EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 4, 2023, 2405-2418 ISSN 1307-5543 – ejpam.com Published by New York Business Global Metrical Fixed Point Results on b-multiplicative metric spaces employing binary relaion Ibtesam Alshammari1,∗, Shahbaz Ali2, Qamrul Haque Khan2, Tawseef Rashid3, Cenap Ozel4 1 Department of Mathematics, University of Hafr Al Batin, Hafr Al Batin, Saudi Arabia 2 Department of Mathematics, Aligarh Muslim University, Aligarh, India 3 Department of Mathematical Sciences, IUST, Awantipora, India 4 Department of Mathematics, King Abdulaziz University, Jeddah-21589, Saudi Arabia Abstract. In this manuscrit, we establish some results on the existence and uniqueness of fixed points by using b-multiplicative metric spaces(MMS) endowed with a binary relation. We also find result on the coincidence of points involving a pair of mappings. Finally some examples are presented to illustrate the suitability of our results. 2020 Mathematics Subject Classifications: 47H10, 54H25,46J10 Key Words and Phrases: Multiplicative metric space, Binary relation, Relation theoretic con- tractions, Coincidence Points 1. Introduction and Prilimaries In 1922, Banach [1] laid the important result of fixed point theory in metric spaces. Later on, several authors generalized the Banach contraction principle, see[2–4]. Inspired by Turinici [5] work, Ran and Reurings [6] in 2004 worked on Banach contraction prin- ciple in ordered metric space and assumed the contractive condition only to hold on the comparable elements instead of the whole space. Fixed point in ordered metric space has been extensively studied in the literature [7–9]. The idea of MMS, which is a generalization of metric space, was first introduced by Bashirov et al. [10] in 2008. The main idea behind introducing MMS was to replace usual triangular inequality by the multiplicative triangle inequality. Later on, many research papers were written on fixed points in MMS [11–16, 18–20]. Czerwik [17] introduced the notion of b-metric space which is a generalization of metric space. There are some fixed point results in b-metric space. Later on, Muhammad Usman et al. [21] introduce the ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4953 Email addresses: iealshamri@uhb.edu.sa (I. Alshammari), shahbazali4786@gmail.com (S. Ali), qhkhan.ssitm@gmail.com (Q.H. Khan), tawseefrashid123@gmail.com (T. Rashid), cenap.ozel@gmail.com (C. Ozel) https://www.ejpam.com 2405 © 2023 EJPAM All rights reserved. I. Alshammari et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2405-2418 2406 new notion of b-multiplicative metric space and proved fixed point theorems for single and multivalued mapping on b-multiplicative metric spaces, endowed with a graph. In this paper we prove fixed point theorems for mapping on b-multiplicative metric space endowed with a binary relation and also prove a coincidence of points involving a pair of mapping and provide some examples to demonstrate our results. Definition 1. [21]. Let Ḧ be a non-empty set and let k ≥ 1 be a given real number. A mapping p : Ḧ×Ḧ → R is called a b-multiplicative metric with coefficient k, if the following conditions hold: (M1) p(ϖ, ρ) ≥ 1 for all ϖ, ρ ∈ Ḧ and p(ϖ, ρ) = 1 if and only if ϖ = ρ; (M2) p(ϖ, ρ) = p(ϖ, ρ) for all ϖ, ρ ∈ Ḧ; (M3) p(ϖ, ρ) ≤ p(ϖ, z)k.p(z, ρ)k for all ϖ, ρ, z ∈ Ḧ. The triplet (Ḧ,p,k) is called a b-multiplicative metric space. Definition 2. [4]. Let (Ḧ, p, k) be any b-MMS, {ϖn} be a sequence in Ḧ and ϖ ∈ Ḧ. If for every multiplicative open ball Bϵ(z) = {ρ : p(ϖ, ρ) < ϵ}, ϵ > 1, there exists a natural number N ∈ N such that n ≥ N and ϖn ∈ Bϵ(ϖ). Then the sequence {ϖn} is said to be multiplicative converging to ϖ. We denote as ϖn → ϖ (n → +∞). Lemma 1. [21] let (Ḧ, p, k) is a b-multiplicative metric space. If a sequence {ϖn} is a multiplicative convergent, then the multiplicative limit point is unique. Let (Ḧ, p) be a MMS, {ϖn} be a sequence in Ḧ and ϖ ∈ Ḧ. Then ϖn → ϖ(n → +∞) ⇔ p(ϖn, ϖ) → 1(n → +∞). Definition 3. [4]. Let (Ḧ, p) be a MMS and {ϖn} be a sequence in Ḧ. • Then {ϖn} is said to be multiplicative Cauchy sequence if for ϵ > 1, there exists a positive integer N ∈ N such that d(ϖm, ϖn) < ϵ for all n,m ≥ N. • Then {ϖn} is said to be multiplicative Cauchy if and only if p(ϖn, ϖm) → 1(n,m → +∞). Definition 4. [4]. If every multiplicative Cauchy sequence in (Ḧ, p) is multiplicative convergent in Ḧ, then MMS (Ḧ, p) is said to be multiplicative complete Definition 5. [22]. Let Ḧ be a nonempty set. A subset R̈ of Ḧ2 is called a binary relation on Ḧ. The subsets, Ḧ2 and ϕ of Ḧ2 are called the universal relation and empty relation respectively. Definition 6. [22]. Let R̈ be a binary relation on a nonempty set Ḧ. For ϖ, ρ ∈ Ḧ, we say that ϖ and ρ are R̈-comparative if either (ϖ, ρ) ∈ R̈ or (ρ,ϖ) ∈ R̈. We denote it by [ϖ, ρ] ∈ R̈ I. Alshammari et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2405-2418 2407 Proposition 1. If (Ḧ, p, k ≥ 1) is a b-metric space, R̈ is a binary relation on Ḧ, F̈ is a self-mapping on Ḧ and λ ∈ [0, 1k ), then these conditions are equivalent. (I) p(F̈ϖ, F̈ρ) ≤ p(ϖ, ρ)λ for all ϖ, ρ ∈ Ḧ with (ϖ, ρ) ∈ R̈, (II) p(F̈ϖ, F̈ρ) ≤ p(ϖ, ρ)λ for all ϖ, ρ ∈ Ḧ with [ϖ, ρ] ∈ R̈. Proof. The implication (II) =⇒ (I) is trivial. Coversely, we assume that (I) holds. Take ϖ, ρ ∈ Ḧ with [ϖ, ρ] ∈ R̈. if (ϖ, ρ) ∈ R̈, then (II) directly follows from (1). But, if (ρ,ϖ) ∈ R̈, then using the symmetry of p and (I), we obtain p(F̈ϖ, F̈ρ) = p(F̈ρ, F̈ϖ) ≤ p(ρ,ϖ)λ = p(ρ,ϖ)λ. which shows that (I) =⇒ (II). Proposition 2. If (Ḧ, p, k ≥ 1) is a b-metric space, R̈ is a binary relation on Ḧ, F̈ and S are self-mapping on Ḧ and λ ∈ [0, 1k ), then these conditions are equivalent. (1) p(F̈ϖ, F̈ρ) ≤ p(Sϖ, Sρ)λ for all ϖ, ρ ∈ Ḧ with (ϖ, ρ) ∈ R̈, (2) p(F̈ϖ, F̈ρ) ≤ p(Sϖ, Sρ)λ for all ϖ, ρ ∈ Ḧ with [ϖ, ρ] ∈ R̈. Definition 7. [23].“Let Ḧ be a non-empty set and R̈ be a binary relation on Ḧ. (1) The inverse, transpose or dual relation of R̈, denoted by R̈−1 is defined by R̈−1 = {(ϖ, ρ) ∈ Ḧ2 : (ρ,ϖ) ∈ R̈} (2) The reflexive closure of R̈, denoted by R̈#, is defined to be the set R̈ ∪ △ϖ (i.e., R̈# := R̈ ∪ △ϖ). (3) The symmetric closure of R̈, denoted by R̈s, is defined to be the set R̈ ∪R̈−1 (i.e., R̈# := R̈ ∪R̈−1). Proposition 3. [24] For a binary relation R̈ defined on a nonempty set Ḧ, (ϖ, ρ) ∈ R̈s ⇐⇒ [ϖ, ρ] ∈ R̈. Definition 8. [24]. Let Ḧ be a non-empty set and R̈ a binary relation on Ḧ. A sequence ϖn ⊂ Ḧ is called R̈- preserving if (ϖn, ϖn+1) ∈ R̈ for all n ∈ N0. Definition 9. [24] Let (Ḧ,p) be a metric space. A binary relation R̈ defined on Ḧ is called p-selfclosed if whenever {ϖn} is an R̈-preserving sequence and ϖn p−→ ϖ then there exists a subsequence {ϖnk } of {ϖn} with [ϖnk , ϖ] ∈ R̈ for all k ∈ N0. I. Alshammari et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2405-2418 2408 Definition 10. [24] Let Ḧ be a nonempty set and F̈ a self-mapping on Ḧ. A binary relation R̈ defined on Ḧ is called F̈-closed if for any ϖ, ρ ∈ Ḧ (ϖ, ρ) ∈ R̈ =⇒ (F̈ϖ, F̈ρ) ∈ R̈. Proposition 4. [24] Let Ḧ, F̈ and R̈ be same as in definition 1.10. R̈s must also be F̈-closed if R̈ is F̈-closed. Definition 11. [24] Let S and V are self mappings on a nonempty set Ḧ. A binary re- lation R̈ on Ḧ is called (S,V)-closed if for all ϖ, ρ ∈ Ḧ, (Vϖ, Vρ) ∈ R̈ yield that (Sϖ, Sρ) belong to R̈. if we take V= identity mapping, then we conclude that R̈ is S-closed. if R̈ is S-closed, then R̈s is also S-closed. Definition 12. [25] Let (Ḧ, p, k ≥ 1) be a b-metric space and let R̈ a binary relation on Ḧ. (i) we say that (ϖ, p) is R̈-complete if every R̈-preserving b-Cauchy sequence in Ḧ con- verges. (ii) A subset G of Ḧ is called R̈-closed if every R̈-preserving b-convergent sequence in G converges to a point of G. Definition 13. [25] Let (Ḧ, p, k ≥ 1) be a b-metric space and let V : Ḧ → Ḧ. A binary relation R̈ defined on Ḧ is called (V, bp)-self closed if, whenever {ϖn} is an R̈-preserving sequence and ϖn →p ϖ, there exists a subsequence {ϖni} of {ϖn} with [V ϖni , V ϖ] ∈ R̈ for all i ∈ N. If V is the identity mapping, then we get the following definitions: Definition 14. [25] Let (Ḧ, p, k ≥ 1) be a b-metric space. A binary relation R̈ defined on Ḧ is called bp-self closed if, whenever {ϖn} is an R̈-preserving sequence and ϖn →p ϖ, there exists a subsequence {ϖnj} of {ϖn} with (ϖnj , ϖ) ∈ R̈ for all j ∈ N. Definition 15. [26] Let Ḧ be a nonempty set and R̈ a binary relation on Ḧ. A subset G of Ḧ is called R̈-directed if for each ϖ, ρ ∈ G, there exists z ∈ Ḧ such that (ϖ, z) ∈ R̈ and (ρ, z) ∈ R̈. Definition 16. [27] Let Ḧ be a nonempty set and R̈ a binary relation on Ḧ. For ϖ, ρ ∈ Ḧ, a path of length k (where k is a natural number) in R̈ from ϖ to ρ is a finite sequence {t0, t1, t2, ....tk} ⊂ Ḧ satisfying the following conditions: (i) t0 = ϖ and tk = ρ (ii) (tj , tj+1) ∈ R̈ for each j (0 ≤ j ≤ k − 1). Note that although they are not necessarily distinct, a path of length k involves k+1 ele- ments of Ḧ”. I. Alshammari et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2405-2418 2409 Definition 17. [25] Let (Ḧ, p, k ≥ 1) be a b-metric space, let R̈ be a binary relation on Ḧ, and let S and V be two self-mappings on Ḧ. we say that S and V are R̈-compatible if, for any sequence {ϖn} ∈ Ḧ such that {Sϖn} and {V ϖn} are R̈-preserving and lim ϖ→+∞ V (ϖn) = lim ϖ→+∞ S(ϖn), we have lim ϖ→+∞ d(V P (ϖn), PV ϖn) = 0. Lemma 2. [28] Let Ḧ be a non empty set and let F̈ be a self mapping on Ḧ. Then there exists a subset G ⊆ Ḧ such that F̈(G) = F̈(Ḧ) and F̈ : G → Ḧ is one-to one. . 2. Main Result In this manuscript, we utilize the following notations: (i) F(F̈) = the set of all fixed points of F̈, (ii) Ḧ(F̈; R̈) := {ϖ ∈ Ḧ : (ϖ, F̈ϖ) ∈ R̈}, (iii) γ (ϖ,ρ,R̈) := the class of all paths in R̈ from ϖ to ρ Theorem 1. Let (Ḧ,p,k ≥ 1) be a b-complete b-multiplicative metric space and R̈ a binary relation on Ḧ. F̈: Ḧ × Ḧ be a self-mapping satisfying the following conditions given below. (i) Ḧ(F̈; R̈) is non-empty. (ii) R̈ is F̈-closed. (iii) Either F̈ is b-continuous or R̈ is bp-self closed. (iv) There exists λ ∈ [0, 1k ) such that. p(F̈ϖ, F̈ρ) ≤ p(ϖ, ρ)λ Then F̈ has a fixed point. i.e., there exists ϖ∗ ∈ Ḧ such that F̈ϖ∗ = ϖ∗. (v) γ(ϖ, ρ, R̈s) is non- empty, for each ϖ, ρ ∈ Ḧ, then F̈ has a unique fixed point. Proof. Let ϖ0 ∈ (F̈; R̈) be an arbitrary element. Now we define the sequence ϖn of picard iterates i.e., ϖn = F̈ϖn−1 = F̈nϖ0 for all n ∈ N. As (ϖ0, F̈ϖ0) ∈ R̈ and R̈ is F̈-closed, we get. (F̈ϖ0, F̈ 2ϖ0), (F̈ 2ϖ0, F̈ 3ϖ0), ......., (F̈ nϖ0, F̈ n+1ϖ0), ....,∈ R̈ I. Alshammari et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2405-2418 2410 so that (ϖn, ϖn+1) ∈ R̈, for all n ∈ N (1) therefore the sequence ϖn is R̈-preserving. Applying the contractivity condition (iv) to (1). We deduce, for all n ∈ N. that p(ϖn, ϖn+1) ≤ p(ϖn−1, ϖn) λ, which by induction yield that p(ϖn, ϖn+1) ≤ p(ϖ0, F̈ϖ0) λn for all n ∈ N (2) By using (2) and multiplicative triangular inequality, for all n ∈ N, r ∈ N, we have p(ϖn, ϖn+r) ≤ p(ϖn, ϖn+1) kn · p(ϖn+1, ϖn+2) kn+1 · · · ·p(ϖn+r−1, ϖn+r) kn+r−1 ≤ p(ϖn, ϖn+1) λnkn · d(ϖn+1, ϖn+2) λn+1kn+1 · · · ·p(ϖn+r−1, ϖn+r) λn+r−1kn+r−1 ≤ p(ϖ0, F̈ϖ0) (λk)n+(λk)n+1+···+(λk)n+r−1 ≤ p(ϖ0, F̈ϖ0) (λk)n 1−(λk) . This implies that p(ϖn, ϖn+r) →b 1, (as n → +∞) Hence, the sequence ϖn is multi- plicative Cauchy sequence in Ḧ. As (Ḧ, p, k ≥ 1) is b-complete, there exists ϖ∗ ∈ Ḧ such that ϖn −→ ϖ∗. Now, in lieu of (iii) assume that F̈ is b-continuous, we have ϖn+1 = F̈ϖn p−→ F̈ϖ∗. owing to the uniqueness of limit, we obtain F̈ϖ∗ = ϖ∗ i.e., ϖ∗ is a fixed point of F̈. Alternately, suppose that R̈ is bp − selfclosed. since ϖn is an R̈-preserving sequence and ϖn p−→ ϖ. by the bp − selfcloseness of R̈, there exists a subsequence {ϖnj} of {ϖn} with [ϖnj ,ϖ] ∈ R̈ for all j ∈ N using (iv), Proposition (1.1), we obtain p(ϖ∗, F̈ϖ∗) ≤ [p(ϖ∗, ϖn+1) · p(ϖn+1, F̈ϖ ∗)]k I. Alshammari et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2405-2418 2411 = [p(ϖ∗, ϖn+1) · p(ϖn+1, F̈ϖ ∗)]k ≤ [p(ϖ∗, ϖn+1) · p(ϖn+1, F̈ϖ ∗)λ]k → 1 as n → +∞. Hence, F̈ϖ∗ = ϖ∗ and ϖ∗ is a fixed point of F̈. suppose that ρ∗ is another fixed point of F̈. By assumption (v), there exists a path (say {t0, t1, t2, .....tk, }) of some finite length k in R̈s from ϖ to ρ so that t0 = ϖ, tk = ρ, [tj , tj+1] ∈ R̈ for each j (0 ≤ j ≤ k − 1). (3) As R̈ is F̈-closed, by using proposition (1.3) , we have [F̈ntj , F̈ ntj+1] ∈ R̈ for each j (0 ≤ j ≤ k − 1) and for each n ∈ N (4) Making use of (3), (4), (5), triangular inequality, assumption (iv) and proposition (1.1), we obtain p(ϖ, ρ) = p(F̈nt0, F̈ ntk) ≤ k−1∏ j=0 (F̈ntj , F̈ ntj+1) ≤ k−1∏ j=0 p(F̈n−1tj , F̈ n−1tj+1) λ ≤ k−1∏ j=0 p(F̈n−2tj , F̈ n−2tj+1) λ2 ≤ · · ·· ≤ k−1∏ j=0 p(tj , tj+1) λn → 1 as n → +∞ (5) so, that ϖ=ρ. Hence F̈ has a unique fixed point. Theorem 2. Let (Ḧ,d,k ≥ 1) be a b-complete b-multiplicative metric space and R̈ a binary relation on Ḧ. S,V: Ḧ → Ḧ be a self-mapping satisfying the following conditions given below. (i) Ḧ(S, V ; R̈) are non-empty and S(Ḧ) ⊆ V (Ḧ); (ii) R̈ is (S,V)-closed. (iii) There exists λ ∈ [0, 1k ) such that d(Sϖ, Sρ) ≤ d(V ϖ, V ρ)λ I. Alshammari et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2405-2418 2412 (iv) Either S is (V,R̈)-continuous or S and V are continuous. or (iv’) S and V are R̈- compatible, V is R̈- continuous, and either S is R̈-continuous or R̈ is (V, bd)− self − closed, Then S and V have a point of coincidence. Proof. let ϖ0 ∈ Ḧ(S,V,R̈) be an arbitrary element. Then (V ϖ0, Sϖ0) ∈ R̈. If V (ϖ0) = S(ϖ0), then ϖ0 is a coincidence point of S and V and, hence, we are through. otherwise, if V (ϖ0) ̸= S(ϖ0), then, in view of S(Ḧ) ⊆ V (Ḧ), we can choose ϖ1 ∈ Ḧ such that V (ϖ1) = S(ϖ0). Again from S(Ḧ) ⊆ V (Ḧ), we can choose ϖ2 ∈ Ḧ such that V (ϖ2) = S(ϖ1). construct the sequence {ϖn} ⊂ Ḧ such that V (ϖn+1) = S(ϖn) for all n ∈ N (6) Now,we claim that {V ϖn} is R̈- preserving sequence, i.e., (V ϖn, V ϖn+1) ∈ R̈ for all n ∈ N (7) we can show this fact by induction. By equation (6) (with n= 0) and fact that ϖ0 ∈ Ḧ(S, V, R̈), We conclude that (V ϖ0, V ϖ1) ∈ R̈. which means that (7) holds for n=0. suppose (7) is true for n = r ≥ 0 i.e., (V ϖr, V ϖr+1) ∈ R̈. As R̈ is (S,V)- closed, we get (Sϖr, Sϖr+1) ∈ R̈. by using , this yield that (V ϖr+1, V ϖr+2) ∈ R̈, i.e., inclusion (7) holds for n=r+1. Hence by induction, inclusion (7) is valid for all n ∈ N. in view of (6) and (7), the sequence {Sϖn} is also an R̈-preserving, i.e., (Sϖn, Sϖn+1) ∈ R̈ for all n ∈ N By using (6), (7)and assumption (iii), we find p(V ϖn, V ϖn+1) = p(Sϖn−1, Sϖn) ≤ p(V ϖn−1, V ϖn) λ for all n ∈ N (8) which by induction yield that p(V ϖn, V ϖn+1) = p(Sϖn−1, Sϖn) ≤ p(V ϖn−1, V ϖn) λn for all n ∈ N (9) By using (9) and multiplicative triangular inequality, for all n ∈ N, r ∈ N, we have I. Alshammari et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2405-2418 2413 p(V ϖn, V ϖn+r) ≤ p(V ϖn, V ϖn+1) kn · p(V ϖn+1, V ϖn+2) kn+1 · · · p(V ϖn+r−1, V ϖn+r) kn+r−1 ≤ p(V ϖn, V ϖn+1) λnkn · p(V ϖn+1, V ϖn+2) λn+1kn+1 · · · · p(V ϖn+r−1, V ϖn+r) λn+r−1kn+r−1 ≤ p(V ϖ0, V ϖ1) (λk)n+(λk)n+1+···+(λk)n+r−1 ≤ p(V ϖ0, V ϖ1) (λk)n 1−(λk) . This implies that p(V ϖn, V ϖn+r) →b 1, (as n → +∞) Hence, the sequence V ϖn is multiplicative Cauchy sequence in Ḧ. By using (3), we have V ϖn ⊆ S(Ḧ) and hence V ϖn is an R̈-preserving b-multiplicative Cauchy sequence in Ḧ. As (Ḧ, p, k ≥ 1) is b-complete, there exists u ∈ V (Ḧ) such that lim ϖ→+∞ V (ϖn) = V (u) (10) By using (6) and (10), we get lim ϖ→+∞ S(ϖn) = V (u) (11) Now we show that u is a coincidence point of S and V. Now, in lieu of (iv) consider that p is (V, R̈)-continuous, Thus utilizing (7) and (10) we obtain lim ϖ→+∞ S(ϖn) = S(u) (12) In view of (11) and (12), we obtain V(u)= S(u). Hence, we are completed. second , we assume that S and V are continuous and owing to the Lemma 1.1, there exists a subset G ⊆ Ḧ such that V (G) = V (Ḧ) and V : G → Ḧ is one to one. Now we define F̈ : V (G) → V (Ḧ) by F̈(V a) = S(a) for all V (a) ∈ V (G) where a ∈ G As V : G → Ḧ is injective and S(Ḧ) ⊆ V (Ḧ), we get to the conclusion that F̈ is well defined. Additionally, F̈ is continuous because S and V are continuous. As V (Ḧ) = V (G) and S(Ḧ) ∈ V (Ḧ), we get S(Ḧ) ∈ V (G). This means that, it is possible to construct {ϖn} ∈ G satisfying relation (6) and we choose u ∈ G. Utilizing equation (10) and (11) and the continuity of F̈, we find S(u) = F̈(V u) = F̈( lim n→+∞ V ϖn) = lim n→+∞ F̈(V ϖn) = lim n→+∞ S(ϖn) = V (u) Hence, u ∈ Ḧ is a point of coincidence of a pair of maps. This end the proof. Owing to (6), we have {V ϖn} ⊆ S(Ḧ) and hence, {V ϖn} is b-multiplicative Cauchy sequence in Ḧ. As Ḧ is b-complete, there exists u ∈ V (Ḧ) such that I. Alshammari et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2405-2418 2414 lim ϖ→+∞ V (ϖn) = V (u). (13) By using (6) and (13), we get lim ϖ→+∞ S(ϖn) = V (u). (14) As V is R̈-continuous, we find lim ϖ→+∞ V (V ϖn) = V ( lim n→+∞ V (ϖn)) = V (V (u)) (15) moreover, we get lim ϖ→+∞ V (Sϖn) = V ( lim n→+∞ S(ϖn)) = V (V (u)) (16) since {Sϖn} and {V ϖn} are R̈-preserving lim ϖ→+∞ S(ϖn) = V (u) = lim n→+∞ V (ϖn) (17) and S and V are R̈-compatible, we obtain lim ϖ→+∞ p(V S(ϖn), SV (ϖn)) = 0. (18) Now, we demonstrate that V(u) is a coincidence point of S and V. We assume that S is R̈-continuous. By using (7), we get lim ϖ→+∞ S(V ϖn) = S lim n→+∞ V (ϖn) == S(V (u)) (19) suppose that V(u) = z, utilizing triangle inequality, we get p(V z, Sz) ≤ [p(V z, V (Sϖn)) · p(V (Sϖn), Sz)] k ≤ p(V z, V (Sϖn)) k · [p(V (Sϖn), S(V ϖn) · p(S(V ϖn), Sz) k2 Making n → +∞ , we get p(Vz,Sz) = 1, which implies Vz = Sz, i.e., z = V(u) is coincidence point of S and V. Alternatively, assume that R̈ is (V, bp)-self closed. Since {V ϖn} is R̈-preserving and V ϖn → V u, in view of the (V, bp)-self closeness of R̈, there exists a subsequence {V ϖni} of {V ϖn} such that [V V ϖni , V V u] belongs to R̈ for all i ∈ N ∪ {0}. Since V ϖni → V u, in the view of proposition 1.3, we get p(SV ϖni , SV u) ≤ p(V V ϖni , V V u)λ for all i ∈ N ∪ {0} I. Alshammari et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2405-2418 2415 we choose Vu = z. By the triangle inequality , we get p(V z, Sz) ≤ [p(V z, V (Sϖni)) · p(V (Sϖni), Sz)] k ≤ p(V z, V (Sϖni)) k · [p(V (Sϖni), S(V ϖni) · p(S(V ϖni), Sz)] k2 ≤ p(V z, V (Sϖni)) k · p(V (Sϖni), S(V ϖni) k2 · p(S(V ϖni), Sz). λk2 Making i → +∞, we get p(Vz,Sz) = 1, which implies Vz = Sz, that is, z = V(u) is a coincidence point of S and V. Now we can give examples in support of theorem 1. Example 1. Let Ḧ = R+ and p = |ϖρ |, then (Ḧ,p) is a complete multiplicative metric space. Define binary relation R̈ = {(ϖ, ρ) ∈ R2 +: ϖ ρ ≥ 1, ϖ, ρ ∈ R+} on Ḧ. consider mapping F̈:Ḧ → Ḧ defined by F̈(ϖ) = ϖ 2 3 obviously, R̈ is F̈ closed and F̈ is continuous. Now, for ϖ, ρ ∈ Ḧ with (ϖ, ρ) ∈ R+. We have p(F̈ϖ, F̈ρ) = ∣∣∣∣ϖ 2 3 ρ 2 3 ∣∣∣∣ = ∣∣∣∣ϖρ ∣∣∣∣ 23 = p(ϖ, ρ) 2 3 < p(ϖ, ρ) 3 4 i.e., F̈ satisfies assumption (iv) of Theorem (2.1) for λ = 3 4 . Consequently, every condi- tions (i)-(iv) of Theorem (2.1) also holds and therefore, F̈ has a unique fixed point (for ϖ = 1). Example 2. Let Ḧ = [0.1, 1] and p = |ϖρ |, then (Ḧ,p) is complete B-MMS. Define binary relation R̈ = {(ϖ, ρ) ∈ [0.1, 1]2: ϖ ρ ≥ 1, ϖ, ρ ∈ R+} on Ḧ. consider mapping F̈ : Ḧ → Ḧ defined by F̈(ϖ) = eϖ−1−ϖ3 10 obviously, R̈ is F̈-closed and F̈ is continuous. Now, for ϖ, ρ ∈ [0.1, 1]. We have p(F̈ϖ, F̈ρ) = ∣∣∣∣ F̈ϖF̈ρ ∣∣∣∣ ≤ ∣∣∣∣ϖρ ∣∣∣∣λ = p(ϖ, ρ)λ for all ϖ, ρ ∈ X where, λ = 0.997, finally , we can say that F̈ has a unique fixed point 0.7411317711 ∈ X. REFERENCES 2416 Example 3. 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