Ratio Mathematica Volume 43, 2022 A Common Fixed Point Theorem For Three Weakly Compatible Selfmaps Of A S-metric Space Kiran Virivinti* Niranjan Goud Javaji† Rajani Devi Katta‡ Abstract Fixed point theorems were established by using contractive condi- tions. In this paper we prove a common fixed point theorem for three weakly compatible selfmaps of a S-metric space by utilizing a contractive condition of rational type.Further we deduce a common fixed point theorem for two weakly compatible selfmaps of a S-metric space. Keywords: S-metric space; Fixed point; Weakly compatible map- pings; Associated sequence of a point relative to three selfmaps. 2020 AMS subject classifications: 54H25,47H10. 1 *Department of Mathematics, Osmania University, Hyderabad, India; kiran- mathou@gmail.com. †Department of Mathematics, M.V.S Government College, Mahaboobnagar, Telangana, India; jngoud1979@gmail.com. ‡Department of Mathematics, K.V.R(W) Government Degree College, Kurnool, Andhra pradesh, India; dr.rajanidevi@gmail.com. 1Received on April 21, 2022. Accepted on September 1, 2022. Published on October 1, 2022. doi: 10.23755/rm.v42i0.767. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. V. Kiran, J. Niranjan Goud, K. Rajani Devi 1 Introduction Fixed point theory is an important branch of non-linear analysis due to its application potential. In proving fixed point theorems, we use completeness, con- tinuity, convergence and various other topological aspects. Banach’s Contraction Principle or Banach’s fixed point theorem is one of the most important results in nonlinear analysis. This theorem has been generalized in many directions by gen- eralizing the underlying space or by viewing it as a common fixed point theorem along with other selfmaps. In the past few years, a number of generalizations of metric spaces like G -metric spaces, partial metric spaces and cone metric spaces were initiated. These general- izations were used to extend the scope of the study of fixed point theory. Recently one more generalization, namely S -metric spaces, was introduced by Sedghi S , Shobe N, Aliouche.A [2012]. Among all generalizations, S-metric spaces evinced a lot of interest in many researchers as they unified, extended, generalized and re- fined several existing results onto these S -metric spaces. Commutativity plays an important role in proving common fixed point theorems. As it is a stronger requirement, Sessa [1982] introduced the notion of weakly commuting maps as a generalization of commuting maps. Afterwards the idea of compatibility was introduced by G. Jungck [1986]. Later on Jungck and Rhoades [1998]introduced the notion of weakly compatibility as a generalization of com- patibility. They also proved that compatible mappings are weakly compatible but not conversely. In this paper we establish a common fixed point theorem for three weakly com- patible selfmaps of a S-metric space using a contractive condition of rational type. Our theorem which is established in the framework of S-metric spaces generalizes the theorem of Sumit Chandok [2018] which is proved in metric space. Now we recall some basic definitions required in the sequel in section 2 and es- tablish main results in section 3. 2 Preliminaries We now recollect the essential definitions which are useful for our discussion. Definition 2.1. Let Y be a nonempty set. A function S : Y 3 → [0,∞) is said to be S − metric if it satisfies the following conditions for each β1,β2,β3,β4 ∈ Y (i) S(β1,β2,β3) ≥ 0, (ii) S(β1,β2,β3) = 0 if and only if β1 = β2 = β3, (iii) S(β1,β2,β3) ≤ S(β1,β1,β4) + S(β2,β2,β4) + S(β3,β3,β4). Then (Y,S) is said to be a S-metric space. A Common Fixed Point Theorem For Three Weakly Compatible Selfmaps Of A S-metric Space Example 2.1. Let Y = R and S : R3 → [0,∞) be defined by S(β1,β2,β3) = |β2 + β3 − 2β1| + |β2 − β3| for β1,β2,β3 ∈ R, then (Y,S) is a S-metric space. Remark 2.1. It is shown in a S-metric space that S(β1,β1,β2) = S(β2,β2,β1) for all β1,β2 ∈ Y . Definition 2.2. Let (Y,S) be an S-metric space. A sequence {tn} in Y said to convergent, if there is a t ∈ Y such that S(tn, tn, t) → 0; that is for each ϵ > 0, there exists an n0 ∈ N such that for all n ≥ n0, we have S(tn, tn, t) < ϵ and we denote this by lim n→∞ tn = t. Definition 2.3. Suppose (Y,S) is an S-metric space. A sequence {tn} in Y is called a Cauchy sequence if to each ϵ > 0, there exists n0 ∈ N such that S(tn, tn, t) < ϵ for each n,m ≥ n0 . Definition 2.4. Let (Y,S) be an S-metric space, If there exists sequences {tn} and {un} such that lim n→∞ tn = t and lim n→∞ un = u then lim n→∞ S(tn, tn,un) = S(t, t,u), then we say that S(t,u,v) is continuous in t and u . Definition 2.5. Suppose ϕ and ψ self maps of a S-metric space (Y,S) such that for every sequence {tn} in Y with lim n→∞ ψtn = lim n→∞ ϕtn = t for some t ∈ X we have lim n→∞ S(ψϕtn,ψϕtn,ϕψtn) = 0, then ψ and ϕ are called compatible mappings. Definition 2.6. In a S-metric space (Y,S),two selfmaps ϕ and ψ of Y are said to be weakly compatible if ϕψt = ψϕt whenever ϕt = ψt for t ∈ Y . Definition 2.7. If ψ, µ and ϕ are self maps of a non empty set Y such that ψ(Y ) ⊆ ϕ(Y ), and µ(Y ) ⊆ ϕ(Y ) then for any t0 ∈ Y , if {tn} is a sequence in Y such that ϕt2n+1 = ψt2n and ϕt2n+2 = µt2n+1 for n ≥ 1 then {tn} is called an associated sequence of t0 relative to three selfmaps ψ, µ and ϕ. 3 Main Theorem We now state our main theorem of the section. Theorem 3.1. Let P be a subset of a S-metric space (Y,S), ψ,µ and ϕ are three selfmaps of P such that (i) ψ(Y ) ∪ µ(Y ) ⊆ ϕ(Y ) and (ϕ(P),S) is complete. V. Kiran, J. Niranjan Goud, K. Rajani Devi (ii) S(µy1,µy1,ψy2) ≤ k1{ S(ϕy1,ϕy1,µy1).S(ϕy2,ϕy2,ψy2) S(ϕy1,ϕy1,ϕy2) + S(ϕy1,ϕy1,ψy2) + S(ϕy2,ψy2,µy1) } + k2S(ϕy1,ϕy1,ϕy2) for every y1,y2 ∈ P and k1,k2 ∈ [0,1) with 2k1 + k2 < 1. (iii) The pairs (ϕ,ψ) and (ϕ,µ) are weakly compatible. Then ψ,µ and ϕ have a unique common fixed point. Proof. Let t0 be a point in Y. Since ψ(Y )∪µ(Y ) ⊆ ϕ(Y ), we obtain an associated sequence {tn} in Y such that ϕt2n+1 = µt2n,ϕt2n+2 = ψt2n+1. From the condition (ii) of Theorem 3.1 we have, S(ϕt2n+1,ϕt2n+1,ϕt2n+2) = S(µt2n,µt2n,ψt2n+1) ≤ k1 [ S(ϕt2n,ϕt2n,µt2n).S(ϕt2n+1,ϕt2n+1,ψt2n+1) S(ϕt2n,ϕt2n,ϕt2n+1) + S(ϕt2n,ϕt2n,ψt2n+1) + S(ϕt2n+1,ϕt2n+1,µt2n) ] + k2S(ϕt2n,ϕt2n,ϕt2n+1) ≤ k1 [ S(ϕt2n,ϕt2n,ϕt2n+1).S(ϕt2n+1,ϕt2n+1,ϕt2n+2) S(ϕt2n,ϕt2n,ϕt2n+1) + S(ϕt2n,ϕt2n,ϕt2n+2) + S(ϕt2n+1,ϕt2n+1,ϕt2n+1) ] + k2S(ϕt2n,ϕt2n,ϕt2n+1) ≤ 2k1 [ S(ϕt2n,ϕt2n,ϕt2n+1).S(ϕt2n+1,ϕt2n+1,ϕt2n+2) S(ϕt2n+1,ϕt2n+1,ϕt2n+1) ] + k2S(ϕt2n,ϕt2n,ϕt2n+1) ≤ (2k1 + k2) S(ϕt2n,ϕt2n,ϕt2n+1). Similarly, we can prove S(ϕt2n,ϕt2n,ϕt2n+1) ≤ (2k1 + k2)S(ϕt2n−1,ϕt2n−1,ϕt2n) Therefore S(ϕtn,ϕtn,ϕtn+1) ≤ (2k1 + k2) S(ϕtn−1,ϕtn−1,ϕtn) ≤ (2k1 + k2)2 S(ϕtn−2,ϕtn−2,ϕtn−1) ≤ (2k1 + k2)3 S(ϕtn−3,ϕtn−3,ϕtn−2) · · · · · · · · · · · · · · · · · · ≤ (2k1 + k2)n S(ϕt0,ϕt0,ϕt1) → 0, A Common Fixed Point Theorem For Three Weakly Compatible Selfmaps Of A S-metric Space since (2k1 + k2)n → 0 as n → ∞. Now we claim that {ϕtn} is a Cauchy sequence. For any m,n ∈ N such that m > n we have, S(ϕtn,ϕtn,ϕtm) ≤ 2[S(ϕtn,ϕtn,ϕtn+1) + S(ϕtn+1,ϕtn+1,ϕtn+2) + · · · + S(ϕtm−1,ϕtm−1,ϕtm)] ≤ 2cnS(ϕt0,ϕt0,ϕt1) + cn+1S(ϕt0,ϕt0,ϕt1) + · · · + cmS(ϕt0,ϕt0,ϕt1) ≤ 2cn(1 + c + c2 + · · · + cm−n)S(ϕt0,ϕt0,ϕt1) ≤ 2cn 1 − cm−n 1 − c S(ϕt0,ϕt0,ϕt1) ≤ 2 cn 1 − c S(ϕt0,ϕt0,ϕt1) → 0, since c < 1 then cn → 0 as n → ∞. Therefore {ϕtn} is a Cauchy sequence in X. Since (ϕ(P),S) is complete, there is a t ∈ P such that ϕtn → ϕt as n → ∞. We now prove that t is a point of coincidence of µ,ψ and ϕ. From the condition (ii) of Theorem 3.1 we have, S(ϕt2n+1,ϕt2n+1,ψt) = S(µt2n,µt2n,ψt) ≤ k1 [ S(ϕt2n,ϕt2n,µt2n).S(ϕt,ϕt,ψt) S(ϕt2n,ϕt2n,ϕt) + S(ϕt2n,ϕt2n,ψt) + S(ϕt,ϕt,µt2n) ] + k2S(ϕt2n,ϕt2n,ϕt) ≤ k1 [ S(ϕt2n,ϕt2n,ϕt2n+1).S(ϕt,ϕt,ψt) S(ϕt2n,ϕt2n,ϕt) + S(ϕt2n,ϕt2n,ψt) + S(ϕt,ϕt,ϕt2n+1) ] + k2S(ϕt2n,ϕt2n,ϕt), which gives S(ϕt,ϕt,ψt) = 0 as n → ∞ and hence ϕt = ψt. Also we have, S(µt,µt,ϕt) = S(µt,µt,ψt) ≤ k1 [ S(ϕt,ϕt,µt).S(ϕt,ϕt,ψt) S(ϕt,ϕt,ϕt) + S(ϕt,ϕt,ψt) + S(ϕt,ϕt,µt) ] + k2S(ϕt,ϕt,ϕt), which implies S(µt,µt,ϕt) = 0 proving µt = ϕt. Therefore we have ϕt = ψt = µt = a(say), V. Kiran, J. Niranjan Goud, K. Rajani Devi proving that t is a coincident point of µ,ψ and ϕ. Since the pairs (ϕ,µ) and (ϕ,ψ) are weakly compatible, we have ϕµt = µϕt and ϕψt = ψϕt which implies ϕa = ψa = µa. Now we have, S(ϕa,ϕa,a) = S(µa,µa,ψt) ≤ k1 [ S(ϕa,ϕa,µa).S(ϕt,ϕt,ψt) S(ϕa,ϕa,ϕt) + S(ϕa,ϕa,µt) + S(ϕt,ϕt,µa) ] + k2S(ϕa,ϕa,ϕt) ≤ k1 [ S(ϕa,ϕa,ϕa).S(a,a,a) S(ϕa,ϕa,a) + S(ϕa,ϕa,a) + S(a,a,ϕa) ] + k2S(ϕa,ϕa,a) ≤ k2 S(ϕa,ϕa,a), leading to a contradiction, giving that S(ϕa,ϕa,a) = 0 implies ϕa = a. Hence ϕa = ψa = µa = a, showing that a is a common fixed point of µ,ψ and ϕ. We now prove that the common fixed point is unique. Suppose a′(̸= a) is another common fixed point of µ,ψ and ϕ. That is a′ = ϕa′ = ψa′ = µa′. We have S(a,a,a′) = S(µa,µa,ψa′) ≤ k1 [ S(ϕa,ϕa,µa).S(ϕa′,ϕa′,ψa′) S(ϕa,ϕa,ϕa′) + S(ϕa,ϕa,ψa′) + S(ϕa′,ϕa′,µa) ] + k2S(ϕa,ϕa,ϕa ′) ≤ k1 [ S(a,a,a).S(a′,a′,a′) S(a,a,a′) + S(a,a,a′) + S(a,a,a′) ] + k2S(a,a,a ′) ≤ k2S(a,a,a′), which is a contradiction since k2 < 1. Therefore S(a,a,a′) = 0 implies a = a′, Proving the uniqueness. Corolary 3.1. Let P be a subset of a S-metric space (Y,S). Suppose that ϕ,µ are two selfmaps of P satisfy (i) µ(Y ) ⊆ ϕ(Y ) and (ϕ(P),S) is complete. (ii) S(µy1,µy1,µy2) ≤ k1{ S(ϕy1,ϕy1,µy1).S(ϕy2,ϕy2,µy2) S(ϕy1,πy1,ϕy2) + S(ϕy1,ϕy1,µy2) + S(ϕy2,ϕy2,µy1) } + k2S(ϕy1,ϕy1,ϕy2) A Common Fixed Point Theorem For Three Weakly Compatible Selfmaps Of A S-metric Space for every y1,y2 ∈ P and k1,k2 ∈ [0,1) with 2k1 + k2 < 1. (iii) The pair (µ,ϕ) is weakly compatible. Then µ and ϕ have a unique common fixed point. Proof. On taking ψ = µ in Theorem 3.1, the corollary follows. 4 Conclusion In this paper, a common fixed theorem for three weakly compatible selfmaps of a S-metric space is established with the aid of an associated sequence of three selfmaps.Moreover, we deduce a common fixed point theorem for two selfmaps. As S-metric space is a robust generalization of metric space, our theorem gener- alizes the theorem in literature. References N. Aliouche.A, Sedghi. A generalization of fixed point theorems in s-metric spaces. Mat. Vesnik, 3:258–266, 2012. S. Chandok. Common fixed theorem for generalized contractions mappings. Thai Journal of Mathematics, 16:305–314, 2018. Jungck and Rhoades. Fixed point for set valued functions with out continuity. Indian j pure Appl.Math.Sci, 29:227–238, 1998. G. Jungck. Compatible mappings and common fixed points. Int j of Math and Math.Sci, 4:771–779, 1986. Sessa. On a weak commutativity condition of mappings in a fixed point consider- ations,. 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