Ratio Mathematica Some fixed point results using (ψ,ϕ)-generalized almost weakly contractive maps in S-metric spaces D.Venkatesh* V.Naga Raju † Abstract Fixed point theorems have been proved for various contractive con- ditions by several authors in the existing literature. In this article, we define an (ψ,ϕ)- generalized almost weakly contractive map in S-metric spaces and prove an existence and uniqueness of fixed point of such maps. And also we deduce some existing results as special cases of our result. Moreover, we give an example in support of the results. Keywords: Fixed point; generalized almost weakly contractive map; S-metric space; 2020 AMS subject classifications: 47H10,54H25 1 *(Department Of Mathematics, Osmania University, Hyderabad, Telangana-500007, India) ; venkat409151@gmail.com. †(Department Of Mathematics, Osmania University, Hyderabad, Telangana-500007, India); viswanag2007@gmail.com. 1 Received on September 7, 2022. Accepted on March 1, 2021. Published on June 30, 2023. doi: 10.23755/rm.v4li0.855. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. Volume 47, 2023 244 D.Venkatesh, V.Naga Raju 1 Introduction Fixed point technique is considered as one of the powerful tools to solve sev- eral problems occur in several fields like Computer science, Economics, Mathe- matics and its allied subjects. In the year 1906, M.Frechet [7] introduced metric spaces. Later, in the year 1922, Stefan Banach [4] proved a very famous theo- rem called ”Banach Fixed Point Theorem”. This theorem has been generalized in many directions by generalizing the underlying space or by viewing it as a common fixed point theorem along with other self maps. 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 generalizations are used to extend the scope of the study of fixed point theory. In 2012, Sedghi, Shobe and Aliouche [13] introduced S-metric spaces and studied some properties of these spaces. We observe that, every G-metric space need not be a S-metric space and vice-versa. For details, see Examples 2.1 and 2.2 in [5]. Generally, in proving fixed point results for a single self map, we utilize completeness and a contractive condition. Nowadays, the study of fixed point theorems for self maps satisfying different con- traction conditions is the center of rigorous research activities. In this direction, Dutta et al. [6] introduced (ψ,ϕ)-weakly contractive maps in 2008 and obtained some fixed point results for such contractions. Later, G.V.R. Babu et al. [1] intro- duced (ψ,ϕ)-almost weakly contractive maps in G-metric spaces in 2014. Fixed points of contractive maps on S-metric spaces were studied by several authors [2], [3] and [11]. Since then, several contractions have been considered for proving fixed point theorems. The main purpose of this paper is to define an (ψ,ϕ)- generalized almost weakly contractive map in S-metric spaces and prove an existence and uniqueness of fixed point of such maps. Furthermore we deduce some results as corollaries to our re- sult and provide an example to validate our result. 2 Preliminaries Definition 2.1. [8] A function ψ : [0,∞) → [0,∞) is said to be an altering distance function if it satisfies (i) ψ is continuous and non decreasing and (ii) ψ(t) = 0 if and only if t = 0. We denote the class of all altering distance functions by Ψ. We denote Φ = {ϕ : [0,∞) → [0,∞) : (i) ϕ is continuous and (ii) ϕ(t)=0 if and only if t=0}. In the following, Dutta and Choudhury [6] established the fixed points of (ψ,ϕ)- 245 Some fixed point results in S-metric spaces weakly contractive maps in complete metric spaces. Theorem 2.1. [6] Let (X,d) be a complete metric space and let h:X→X be a self- maps of X. If there exist ψ,ϕ ∈ Ψ such that ψ(d(hξ,hϑ)) ≤ ψ(d(ξ,ϑ)) − ϕ(d(ξ,ϑ)) for all ξ,ϑ ∈ X. Then h has a unique fixed point. Definition 2.2. [10] Let X be a non-empty set, G:X3 → [0,∞) be a function satisfying the following properties: (i) G(ξ,ϑ,w) = 0 if ξ = ϑ = w, (ii) G(ξ,ξ,ϑ) > 0 for all ξ,ϑ ∈X with ξ ̸= ϑ, (iii) G(ξ,ξ,ϑ) ≤ G(ξ,ϑ,w) for all ξ,ϑ,w ∈X, (iv) G(ξ,ϑ,w) = G(ξ,w,ϑ) = G(w,ξ,ϑ) =...(symmetriy in all three variables), (v) G(ξ,ϑ,w) ≤ G(ξ,a,a) + G(a,ϑ,w) for all ξ,ϑ,w,a ∈X. Then the function G is called a generalized metric(G-metric) and the pair (X,G) is called a G-metric space. Definition 2.3. [14] Let (X,G) be a G-metric space. A self mapping h of X is said to be weakly contractive if for all ξ,ϑ,w ∈ X G(hξ,hϑ,hw) ≤ G(ξ,ϑ,w) − ψ(G(ξ,ϑ,w)) where ψ is an altering distance function. In 2012, Khandaqji, Al-Sharif and Al-Khaleel [9] proved the following for weakly contractive maps in G-metric spaces. Theorem 2.2. [9] Let (X,G) be a complete G-metric space and h:X→X be a self map. If there exist ψ ∈ Ψ and ϕ ∈ Φ such that ψ(G(hξ,hϑ,hw)) ≤ ψ(max{G(ξ,ϑ,w),G(ξ,hξ,hξ),G(ϑ,hϑ,hϑ),G(w,hw,hw), αG(hξ,hξ,ϑ) + (1 − α)G(hϑ,hϑ,w),βG(ξ,hξ,hξ) + (1 − β)G(ϑ,hϑ,hϑ)}) − ϕ(max{G(ξ,ϑ,w),G(ξ,hξ,hξ), G(ϑ,hϑ,hϑ),G(w,hw,hw),αG(hξ,hξ,ϑ) + (1 − α)G(hϑ,hϑ,w),βG(ξ,hξ,hξ) + (1 − β)G(ϑ,hϑ,hϑ)}) (1) for all ξ,ϑ,w ∈X,where α,β ∈ (0,1). Then h has a unique fixed point u(say) and h is G-continuous at u. Definition 2.4. [13] Let a nonempty set X, then we say that a function S:X3 → [0,∞) is S-metric on X if: (S1) S(ξ,ϑ,w) > 0 for all ξ,ϑ,w ∈X with ξ ̸= ϑ ̸= 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)]. for all ξ,ϑ,w,a ∈X. Then (X,S) is called an S-metric space. 246 D.Venkatesh, V.Naga Raju Example 2.1. [13] Let (X,d) be a metric space. Define S:X3 → [0,∞) by S(ξ,ϑ,w) = d(ξ,ϑ) + d(ξ,w) + d(ϑ,w) for all ξ,ϑ,w ∈X. Then S is an S-metric on X and S is called the S-metric induced by the metric d. Example 2.2. [5] Let X=R, the set 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] Let X=R, the set 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. Let X=[0,1] and we define S:X3 → [0,∞) by S(ξ,ϑ,w) = { 0 if ξ = ϑ = w max{ξ,ϑ,w} otherwise . Then S is an S-metric on X. The following lemmas are useful in our main results. Lemma 2.1. [13] In an S-metric space, we have S(ξ,ξ,ϑ) = S(ϑ,ϑ,ξ). Lemma 2.2. [5] In an S-metric space, we have (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 2.5. [13] Let (X,S) be an S-metric space. We define the following: (i) a sequence {ξn} ∈X converges to a point ξ ∈X if S(ξn,ξn,ξ) → 0 as n → ∞. That is, for each ϵ > 0, there exists n0 ∈ N such that for all n≥ n0, S(ξn,ξn,ξ) < ϵ and we denote it by limn→∞ ξn = ξ. (ii) a sequence {ξn} ∈X is called a Cauchy sequence if for each ϵ > 0, there exists n0 ∈ N such that S(ξn,ξn,ξm) < ϵ for all n,m≥ n0. (iii) (X,S) is said to be complete if each Cauchy sequence in X is convergent. Definition 2.6. Let (X,S) and (Y,S’) be two S-metric spaces. Then a function h:X→Y is S-continuous at a point ξ ∈X if it is S-sequentially continuous at ξ, that is, whenever {ξn} is S-convergent to ξ, we have h(ξn) is S’-convergent to h(ξ). Lemma 2.3. [13] Let (X,S) be an S-metric space. If the sequences {ξn} in X converges to ξ, then ξ is unique. Lemma 2.4. [13] Let (X,S) be an S-metric space. If there exist sequences {ξn} and {ϑn} in X such that limn→∞ ξn = ξ and limn→∞ ϑn = ϑ, then limn→∞ S(ξn,ξn,ϑn) = S(ξ,ξ,ϑ). 247 Some fixed point results in S-metric spaces Definition 2.7. [13] Let (X,S) be an S-metric space. A map h:X→X is said to be an S-contraction if there exists a constant 0 ≤ λ < 1 such that S(h(ξ),h(ξ),h(ϑ)) ≤ λS(ξ,ξ,ϑ) for all ξ,ϑ ∈ X. We now introduce the following definition and support it with a subsequent ex- ample. Definition 2.8. Let (X,S) be an S-metric space. A map h:X → X is called (ψ,ϕ) -generalized almost weakly contractive if it satisfies the inequality ψ(S(hξ,hϑ,hw)) ≤ ψ(M(ξ,ϑ,w)) − ϕ(M(ξ,ϑ,w)) + L.θ(ξ,ϑ,w) (2) for all ξ,ϑ,w ∈ X, ψ ∈ Ψ, ϕ ∈ Φ and L ≥ 0, where M(ξ,ϑ,w) = max{S(ξ,ϑ,w),S(ξ,ξ,hξ),S(ϑ,ϑ,hϑ), 1 2 [S(ξ,ξ,hϑ)+S(ϑ,ϑ,hξ)]}, θ(ξ,ϑ,w) = min{S(ξ,ξ,hξ),S(ϑ,ϑ,hϑ),S(w,w,hξ),S(ξ,ξ,hw)}. Example 2.5. Let X = [0, 8 7 ] and we define h : X → X by hξ = { ξ 10 if ξ ∈ [0,1] ξ − 4 5 if ξ ∈ (1, 8 7 ] . We define S: X3 → [0,∞) by S(ξ,ϑ,w) = |ξ − w| + |ϑ − w| for all ξ,ϑ,w ∈ X. Then (X,S) is a complete S-metric space. We now define functions ψ,ϕ : [0,∞) → [0,∞) by ψ(t) = t, for all t≥ 0 and ϕ(t) = { t 2 if t ∈ [0,1] t t+1 if t ≥ 1. . We now show that h satisfies the inequality (2). Case(i): Let ξ,ϑ,w ∈ [0,1]. Without loss of generality, we assume that ξ > ϑ > w. S(hξ,hϑ,hw) = S( ξ 10 , ϑ 10 , w 10 ) = 1 10 (|ξ − w| + |ϑ − w|) and S(ξ,ϑ,w) = |ξ − w| + |ϑ − w|. sub case (i): If |ξ − w| + |ϑ − w| ∈ [0,1]. In this case, S(hξ,hϑ,hw) = 1 10 (|ξ − w| + |ϑ − w|) ≤ 1 2 (|ξ − w| + |ϑ − w|) = 1 2 S(ξ,ϑ,w) ≤ 1 2 M(ξ,ϑ,w) = M(ξ,ϑ,w) − 1 2 M(ξ,ϑ,w) = ψ(M(ξ,ϑ,w)) − ϕ(M(ξ,ϑ,w)). 248 D.Venkatesh, V.Naga Raju Sub case(ii): If |ξ − ϑ| + |ϑ − w| ≥1. In this case, S(hξ,hϑ,hw) = 1 10 (|ξ − ϑ| + |ϑ − w|) ≤ |ξ − ϑ| + |ϑ − w| − |ξ − ϑ| + |ϑ − w| 1 + |ξ − ϑ| + |ϑ − w| = S(ξ,ϑ,w) − S(ξ,ϑ,w) 1 + S(ξ,ϑ,w) = (S(ξ,ϑ,w))2 1 + S(ξ,ϑ,w) ≤ (M(ξ,ϑ,w))2 1 + M(ξ,ϑ,w) = M(ξ,ϑ,w) − M(ξ,ϑ,w) 1 + M(ξ,ϑ,w) = ψ(M(ξ,ϑ,w)) − ϕ(M(ξ,ϑ,w)). Case(ii): Let ξ,ϑ,w ∈ (1, 8 7 ]. Without loss of generality, we assume that ξ > ϑ > w. S(hξ,hϑ,hw) = S(ξ − 4 5 ,ϑ − 4 5 ,w − 4 5 ) = |ξ − w| + |ϑ − w| ≤ 2 7 ≤ 64 65 = 8 5 − 8 13 = S(ξ,ξ,hξ) − S(ξ,ξ,hξ) 1 + S(ξ,ξ,hξ) = (S(ξ,ξ,hξ))2 1 + S(ξ,ξ,hξ) ≤ (M(ξ,ϑ,w))2 1 + M(ξ,ϑ,w) = M(ξ,ϑ,w) − M(ξ,ϑ,w) 1 + M(ξ,ϑ,w) = ψ(M(ξ,ϑ,w)) − ϕ(M(ξ,ϑ,w)). Case(iii): Let ϑ,w ∈ [0,1] and ξ ∈ (1, 8 7 ]. Without loss of generality, we assume that ϑ >w. S(hξ,hϑ,hw) = S(ξ − 4 5 , ϑ 10 , w 10 ) = |ξ − 4 5 − w 10 | + | ϑ 10 − w 10 | = ξ − w 10 − 4 5 + ϑ − w 10 = ξ + ϑ 10 − w 5 − 4 5 = 31 70 ≤ 64 65 = 8 5 − 8 13 = S(ξ,ξ,hξ) − S(ξ,ξ,hξ) 1 + S(ξ,ξ,fξ) = (S(ξ,ξ,hξ))2 1 + S(ξ,ξ,hξ) ≤ (M(ξ,ϑ,w))2 1 + M(ξ,ϑ,w) = M(ξ,ϑ,w) − M(ξ,ϑ,w) 1 + M(ξ,ϑ,w) = ψ(M(ξ,ϑ,w)) − ϕ(M(ξ,ϑ,w)). 249 Some fixed point results in S-metric spaces Case(iv): Let w∈ [0,1] and ξ,ϑ ∈ (1, 8 7 ]. Without loss of generality, we assume that ξ > ϑ. S(hξ,hϑ,hw) = S(ξ − 4 5 ,ϑ − 4 5 , w 10 ) = |ξ − 4 5 − w 10 | + |ϑ − 4 5 − w 10 | = ξ + ϑ − w 5 − 8 5 = 12 35 ≤ 64 65 = 8 5 − 8 13 = S(ϑ,ϑ,hϑ) − S(ϑ,ϑ,hϑ) 1 + S(ϑ,ϑ,hϑ) = (S(ϑ,ϑ,hϑ))2 1 + S(ϑ,ϑ,hϑ) ≤ (M(ξ,ϑ,w))2 1 + M(ξ,ϑ,w) = M(ξ,ϑ,w) − M(ξ,ϑ,w) 1 + M(ξ,ϑ,w) = ψ(M(ξ,ϑ,w)) − ϕ(M(ξ,ϑ,w)). Case (v): Let ξ,ϑ ∈ [0,1] and w ∈ (1, 8 7 ]. Without loss of generality, we assume that ξ > ϑ. S(hξ,hϑ,hw) = S( ξ 10 , ϑ 10 ,w − 4 5 ) = | ξ 10 − w + 4 5 | + | ϑ 10 − w + 4 5 | = | 4 5 − (w − ξ 10 )| + | 4 5 − (w − ϑ 10 )| = w − ξ 10 − 4 5 + w − ϑ 10 − 4 5 = 2w − ξ + ϑ 10 − 8 5 = 41 70 ≤ 64 65 = 8 5 − 8 13 = S(ξ,ξ,hξ) − S(ξ,ξ,hξ) 1 + S(ξ,ξ,hξ) = (S(ξ,ξ,hξ))2 1 + S(ξ,ξ,hξ) ≤ (M(ξ,ϑ,w))2 1 + M(ξ,ϑ,w) = M(ξ,ϑ,w) − M(ξ,ϑ,w) 1 + M(ξ,ϑ,w) = ψ(M(ξ,ϑ,w)) − ϕ(M(ξ,ϑ,w)). Case (vi): Let ξ ∈ [0,1] and w,ϑ ∈ (1, 8 7 ]. Without loss of generality, we assume that w > ϑ. S(hξ,hϑ,hw) = S( ξ 10 ,ϑ − 4 5 ,w − 4 5 ) = | ξ 10 − w + 4 5 | + |ϑ − w| = w − ξ 10 − 4 5 + w − ϑ = 2w − ξ 10 − 4 5 − ϑ 250 D.Venkatesh, V.Naga Raju ≤ 27 70 ≤ 64 65 = 8 5 − 8 13 = S(ϑ,ϑ,hϑ) − S(ϑ,ϑ,hϑ) 1 + S(ϑ,ϑ,hϑ) = (S(ϑ,ϑ,hϑ))2 1 + S(ϑ,ϑ,hϑ) ≤ (M(ξ,ϑ,w))2 1 + M(ξ,ϑ,w) = M(ξ,ϑ,w) − M(ξ,ϑ,w) 1 + M(ξ,ϑ,w) = ψ(M(ξ,ϑ,w)) − ϕ(M(ξ,ϑ,w)). From all the above cases, we conclude that h is an (ψ,ϕ)-generalized almost weakly contraction map on X. Lemma 2.5. [5] Let (X,S) be an S-metric space and {ξn} be a sequence in X such that limn→∞ Sb(ξn,ξn,ξn+1) = 0. If {ξn} is not a Cauchy sequence, then there exist an ϵ > 0 and two sequences {mk} and {nk} of natural numbers with nk > mk > k such that S(ξmk,ξmk,ξnk) ≥ ϵ, S(ξmk−1,ξmk−1,ξnk) < ϵ and (i)limk→∞ Sb(ξmk,ξmk,ξnk) = ϵ. (ii) limk→∞ Sb(ξmk−1,ξmk−1,ξnk) = ϵ. (iii) limk→∞ Sb(ξmk,ξmk,ξnk−1) = ϵ. (iv) limk→∞ Sb(ξmk−1,ξmk−1,ξnk−1) = ϵ. 3 Main Results Theorem 3.1. Let (X,S) be a complete S-metric space and h: X → X be a (ψ,ϕ)- generalized almost weakly contractive mapping. Then h has a unique fixed point in X. Proof. Let ξ0 ∈ X be arbitrary. We define a sequence {ξn} by hξn = ξn+1, for n = 0,1,2,.... If ξn = ξn+1, for some n∈ N, then ξn is a fixed point of h. Suppose ξn ̸= ξn+1, for all n∈ N. Consider, ψ(S(ξn+1,ξn+1,ξn)) = ψ(S(hξn,hξn,hξn−1)) ≤ ψ(max{S(ξn,ξn,ξn−1),S(ξn,ξn,hξn),S(ξn,ξn,hξn), 1 2 [S(ξn,ξn,hξn) + S(ξn,ξn,hξn)]}) − ϕ(max{S(ξn,ξn,ξn−1),S(ξn,ξn,hξn),S(ξn,ξn,hξn), 1 2 [S(ξn,ξn,hξn) + S(ξn,ξn,hξn)]}) + L.min{S(ξn,ξn,hξn),S(ξn,ξn,hξn),S(ξn−1,ξn−1,hξn),S(ξn,ξn,hξn−1)} 251 Some fixed point results in S-metric spaces = ψ(max{S(ξn,ξn,ξn−1),S(ξn,ξn,ξn+1),S(ξn,ξn,ξn+1), 1 2 [S(ξn,ξn,ξn+1) + S(ξn,ξn,ξn+1)]}) − ϕ(max{S(ξn,ξn,ξn−1),S(ξn,ξn,ξn+1),S(ξn,ξn,ξn+1), 1 2 [S(ξn,ξn,ξn+1) + S(ξn,ξn,ξn+1)]}) + L.min{S(ξn,ξn,ξn+1),S(ξn,ξn,ξn+1),S(ξn−1,ξn−1,ξn+1),S(ξn,ξn,ξn)} = ψ(max{S(ξn,ξn,ξn−1),S(ξn,ξn,ξn+1)}) − ϕ(max{S(ξn,ξn,ξn−1), S(ξn,ξn,ξn+1)}) + L.0 If max{S(ξn,ξn,ξn−1),S(ξn,ξn,ξn+1)} = S(ξn,ξn,ξn+1), then we get ψ(S(ξn+1,ξn+1,ξn)) ≤ ψ(S(ξn+1,ξn+1,ξn)) − ϕ(S(ξn+1,ξn+1,ξn)) that is, ϕ(S(ξn+1,ξn+1,ξn)) ≤ 0, which implies that S(ξn+1,ξn+1,ξn) = 0. Then we get ξn+1 = ξn, which is a contradiction to our assumption that ξn ̸= ξn+1, for each n. Therefore, max{S(ξn,ξn,ξn−1),S(ξn,ξn,ξn+1)} = S(ξn,ξn,ξn−1), then we get ψ(S(ξn+1,ξn+1,ξn)) ≤ ψ(S(ξn,ξn,ξn−1)) − ϕ(S(ξn,ξn,ξn−1)) (3) that is ψ(S(ξn+1,ξn+1,ξn)) ≤ ψ(S(ξn,ξn,ξn−1)) Therefore we get, S(ξn+1,ξn+1,ξn) ≤ S(ξn,ξn,ξn−1), for all n and the sequence {S(ξn+1,ξn+1,ξn)} is decreasing and bounded. So, there exists r ≥ 0 such that lim n→∞ S(ξn+1,ξn+1,ξn) = r. Letting n → ∞ in equation (3), we get ψ(r) ≤ ψ(r) − ϕ(r), which is a contradiction unless r = 0. Hence, lim n→∞ S(ξn+1,ξn+1,ξn) = 0. (4) Now we prove that {ξn} is a Cauchy sequence. If not, then there exists an ϵ > 0 for which we can find subsequences {ξm(k)} and {ξn(k)} of {ξn} and increasing sequence of integers {m(k)} and {n(k)} such that n(k) is the smallest index for which n(k) > m(k) > k, S(ξm(k),ξm(k),ξn(k)) ≥ ϵ (5) 252 D.Venkatesh, V.Naga Raju Then, we have S(ξm(k),ξm(k),ξn(k)−1) < ϵ (6) Now, ϵ ≤ S(ξm(k),ξm(k),ξn(k)) = S(ξn(k),ξn(k),ξm(k)) ≤ 2S(ξn(k),ξn(k),ξn(k)−1) + S(ξm(k),ξm(k),ξn(k)−1) ≤ ϵ + 2S(ξn(k),ξn(k),ξn(k)−1) (Using equation 6) Letting k→ ∞, we get lim k→∞ S(ξm(k),ξm(k),ξn(k)) = ϵ. (7) Also, S(ξm(k),ξm(k),ξn(k)) ≤ 2S(ξm(k),ξm(k),ξm(k)−1) + S(ξn(k),ξn(k),ξm(k)−1) ≤ 2S(ξm(k),ξm(k),ξm(k)−1) + 2S(ξn(k),ξn(k),ξn(k)−1) + S(ξm(k)−1,ξm(k)−1,ξn(k)−1) (8) and S(ξm(k)−1,ξm(k)−1,ξn(k)−1) ≤ 2S(ξm(k)−1,ξm(k)−1,ξm(k))+S(ξn(k)−1,ξn(k)−1,ξm(k)) = 2S(ξm(k),ξm(k),ξm(k)−1) + S(ξm(k),ξm(k),ξn(k)−1) (9) Letting k → ∞ in equation (9) and using equations (4), (6), (7) and (8) we get lim k→∞ S(ξm(k)−1,ξm(k)−1,ξn(k)−1) = ϵ (10) Setting ξ = ξm(k)−1,y = ξm(k)−1 and z = ξn(k)−1 in equation (2), we obtain ψ(ϵ) ≤ ψ(S(ξm(k),ξm(k),ξn(k))) = ψ(S(hξm(k)−1,hξm(k)−1,hξn(k)−1)) ≤ ψ(max{S(ξm(k)−1,ξm(k)−1,ξn(k)−1),S(ξm(k)−1,ξm(k)−1,hξm(k)−1), S(ξm(k)−1,ξm(k)−1,hξm(k)−1), 1 2 [S(ξm(k)−1,ξm(k)−1,hξm(k)−1) + S(ξm(k)−1,ξm(k)−1,hξm(k)−1)]}) − ϕ(max{S(ξm(k)−1,ξm(k)−1,ξn(k)−1),S(ξm(k)−1,ξm(k)−1,hξm(k)−1), S(ξm(k)−1,ξm(k)−1,hξm(k)−1), 1 2 [S(ξm(k)−1,ξm(k)−1,hξm(k)−1) + S(ξm(k)−1,ξm(k)−1,hξm(k)−1)]}) 253 Some fixed point results in S-metric spaces + L.min{S(ξm(k)−1,ξm(k)−1,hξm(k)−1),S(ξm(k)−1,ξm(k)−1,hξm(k)−1), S(ξn(k)−1,ξn(k)−1,hξm(k)−1),S(ξm(k)−1,ξm(k)−1,hξn(k)−1)} ≤ ψ(max{S(ξm(k)−1,ξm(k)−1,ξn(k)−1),S(ξm(k)−1,ξm(k)−1,ξm(k)), S(ξm(k)−1,ξm(k)−1,ξm(k)), 1 2 [S(ξm(k)−1,ξm(k)−1,ξm(k)) + S(ξm(k)−1,ξm(k)−1,ξm(k))]}) − ϕ(max{S(ξm(k)−1,ξm(k)−1,ξn(k)−1),S(ξm(k)−1,ξm(k)−1,ξm(k)), S(ξm(k)−1,ξm(k)−1,ξm(k)), 1 2 [S(ξm(k)−1,ξm(k)−1,ξm(k)) + S(ξm(k)−1,ξm(k)−1,ξm(k))]}) + L.min{S(ξm(k)−1,ξm(k)−1,ξm(k)),S(ξm(k)−1,ξm(k)−1,ξm(k)), S(ξn(k)−1,ξn(k)−1,ξm(k)),S(ξm(k)−1,ξm(k)−1,ξn(k))} Letting k → ∞ and using equation (10) we get ψ(ϵ) ≤ ψ(max{ϵ,0,0,0}) − ϕ(max{ϵ,0,0,0}) + L.min{0,0,0, ϵ} ψ(ϵ) ≤ ψ(ϵ) − ϕ(ϵ) + L.0 This is a contradiction, since ϵ > 0. This shows that {ξn} is a Cauchy sequence in the complete S-metric space (X,S). There exists κ ∈X such that {ξn} → κ as n → ∞. Now we prove that hκ = κ. Put ξ = ξn,ϑ = ξn and w = κ in equation (2), then we get ψ(S(ξn+1,ξn+1,fκ)) = ψ(S(hξn,hξn,hκ)) ≤ ψ(max{S(ξn,ξn,κ),S(ξn,ξn,hξn),S(ξn,ξn,hξn), 1 2 [S(ξn,ξn,hξn) + S(ξn,ξn,hξn)]}) − ϕ(max{S(ξn,ξn,κ),S(ξn,ξn,hξn),S(ξn,ξn,hξn), 1 2 [S(ξn,ξn,hξn) + S(ξn,ξn,hξn)]}) + L.min{S(ξn,ξn,hξn),S(ξn,ξn,hξn),S(κ,κ,hξn),S(ξn,ξn,hκ)} = ψ(max{S(ξn,ξn,κ),S(ξn,ξn,ξn+1),S(ξn,ξn,ξn+1), 1 2 [S(ξn,ξn,ξn+1) + S(ξn,ξn,ξn+1)]}) − ϕ(max{S(ξn,ξn,κ),S(ξn,ξn,ξn+1), S(ξn,ξn,ξn+1), 1 2 [S(ξn,ξn,ξn+1) + S(ξn,ξn,ξn+1)]}) + L.min{S(ξn,ξn,ξn+1),S(ξn,ξn,ξn+1),S(κ,κ,ξn+1),S(ξn,ξn,hκ)} Letting n → ∞, we get ψ(S(κ,κ,hκ)) ≤ ψ(S(κ,κ,κ)) − ϕ(S(κ,κ,κ)+L.0 ψ(S(κ,κ,hκ)) ≤ 0. So, we get S(κ,κ,hκ)=0. 254 D.Venkatesh, V.Naga Raju Hence hκ = κ. That is κ is a fixed point of h. To prove the uniqueness of κ, let j be a fixed point of h with κ ̸=j. Using equation (2),we consider ψ(S(κ,κ,j)) = ψ(S(hκ,hκ,hj)) ≤ ψ(max{S(κ,κ,j),S(κ,κ,hκ),S(κ,κ,hκ), 1 2 [S(κ,κ,hκ) + S(κ,κ,hκ)]}) − ϕ(max{S(κ,κ,j),S(κ,κ,hκ),S(κ,κ,hκ), 1 2 [S(κ,κ,hκ) + S(κ,κ,hκ)]}) + L.min{S(κ,κ,hκ),S(κ,κ,hκ),S(j,j,hκ),S(κ,κ,hj)} That is, ψ(S(κ,κ,j)) ≤ ψ(S(κ,κ,j)) − ϕ(S(κ,κ,j)) is a contradiction, unless S(κ,κ,j) = 0. Hence we get κ = j. This shows that the fixed point of h is unique. 2 If L=0 in the Theorem 3.1, then we get the following. Corollary 3.1. Let (X,S) be a complete S-metric space and h:X→X be a mapping. Suppose there exist ψ ∈ Ψ and ϕ ∈ Φ such that S(hξ,hϑ,hw) ≤ ψ(max{S(ξ,ϑ,w),S(ξ,ξ,hξ),S(ϑ,ϑ,hϑ), 1 2 [S(ξ,ξ,hϑ) + S(ϑ,ϑ,hξ)]}) − ϕ(max{S(ξ,ϑ,w),S(ξ,ξ,hξ),S(ϑ,ϑ,hϑ), 1 2 [S(ξ,ξ,hϑ) + S(ϑ,ϑ,hξ)]}), for all ξ,ϑ,w ∈ X. Then h has a unique fixed point κ in X. If ψ is the identity map in the above Corollary (3.1), then we get the following. Corollary 3.2. Let (X,S) be a complete S-metric space and h:X→X be a mapping. Suppose there exist ϕ ∈ Φ such that S(hξ,hϑ,hw) ≤ max{S(ξ,ϑ,w),S(ξ,ξ,hξ),S(ϑ,ϑ,hϑ), 1 2 [S(ξ,ξ,hϑ) + S(ϑ,ϑ,hξ)]} − ϕ(max{S(ξ,ϑ,w),S(ξ,ξ,hξ),S(ϑ,ϑ,hϑ), 1 2 [S(ξ,ξ,hϑ) + S(ϑ,ϑ,hξ)]}) for all ξ,ϑ,w ∈ X. Then h has a unique fixed point κ in X. The following example is in support of Theorem 3.1. Example 3.1. Let X = [0, 7 6 ]. We define S:X3 → [0,∞) by S(ξ,ϑ,w) = max{|ξ− w|, |ϑ − w|}, for all ξ,ϑ,w ∈ X. Then S is an S-metric on X. We define h:X → X by 255 Some fixed point results in S-metric spaces hξ = { 1 2 if ξ ∈ [0,1] 4 3 − ξ if ξ ∈ (1, 7 6 ] . We define ψ,ϕ : [0,∞) → [0,∞) by ψ(t) = t, for all t≥ 0 and ϕ(t) = t 1+t for all t≥0. We now show that h satisfies inequality (2). Case(i) Let ξ,ϑ,w ∈[0,1]. Without loss of generality, we assume that ξ > ϑ > w. S(hξ,hϑ,hw) = S(1 2 , 1 2 , 1 2 ) = 0. Then trivially the inequality (2) holds. Case(ii) Let ξ,ϑ,w ∈ (1, 7 6 ]. Without loss of generality, we assume that ξ > ϑ > w. S(hξ,hϑ,hw) = S( 4 3 − ξ, 4 3 − ϑ, 4 3 − w) = max{| 4 3 − ξ − ( 4 3 − w)|, | 4 3 − ϑ − ( 4 3 − w)|} = max{|w − ξ|, |w − ϑ|} = ξ − w ≤ 1 6 ≤ 4 15 = 2 3 − 2 5 ≤ S(ξ,ξ,hξ) − S(ξ,ξ,hξ) 1 + S(ξ,ξ,hξ) = (S(ξ,ξ,hξ))2 1 + S(ξ,ξ,hξ) ≤ (M(ξ,ϑ,w))2 1 + M(ξ,ϑ,w) = M(ξ,ϑ,w) − M(ξ,ϑ,w) 1 + M(ξ,ϑ,w) = ψ(M(ξ,ϑ,w)) − ϕ(M(ξ,ϑ,w)). Case(iii) Let ϑ,w ∈[0,1] and ξ ∈ (1, 7 6 ]. Without loss of generality, we assume that ϑ > w. S(hξ,hϑ,hw) = S( 4 3 − ξ, 1 2 , 1 2 ) = max{| 4 3 − ξ − 1 2 |, | 1 2 − 1 2 |} = ξ − 5 6 ≤ 1 6 ≤ 4 15 = 2 3 − 2 5 ≤ S(ξ,ξ,hξ) − S(ξ,ξ,hξ) 1 + S(ξ,ξ,hξ) = (S(ξ,ξ,hξ))2 1 + S(ξ,ξ,hξ) ≤ (M(ξ,ϑ,w))2 1 + M(ξ,ϑ,w) = M(ξ,ϑ,w) − M(ξ,ϑ,w) 1 + M(ξ,ϑ,w) = ψ(M(ξ,ϑ,w)) − ϕ(M(ξ,ϑ,w)). Case(iv) Let w ∈[0,1] and ξ,ϑ ∈ (1, 7 6 ]. Without loss of generality, we assume that ϑ > ξ. S(hξ,hϑ,hw) = S( 4 3 − ξ, 4 3 − ϑ, 1 2 ) = max{| 4 3 − ξ − 1 2 |, | 4 3 − ϑ − 1 2 |} 256 D.Venkatesh, V.Naga Raju = max{| 5 6 − ξ|, | 5 6 − ϑ|} = ξ − 5 6 ≤ 1 6 ≤ 4 15 = 2 3 − 2 5 ≤ S(ϑ,,ϑ,hϑ) − S(ϑ,,ϑ,hϑ) 1 + S(ϑ,,ϑ,hϑ) = (S(ϑ,,ϑ,hϑ))2 1 + S(ϑ,,ϑ,hϑ) ≤ (M(ξ,ϑ,w))2 1 + M(ξ,ϑ,w) = M(ξ,ϑ,w) − M(ξ,ϑ,w) 1 + M(ξ,ϑ,w) = ψ(M(ξ,ϑ,w)) − ϕ(M(ξ,ϑ,w)). Case(v) Let ξ,ϑ ∈[0,1] and w ∈ (1, 7 6 ]. Without loss of generality, we assume that ξ > ϑ. S(hξ,hϑ,hw) = ( 1 2 , 1 2 , 4 3 − w) = max{| 1 2 − ( 4 3 − w)|, | 1 2 − ( 4 3 − w)|} = w − 5 6 ≤ 1 6 ≤ 4 15 = 2 3 − 2 5 ≤ S(ξ,ξ,hξ) − S(ξ,ξ,hξ) 1 + S(ξ,ξ,hξ) = (S(ξ,ξ,hξ))2 1 + S(ξ,ξ,hξ) ≤ (M(ξ,ϑ,w))2 1 + M(ξ,ϑ,w) = M(ξ,ϑ,w) − M(ξ,ϑ,w) 1 + M(ξ,ϑ,w) = ψ(M(ξ,ϑ,w)) − ϕ(M(ξ,ϑ,w)). case(vi) Let ξ ∈[0,1] and ϑ,w ∈ (1, 7 6 ]. Without loss of generality, we assume that w > ϑ. S(hξ,hϑ,hw) = S( 1 2 , 4 3 − ϑ, 4 3 − w) = max{| 1 2 − ( 4 3 − w)|, | 4 3 − ϑ − ( 4 3 − w)|} = max{w − 5 6 , |w − ϑ|} = w − 5 6 ≤ 1 6 ≤ 4 15 = 2 3 − 2 5 = S(ϑ,ϑ,hϑ) − S(ϑ,ϑ,hϑ) 1 + S(ϑ,ϑ,hϑ) = (S(ϑ,ϑ,hϑ))2 1 + S(ϑ,ϑ,hϑ) ≤ (M(ξ,ϑ,w))2 1 + M(ξ,ϑ,w) = M(ξ,ϑ,w) − M(ξ,ϑ,w) 1 + M(ξ,ϑ,w) = ψ(M(ξ,ϑ,w)) − ϕ(M(ξ,ϑ,w)). case(vii) Let ϑ ∈[0,1] and ξ,w ∈ (1, 7 6 ]. Without loss of generality, we assume that w > ξ. S(hξ,hϑ,hw) = S( 4 3 − ξ, 1 2 , 4 3 − w) = max{| 4 3 − ξ − ( 4 3 − w)|, | 1 2 − ( 4 3 − w)|} 257 Some fixed point results in S-metric spaces = max{|w − ξ|,w − 5 6 } = w − 5 6 ≤ 1 6 ≤ 4 15 = 2 3 − 2 5 = S(ϑ,ϑ,hhϑ) − S(ϑ,ϑ,hϑ) 1 + S(ϑ,ϑ,hϑ) = (S(ϑ,ϑ,hϑ))2 1 + S(ϑ,ϑ,hϑ) ≤ (M(ξ,ϑ,w))2 1 + M(ξ,ϑ,w) = M(ξ,ϑ,w) − M(ξ,ϑ,w) 1 + M(ξ,ϑ,w) = ψ(M(ξ,ϑ,w)) − ϕ(M(ξ,ϑ,w)). From all the above cases, we conclude that h is an (ψ,ϕ)-generalized almost weakly contraction map on X and 1 2 is the unique fixed point of h. 4 Conclusion In this paper, we establish an existence and uniqueness of a fixed point the- orem for (ψ,ϕ)-generalized almost weakly contraction maps in S-metric spaces. As S-metric space is a generalization of metric space, our result in this article ex- tends and improves the result of Khandaqji, Al-Sharif and Al-Khaleel [9] and also generalize several well-known comparable results in the literature. Further, the result in this paper can be extended to several spaces like Sb-metric space, partial Sb-metric spaces and other spaces. References [1] G.V.R.Babu, D.R. Babu, Kanuri Nageswara Rao, Bendi Venkata Siva Kumar, Fixed Points of (ψ,ϕ)-Almost Weakly Contractive maps in G-Metric Spaces, Applied Mathematics E-Notes, 4 (2014) 69–85. [2] G.V.R.Babu and B.K.Leta, Fixed Points of (α,ψ,ϕ)- Generalized Contrac- tive Maps and Property(P) in S-Metric spaces, Filomat 1:14 (2017) 4469– 4481.DOI 10.2298/FIL1714469B. [3] G.V.R.Babu, P.D.Sailaja, G.Srichandana, Common fixed points of (α,ψ,ϕ)- almost generalized weakly contractive maps in S-metric spaces, Commun. Non- linear Anal. 7(1)(2019), 17–35. [4] S. Banach, Sur les operations dans les ensembles abstraits et leur applications aux equations integrales, Fund. Math. 3 (1922) 133–181. [5] N.V.DUNG, N.T.HIEU and S.RADOJEVIC,Fixed Point Theorems for g- Monotone Maps on Partially Ordered S-Metric Space, J. Published by Faculty 258 D.Venkatesh, V.Naga Raju of Science and Mathematics, University of Nis. Serbia, Filomat 28:9 (2014) 1885– 1898.DOI 10.2298/FIL1409885D [6] P.N.Dutta and B.S.Choudhury, A Generalisation of Contraction Principle in Metric Spaces, J. Fixed Point Theory and Applications Article ID 406368, (2008) 8 pages. [7] M. Frechet Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Matematico di Palermo 22 (1906) 1–72. [8] M.S.Khan, M.Swaleh and S.Sessa, Fixed point theorems by altering distance between points, J. Bull. Aust. Math. Soc.. 30 (1) (1984) 1–9. [9] M.Khandaqji, S.Al-Sharif and M.Al-Khaleel, Property(P) and Some Fixed Point Results on (ψ,ϕ) Weakly Contractive G-Metric Spaces, Interna- tional journal of Mathematics and Mathematical Sciences, (2012), Article ID 675094,doi:10.1155/2012/675094,11 pages. [10] Z.Mustafa, B.Sims, A new approach to generalized metric spaces, J.Nonlinear Convex Anal. 7 (2006) 289–297. [11] N.Y.Ozgur and N.Tas, Some Fixed Point Theorems on S-Metric Spaces, Math. Vesnik, 69 (1) (2017) 39–52. [12] S.Sedghi and 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. [14] N.Surender and B.K.Reddy, Common Fixed Point Theorems for Weakly Compatible Mappings Satisfying Generalized Contraction Principle in Complete G-Metric Spaces, Annals of Pure and Applied Mathematics,10(2)(2015),179–190. 259 Introduction Preliminaries Main Results Conclusion