝑤. On using condition (𝒞 − 2) with 𝑥 = u and γ = w then we get MKM(𝔓u, 𝔔w, k𝓉) ≥ MKM(𝔄u, 𝔖w, 𝓉) ∗ MKM(𝔓u, 𝔄u, 𝓉) ∗ MKM(𝔔w, 𝔖w, 𝓉) ∗ MKM(𝔓u, 𝔖w, 𝓉). This implies 𝔔w = p ∗.This gives 𝔖w = 𝔔w = p ∗. Since the pair (𝔔, 𝔖) is 𝒮 − intimate this gives MKM(𝔖p ∗, p ∗, 𝓉) ≥ MKM(𝔔p ∗, p ∗, 𝓉). 182 Some fixed point results in fuzzy metric space using intimate mappings Suppose that 𝔖𝑝 ∗≠ 𝑝 ∗. Put 𝑥 = u and γ = p ∗ in contraction condition (𝒞 − 2) M𝐾𝑀(𝔓u, 𝔔𝑝 ∗, k𝓉) ≥ M𝐾𝑀 (𝔄u, 𝔖𝑝 ∗, 𝓉) ∗ M𝐾𝑀(𝔓u, 𝔄u, 𝓉) ∗ M𝐾𝑀(𝔔𝑝 ∗, 𝔖𝑝 ∗, 𝓉) ∗ M𝐾𝑀 (𝔓u, 𝔖𝑝 ∗, 𝓉) implies 𝔔p ∗= p ∗. Using MKM(𝔖p ∗, p ∗, t) ≥ MKM(p ∗, p ∗, 𝓉) we get 𝔖p ∗= p ∗. Therefore 𝔔p ∗= 𝔖p ∗= p ∗…. . . . . . (ψ − 1). Since 𝔓u = 𝔄u = p ∗ and using (𝔓, 𝔄) is 𝒜 −intimate then we get 𝔄p ∗= p ∗. By putting 𝑥 = γ = p ∗ we get MKM(𝔓p ∗, 𝔔p ∗, k𝓉) ≥ MKM(𝔄p ∗, 𝔖p ∗, 𝓉) ∗ MKM(𝔓p ∗, 𝔄p ∗, 𝓉) ∗ MKM(𝔔p ∗, 𝔖p ∗, 𝓉) ∗ MKM(𝔓p ∗, 𝔖p ∗, 𝓉). This implies 𝔓p ∗= p ∗ and this gives 𝔄p ∗= 𝔓p ∗= p ∗. . . . . . (ψ − 2). From (ψ − 1) and (ψ − 2) we conclude that 𝔄p ∗= 𝔓p ∗= 𝔔p ∗= 𝔖p ∗= p ∗. We can prove the uniqueness of the fixed point easily. Example 3.3.1: Suppose (𝕏, MKM, *) is a standard FMS with 𝒶 ∗ 𝒶  𝒶 ∀𝒶 ∈ [1,20], where 𝔄, 𝔖, 𝔓 and 𝔔:𝕏→𝕏 as 𝔓(𝑥) = 𝔔(𝑥) = { 1 if 𝑥 = 1, 2 ≤ 𝑥 < 20 𝑥 if 1 ≤ 𝑥 < 2 𝔖(𝑥) = { 1 if 𝑥 = 1 12 if 1 < 𝑥 < 2 𝑥 + 1 3 if 2 ≤ 𝑥 ≤ 20 𝔄(x) = { 1 if 𝑥 = 1 7 if 1 < 𝑥 < 2 2𝑥 + 5 9 if 2 ≤ 𝑥 ≤ 20 𝔓(𝕏) = 𝔔(𝕏) = {1} ∪ (1,2) , 𝔖(𝕏) = {1} ∪ {12} ∪ [1,5] and 𝔄(𝕏) = {1} ∪ {7} ∪ [1,9] these sets satisfy the conditions (𝒞 − 1) and (𝒞 − 3). Now assume 〈𝓍𝓃 〉 = {2 + 1 𝓃 } and 〈γ𝓃 〉 = {1} then lim 𝓃→∞ 𝔄𝓍𝓃 = lim 𝓃→∞ 𝔓𝓍𝓃 = lim 𝓃→∞ 𝔖γ𝓃 = lim 𝓃→∞ 𝔔γ𝓃 = 1. This implies the pairs (𝔓, 𝔄) and (𝔖,𝔔) share the common E. A property and also we have, lim 𝓃→∞ 𝔄𝔓𝓍𝓃 = lim 𝓃→∞ 𝔓𝔓𝓍𝓃 = 1 this gives lim 𝓃→∞ MKM(𝔄𝔓𝓍𝓃, 𝔄𝓍𝓃 , 𝓉) lim 𝓃→∞ MKM(𝔓𝔓𝓍𝓃, 𝔓𝓍𝓃 , 𝓉), for 𝓉 > 0. Thus, the pair (𝔄,𝔓) is 𝒜-intimate. Since lim 𝓃→∞ 𝔖𝓍𝓃 = lim 𝓃→∞ 𝔔𝓍𝓃 = 1 and lim 𝓃→∞ 𝔖𝔔𝓍𝓃 = lim 𝓃→∞ 𝔔𝔔𝓍𝓃 = 1 this gives lim 𝓃→∞ MKM(𝔖𝔔𝓍𝓃 , 𝔖𝓍𝓃 , 𝓉) lim 𝓃→∞ MKM(𝔔𝔔𝓍𝓃 , 𝔔𝓍𝓃 , 𝓉). Thus, the pair (𝔖,𝔔) is 𝒮-intimate. Moreover, it satisfies the contraction condition of the theorem. Clearly 1 is the unique common fixed point for these four mappings. 183 Vijayabaskerreddy Bonuga and Srinivas Veladi 4 Conclusion This paper aimed to prove three common fixed point theorems to generalize the class of compatible mappings by using the calss of non compatible mappings like different forms of E.A properties along with intimate mappings in fuzzy metric space. 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