Ratio Mathematica Volume To be assigned, 2023 Common fixed point theorem for weakly compatible mappings in Sm metric space Mallaiah Katta * Srinivas Veladi† Abstract In the present paper, at first, we study the structure of the newly Sm- metric space, which is a combination of S-metric space and mul- tiplicative metric space. We have proved a common fixed point the- orem for four self-maps in Sm metric space with a new contraction condition by applying the concepts of weakly compatible mappings, semi-compatible mappings, and reciprocally continuous mappings. Further, we also provide some examples to support our results. Keywords: Multiplicative metric space, S-metric space,Sm-metric space, weakly compatible mappings, reciprocally continuous map- pings, and semi-compatible mappings. 2020 AMS subject classifications: 54H25 1 *JN Government Polytechnic, Hyderabad, India; kamanilayam95l@ gmail.com. †University college of science OU, Hyderabad,India; srinivasmaths4141@gmail.com. 1Received on August 8, 2022. Accepted on December 1, 2022. Published on January 2, 2023. DOI: 10.23755/rm.v39i0.808. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. Mallaiah K and Srinivas V 1 Introduction The notion of multiplicative metric space (MMS) was first developed by Bashirove [1]. Following that, several theorems came to light in this area of MMS [2] ,[3] and [4]. On the other side, Sedghi.S et al.[5] presented a new structure to S-metric space which modified D-metric and G-metric spaces, and then several fixed point theorems [6] and [7] were obtained. Pant et al. [8] generalized the notion of reciprocally continuous mapping which is weaker than continuous and compatible mappings. Recently, Mukesh Kumar Jain [9] introduced a more gen- eral form of semi-compatible mappings and proved many fixed point theorems in metric space. In this article, we use a new generalized metric space referred to as Sm -metric space, which is a combination of both MMS and S -metric space. Using this con- cept, we establish a common fixed point theorem by applying weakly compatible mappings(WCM), reciprocally continuous mappings, and semi-compatible map- pings. Furthermore, some examples are also discussed to support our conclusions. 2 Preliminaries: Now we give some definitions and examples which are used in this theorem. Definition 2.1. [1] “Let χ be a non-empty set and δ : χ2 → R+ be a multiplicative metric space (MMS) satisfying the properties : (i) δ(ψ,ϕ) ≥ 1 and δ(ψ,ϕ) = 1 ⇐⇒ ψ = ϕ (ii) δ(ψ,ϕ) = δ(ϕ,ψ) (iii) δ(ψ,ϕ) ≤ δ(ψ,σ)δ(σ,ϕ),∀ψ,ϕ,σ ∈ χ.” Definition 2.2. [5] “ Let χ be a non-empty set defined S : χ3 → [0,∞) satisfying: (i) S(ψ,ϕ,σ) ≥ 0 (ii) S(ψ,ϕ,σ) = 0 ⇐⇒ ψ = ϕ = σ (iii) S(ψ,ϕ,σ) ≤ S(ψ,ψ,ρ) + S(ϕ,ϕ,ρ) + S(σ,σ,ρ),∀ψ,ϕ,σ,ρ ∈ χ. A mapping S together with χ,(χ,S) is called a S-metric space.” Definition 2.3. [10] “ Let χ be a non-empty set .A function Sm : χ3 → R+ satisfying the conditions : Sm metric space (i) Sm(ψ,ϕ,σ) ≥ 1 (ii) Sm(ψ,ϕ,σ) = 1 ⇐⇒ ψ = ϕ = σ (iii) Sm(ψ,ϕ,σ) ≤ Sm(ψ,ψ,ρ)Sm(ϕ,ϕ,ρ)Sm(σ,σ,ρ),∀ψ,ϕ,σ,ρ ∈ χ. The pair (χ,Sm) is called as Sm-metric space”. Definition 2.4. [10] “ Let (χ,Sm) be a Sm-metric space, a sequence {ψθ} ∈ χ is said to be (i) cauchy sequence ⇐⇒ Sm(ψθ,ψθ,ψl) → 1, for all θ, l → ∞; (ii) convergent ⇐⇒ ∃ψ ∈ χ such that Sm(ψθ,ψθ,ψ) → 1 as θ → ∞; (iii) is complete if every cauchy sequence is convergent.” Definition 2.5. [11] ” Two self-maps M and K of a Sm metric space are said to be (i) Compatible: if lim θ→∞ Sm(MKψθ,MKψθ,KMψθ) = 1, whenever there exist a sequence {ψθ} ∈ χ such that lim θ→∞ Sm(Mψθ,Kψθ,ω) = 1 for some ω ∈ χ. (ii) Weakly- compatible mappings: if they commute at their coincidence points, i.e.ω ∈ χ,Sm(Mω,Mω,Kω) = 1, =⇒ Sm(MKω.MKω,KMω) = 1.” Definition 2.6. [9] “Two self maps M and K of Sm-metric space are said to be Semi- compatible: if lim θ→∞ Sm(MKψθ,MKψθ,Kω) = 1 whenever there exists a sequence {ψθ} ∈ X such that lim θ→∞ Sm(Mψθ,Kψθ,ω) = 1 for all ω ∈ χ.” Mallaiah K and Srinivas V Now we present an example in which semi-compatible is weaker than com- patible. Example 2.6.1 Consider χ = [0,∞) with Sm(ψ,ϕ,σ) = e|ψ−ϕ|+|ϕ−σ|+|σ−ψ|, for every ψ,ϕ,σ ∈ χ. Define two self maps M and K as M(ψ) = { cos2(πψ)+1 2 if 0 < ψ ≤ 1 2 ; sin(πψ) if 1 2 < ψ ≤ 3. and K(ψ) = { 2sin(πψ)−1 2 if 0 < ψ ≤ 1 2 ; 1 − sin(πψ) if 1 2 < ψ ≤ 3. Consider a sequence {ψθ} as ψθ = {π2 − 1 θ } for θ ≥ 0. Then lim θ→∞ M(ψθ) = lim θ→∞ M( 1 2 − 1 θ ) = lim θ→∞ cos2π(1 2 − 1 θ ) + 1 2 = lim θ→∞ sin2(π θ ) + 1 2 = 1 2 and lim θ→∞ K(ψθ) = lim θ→∞ K( 1 2 − 1 θ ) = lim θ→∞ 2sinπ(1 2 − 1 θ ) − 1 2 = lim θ→∞ 2cos(π θ ) − 1 2 = 1 2 . Therefore lim θ→∞ Mψθ = lim θ→∞ Kψθ = 1 2 = ω (say). Now lim θ→∞ MK(ψθ) = lim θ→∞ M( 2cosπ θ − 1 2 ) = lim θ→∞ cos2π( 2cosπ θ −1 2 ) + 1 2 = cos2 π 2 + 1 2 = 1 2 and lim θ→∞ KM(ψθ) = lim θ→∞ K( sin2 π θ + 1 2 ) = lim θ→∞ [1 − sinπ( sin2 π θ + 1 2 )] = 0. ∴ lim θ→∞ Sm(MKψθ,MKψθ,KMψθ) ̸= 0. This implies these two self-maps M and K are not compatible. But K(ω) = K(1 2 ) = 1 2 . Therefore lim θ→∞ Sm(MKψθ,MKψθ,Kω) = lim θ→∞ Sm( 1 2 , 1 2 , 1 2 ) = 1. Hence these two self maps M and K are semi-compatible but not compatible. Sm metric space Definition 2.7. [8] “Two self-maps M, K of Sm-metric space are said to be re- ciprocally continuous if lim θ→∞ Sm(MKψθ,MKψθ,Mω) = 1 and lim θ→∞ Sm(KMψθ,KMψθ,Kω) = 1, whenever there exist a sequence {ψθ} ∈ χ such that lim θ→∞ Sm(Mψθ,Kψθ,ω) = 1 some ω ∈ χ.” Now we present an example in which satisfies reciprocally continuous is weaker but not compatible. Example 2.7.1 Consider χ = (0,∞) with Sm(ψ,ϕ,σ) = e|ψ−ϕ|+|ϕ−σ|+|σ−ψ|, for every ψ,ϕ,σ ∈ χ. Define two self maps M and K as M(ψ) = { ψ2 + 2 if 0 < ψ ≤ 1; 4 − ψ if 1 < ψ ≤ 3. and K(ψ) = { 1 − 2ψ if 0 < ψ ≤ 1; ψ − 2 if 1 < ψ ≤ 3. Consider a sequence {ψθ} as ψθ = {3 − 1θ}, for θ ≥ 0. Now lim θ→∞ M(ψθ) = lim θ→∞ [4 − (3 − 1 θ )] = 1 and lim θ→∞ K(ψθ) = lim θ→∞ [(3 + 1 θ ) − 2] = 1 ∴ lim θ→∞ Mψθ = lim θ→∞ Kψθ = 1 = ω1 ̸= ϕ. Also lim θ→∞ MK(ψθ) = lim θ→∞ M[(3 − 1 θ ) − 2] = lim θ→∞ M(1 − 1 θ ) = 3 and lim θ→∞ KM(ψθ) = lim θ→∞ K(4 − (3 − 1 θ ) = lim θ→∞ K(1 + 1 θ ) = −1. ∴ lim θ→∞ Sm(MKψθ,MKψθ,KMψθ) = Sm(3,3,−1) ̸= 1. This gives the self maps M and K are not compatible in Sm- metric space. Moreover,M(ω1) = 3 and K(ω1) = −1. Which gives lim θ→∞ Sm(MKψθ,MKψθ,Mω1) = Sm(3,3,3) = 1, Mallaiah K and Srinivas V and lim θ→∞ Sm(KMψθ,KMψθ,Kω1) = Sm(−1,−1,−1) = 1. This implies the self-maps M and K are reciprocally continuous but not compati- ble in Sm metric space. Now we proceed to the main theorem. 3 Main Theorem Theorem 3.1. Let M, H, K, and J be self-mapping of a complete Sm-metric space satisfying the following (3.1.1) M(χ) ⊆ J(χ) and H(χ) ⊆ K(χ) (3.1.2) Sm(Mψ,Mψ,Hϕ) ≤ { max[Sm(Mψ,Mψ,Kψ)Sm(Hϕ,Hϕ,Jϕ), Sm(Mψ,Mψ,Jϕ)Sm(Kψ,Kψ,Hϕ), Sm(Mψ,Mψ,Jϕ)Sm(Hϕ,Hϕ,Jϕ), Sm(Mψ,Mψ,Kψ)Sm(Hϕ,Hϕ,Kψ)] }λ where λ ∈ (0, 1 2 ) (3.1.3) the pair M and K are reciprocally continuous and semi-compatible, (3.1.4) the pair H and J are weakly compatible. Then the self-maps M, H, K, and J have a unique common fixed point in χ. Proof: Let there is a point ψ0 ∈ χ, and the sequence {ψθ} be defined as Mψ0 = Jψ1 = ϕ0. For this point ψ1 then there exists ψ2 ∈ χ such that Hψ1 = Kψ2 = ϕ1. In general, by induction choose ψθ+1 , construct a sequence {ϕθ} ∈ χ such that ϕ2θ = Mψ2θ = Jψ2θ+1 and ϕ2θ+1 = Hψ2θ+1 = Kψ2θ+2, for θ ≥ 0. Sm metric space On putting ψ = ψ2θ and ϕ = ϕ2θ+1 in ( 3.1.2) we get. Sm(ϕ2θ,ϕ2θ,ϕ2θ+1) = Sm(Mψ2θ,Mψ2θ,Hψ2θ+1) ≤ max { Sm(Mψ2θ,Mψ2θ,θψ2θ)Sm(Hψ2θ+1,Hψ2θ+1,Jψ2θ+1), Sm(Mψ2θ,Mψ2θ,Jψ2θ+1)Sm(Hψ2θ+1,Hψ2θ+1,θψ2θ), Sm(Mψ2θ,Mψ2θ,Jψ2θ+1)Sm(Hψ2θ+1,Hψ2θ+1,Jψ2θ+1), Sm(Mψ2θ,Mψ2θ,Kψ2θ)Sm(Hψ2θ+1,Hψ2θ+1,Kψ2θ) }λ Sm(ϕ2θ,ϕ2θ,ϕ2θ+1) ≤ max { Sm(ϕ2θ,ϕ2θ,ϕ2θ−1)Sm(ϕ2θ+1,ϕ2θ+1,ϕ2θ), Sm(ϕ2θ,ϕ2θ,ϕ2θ)Sm(ϕ2θ+1,ϕ2θ+1,ϕ2θ−1), Sm(ϕ2θ,ϕ2θ,ϕ2θ)Sm(ϕ2θ+1,ϕ2θ+1,ϕ2θ), Sm(ϕ2θ,ϕ2θ,ϕ2θ−1)Sm(ϕ2θ+1,ϕ2θ+1,ϕ2θ−1) }λ this implies that Sm(ϕ2θ,ϕ2θ,ϕ2θ+1) ≤ Sm(ϕ2θ−1,ϕ2θ−1,ϕ2θ+1)λ. Sm(ϕ2θ,ϕ2θ,ϕ2θ+1) ≤ {Sm(ϕ2θ−1,ϕ2θ−1,ϕ2θ)Sm(ϕ2θ,ϕ2θ,ϕ2θ+1)}λ. S1−λm (ϕ2θ,ϕ2θ,ϕ2θ+1) ≤ S λ m(ϕ2θ−1,ϕ2θ−1,ϕ2θ). Sm(ϕ2θ,ϕ2θ,ϕ2θ+1) ≤ S λ 1−λ m (ϕ2θ−1,ϕ2θ−1,ϕ2θ). Sm(ϕ2θ,ϕ2θ,ϕ2θ+1) ≤ Spm(ϕ2θ−1,ϕ2θ−1,ϕ2θ). where p = λ 1 − λ . Now this gives Sm(ϕθ,ϕθ,ϕθ+1) ≤ Spm(ϕθ−1,ϕθ−1,ϕθ) ≤ S p2 m (ϕθ−2,ϕθ−2,ϕθ−1) ≤ · · ·S pn m (ϕ0,ϕ0,ϕn). By using triangular inequality Sm(ϕθ,ϕθ,ϕn) ≤ Sp θ m (ϕ0,ϕ0,ϕl) ≤ S pθ+1 m (ϕ0,ϕ0,ϕn) ≤ · · ·S pn−1 m (ϕ0,ϕ0,ϕn) Mallaiah K and Srinivas V Sm(ϕθ,ϕθ,ϕn) ≤ S pθ 1−p m (ϕ0,ϕ0,ϕl) for all θ ≥ 1. Hence {ϕθ} is a cauchy sequence in Sm-metric space. Since the self-maps, M and K are weakly reciprocally continuous. lim θ→∞ Sm(MKψθ,MKψθ,Mω) = 1 or lim θ→∞ Sm(KMψθ,KMψθ,θω) = 1. (1) Also, the pair ( M, K) is semi compatible, we have lim θ→∞ Sm(MKψθ,MKψθ,Kω) = 1. (2) From (1) and (2) we get Sm(Mω,Mω,Kω) = 1. (3) Since M(χ) ⊆ J(χ) which gives then there exists ν ∈ χ such that Jν = Mψθ, since Mψθ → ω as θ → ∞. Which implies Sm(Jν,Jν,ω) = 1. (4) Now, we have to prove Sm(Jν,Hν,ω) = 1. Substitute ψ = ψθ and ϕ = ν in (3.1.2) we have Sm(Mψθ,Mψθ,Hν) ≤ { max[Sm(Mψθ,Mψθ,Kψθ)Sm(Hν,Hν,Jν), Sm(Mψθ,Mψθ,Jν)Sm(Kψ1,Kψ1,Hν), Sm(Mψθ,Mψθ,Jν)Sm(Hν,Hν,Jν), Sm(Mψθ,Mψθ,Kψθ)Sm(Hν,Hν,Kψθ)] }λ Sm(ω,ω,Hν) ≤{ max[Sm(ω,ω,ω)Sm(Hν,Hν,ω),Sm(ω,ω,ω)Sm(ω,ω,Hν), Sm(ω,ω,ω)Sm(Hν,Hν,ω),Sm(ω,ω,ω)Sm(Hν,Hν,ω)] }λ Sm(ω,ω,Hν) ≤ {(Sm(ω,ω,Hν)}λ S(1−λ)m (ω,ω,Hν) ≤ 1 =⇒ Sm(Hν,Hν,ω) = 1. ∴ Sm(Jν,Hν,ω) = 1. Sm metric space Since the pair (H.J) is WCM and ν is a coincidence point then HJν = JHν Sm(Hω,Hω,Jω) = 1. (5) Substitute ψ = ψθ and ϕ = ω in (3.1.2) we have Sm(Mψθ,Mψθ,Hω) ≤{ max[Sm(Mψθ,Mψθ,Kψθ)Sm(Hω,Hω,Jω), Sm(Mψθ,Mψθ,Jω)Sm(Kψ1,Kψ1,Hω), Sm(Mψθ,Mψθ,Jω)Sm(Hω,Hω,Jω), Sm(Mψθ,Mψθ,Kψθ)Sm(Hω,Hω,Kψθ)] }λ also Sm(Hω,ω,ω) ≤{ max[Sm(ω,ω,ω)Sm(Hω,Hω,ω),Sm(ω,ω,ω)Sm(ω,ω,Hω), Sm(ω,ω,ω)Sm(Hω,Hω,ω),Sm(ω,ω,ω)Sm(Hω,Hω,ω)] }λ and this gives Sm(Hω,ω,ω) ≤ Sm(Hω,ω,ω)λ S(1−λ)m (Hω,ω,ω) ≤ 1 =⇒ Hω = ω ∴ Sm(Hω,Jω,ω) = 1. (6) Replace ψ = ω and ϕ = ν in (3.1.2) then we have Sm(Mω,Mω,Hν) ≤ { max[Sm(Mω,Mω,Kω)Sm(Jν,Hν,Hν), Sm(Mω,Mω,Jν)Sm(Kω,Kω,Hν), Sm(Mω,Mω,Jν)Sm(Jν,Jν,Hν), Sm(Mω,Mω,Kω)Sm(Hν,Hν,Kω)] }λ Mallaiah K and Srinivas V Sm(Mω,Mω,ω) ≤ { max[Sm(Mω,Mω,Mω)Sm(ω,ω,ω), Sm(Mω,Mω,ω)Sm(Mω,Mω,ω), Sm(Mω,Mω,ω)Sm(ω,ω,ω), Sm(Mω,Mω,Mω)Sm(ω,ω,Mω)] }λ Sm(Mω,Mω,ω) ≤ {Sm(Mω,Mω,ω)}λ S(1−λ)m (Mω,Mω,ω) ≤ 1 =⇒ Mω = ω ∴ Sm(Mω,Jω,ω) = 1. (7) From (6) and (7) we get Mω = Jω = Hω = Kω = ω. (8) Therefore “ω” is a common fixed point of M, H, K, and J. Uniqueness Let ρ be one more fixed point, we assume that ρ ̸= ω then we have Mρ = Kρ = Hρ = Jρ = ρ. In the condition (3.1.2) put ψ = ω and ϕ = ρ we get Sm(Mω,Mω,Hρ) ≤ { max[Sm(Mω,Mω,Kω)Sm(Hρ,Hρ,Jρ), Sm(Mω,Mω,Jρ)Sm(Kω,Kω,Hρ), Sm(Mω,Mω,Jρ)Sm(Hρ,Hρ,Jρ), Sm(Mω,Mω,Kω)Sm(Hρ,Hρ,Kω)] }λ Sm(ω,ω,ρ) ≤{ max[Sm(ω,ω,ω)Sm(ρ,ρ,ρ),Sm(ω,ω,ρ)Sm(ω,ω,ρ), Sm(ω,ω,ρ)Sm(ρ,ρ,ρ),Sm(ω,ω,Kω)Sm(ρ,ρ,ω)] }λ Sm(ω,ω,ρ) ≤ { Sm(ω,ω,ρ) }λ this implies that Sm(ω,ω,ρ) = 1 =⇒ ω = ρ. This shows that “ω” is the unique common fixed point of M.H.J and K. Sm metric space Now, the following example substantiates our theorem. Example 3.2 Suppose χ = (0,1),Sm- metric space by Sm(ψ,ϕ,σ) = e|ψ−ϕ|+|ϕ−σ|+|σ−ψ|, when ψ,ϕ,σ ∈ χ. Define M ,K ,H J:χXχ → χ as follows M(ψ) = { 2−ψ 5 if 0 < ψ ≤ 1 3 ; ψ if 1 3 < ψ < 1. K(ψ) = { 1 − 2ψ if 0 < ψ ≤ 1 3 ; 1+ψ 2 if 1 3 < ψ < 1. H(ψ) = { 3ψ2 − 3ψ + 1 if 0 < ψ ≤ 1 3 ; 2+ψ 7 if 1 3 < ψ < 1. J(ψ) = { 1 − 6ψ2 if 0 < ψ ≤ 1 3 ; 1 − ψ if 1 3 < ψ < 1. Then M(χ) = (1 3 ,1] ⊆ J(χ) = (0,1] and H(χ) = (1 3 ,1] ⊆ K(χ) = (1 3 ,1]. Therefore the condition (3.1.1 ) holds. Consider a sequence {ψθ} as ψθ = {13 − 1 θ } as θ ≥ 0. Then lim θ→∞ M(ψθ) = lim θ→∞ M( 1 3 − 1 θ ) = lim θ→∞ 2 − (1 3 − 1 θ ) 5 = 1 3 and lim θ→∞ K(ψθ) = lim θ→∞ K( 1 3 − 1 θ ) = lim θ→∞ [1 − 2( 1 3 − 1 θ )] = 1 3 . Therefore lim θ→∞ M(ψθ) = lim θ→∞ K(ψθ) = 1 3 = ω1. Further lim θ→∞ H(ψθ) = lim θ→∞ H( 1 3 − 1 θ ) = lim θ→∞ [3( 1 3 − 1 θ )2 − 3( 1 3 − 1 θ ) + 1] = 1 3 and lim θ→∞ J(ψθ) = lim θ→∞ J( 1 3 − 1 θ ) = lim θ→∞ [1 − 6( 1 3 − 1 θ )2] = 1 3 . Therefore lim θ→∞ H(ψθ) = lim θ→∞ J(ψθ) = 1 3 = ω1. Mallaiah K and Srinivas V Moreover lim θ→∞ MK(ψθ) = lim θ→∞ M[1 − ( 2 3 − 2 θ )] = lim θ→∞ M( 1 3 + 2 θ ) = 1 3 and lim θ→∞ KM(ψθ) = lim θ→∞ K( 1 3 + 1 5θ ) = lim θ→∞ 1 + 2(1 3 + 1 5θ 2 ) = 2 3 . ∴ lim θ→∞ Sm(MKψθ,MKψθ,KMψθ) = Sm( 1 3 , 1 3 , 2 3 ) ̸= 1 which implies that the pair ( M, K) is not compatible. Furthermore lim θ→∞ HJ(ψθ) = lim θ→∞ H( 1 3 + 4 θ − 1 θ2 ) = lim θ→∞ ( 2 + (1 3 + 4 θ − 1 θ2 7 ) = 1 3 and lim θ→∞ JH(ψθ) = lim θ→∞ J( 1 3 + 4 θ − 1 θ2 ) = lim θ→∞ [1 − ( 1 3 + 4 θ − 1 θ2 )] = 2 3 Therefore lim θ→∞ Sm(HJψθ,HJψθ,JHψθ) = Sm( 1 3 , 1 3 , 2 3 ) ̸= 1. Which shows that the pair( H,J) is not compatible . Also M(1 3 ) = 1 3 ,K(1 3 ) = 1 3 . This implies lim θ→∞ Sm(MKψθ,MKψθ,Mω1) = Sm( 1 3 , 1 3 , 1 3 ) = 1 and lim θ→∞ Sm(KMψθ,KMψθ,Kω1) = Sm( 1 3 , 1 3 , 1 3 ) = 1. This shows that the pair (M, K ) is reciprocally continuous in Sm metric space. Also lim θ→∞ Sm(MKψθ,MKψθ,Kω1) = Sm( 1 3 , 1 3 , 1 3 ) = 1. This shows that the pair (M, K ) is semi-compatible in Sm metric space. Hence the inequality (3.1.3)holds. Further Sm(H( 1 3 ),J( 1 3 ), 1 3 ) = 1 and Sm(HJ( 1 3 ),JH( 1 3 ), 1 3 ) = 1. Sm metric space This implies that Sm(HJ( 1 3 ),HJ(1 3 ),JH(1 3 )) = Sm( 1 3 , 1 3 , 1 3 ) = 1. Which indicates that the pair ( H, J) is weakly compatible. Now, we prove the condition (3.1.2 ) in various cases CASE-I Let ψ,ϕ ∈ [0, 1 2 ],while we have Sm(ψ,ϕ,σ) = e|ψ−σ|+|ϕ−σ|. Take ψ = 1 4 and ϕ = 1 5 then M(1 4 ) = 7 20 ,K(1 4 ) = 1 2 ,H(1 5 ) = 13 25 and J(1 5 ) = 19 25 substitute the above values in (3.1.2) Sm( 7 20 , 7 20 , 13 25 ) ≤{ max[Sm( 7 20 , 7 20 , 1 2 )Sm( 13 25 , 13 25 , 19 25 ),Sm( 7 20 , 7 20 , 19 25 )Sm( 13 25 , 13 25 , 1 2 ), Sm( 7 20 , 7 20 , 19 25 )Sm( 13 25 , 13 25 , 19 25 ),Sm( 7 20 , 7 20 , 1 2 )Sm( 13 25 , 13 25 , 1 2 )] }λ wehavee0.34 ≤ { max[e0.3e0.48,e0.82e0.34,e0.3e0.04,e0.82e0.48] }λ e0.34 ≤ { max[e0.78,e1.16,e0.0.34,e1.3]}λ =⇒ e0.34 ≤ e1.16λ which gives λ = 0.2 where λ ∈ (0, 1 3 ). CASE-II Let ψ,ϕ ∈ (1 2 ,1], then Sm(ψ,ϕ,σ) = e|ψ−σ|+|ϕ−σ|. Take ψ = 1 2 and ϕ = 1 2 then M(1 2 ) = 1 2 ,K(1 2 ) = 3 4 ,H(1 2 ) = 5 14 and J(1 2 ) = 1 2 substitute the above values in (3.1.2) Sm( 1 2 , 1 2 , 5 14 ) ≤{ max[Sm( 1 2 , 1 2 , 3 4 )Sm( 5 14 , 5 14 , 1 2 ),Sm( 1 2 , 1 2 , 1 2 )Sm( 5 14 ., 5 14 , 3 4 ), Sm( 1 2 , 1 2 , 1 2 )Sm( 5 14 , 5 14 , 1 2 ),Sm( 1 2 , 1 2 , 3 4 )Sm( 5 14 , 5 14 , 3 4 )] }λ which implies that e0.285 ≤ { max[e0.5e0.285,e0.0e0.786,e0.0e0.28,e0.5e0.786] }λ e0.285 ≤ { max[e0.785,e0.786,e0.28,e1.286]}λ =⇒ e0.285 ≤ e1.286λ Mallaiah K and Srinivas V which gives λ = 0.22 where λ ∈ (0, 1 2 ). CASE-III Let ψ,ϕ ∈ (1 2 ,1], then Sm(ψ,ϕ,σ) = e|ψ−σ|+|ϕ−σ| Take ψ = 1 4 and ϕ = 1 2 then M(1 4 ) = 7 20 ,K(1 4 ) = 1 2 ,H(1 5 ) = 5 14 and J(1 5 ) = 1 2 substitute the above values in (3.1.2) Sm( 7 20 ,0 7 20 , 5 14 ) ≤{ max[Sm( 7 20 , 7 20 , 1 2 )Sm( 5 14 , 5 14 , 1 2 ),Sm( 7 20 , 7 20 , 1 2 )Sm( 5 14 , 5 14 , 1 2 ), Sm( 7 20 , 7 20 , 1 2 )Sm( 5 14 , 5 14 , 1 2 ),Sm( 7 20 , 7 20 , 1 2 )Sm( 5 14 , 5 14 , 1 2 )] }λ which implies that e0.014 ≤ { max[e0.3e0.28,e0.3e0.28,e0.3e0.28,e0.3e0.28] }λ e0.014 ≤ { max[e0.58,e0.58,e0.58,e0.58]}λ =⇒ e0.014 ≤ e0.5.8λ this gives that λ = 0.14 where λ ∈ (0, 1 2 ). Hence the inequality (3.3.2) holds. It can be seen that “1 2 ” is a unique common fixed point for four self mappings M, K H, and J. 4 Conclusions In this article, we established a common fixed point theorem in Sm-metric space by using weakly-compatible mappings, semi-compatible mappings, and re- ciprocally continuous mappings for four self-maps. Furthermore, our results are also justified with suitable examples. References [1] Agamirza E Bashirov, Emine Mısırlı Kurpınar, and Ali Özyapıcı. Multi- plicative calculus and its applications. Journal of mathematical analysis and applications, 337(1):36–48, 2008. Sm metric space [2] Afrah AN Abdou. Common fixed point results for compatible-type map- pings in multiplicative metric spaces. J. Nonlinear Sci. Appl, 9:2244–2257, 2016. [3] Nihal Yılmaz Özgür and Nihal Taş. Some generalizations of fixed-point theorems on s-metric spaces. In Essays in Mathematics and its Applications, pages 229–261. Springer, 2016. [4] Ravi P Agarwal, Erdal Karapınar, and Bessem Samet. An essential remark on fixed point results on multiplicative metric spaces. Fixed point theory and applications, 2016(1):1–3, 2016. [5] Shaban Sedghi, N Shobkolaei, M Shahraki, and T Došenović. Com- mon fixed point of four maps in s-metric spaces. Mathematical Sciences, 12(2):137–143, 2018. [6] Somkiat Chaipornjareansri. Fixed point theorems for generalized weakly contractive mappings in s-metric space. Thai Journal of Mathematics, pages 50–62, 2018. [7] Nguyen Van Dung. On coupled common fixed points for mixed weakly monotone maps in partially ordered s-metric spaces. Fixed Point Theory and Applications, 2013(1):1–17, 2013. [8] RP Pant. Common fixed points of four mappings. Bull. Cal. Math. Soc., 90:281–286, 1998. [9] Mukesh Kumar Jain and Mohammad Saeed Khan. Generalization of semi compatibility with some fixed point theorems under strict contractive condi- tion. Applied Mathematics E-Notes, 17:25–35, 2017. [10] K Mallaiah and V Srinivas. Common fixed point of four maps in sm-metric space. International Journal of Analysis and Applications, 19(6):915–928, 2021. [11] MA Al-Thagafi and Naseer Shahzad. Generalized i-nonexpansive selfmaps and invariant approximations. Acta Mathematica Sinica, English Series, 24(5):867–876, 2008.