EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6810 ISSN 1307-5543 – ejpam.com Published by New York Business Global Fixed Point Theorems for Mappings Contracting Perimeter of Triangles Embedded with F-Contractions in b-Metric Spaces Samina Batul1, Haitham Qawaqneh2, Hira Ulfat1, Dur-e-Shehwar Sagheer1, Hassen Aydi3,4,∗ 1 Department of Mathematics, Capital University of Science and Technology, Islamabad, Pakistan 2 Al-Zaytoonah University of Jordan, Amman 11733, Jordan 3 Université de Sousse, Institut Supérieur d’Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia 4 Department of Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa Abstract. In this article, the concept of a mapping contracting perimeter of triangles embedded with F-contractions in the framework of b-metric spaces is introduced. Some related fixed point results are established. Banach Contraction Principle is derived as a corollary of main result. Additionally, we construct examples of mappings contracting perimeters of triangles embedded with F-contractions which are not contraction mappings in the framework of b-metric spaces. The results of this article are the extensions of some already established results in literature. 2020 Mathematics Subject Classifications: 47H10, 54H25 Key Words and Phrases: Fixed point (FP), Banach contraction principle (BCP), metric space (MS), b-metric space (b-MS), mapping contracting perimeters of triangle (MCPT) 1. Introduction and Preliminaries Fixed point (FP) theory is a significant and highly active area within functional anal- ysis. It offers crucial methods for addressing problems encountered across multiple fields of mathematical analysis. This theory plays a key role in ensuring both the existence and uniqueness of solutions to integral and differential equations. For more details, see [1–9]. In 1922, the Polish mathematician Banach [10] introduced the contraction principle, which has become one of the most renowned and influential results in mathematics. In ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6810 Email addresses: samina.batul@cust.edu.pk (S. Batul), h.alqawaqneh@zuj.edu.jo (H. Qawaqneh), hiraulfat786@gmail.com (H. Ulfat), d.e.shehwar@cust.edu.pk (D. e. Shehwar Sagheer), hassen.aydi@isima.rnu.tn (H. Aydi) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 2 of 23 the existing literature, the Banach contraction principle (BCP) has been generalized in two main ways: either by modifying the contraction condition, or by altering the structure of the metric space (MS). Within FP theory, numerous types of contractions have been formulated in a MS, including Boyd and Wong nonlinear contraction [11], the Meir-Keeler contraction [12, 13], Suzuki contraction [14], Kannan contraction [15], Ćirić generalized contraction [16] and quasi contraction [17], weak contraction [18], Chatterjea contraction [19], Zamfirescu contraction [20], the F-Suzuki contraction [21], and among others [22, 23]. A MS is a vast concept, and even small modifications to its axioms can lead to the creation of different structures, such as a 2-MS [24], a cone MS [25], and many others. The notion of a b-metric space (b-MS) was pioneered by Bakhtin [26] in 1989, and later refined by Czerwik [27] in 1993. This innovation introduced a new coefficient in the triangular in- equality of a MS, laying the groundwork for the development of b-MSs. Researchers have developed a wide range of FP results utilizing the framework of b-MS. Karapinar [28] discusses foundational aspects and key contributions in FP theory within the framework of b-MS. In 2022, Berinde and Păcurar [29] survey the early progress and key issues in FP theory within b-MS. Ma et al. [30] introduced the concept of C∗-algebra-valued con- traction mappings. Building on this, Batul et al. [31] generalized the idea by relaxing the contraction condition initially proposed in [30]. In another development, Shehwar et al. [32] extended Caristi FP theorem to mappings defined on C∗-algebra-valued MSs. They demonstrated the existence of FPs by employing the concept of minimal elements within these spaces and introduced a partial order on the set U . Recently, Pasicki [33] explores the characteristics of Cauchy sequences in b-MS, contributing to a deeper understanding of their convergence properties. In 2012, Wardowski [34] introduced a novel type of contractions, known as an F-contraction, for real-valued functions defined on the set of positive real numbers and satisfying specific conditions. He also established a fixed point theorem for this class of contractions. Since then, numerous researchers have extended and explored F-contraction mappings within various types of MS. Fabiano et al. [35] present an in-depth overview of F-contractions, focusing on their origins, theoretical advancements and applications in generalized MS. Petrov [36] obtained some FP theorems for “mappings contracting perimeters of triangles” (MCPTs) in the framework of MSs. In this paper, the author [36] introduced a new type of mappings in MSs which can be characterized as a MCPT. Influenced by the work of Petrov, we bring to light some FP theorems for MCPTs embed- ded with F-contractions in the framework of b-MSs. Standard contraction mappings represent a notable subclass within this broader frame- work, enabling us to recover Banach classical result as a straightforward corollary. Fur- thermore, we provide examples of mappings that contract the perimeters of triangles embedded with F-contractions in b-MSs, but do not qualify as contraction mappings in the traditional sense. The following are some definitions and results which are useful for the proof of main theorems. Definition 1. [26] Let U be a nonempty set and let s ≥ 1 be a given real number. A S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 3 of 23 function σb : U × U → R+ is called a b-metric provided that, for all η, ξ, ζ ∈ U , (Mb1): σb(η, ξ) ≥ 0, (Mb2): σb(η, ξ) = 0 if and only if η = ξ, (Mb3): σb(η, ξ) = σb(ξ, η), (Mb4): σb(η, ζ) ≤ s [σb(η, ξ) + σb(ξ, ζ)]. The pair (U , σb) is called a b-MS. In general, a b-metric is not a continuous function. However, throughout the article, we will assume that the b-metric is continuous. Example 1. Let U = N. Define σb : U × U → [0,+∞) by σb(η, ξ) =  0, if η = ξ, 4α, if η, ξ ∈ {1, 2}, α, if η or ξ /∈ {1, 2} and η 6= ξ, where α > 0 is a constant. Here (U , σb) is a b-MS with s = 3. Definition 2. [34] Suppose F : R+ → R is a function that satisfies the following: (F-1): F is increasing, i.e., for all η, ξ ∈ R+ such that η < ξ, ⇒ F(η) < F(ξ). (F-2): For any sequence {ηn}∞n=1 of positive real numbers, lim n→+∞ ηn = 0 if and only if lim n→+∞ F(ηn) = −∞. (F-3): There exists k ∈ (0, 1) such that lim α→0+ αkF(α) = 0. Definition 3. [34] Let (U , σ) be a MS. A mapping Υ : U → U is said to be a Wardowski F-contraction if there are F ∈ F and τ > 0 such that η, ξ ∈ U , σ(Υη,Υξ) > 0 ⇒ τ + F(σ(Υη,Υξ)) ≤ F(σ(η, ξ)). In 2015, Cosentine et al. [37] introduced a new condition in Definition 2 to derive certain fixed point results in b-MSs. In this article, we further extend this definition by incorporating an additional condition into Definition 2. (F − 4): Let s ≥ 1 be a real number. For each sequence {βn}n∈N of positive real numbers such that τ + F(s2βn) ≤ F(βn−1) (1) for all n ∈ N and some τ > 0, then τ + F(snβn) ≤ F(sn−2βn−1). (2) Throughout the paper, F denotes the collection of mappings that satisfy (F−1) to (F−4). S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 4 of 23 2. Main Results The following section is concerned with the principal results of this paper. Definition 4. Consider a b-MS (U , σb) and (s ≥ 1) with at least three elements, i.e., |U| ≥ 3. A mapping Υ : U −→ U is said to be a MCPT embedded with an F-contraction on U if there exist F ∈ F and τ > 0 such that the following inequality τ + F(σb(Υη,Υξ) + σb(Υξ,Υζ) + σb(Υη,Υζ)) ≤ F ( 1 s2 (σb(η, ξ) + σb(ξ, ζ) + σb(η, ζ)) ) , (3) holds for all possible combinations of three pairwise distinct points η, ξ, ζ in U . Proposition 1. Let (U , σb) be a complete b-MS and Υ : U −→ U be a MCPT embedded with an F-contraction. Then Υ is continuous. Proof. Suppose that (U , σb) is a b-MS with |U| ≥3, Υ : U −→ U is a MCPT embedded with an F-contraction on U and let η0 be an isolated point in U . Then, clearly, Υ is continuous at η0. Let suppose that η0 be a limit point of U . Now, we show that for every ε > 0, there exists δ > 0 such that σb(Υη0,Υη) < ε whenever σb(η0, η) < δ. Since η0 is a limit point, for every δ > 0 there exists ξ ∈ U such that σb(η0, ξ) < δ. Using (3), we have F(σb(Υη0,Υη)) ≤ τ + F(σb(Υη0,Υη)), ≤ τ + F(σb(Υη0,Υη) + σb(Υη0,Υξ) + σb(Υη,Υξ)) ≤ F ( 1 s2 (σb(η0, η) + σb(η0, ξ) + σb(η, ξ)) ) ≤ F ( 1 s2 (σb(η0, η) + σb(η0, ξ) + s(σb(η0, η) + σb(η0, ξ))) ) ≤ F ( 1 s2 (1 + s)(σb(η0, η) + σb(η0, ξ)) ) < F ( 1 s2 (1 + s)(δ + δ) ) = F ( 2 1 s2 (1 + s)δ ) . (4) Setting δ = εs2 2(1 + s) , then equation (4) becomes F(σb(Υη0,Υη)) < F(ε). Since F in increasing, one has σb(Υη0,Υη)) < ε. Hence, the MCPT embedded with an F-contraction is continuous. S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 5 of 23 Definition 5. Consider a mapping Υ on the b-MS U . A point η ∈ U is said to be a periodic point of period n if Υn(η) = η, where n is the least positive integer for which Υn(η) = η, such a positive integer n is called the prime period of η. Theorem 1. Consider a complete b-MS (U , σb) with at least three elements, i.e., |U| ≥ 3. Assume that the mapping Υ : U −→ U satisfies a MCPT embedded with F-contraction condition on U . Then the following statements are true: i) The mapping Υ has a FP if and only if it does not have periodic points with a prime period 2. ii) Υ possesses at most two FPs. Proof. Suppose that no point is periodic with prime period 2 under the mapping Υ. Our objective is to show that Υ has a FP. Let η0 ∈ Υ and, Υη0 = η1,Υη1 = η2, · · · ,Υηn = ηn+1, · · · . Assume that, for all i = 0, 1, 2, · · · , there are no FP of the mapping Υ among the points ηi. Our goal is to demonstrate the distinctness of every point ηi. We have ηi 6= ηi+1 = Υηi because ηi is not a FP. We also know that ηi+2 = Υ(Υ(ηi)) 6= ηi since Υ lacks any periodic points of prime period 2. Moreover, ηi+1 6= ηi+2 = Υηi+1 since ηi+1 is not a FP. As a result, pairwise distinct points are ηi, ηi+1, and ηi+2. Furthermore, suppose that γ0 = σb(η0, η1) + σb(η1, η2) + σb(η2, η0), γ1 = σb(η1, η2) + σb(η2, η3) + σb(η3, η1), ... γn = σb(ηn, ηn+1) + σb(ηn+1, ηn+2) + σb(ηn+2, ηn), ... Applying the contraction condition to the pairwise distinct points ηi, ηi+1, and ηi+2, we obtain F (σb(η1, η2) + σb(η2, η3) + σb(η1, η3)) = F (σb(Υη0,Υη1) + σb(Υη1,Υη2) + σb(Υη0,Υη2)) , ≤ F ( 1 s2 ((σb(η0, η1) + σb(η1, η2) + σb(η0, η2)) ) − τ, F(γ1) ≤ F ( 1 s2 (γ0) ) − τ. Since F is increasing, we can write above equation as F(s2γ1) ≤ F(γ0)− τ. Similarly, F(s2γ2) ≤ F(γ1)− τ, F(s2γ3) ≤ F(γ2)− τ, ... F(s2γn) ≤ F(γn−1)− τ, S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 6 of 23 F(s2γn+1) ≤ F(γn)− τ. (5) Since s ≥ 1, one has γ0 > γ1 > · · · > γn > · · · . (6) Assume that j ≥ 3 is the smallest natural number such that ηj = ηi for some i satisfying 0 ≤ i < j − 2. Then, we have ηj+1 = ηi+1 and ηj+2 = ηi+2. Consequently, γi = γj which contradicts (6), This shows that all ηi’s are distinct. Further, we have to show that {ηn} is a Cauchy sequence. It is clear that F(s2γn+1) ≤ F(γn)− τ. Using (F − 2), F(sn+2γn+1) ≤ F(snγn)− τ. (7) It follows by induction that F(snγn) ≤ F(sn−2γn−1)− τ, ≤ F(sn−4γn−2)− 2τ, ≤ F(sn−6γn−3)− 3τ. By continuing this process, one can obtain F(snγn) ≤ F(γ0)− nτ. (8) Taking limit as n → +∞ in (8) to obtain lim n→+∞ F(snγn) → −+∞ which together with (F − 2) yield that lim n→+∞ snγn = 0. According to (F − 3), there is k ∈ (0,1) such that lim n→+∞ (snγn) kF(snγn) = 0. Multiplying (8) by (snγn) k leads to 0 ≤ (snγn) kF(snγn) + (snγn) knτ ≤ (snγn) kF(γ0). Taking limit as n → +∞, we get lim n→+∞ (snγn) kn = 0. As a result, it can be concluded that there is n1 ∈ N such that (snγn) kn ≤ 1, forall n ≥ n1. S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 7 of 23 Therefore, (snγn) k ≤ 1 n , forall n ≥ n1. It implies snγn ≤ 1 n 1 k , forall n ≥ n1. (9) This implies that the series +∞∑ i=1 siγi converges. Now, by the triangular inequality, for all n, p ∈ N, σb(ηn, ηn+p) ≤s(σb(ηn, ηn+1) + σb(ηn+1, ηn+p)). That is, σb(ηn, ηn+p) ≤sσb(ηn, ηn+1) + s2(σb(ηn+1, ηn+2) + σb(ηn+2, ηn+p)). Continuing in this way, σb(ηn, ηn+p) ≤ sσb(ηn, ηn+1) + s2σb(ηn+1, ηn+2) + s3σb(ηn+2, ηn+3) + · · ·+ spσb(ηn+p−1, ηn+p). (10) Putting n =0 in (7), F(s2γ1) ≤ F(γ0)− τ. (11) As γ1 = σb(η1, η2) + σb(η2, η3) + σb(η3, η1), (11) becomes, F(s2(σb(η1, η2) + σb(η2, η3) + σb(η3, η1))) ≤ F(γ0)− τ. That is, F(s2(σb(η1, η2) + σb(η2, η3) + σb(η3, η1))) ≤ F(γ0). Since F is increasing and s ≥1, σb(η1, η2) + σb(η2, η3) + σb(η3, η1)) ≤ γ0. Thus, σb(η1, η2) ≤ γ0. Continuing the same process, we obtain σb(η2, η3) ≤ γ1, σb(η3, η4) ≤ γ2, ... σb(ηn, ηn+1) ≤ γn−1, σb(ηn+1, ηn+2) ≤ γn, ... S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 8 of 23 Substituting in (10), σb(ηn, ηn+p) ≤ sγn−1 + s2γn + s3γn+1 + · · ·+ spγn+p−2, ≤ s(γn−1 + sγn + s2γn+1 + · · ·+ sp−1γn+p−2), ≤ s sn−1 (sn−1γn−1 + snγn + sn+1γn+1 + · · ·+ sn+p−2γn+p−2), ≤ 1 sn−2 ( n+p−2∑ i=n−1 siγi ) , ≤ 1 sn−2 ( +∞∑ i=n−1 siγi ) . Therefore, for all n ≥ n1 and p ∈ N, (9) implies that σb(ηn, ηn+p) ≤ 1 sn−2 ( +∞∑ i=n−1 siγi ) ≤ 1 sn−2 ( +∞∑ i=n−1 1 i 1 k ) . Taking limit as n −→ +∞, σb(ηn, ηn+p) −→ 0. It follows that {ηn} is a Cauchy sequence in U . As (U , σb) is complete, {ηn} has a limit η∗ in U . To show that Υη∗ = η∗, we apply the triangular inequality and inequality (3). For this, σb(η ∗,Υη∗) ≤s(σb(η ∗, ηn) + σb(ηn,Υη∗)), =sσb(η ∗, ηn) + sσb(Υηn−1,Υη∗), ≤sσb(η ∗, ηn) + s(σb(Υηn−1,Υη∗) + σb(Υηn−1,Υηn) + σb(Υηn,Υη∗)), ≤sσb(η ∗, ηn) + s( 1 s2 (σb(ηn−1, η ∗) + σb(ηn−1, ηn) + σb(ηn, η ∗))), ≤sσb(η ∗, ηn) + s( 1 s2 (σb(ηn−1, η ∗) + σb(ηn−1, ηn) + σb(ηn, η ∗)), ) ≤sσb(η ∗, ηn) + ( 1 s (σb(ηn−1, η ∗) + σb(ηn−1, ηn) + σb(ηn, η ∗))). Taking the limit as n → +∞, we note that each term in the preceding sum vanishes, and so σb(η ∗,Υη∗) = 0. Therefore, we conclude that Υη∗ = η∗. In order to prove there exists at most two FPs. Assume by contradiction that Υ has at least three pairwise distinct FPs, say η, ξ, and ζ. That is, Υη = η, Υξ = ξ and Υζ = ζ. Then by contraction condition, τ + F (σb(η, ξ) + σb(ξ, ζ) + σb(η, ζ)) = τ + F (σb(Υη,Υξ) + σb(Υξ,Υζ) + σb(Υη,Υζ)) , ≤ F ( 1 s2 (σb(η, ξ) + σb(ξ, ζ) + σb(η, ζ)) ) . S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 9 of 23 Since F is increasing, σb(η, ξ) + σb(ξ, ζ) + σb(η, ζ) ≤ 1 s2 (σb(η, ξ) + σb(ξ, ζ) + σb(η, ζ)), which is a contradiction, since s2 ≥ 1. Thus, we conclude that Υ possesses at most two FPs. Conversely, Suppose that Υ possesses a FP η∗. We have to prove that there are no periodic point in Υ with a prime period 2. For this, suppose by contradiction that Υ has a periodic point η of prime period 2, that is, Υ(Υη) = η. Define ξ = Υη and η = Υξ. Then τ + F(σb(Υη,Υξ) + σb(Υξ,Υη∗) + σb(Υη,Υη∗)) = F(σb(ξ, η) + σb(η, η ∗) + σb(ξ, η ∗)), which contradicts (3). Thus, Υ does not have periodic points with a prime period 2. Remark 1. In the assumption of Theorem 1, if we add an extra condition that the FP η∗ is the limit of some iterative sequence, then Υ has a unique FP. Now, we show that ηn 6= η∗foralln = 1,2,· · · . For this, let η0 be an initial point, and iterative sequence will be η1 = Υη0, η2 = Υη1, · · · . In that case, there is only one FP η∗. Assume in fact that Υ has another FP x∗∗. For every n = 1,2,· · · , it is evident that xn 6= η∗∗. Thus, for every n = 1,2,· · · , we have that the points η∗, η∗∗, and ηn are pairwise distinct. Now, by contraction condition, τ + F(σb(η ∗, η∗∗) + σb(η ∗, ηn+1) + σb(η ∗∗, ηn+1)) = τ + F(σb(Υη∗,Υη∗∗) + σb(Υη∗,Υηn) + σb(Υη∗∗,Υηn)) ≤ F ( 1 s2 (σb(η ∗, η∗∗) + σb(η ∗, ηn) + σb(η ∗∗, ηn)) ) . As n −→0, we get σb(η∗, ηn+1) −→ 0, σb(η∗, ηn) −→ 0 , σb(η∗∗, ηn+1) −→ σb(x ∗∗, x∗) and σb(η ∗∗, ηn) −→ σb(η ∗∗, η∗). Hence, τ + F(2σb(η ∗, η∗∗)) ≤ F( 1 s2 2σb(η ∗, η∗∗)), implies F(2σb(η ∗, η∗∗)) ≤ F( 2 s2 σb(η ∗, η∗∗)). Since F is increasing, 2σb(η ∗, η∗∗) ≤ 2 s2 σb(η ∗, η∗∗). which is a contradiction as s ≥ 1. This shows that η∗ = η∗∗. Therefore, Υ has a unique FP. The followings are examples of a MCPT embedded with F-contraction with exactly two FPs. Example 2. Let U = {2, 3, 10}. Let σb : U × U −→ R be defined as: σb(η, ξ) = (η − ξ)2, for all η, ξ ∈ U . S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 10 of 23 Then (U , σb) is a b-MS with s = 2. Define Υ : U −→ U by Υη = η,Υξ = ξ and Υζ = η. Consider τ + F(σb(Υη,Υξ) + σb(Υξ,Υζ) + σb(Υη,Υζ)) = τ + F(σb(η, ξ) + σb(ξ, ζ) + σb(η, η)), = τ + F(σb(η, ξ) + σb(ξ, η) + σb(η, η)), = τ + F(2σb(η, ξ)), = τ + F(2(η − ξ)2), = τ + F(2(2− 3)2), τ + F(σb(Υη,Υξ) + σb(Υξ,Υζ) + σb(Υη,Υζ)) = τ + F(2). (12) Now, F ( 1 s2 (σb(η, ξ) + σb(ξ, ζ) + σb(η, ζ)) ) =F ( 1 s2 ((η − ξ)2 + (ξ − ζ)2 + (η − ζ)2) ) , =F ( 1 22 ((2− 3)2 + (3− 10)2 + (2− 10)2 ) . By simplifying, one can get F ( 1 s2 (σb(η, ξ) + σb(ξ, ζ) + σb(η, ζ)) ) = F(28.5). (13) Hence, by (12) and (13), we conclude that τ + F(σb(Υη,Υξ) + σb(Υξ,Υζ) + σb(Υη,Υζ)) ≤ F ( 1 s2 (σb(η, ξ) + σb(ξ, ζ) + σb(η, ζ)) ) , with F(η) = ln(η) and τ = 1. Also, Υ has no periodic point of prime period 2. Take Υζ = η, Υ(Υζ) = Υ(η), Υ2(ζ) = η. Hence, Υ has no periodic point with prime period 2. Therefore, all the assumptions of Theorem (1) are true. Thus, Υ has exactly two FPs, namely η and ξ. Note: In the previous example, we verified the conditions of Theorem (1) using a continuous b-MS. Now, in the next example, we will use a discontinuous b-MS to verify Theorem 1. S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 11 of 23 Example 3. Let U = {2, 3, 10}. Let σb : U × U −→ R be defined for all η, ξ ∈ U as follows: σb =  0 if η = ξ, 1 if |η − ξ| = 1, 5 if |η − ξ| > 1. Then, one can easily verify that (U , σb) is a b-MS with s = 2. Define Υ : U −→ U by Υη = η,Υξ = ξ and Υζ = η. Consider τ + F(σb(Υη,Υξ) + σb(Υξ,Υζ) + σb(Υη,Υζ)) = τ + F(σb(η, ξ) + σb(ξ, ζ) + σb(η, η)), = τ + F(σb(η, ξ) + σb(ξ, η) + σb(η, η)), = τ + F(2σb(η, ξ)), = τ + F(2(1)), τ + F(σb(Υη,Υξ) + σb(Υξ,Υζ) + σb(Υη,Υζ)) = τ + F(2). (14) Now, F ( 1 s2 (σb(η, ξ) + σb(ξ, ζ) + σb(η, ζ)) ) =F ( 1 22 ((1) + (5) + (5) ) , =F ( 11 4 ) . By simplifying, one can get F ( 1 s2 (σb(η, ξ) + σb(ξ, ζ) + σb(η, ζ)) ) = F(2.75). (15) Hence, by (14) and (15), we conclude that τ + F(σb(Υη,Υξ) + σb(Υξ,Υζ) + σb(Υη,Υζ)) ≤ F ( 1 s2 (σb(η, ξ) + σb(ξ, ζ) + σb(η, ζ)) ) , with F(η) = ln(η) and τ = 0.01. Also, Υ has no periodic point of prime period 2. Take Υζ = η, Υ(Υζ) = Υ(η), Υ2(ζ) = η. Hence, Υ has no periodic point with prime period 2. Therefore, all the assumptions of Theorem (1) are true. Thus, Υ has exactly two FPs, namely η and ξ. In next example, we prove that if Υ has periodic points of prime period 2, then Υ has no FP. S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 12 of 23 Example 4. Let U = {η, ξ, ζ}. Define σb : U × U −→ R as σb(η, ξ) = (η − ξ)2, for all η, ξ ∈ U . Then one can prove that (U , σb) is a b-MS with s= 2. Now, define Υ : U −→ U by Υη = ξ,Υξ = η and Υζ = η. Then Υ has no FP. Here, η and ξ are periodic points with prime period 2. Consider Υη = ξ, Υ(Υη) = Υ(ξ), Υ2(η) = η. Also, Υξ = η, Υ(Υξ) = Υ(η), Υ2(ξ) = ξ. Definition 6. Let (U , σb) be a b-MS. Then a F-mapping Υ : U −→ U is called an F- contraction mapping on U if there exist a positive real number s ≥ 1 and τ > 0 such that τ + F(σb(Υη,Υξ)) ≤ F ( 1 s2 σb(η, ξ) ) , where F ∈ F and for all η, ξ ∈ U . (16) The following corollary provides a simple and direct proof of Banach FP theorem, in the framework of a b-MS. Corollary 1. Suppose that (U , σb) is a complete b-MS, where U 6= ∅. Then the F- contraction mapping Υ : U → U guarantees that Υ has a unique FP. Proof. Suppose that U is a complete b-MS and |U| = 1. Let U = {u}. In this case, since U has only one element η, the mapping Υ must map η to itself. That is, Υη = η. This is because there are no other element in U for Υη to map. So, we can see that η is indeed a FP of Υ, and it is unique. Therefore, the Banach FP theorem holds trivially for a set U of order 1. Now, if |U| = 2, suppose that U = {u, v} and Υ : U −→ U is an F-contraction mapping. Assume, if possible, that Υ has two distinct FPs η and ξ. Then, Υη = η and Υξ = ξ . By the definition of an F-contraction mapping, we have τ + F(σb(Υη,Υξ)) ≤ F( 1 s2 σb(η, ξ)) That is, F(σb(η, ξ)) ≤ F( 1 s2 σb(η, ξ))− τ. S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 13 of 23 Hence, F(σb(η, ξ)) ≤ F( 1 s2 σb(η, ξ)). F is increasing, so σb(η, ξ) ≤ 1 s2 σb(η, ξ). This is a contradiction since s ≥ 1. Therefore, our assumption that Υ has two distinct FPs is false. Hence, Υ can have at most one FP. So, for |U| = 1, 2 the proof is complete. Assume U has at least three elements, i.e., |U| ≥3, if Υ has some η ∈ U with prime period 2, i.e., Υ(Υ(η)) = η, then σb(η,Υη) = σb(Υη, η) = σb(Υη,Υ(Υη)). which is contradiction with (16). It follows that Υ has no periodic point with prime period 2. Considering pairwise distinct elements η, ξ, ζ ∈ U , and applying (16), we have τ + F(σb(Υ(η),Υ(ξ))) ≤ F( 1 s2 σb(η, ξ)). Let F(η) = ln(η), so τ + ln(σb(Υ(η),Υ(ξ))) ≤ ln( 1 s2 σb(η, ξ)). That is, eτσb(Υ(η),Υ(ξ)) ≤ 1 s2 σb(η, ξ). That is, σb(Υ(η),Υ(ξ)) ≤ e−τ s2 σb(η, ξ). (17) Similarly, one can get τ + F(σb(Υ(ξ),Υ(ζ))) ≤ F( 1 s2 σb(ξ, ζ)) implying σb(Υ(ξ),Υ(ζ)) ≤ e−τ s2 σb(ξ, ζ). (18) and τ + F(σb(Υ(η),Υ(ζ))) ≤ F( 1 s2 σb(η, ζ)). Thus, σb(Υ(η),Υ(ζ)) ≤ e−τ s2 σb(η, ζ). (19) S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 14 of 23 Adding (17), (18) and (19), one has σb(Υ(η),Υ(ξ)) + σb(Υ(ξ),Υ(ζ)) + σb(Υ(η),Υ(ζ)) ≤ e−τ s2 (σb(η, ξ) + σb(ξ, ζ) + σb(η, ζ)), with α = e−τ . Hence, Υ is a MCPT embedded with an F-contraction on U . By Theorem 1, a FP exists for the mapping Υ. For the uniqueness, suppose that Υ has two FPs η and η∗, i.e, Υη = η and Υη∗ = η∗. Now, by the definition of a b-metric and the given assumption, one writes 0 < F(σb(η, η ∗)) = F(σb(Υη,Υη∗)), < τ + F(σb(Υη,Υη∗)), F(σb(η, η ∗)) ≤ F( 1 s2 σb(η, η ∗)), since F is increasing, σb(η, η ∗) ≤ ( 1 s2 σb(η, η ∗)), where s ≥ 1, which is only possible when σb(η, η ∗) = 0. Thus, η = η∗. Hence, Υ has a unique FP. Proposition 2. Consider a b-MS (U , σb) with at least three elements, i.e, |U| ≥3, and Υ : U → U is a MCPT embedded with an F-contraction. Then, for all points ξ ∈ U , Υ is an F-contraction mapping if η is a limit point of U . Proof. Consider an accumulation point η ∈ U and any point ξ ∈ U . If ξ = η, then (16) is obviously satisfied. Now, consider the case where ξ 6= η. As η is a limit point, which implies the existence of a sequence {ζn} converging to η, satisfying ζn 6= x, ζn 6= ξ with all distinct elements ζn. Consequently, applying (3) establishes the following τ + F(σb(Υη,Υξ) + σb(Υξ,Υζn) + σb(Υη,Υζn)) ≤ F( 1 s2 (σb(η, ξ) + σb(ξ, ζn) + σb(η, ζn))), (20) which is satisfied for all n ∈ N. As σb(η, ζn) −→ 0, ζn −→ x and the continuity of b-MS implies σb(ξ, ζn) → σb(η, ξ). By continuity of Υ, σb(Υη,Υζn) → σb(Υη,Υη) = 0 and σb(Υξ,Υζn) → σb(Υη,Υξ). Taking limit n → +∞ in (20) gives τ + F(σb(Υη,Υξ) + σb(Υη,Υξ)) ≤ F( 1 s2 (σb(η, ξ) + σb(η, ξ))), τ + F(2σb(Υη,Υξ)) ≤ F( 2 s2 σb(η, ξ)), τ + F(σb(Υη,Υξ)) ≤ F( 1 s2 σb(η, ξ)). Hence, if η is a limit point of U , then Υ is an F-contraction. S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 15 of 23 Corollary 2. Consider Υ : U −→ U is a MCPT embedded with an F-contraction, and (U , σb) is a b-MS with at least three points, i.e, |U | ≥3. Then Υ is an F-contracting mapping whenever every element of U is an accumulation point of U . In a MS (U , σb), ξ is an intermediate point for η and ζ, whenever σb(η, ζ) = σb(η, ξ) + σb(ξ, ζ) where η, ξ, ζ ∈ U . (21) Let us develop an example demonstrating the distinction between a MCPT embedded with an F-contraction and an F-contraction in the framework of a b-MS. Example 5. Suppose U has countably infinite elements, |U| = ℵ0, specifically U = {η∗, η0, η1, . . .}. Consider a mapping Υ : U −→ U , that is, a MCPT embedded with an F-contraction, but exhibits non-contracting behavior in a b-MS U . Figure 1: The points of the space (U , σb) with consecutive distances between them Let a be a positive real number. Define σb on U × U as follows: σb(η, ξ) =  a2/2bi/2c, if η = ηi, ξ = ηi+1, i = 1, 2, 3, · · · ., σb(ηi, ηi+1) + · · ·+ σb(ηj−1, ηj), if η = ηi, ξ = ηj, i+ 1 < j, 4a2 − σb(η0, ηi), if η = ηi, ξ = η∗, 0, if η = ξ, where b·c is the floor function defined as the greatest integer less than or equal to a given number. Clearly, for every triplet of distinct points in U , one point is situated between the remaining two, we can see in Fig.1. Furthermore, the space has a single accumulation point η∗, and so it is complete. Define the mapping Υ : U → U given by Υ(ηi) = ηi+1 for all i ∈ N ∪ {0}, and Υ(η∗) = η∗. We can see that Υ is not an F-contraction mapping. Indeed, σb(η2n, η2n+1) = σb(Υη2n,Υη2n+1), for all n = 0,1,2,· · · . Now, we prove that Υ is a MCPT embedded with an F-contraction . Consider the first triplets of points ηi, ηj, η ∗ ∈ U with 0 ≤ i < j. According to the definition of the σb given above σb(ηi, ηj) + σb(ηj, η ∗) = σb(ηi, η ∗). and adding σb(ηi, η ∗) in both sides, one writes σb(ηi, ηj) + σb(ηj, η ∗) + σb(ηi, η ∗) = 2σb(ηi, η ∗), = 2(4a2 − σb(η0, ηi)), = 8a2 − 2σb(η0, ηi). S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 16 of 23 Also, σb(Υηi,Υηj) + σb(Υηj,Υη∗) =σb(Υηi,Υη∗). Adding σb(Υηi,Υη∗) in both sides, σb(Υηi,Υηj) + σb(Υηj,Υη∗) + σb(Υηi,Υη∗) =2σb(Υηi,Υη∗), =2(σb(ηi+1, η ∗)), =2(4a2 − σb(η0, ηi+1)), =8a2 − 2σb(η0, ηi+1). Using the formula for the sum of a geometric series with n terms, we get σb(η0, ηi) =  4a2 ( 1− ( 1 2 )n) , if i = 2n, 4a2 ( 1− ( 1 2 )n) − a2 2n−1 , if i = 2n− 1. n = 1, 2, · · · . By the equation (21), σb(η0, ηi+1) = σb(η0, ηi) + σb(ηi, ηi+1), = σb(η0, ηi) + a2 (2bi/2c) . Now, σb(Υηi,Υηj) + σb(Υηj,Υη∗) + σb(Υηi,Υη∗) σb(ηi, ηj) + σb(ηj, η∗) + σb(ηi, η∗) = 8a2 − 2σb(η0, ηi+1) 8a2 − 2σb(η0, ηi) , = 4a2 − σb(η0, ηi)− a2 (2bi/2c) 4a− σb(η0, ηi) , =  4a2 − 4a2 ( 1− ( 1 2 )n) − a2 (2bi/2c) 4a2 − 4a2(1− ( 1 2 )n) , if i = 2n, 4a2 − 4a2 ( 1− ( 1 2 )n) + a2 2n−1 − a2 (2bi/2c) 4a2 − 4a2(1− ( 1 2 )n) + a2 2n−1 , if i = 2n− 1, =  3 4 , if i = 2n, 2 3 , if i = 2n− 1. (22) S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 17 of 23 Considering ηi, ηj, ηk ∈ U with 0 ≤ i < j < k, Fig.1 illustrates that σb(ηi, ηj) = σb(ηi, ηi+1) + σb(ηi+1, ηi+2) + · · ·+ σb(ηj−1, ηj), (23) σb(ηj, ηk) = σb(ηj, ηj+1) + σb(ηj+1, ηj+2) + · · ·+ σb(ηk−1, ηk), (24) and σb(ηi, ηk) = σb(ηi, ηi+1) + · · ·+ σb(ηj−1, ηj) + · · ·+ σb(ηk−1, ηk). (25) Adding (23), (24) and (25) yields that σb(ηi, ηj) + σb(ηj, ηk) + σb(ηi, ηk) = 2(σb(ηi, ηi+1) + σb(ηi+1, ηi+2) + · · ·+ σb(ηk−1, ηk)). (26) Now, by the definition of σb, σb(Υηi,Υηj) = σb(ηi+1, ηj+1) = σb(ηi+1, ηi+2) + · · ·+ σb(ηj−1, ηj), (27) σb(Υηj,Υηk) = σb(ηj+1, ηk+1) = σb(ηj+1, ηj+2) + · · ·+ σb(ηk, ηk+1), (28) and σb(Υηi,Υηk) = σb(ηi+1, ηk+1) = σb(ηi+1, ηi+2) + · · ·+ σb(ηk, ηk+1). (29) Adding (27), (28) and (29) leads to σb(Υηi,Υηj) + σb(Υηj,Υηk) + σb(Υηi,Υηk) = 2(σb(ηi+1, ηi+2) + · · ·+ σb(ηk−1, ηk) + σb(ηk, ηk+1)). (30) By subtracting (26) and (30), one gets σb(ηi, ηj) + σb(ηj, ηk) + σb(ηi, ηk)− (σb(Υηi,Υηj) + σb(Υηj,Υηk) + σb(Υηi,Υηk)), = 2(σb(ηi, ηi+1) + σb(ηk, ηk+1)), = 2 ( a2 (2bi/2c) − a2 (2bk/2c) ) . Rearranging this equation, σb(Υηi,Υηj)+σb(Υηj,Υηk)+σb(Υηi,Υηk) = σb(ηi, ηj)+σb(ηj, ηk)+σb(ηi, ηk)−2 ( a2 (2bi/2c) − a2 (2bk/2c) ) . (31) We can see that i+ 1 < k , which implies that 2bi/2+1c < 2bk/2c, ⇒ 2.2bi/2c < 2bk/2c, ⇒ a2 2bk/2c ≤ a2 (2.2bi/2c) . Using this in (31) to obtain σb(Υηi,Υηj) + σb(Υηj,Υηk) + σb(Υηi,Υηk) ≤ σb(ηi, ηj) + σb(ηj, ηk) + σb(ηi, ηk)− 2a2 2bi/2c + a2 2bi/2c S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 18 of 23 ⇒ σb(Υηi,Υηj) + σb(Υηj,Υηk) + σb(Υηi,Υηk) ≤ σb(ηi, ηj) + σb(ηj, ηk) + σb(ηi, ηk)− a2 2bi/2c . (32) One can show that σb(ηi, η∗) ≤ 4σb(ηi, ηi+1). We have σb(ηi, ηk) ≤ σb(ηi, η ∗). consequently, we obtain σb(ηi, ηk) ≤ 4σb(ηi, ηi+1). Using equality (23) and the preceding inequality, we obtain σb(ηi, ηj) + σb(ηj, ηk) + σb(ηi, ηk) = 2σb(ηi, ηk), ≤ 8σb(ηi, ηi+1), = 8 a2 (2bi/2c) . By putting this inequality in (32), yields σb(Υηi,Υηj) + σb(Υηj,Υηk) + σb(Υηi,Υηk) ≤ σb(ηi, ηj) + σb(ηj, ηk) + σb(ηi, ηk)− 1 8 (σb(ηi, ηj) + σb(ηj, ηk) + σb(ηi, ηk)) σb(Υηi,Υηj) + σb(Υηj,Υηk) + σb(Υηi,Υηk) ≤ 7 8 (σb(ηi, ηj) + σb(ηj, ηk) + σb(ηi, ηk)). (33) By equations (22) and (33), inequality (3) satisfies for any three pairwise distinct points from the space U with F(η) = ln(η) and e−τ s2 = 7 8 = max { 2 3 , 3 4 , 7 8 } . It should be noted that the sequence of iterates of any two points, ηi and ηj, in the preceding example overlap sets. Let’s create an example of a mapping Υ : U −→ U that is a MCPT embedded with an F-contraction and is not a an F-contraction mapping. It has the feature that there are an infinite number of points such that the iteration sequences of these points are disjoint sets. Example 6. Consider the subset U ⊆ R consisting of {η0, η1, ...}∪[0, 1], where η2k = −4 2k and η2k+1 = −3 2k for k ≥ 0, illustrated in Fig.2. Figure 2: The b-MS (U , σb) . Let Υ : U −→ U be defined by Υηi = ηi+1 for all i ∈ {0} ∪ N and Υη = η 2 for η ∈ [0, 1]. S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 19 of 23 The mapping Υ satisfies the required condition for sequences of iterates of points in [0,1] of the form p 2k , where p is a prime number greater than or equal to 3 and k is the smallest natural number ensuring p 2k ⊆ [0, 1]. Setting s = 1 establishes an isometry between the previous example’s b-MS and the sub- space ({0, η0, η1, · · · }, σb) within (U , σb). The F-mapping Υ is defined in a similar manner for this subspace, and it follows that Υ is not an F-contraction mapping. We will demon- strate that for each of the three pairwise distinct points from the space (U , σb), inequality (3) is satisfied. The validity of this property for all distinct triplets in ({0, η0, η1, · · · }, σb) has been previously established. Since the b-metric σb is contractive on ([0, 1], σb), and every F-contraction reduces to triangle perimeters, we only need to prove inequality (3) for three pairwise distinct points η, ξ, ζ ∈ U satisfying: η < ξ < ζ where, η ∈ {η0, η1, · · · } and ζ ∈ (0, 1]. We begin by considering η = η2k = −4 2k . Subsequently, σb(η, ξ) + σb(ξ, ζ) + σb(η, ζ) = 2σb(η, ζ) = 2 ( 4 2k + ζ )2 . (34) From Υη = Υη2k, it follows that Υη2k = η2k+1 = −3 2k . which implies that σb(Υη,Υξ) + σb(Υξ,Υζ) + σb(Υη,Υζ) = 2σb(Υη,Υζ) = 2 ( 3 2k + ζ 2 )2 , = 32 × 2 ( 1 2k + ζ 6 )2 , = 9 16 × 2 ( 4 2k + 4ζ 6 )2 , ≤ 9 16 × 2 ( 4 2k + ζ )2 . By equation (34), one writes σb(Υη,Υξ) + σb(Υξ,Υζ) + σb(Υη,Υζ) ≤ 9 16 (σb(η, ξ) + σb(ξ, ζ) + σb(η, ζ)). We can see that it satisfies the inequality (3). Similarly, for η = η2k+1 = −3 2k , we have σb(η, ξ) + σb(ξ, ζ) + σb(η, ζ) = 2σb(η, ζ) = 2 ( 3 2k + ζ )2 . (35) S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 20 of 23 Applying Υ to η2k+1 yields Υη2k+1 = η2(k+1) = −4 2k+1 . We get σb(Υη,Υξ) + σb(Υξ,Υζ) + σb(Υη,Υζ) = 2σb(Υη,Υζ), = 2 ( 4 2k+1 + ζ 2 )2 , = 2 ( 4 2.2k + ζ 2 )2 , = 4× 2 ( 1 2k + ζ 4 )2 , = 4 9 × 2 ( 3 2k + 3ζ 4 )2 , ≤ 4 9 × 2 ( 3 2k + 2ζ )2 . Now, by equation (35), we obtain σb(Υη,Υξ) + σb(Υξ,Υζ) + σb(Υη,Υζ) ≤ 2 3 (σb(η, ξ) + σb(ξ, ζ) + σb(η, ζ)), which implies that inequality (3) holds with F(η) = ln(η). 3. Conclusion Using some well established results for “mappings contracting perimeter of triangles” the notion of a “mapping contracting perimeters of triangles (MCPT) embedded with an F-contraction in a b-metric space (b-MS)” has been introduced. The FP theorem has been proved and classical Banach FP theorem is derived as a simple corollary. Examples of a MCPT embedded with an F-contraction which are not contraction mappings in the framework of a b-MS have been established. These results open avenues for further re- search. Future work may explore the application of this framework in controlled, double controlled, partial and cone b-MSs. Authors’ Contributions All authors contribute equally in this paper. Acknowledgements We acknowledge the support of this research from Al-Zaytoonah University. S. Batul et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6810 21 of 23 Conflict of interest The authors declare that they have no conflict of interest. References [1] D. Judeh and M. Abu Hammad. Applications of conformable fractional pareto prob- ability distribution. International Journal of Advances in Soft Computing and Its Applications, 14(2):116–124, 2022. [2] T. Kanan, M. Elbes, K. Abu Maria, and M. Alia. 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