Int. J. Anal. Appl. (2023), 21:8 Some Invariant Point Results Using Simulation Function Venkatesh∗, Naga Raju Department of Mathematics, Osmania University, Hyderabad, Telangana-500007, India ∗Corresponding author: venkat409151@gmail.com Abstract. Through this article, we establish an invariant point theorem by defining generalized Zs- contractions in relation to the simulation function in S-metric space. In this article, we generalized the results of Nihal Tas, Nihal Yilmaz Ozgur and N.Mlaiki. In addition to that, we bestow an example which supports our results. 1. Introduction Fixed point is also known as an invariant point. Banach principle of contraction [2] on metric space plays very important role in the field of invariant point theory and non linear analysis. In 1922, Stefan Banach initiated the concept of contraction and established well known Banach contraction theorem. In the year 2006, B Sims and Mustafa [9], established theory on G-metric spaces, that is an extension of metric spaces and established some properties. Later, A.Aliouche, S.Sedghi and N.Shobe [13] initiated S-metric spaces, it is a generalization of G-metric spaces in the year 2012. In 2014, S.Radojevic, N.V.Dung and N.T.Hieu [4] proved by examples that S-metric space is not a generalization of G- metric space and vice versa. Invariant points of various contractive maps on S-metric spaces were studied in [ [1], [3], [6]- [8], [11]]. In 2015, F.Khajasteh, Satish Shukla and S.Radenovic [5] introduced simulation function and the concept of Z-contration in relation to simulation function and proved an invariant point theorem which generalizes the Banach Contraction principle. Very recently, Murat Olgun, O.Bicer and T.Alyildiz [10] defined generalized Z-contraction in relation to the simulation function and proved an invariant point theorem. Received: Dec. 3, 2022. 2020 Mathematics Subject Classification. 54H25, 47H09, 47H10. Key words and phrases. Simulation function; Z-contraction; Fixed point; S-metric space. https://doi.org/10.28924/2291-8639-21-2023-8 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-8 2 Int. J. Anal. Appl. (2023), 21:8 In the year 2019, Nihal Tas, Nihal Ylimaz Ozgur and Nabil Mlaiki [8] proved an invariant point theorem by employing the collection of simulation mappings on S-metric spaces. In this article, we generalized the results of Nihal Tas , Nihal Yilmaz Ozgur and N.Mlaiki. 2. Preliminaries Definition 2.1. [13] Let X 6= ∅, then a mapping S:X3 → [0,∞) is said to be an S-metric on X if: (S1) S(ξ,ϑ,w) > 0 for all ξ,ϑ,w ∈ X with ξ 6= ϑ 6= w. (S2) S(ξ,ϑ,w) = 0 if ξ = ϑ = w. (S3) S(ξ,ϑ,w)≤ [S(ξ,ξ,a)+S(ϑ,ϑ,a)+S(w,w,a)] ∀ξ,ϑ,w,a ∈ X. Then we call (X,S) is an S-metric space. Example 2.1. [13] Define S:X3 → [0,∞) by S(ξ,ϑ,w) = d(ξ,ϑ) + d(ξ,w) + d(ϑ,w) for any ξ,ϑ,w ∈ X, where (X,d) be a metric space. Then (X,S) is an S-metric space. Example 2.2. [4] Suppose X=R, Collection of all real numbers and let S(ξ,ϑ,w)= |ϑ+w −2ξ|+ |ϑ−w| for all ξ,ϑ,w ∈ X. Then (X,S) is an S-metric space. Example 2.3. [12] Suppose X=R, Collection of all real numbers and let S(ξ,ϑ,w)= |ξ−w|+|ϑ−w| for all ξ,ϑ,w ∈ X. Then (X,S) is an S-metric space. Example 2.4. Suppose X=[0,1] and S:X3 → [0,∞) be defined by S(ξ,ϑ,w)=  0 if ξ = ϑ = w max{ξ,ϑ,w} otherwise . Then (X,S) is an S-metric space. Lemma 2.1. [13] In the S-metric space, we observe S(ξ,ξ,ϑ)= S(ϑ,ϑ,ξ). Lemma 2.2. [4] In the S-metric space, we observe (i) S(ξ,ξ,ϑ)≤ 2S(ξ,ξ,w)+S(ϑ,ϑ,w) and (ii) S(ξ,ξ,ϑ)≤ 2S(ξ,ξ,w)+S(w,w,ϑ) Definition 2.2. [13] Let (X,S) be a S-metric space. We have: (i) If S(ξn,ξn,ξ)→ 0 as n →∞. ,then we say sequence {ξn}∈ X converges to ξ ∈ X. i.e., for every � > 0, it can be found a natural number n0 so that to each n≥ n0, S(ξn,ξn,ξ) < � and we indicate it by limn→∞ξn = ξ. (ii) a sequence {ξn} ∈ X is known as Cauchy sequence if to each � > 0, it can be found n0 ∈ N so that S(ξn,ξn,ξm) < � for every n,m≥ n0. (iii)If each Cauchy sequence of X is convergent, then say X is complete. Definition 2.3. [13] A self map h is defined on S-metric space (X,S) is known as an S-contraction if we get a constant 0≤ τ < 1 so that S(h(ξ),h(ξ),h(ϑ))≤ τS(ξ,ξ,ϑ) for all ξ,ϑ ∈ X. Int. J. Anal. Appl. (2023), 21:8 3 Definition 2.4. [5] We say that a mapping γ : [0,∞)× [0,∞)→R is a simulation mapping if: (γ1) γ(0,0) = 0 (γ2) γ(p,q) < q −p for p,q > 0 (γ3) If {pn},{qn} are sequences of (0,∞) so that limn→∞pn = limn→∞qn > 0, then limn→∞sup γ(pn,qn) < 0. We indicate Z as the collection of all simulation mappings. For example, γ(p,q) = τq − p for 0≤ τ <1 belongning to Z. Definition 2.5. [5] Let h be a self map on a metric space (X,d) and γ ∈Z. Then h is known as a Z-contraction in relation to γ if: γ(d(hξ,hϑ),d(ξ,ϑ))≥ 0 for all ξ,ϑ ∈ X. By considering the Definition 2.5. It is concluded that each Banach contraction becomes Z- contraction in relation to γ(p,q) = τq − p with 0 ≤ τ < 1. Further, it can be established from the definition of the simulation mapping that γ(p,q) < 0 for each p ≥ q > 0. Hence, assume that h is a Z-contraction in relation to γ ∈ Z then d(hξ,hϑ) < d(ξ,ϑ) for all distinct ξ,ϑ ∈ X. Theorem 2.1. [5] In complete metric space (X,d), each Z-contraction has a unique invariant point and furthermore the invariant point is the limit of every Picard’s sequence. 3. Main Results Definition 3.1. [13] Let h be a self map on an S-metric space X and γ ∈Z. We say that h is a contraction if we find a constant 0≤ L < 1 such that S(hξ,hξ,hϑ)≤ LS(ξ,ξ,ϑ) for all ξ,ϑ ∈ X. Nihal Tas, N.Y.Ozgur and Nabil Mlaiki [8] defined the Zs-contraction as follows. Definition 3.2. [8] Let h be a self map on an S-metric space (X, S) and γ ∈Z. Then h is said to be a Zs-contraction in relation to γ if γ(S(hξ,hξ,hϑ),S(ξ,ξ,ϑ))≥ 0 for all ξ,ϑ ∈ X Nihal Tas, N.Y.Ozgur and Nabil Mlaiki [8] proved the following theorem. Theorem 3.1. [8] Let h be a self map on an S-metric space (X, S). Then h has a unique invariant point a∈ X and the invariant point is the limit of the Picard sequence {ξn}, whenever h is a Zs-contraction in relation to γ. 4 Int. J. Anal. Appl. (2023), 21:8 Definition 3.3. Let h be a self map on an S-metric space (X, S) and γ ∈ Z. Then h is said to be generalized Zs-contraction in relation to γ if γ(S(hξ,hξ,hϑ),M(ξ,ξ,ϑ))≥ 0 f or all ξ,ϑ ∈ X (3.1) where M(ξ,ξ,ϑ) = max{S(ξ,ξ,ϑ),S(ξ,ξ,hξ),S(ϑ,ϑ,hϑ), 1 2 [S(ξ,ξ,hϑ)+S(ϑ,ϑ,hξ)]} Example 3.1. Let h be a contraction on (X,S). If we take L∈[0,1) and γ(p,q) = Lq-p for all 0 ≤ p,q < ∞, then a contraction h is a Zs-contraction in relation to γ. In fact, consider p = S(hξ,hξ,hϑ) and q = M(ξ,ξ,ϑ). Since h is a contraction, we obtain : S(hξ,hξ,hϑ)≤ LS(ξ,ξ,ϑ)≤ LM(ξ,ξ,ϑ) =⇒ LM(ξ,ξ,ϑ)−S(hξ,hξ,hϑ)≥ 0 =⇒ γ(S(hξ,hξ,hϑ),M(ξ,ξ,ϑ))≥ 0. for all ξ,ϑ ∈ X. Therefore, h is a generalized Zs-contraction in relation to γ. Example 3.2. Consider a complete S-metric space (X,S), where X = [0,1] and S : X3 → [0,∞) by S(ξ,ϑ,w)= |ξ−w|+ |ϑ−w|. Define h:X → X by hξ =   2 5 , for ξ ∈ [0, 2 3 ) 1 5 , for ξ ∈ [2 3 ,1) Now we prove that h be a generalized Zs-contraction in relation to γ, where γ is defined by γ(p,q)= 6 7 q −p. Now we get S(hξ,hξ,hϑ)≤ 3 7 [S(ξ,ξ,hξ)+S(ϑ,ϑ,hϑ)] ≤ 6 7 max{S(ξ,ξ,hξ),S(ϑ,ϑ,hϑ)} ≤ 6 7 M(ξ,ξ,ϑ) for all ξ,ϑ ∈ X. That is, we have γ(S(hξ,hξ,hϑ),M(ξ,ξ,ϑ))= 6 7 M(ξ,ξ,ϑ)−d(hξ,hξ,hϑ)≥ 0. for all ξ,ϑ ∈ X. Definition 3.4. Let (X,S) be an S-metric space. Then we say that a mapping h:X → X is asymp- totically regular at ξ ∈ X if limn→∞S(hnξ,hnξ,hn+1ξ)=0 By the following lemma, we can conclude that a generalized Zs-contraction is asymptotically regular at each point of X. Lemma 3.1. If h : X → X is a generalized Zs-contraction in relation to γ, then h is an asymptotically regular at each point ξ ∈ X. Int. J. Anal. Appl. (2023), 21:8 5 Proof. Let ξ ∈ X. If for some m∈N, we have hmξ = hm−1ξ, that is, hϑ = ϑ, where ϑ = hm−1ξ, then hnϑ = hn−1hϑ = hn−1ϑ = ... = hϑ = ϑ for each n∈N. Therefore, we have: S(hnξ,hnξ,hn+1ξ)= S(hn−m+1hm−1ξ,hn−m+1hm−1ξ,hn−m+2hm−1ξ) = S(hn−m+1ϑ,hn−m+1ϑ,hn−m+2ϑ) = S(ϑ,ϑ,ϑ) =0 Hence lim n→∞ S(hnξ,hnξ,hn+1ξ)=0 Now, we assume that hnξ 6= hn+1ξ, for each n∈N. From the condition(γ2) and the generalized Zs-contraction property, we get: 0≤ γ(S(hn+1ξ,hn+1ξ,hnξ),M(hnξ,hnξ,hn−1ξ)) (3.2) Where M(hnξ,hnξ,hn−1ξ)= max{S(hnξ,hnξ,hn−1ξ),S(hnξ,hnξ,hhnξ),S(hn−1ξ,hn−1ξ,hhn−1ξ), 1 2 [S(hnξ,hnξ,hhn−1ξ)+S(hn−1ξ,hn−1ξ,hhnξ)]} = max{S(hnξ,hnξ,hn−1ξ),S(hnξ,hnξ,hn+1ξ),S(hn−1ξ,hn−1ξ,hnξ), 1 2 [S(hnξ,hnξ,hnξ)+S(hn−1ξ,hn−1ξ,hn+1ξ)} = max{S(hnξ,hnξ,hn−1ξ),S(hn+1ξ,hn+1ξ,hnξ)} If S(hn+1ξ,hn+1ξ,hnξ) > S(hnξ,hnξ,hn−1ξ) then, we get M(hnξ,hnξ,hn−1ξ) = S(hn+1ξ,hn+1ξ,hnξ) From equation (3.2) we have, 0≤ γ(S(hn+1ξ,hn+1ξ,hnξ),S(hn+1ξ,hn+1ξ,hnξ)) < S(hn+1ξ,hn+1ξ,hnξ)−S(hn+1ξ,hn+1ξ,hnξ)=0 which is a contradiction. Hence M(hnξ,hnξ,hn−1ξ) = S(hnξ,hnξ,hn−1ξ). Using generalized Zs-contractive property, we get 0≤ γ(S(hn+1ξ,hn+1ξ,hnξ),M(hnξ,hnξ,hn−1ξ)) = γ(S(hn+1ξ,hn+1ξ,hnξ),S(hnξ,hnξ,hn−1ξ)) < S(hnξ,hnξ,hn−1ξ)−S(hn+1ξ,hn+1ξ,hnξ) 6 Int. J. Anal. Appl. (2023), 21:8 i.e., S(hn+1ξ,hn+1ξ,hnξ) < S(hnξ,hnξ,hn−1ξ) for all n∈N. Then {S(hnξ,hnξ,hn−1ξ)} is a nonnegative reals of decreasing sequence and so it should be conver- gent. Suppose limn→∞S(hnξ,hnξ,hn+1ξ) = η ≥ 0. If η > 0, then from the condition (γ3) and the generalized Zs-contraction property, we get 0≤ lim n→∞ sup γ(S(hn+1ξ,hn+1ξ,hnξ),M(hnξ,hnξ,hn−1ξ) = lim n→∞ sup γ(S(hn+1ξ,hn+1ξ,hnξ),S(hnξ,hnξ,hn−1ξ) < 0 which is a contradiction. It should be η = 0. Therefore limn→∞S(hnξ,hnξ,hn+1ξ)=0. Hence, h is asymptotically regular at each point ξ ∈ X. � Lemma 3.2. The Picard sequence {ξn} so that hξn−1 = ξn, to each n∈N the initial point ξ0 ∈ X is a bounded sequence, whenever h is a generalized Zs-contraction in relation to γ. Proof. Consider {ξn} be the Picard sequence in X with initial value ξ0. Now we claim that {ξn} is a bounded sequence. Assume that {ξn} is unbounded. Let ξn+m 6= ξn, for each m,n∈N. Since {ξn} is unbounded, we can find a subsequence {ξnk} of {ξn} so that n1 =1 and to each k∈N, nk+1 is the smallest integer so that S(ξnk+1,ξnk+1,ξnk) > 1 and S(ξm,ξm,ξnk)≤ 1 for nk ≤ m ≤ nk+1 −1 Hence, from the lemma (2.2), we obtain 1 < S(ξnk+1,ξnk+1,ξnk) ≤ 2S(ξnk+1,ξnk+1,ξnk+1−1)+S(ξnk,ξnk,ξnk+1−1) ≤ 2S(ξnk+1,ξnk+1,ξnk+1−1)+1 Letting k→∞ and using lemma (3.1), we have lim n→∞ S(ξnk+1,ξnk+1,ξnk)=1 1 < S(ξnk+1,ξnk+1,ξnk)≤ M(ξnk+1−1,ξnk+1−1,ξnk−1) = max{S(ξnk+1−1,ξnk+1−1,ξnk−1),S(ξnk+1−1,ξnk+1−1,ξnk+1),S(ξnk−1,ξnk−1,ξnk), 1 2 [S(ξnk+1−1,ξnk+1−1,ξnk)+S(ξnk−1,ξnk−1,ξnk+1)]} = max{S(ξnk−1,ξnk−1,ξnk+1−1),S(ξnk+1−1,ξnk+1−1,ξnk+1),S(ξnk−1,ξnk−1,ξnk), 1 2 [S(ξnk+1−1,ξnk+1−1,ξnk)+S(ξnk−1,ξnk−1,ξnk+1)]} Int. J. Anal. Appl. (2023), 21:8 7 ≤ max{2S(ξnk−1,ξnk−1,ξnk)+S(ξnk+1−1,ξnk+1−1,ξnk),S(ξnk+1−1,ξnk+1−1,ξnk+1), S(ξnk−1,ξnk−1,ξnk), 1 2 [S(ξnk+1−1,ξnk+1−1,ξnk)+S(ξnk−1,ξnk−1,ξnk+1)]} ≤ max{2S(ξnk−1,ξnk−1,ξnk)+1,S(ξnk+1−1,ξnk+1−1,ξnk+1), S(ξnk−1,ξnk−1,ξnk), 1 2 [1+2S(ξnk−1,ξnk−1,ξnk)+S(ξnk,ξnk,ξnk+1)]} Letting n→ ∞, we get 1≤ lim k→∞ M(ξnk+1−1,ξnk+1−1,ξnk−1)≤ 1. That is limk→∞M(ξnk+1−1,ξnk+1−1,ξnk−1)=1 From the condition(γ3) and the generalized Zs-contraction property, we obtain 0≤ lim k→∞ sup γ(S(ξnk+1,ξnk+1,ξnk),M(ξnk+1−1,ξnk+1−1,ξnk−1)) = lim k→∞ sup γ(S(ξnk+1,ξnk+1,ξnk),S(ξnk+1−1,ξnk+1−1,ξnk−1)) < 0 which is a contradiction. Hence our assumption is wrong. Therefore {ξn} is bounded. � Theorem 3.2. Let h be a self map defined on complete S-metric space (X, S). Then h has a unique invariant point a ∈ X and Picard sequence {ξn} converges to the invariant element a, whenever h is a generalized Zs-contraction in relation to γ. Proof. Let the Picard sequence {ξn} be defined as hξn−1 = ξn, ∀n ∈N and ξ0 ∈ X. Now, we claim that {ξn} be a cauchy sequence. To get this, Consider Tn = sup{S(ξi,ξi,ξj) : i, j ≥ n}. Clearly {Tn} be a nonnegative reals of decreasing sequence. Hence, we can find τ ≥ 0 so that limn→∞Tn = τ. Now we prove that τ = 0. If possible suppose that τ > 0. From the definition of Tn, for each k∈N, we can find mk,nk so that k ≤ nk < mk and Tk − 1 k < S(ξmk,ξmk,ξnk)≤ Tk Therefore, we get limn→∞S(ξmk,ξmk,ξnk)= τ. From the lemma (2.2), lemma (3.1) and generalized Zs-contraction property, we get S(ξmk,ξmk,ξnk)≤ S(ξmk−1,ξmk−1,ξnk−1) ≤ 2S(ξmk−1,ξmk−1,ξmk)+S(ξnk−1,ξnk−1,ξmk) ≤ 2S(ξmk−1,ξmk−1,ξmk)+2S(ξnk−1,ξnk−1,ξnk)+S(ξmk,ξmk,ξnk) Letting as k→ ∞, we have lim k→∞ S(ξmk−1,ξmk−1,ξnk−1)= τ 8 Int. J. Anal. Appl. (2023), 21:8 S(ξmk−1,ξmk−1,ξnk−1)≤ M(ξmk−1,ξmk−1,ξnk−1) = max{S(ξmk−1,ξmk−1,ξnk−1),S(ξmk−1,ξmk−1,hξmk−1),S(ξnk−1,ξnk−1,hξnk−1), 1 2 [S(ξmk−1,ξmk−1,hξnk−1)+S(ξnk−1,ξnk−1,hξmk−1)]} = max{S(ξmk−1,ξmk−1,ξnk−1),S(ξmk−1,ξmk−1,ξmk),S(ξnk−1,ξnk−1,ξnk), 1 2 [S(ξmk−1,ξmk−1,ξnk)+S(ξnk−1,ξnk−1,ξmk)]} ≤ max{S(ξmk−1,ξmk−1,ξnk−1),S(ξmk−1,ξmk−1,ξmk),S(ξnk−1,ξnk−1,ξnk), 1 2 [2S(ξmk−1,ξmk−1,ξmk)+S(ξmk,ξmk,ξnk)+ 2S(ξnk−1,ξnk−1,ξnk)+S(ξnk,ξnk,ξmk)]} Letting k → ∞, we get lim k→∞ M(ξmk−1,ξmk−1,ξnk−1)= τ. From the condition (γ3) and the generalized Zs-contraction property, we have 0≤ lim k→∞ sup γ(S(ξmk,ξmk,ξnk),M(ξmk−1,ξmk−1,ξnk−1)) < 0 This is a contraction, Hence, τ = 0. That is {ξn} is a cauchy sequence in the complete S-metric space X, we can find η ∈ X so that limn→∞ξn = η. Now we verify that, η is an invariant point of h. If suppose hη 6= η, then S(η,η,hη)= S(hη,hη,η) > 0. Now, M(ξn,ξn,η)=max{S(ξn,ξn,η),S(ξn,ξn,hξn),S(η,η,hη), 1 2 [S(ξn,ξn,hη)+S(η,η,hξn)]} lim n→∞ M(ξn,ξn,η)= max{S(η,η,η),S(η,η,η),S(η,η,hη), 1 2 [S(η,η,hη)+S(η,η,η)]} = S(η,η,hη) From the conditions (γ2),(γ3) and Zs-contraction property, we get 0≤ lim n→∞ sup γ(S(hξn,hξn,hη),M(ξn,ξn,η)) < 0 This is contradiction. Hence S(η,η,hη)=0 =⇒ hη = η. Hence, η is a invariant point of h. Now we claim that η is unique. Suppose α is an element in X such that α 6= η and hα = α. Int. J. Anal. Appl. (2023), 21:8 9 Now, M(η,η,α)= max{S(η,η,α),S(η,η,hη),S(α,α,hα), 1 2 [S(η,η,hα)+S(α,α,hη]} = max{S(η,η,α),S(η,η,η),S(α,α,α), 1 2 [S(η,η,α)+S(α,α,η)]} = S(η,η,α) From the condition (γ2) and Zs-contraction property, we get 0≤ γ(S(hη,hη,hα),M(η,η,α))= γ(S(hη,hη,hα),S(η,η,α)) < S(η,η,α)−S(η,η,α)=0, This is a contradiction. It should be η = α. � Example 3.3. Consider a complete S-metric space (X, S), where X = [0, 1 4 ] and S : X3 → [0,∞) by S(ξ,ϑ,w) = |ξ −w|+ |ξ −2ϑ+w|. Define h: X → X by hξ = ξ 1+ξ . From example 2.9 in [5], we have h be a Z-contraction in relation to γ ∈ Z, where γ(p,q)= q q+1 4 −p, for any p,q∈ [0,∞) Therefore for all ξ,ϑ ∈ X, we get 0≤ γ(S(hξ,hξ,hϑ),S(ξ,ξ,ϑ)) = S(ξ,ξ,ϑ) S(ξ,ξ,ϑ)+ 1 4 −S(hξ,hξ,hϑ) ≤ M(ξ,ξ,ϑ) M(ξ,ξ,ϑ)+ 1 4 −S(hξ,hξ,hϑ) = γ(S(hξ,hξ,hϑ),M(ξ,ξ,ϑ)) Thus, h is generalized Zs-contraction in relation to γ, for some γ ∈ Z. So, by using Theorem 3.2, h has a unique invariant point a=0. References [1] V.R.B. Guttia, L.B. Kumssa, Fixed Points of (α,ψ,φ)-Generalized Weakly Contractive Maps and Property(P) in S-metric spaces, Filomat. 31 (2017,) 4469–4481. https://doi.org/10.2298/fil1714469b. [2] S. Banach, Sur les Opérations dans les Ensembles Abstraits et leur Application aux Équations Intégrales, Fund. Math. 3 (1922), 133-181. http://eudml.org/doc/213289. [3] T. Dosenovic, S. Radenovic, A. Rezvani, S. Sedghi, Coincidence Point Theorems in S-Metric Spaces Using Integral Type of Contractions, U.P.B. Sci. Bull., Ser. A, 79 (2017), 145-158. [4] D.V. Nguyen, H.T. Nguyen, S. Radojevic, Fixed Point Theorems for G-Monotone Maps on Partially Ordered S- Metric Spaces, Filomat. 28 (2014), 1885–1898. https://doi.org/10.2298/fil1409885d. [5] F. Khojasteh, S. Shukla, S. Radenovic, A New Approach to the Study of Fixed Point Theory for Simulation Functions, Filomat. 29 (2015), 1189–1194. https://doi.org/10.2298/fil1506189k. [6] A. Gupta, Cyclic Contraction on S- Metric Space, Int. J. Anal. Appl. 3 (2013), 119-130. [7] J.K. Kim, S. Sedghi, A. Gholidahneh, M.M. Rezaee, Fixed Point Theorems in S-Metric Spaces, East Asian Math. J. 32 (2016), 677–684. https://doi.org/10.7858/EAMJ.2016.047. https://doi.org/10.2298/fil1714469b http://eudml.org/doc/213289 https://doi.org/10.2298/fil1409885d https://doi.org/10.2298/fil1506189k https://doi.org/10.7858/EAMJ.2016.047 10 Int. J. Anal. Appl. (2023), 21:8 [8] N. Mlaiki, N.Y. Özgür, N. Taş, New Fixed-Point Theorems on an S-metric Space via Simulation Functions, Math- ematics. 7 (2019), 583. https://doi.org/10.3390/math7070583. [9] Z. Mustafa, B. Sims, A New Approach to Generalized Metric Spaces, J. Nonlinear Convex Anal. 7 (2006), 289-297. [10] M. Olgun, O. Bicer, T. Alyildiz, A New Aspect to Picard Operators With Simulation Functions, Turk. J. Math. 40 (2016), 832–837. https://doi.org/10.3906/mat-1505-26. [11] N.Y. Ozgur, N. Tas, Some Fixed Point Theorems on S-Metric Spaces, Mat. Vesnik, 69 (2017), 39-52. https: //hdl.handle.net/20.500.12462/6682. [12] S. Sedghi, N.V. Dung, Fixed Point Theorems on S-Metric Spaces, Math. Vesnik, 66 (2014), 113-124. [13] S. Sedghi, N. Shobe, A. Aliouche, A Generalization of Fixed Point Theorem in S-Metric Spaces, Math. Vesnik, 64 (2012), 258-266. https://doi.org/10.3390/math7070583. https://doi.org/10.3906/mat-1505-26 https://hdl.handle.net/20.500.12462/6682 https://hdl.handle.net/20.500.12462/6682 1. Introduction 2. Preliminaries 3. Main Results References