EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 2384-2404 ISSN 1307-5543 – ejpam.com Published by New York Business Global Some Geraghty type Inequalities in b-Fuzzy Metric Spaces with an Application Vineeta Chandra1, Uma Devi Patel1,∗, Stojan Radenović2 1Department of Mathematics, Guru Ghasidas Vishwavidyalaya (A Central University), Koni, Bilaspur-495009, Chhattisgarh, India 2Faculty of Mechanical Engineering, University of Belgrade, 11120 Belgrade, Serbia Abstract. In this note, we introduce novel Geraghty-type inequalities within the framework of a b-fuzzy metric space and develop new fixed point theorems for such mappings in a G-complete b-fuzzy metric space. To substantiate our findings, we present several illustrative examples using graphical methods. Additionally, we demonstrate the application of our introduced theorems by solving a non-linear integral equation, showing the practical utility of our results. 2020 Mathematics Subject Classifications: 54H25, 47H10 Key Words and Phrases: b-fuzzy metric space, Geraghty type mapping, α-Suzuki Geraghty type contraction 1. Introduction and Preliminaries For the first time, the traditional metric space framework was extended by incorpo- rating fuzzy logic to address uncertainties in distance measurements by Kramosil and Michálek [7]. In a classical metric space, the distance between two points is precisely defined by a real number, adhering to strict metric properties. However, in a fuzzy metric space, distances are represented by fuzzy sets, allowing for a range of values that reflect varying degrees of proximity. Later, George and Veermani [4] modified the definition of fuzzy metric space given by Kramosil and Michálek [7] and proved some fixed point re- sults. Inspired by this, concept of fuzzy b-metric space was introduced by Sedghi et al. [15], where the triangle inequality is replaced by a weaker one by involving b > 1, with this weaker inequality, The researchers introduced many contractive inequalities to obtain fixed point, see([1], [3], [6], [8]). In the line of this, We introduce the concepts of Ger- aghty type inequalities in this b-fuzzy metric spaces and we introduce the notion of fuzzy α-Geraghty type mapping within the context of b-fuzzy metric space. Additionally, we ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5428 Email addresses: umadevipatel@yahoo.co.in (U. D. Patel), vineetachandra4@gmail.com (V. Chandra), radens@beotel.net (S. Radenović) https://www.ejpam.com 2384 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) V. Chandra, U. D. Patel, S. Radenović / Eur. J. Pure Appl. Math, 17 (4) (2024), 2384-2404 2385 introduce the idea of α-Suzuki Geraghty type mapping in the framework of G-complete b-fuzzy metric space, and we investigate specific fixed point problems associated with this generalizations. We offer several illustrative examples with the graphical approach in the support of our findings. In last, as an application, we discuss a solution to a non-linear integral equation via fixed point tools. We must need the following: Definition 1. ([14]). A function ⋄ : [0, 1]2 → [0, 1] is called a continuous triangular-norm if • ⋄ is commutative and associative; • ⋄ is continuous; • 1 ⋄ a = a; • a ⋄ b ≥ c ⋄ d, whenever a ≥ c and b ≥ d. for all a, b, c, d ∈ [0, 1]. Some of the l-norms are a ⋄m b = min{a, b} (minimum), a ⋄p b = ab (product), a ⋄L b = max{a+ b− 1, 0}. Definition 2. [6]. A b-fuzzy metric space is an ordered triple (Ȳ ≠ ϕ,Mz, ⋄), where Mz : Ȳ2 × (0,+∞) → [0, 1] satisfying (i) Mz(δ, γ, l) > 0; (ii) Mz(δ, γ, l) = 1 if and only if δ = γ; (iii) Mz(δ, γ, l) = Mz(γ, δ, l); (iv) Mz(δ, γ, b(l + r)) ≥ Mz(δ, η, l) ⋄Mz(η, γ, r), where b ≥ 1; (v) Mz(δ, γ, .) : (0,+∞) → (0, 1] is continuous from left and lim l→+∞ Mz(δ, γ, l) = 1. for all δ, γ, η ∈ Ȳ and l, r > 0. Note: If b = 1 then definition (2) will become a fuzzy metric space. Example 1. Let Mz(δ, γ, l) = e− |δ–γ|p l , where p > 1 is a real number, Mz is a b-fuzzy metric with b = 2p–1 but not fuzzy metric space. Definition 3. [8] Suppose (Ȳ,Mz, ⋄) is a b-fuzzy metric space. Then (i) {δn} is called G-convergent sequence if there exists δ ∈ Ȳ such that lim n→+∞ Mz(δn, δ, l) = 1. V. Chandra, U. D. Patel, S. Radenović / Eur. J. Pure Appl. Math, 17 (4) (2024), 2384-2404 2386 (ii) {δn} in Ȳ is called a G-Cauchy sequence if lim n,m→+∞ Mz(δn, δm, l) = 1 for all m,n ∈ N and l > 0. (iii) The space is called complete if every Cauchy sequence is convergent in Ȳ. Geraghty ([5]) introduced a category denoted as B which is a collection of maps defined as β : [0,+∞) → [0, 1) satisfying β(tn) → 1 as n → +∞ ⇒ tn → 0 as n → +∞. Researchers introduced many contractive inequalities to obtain fixed points in this fuzzy space. In the line of this, we introduce the concepts of Geraghty type inequalities in this b-fuzzy metric space and we inject the notion of fuzzy α-Geraghty type mapping within the context of b-fuzzy metric space. Additionally, we introduce the idea of α-Suzuki-Geraghty type mapping in G-complete b-fuzzy metric space, and we investigate specific fixed point problems associated with these generalizations. We offer several illustrative examples with the graphical approach in support of our findings. In last, as an application, we discuss a solution to a non-linear integral equation via fixed point tools. 2. Main Results We must require to introduce the following definitions. Definition 4. A b-fuzzy metric Mz is said to be C-triangular, if for all δ, γ, η ∈ Ȳ and l > 0, Mz(δ, γ, l) ≥ Mz(δ, η, l) +Mz(η, γ, l)− 1 (1) holds. Definition 5. A self map L defined on a G-complete b-fuzzy metric space (Ȳ,Mz, ⋄) is called a Geragthy type-I contractive if 1−Mz(Lδ,Lγ, l) ≤ (1−Mz(δ, γ, l)) · β(1−Mz(δ, γ, l)) (2) where β ∈ B, for all δ, γ ∈ Ȳ and l > 0. Now we write a theorem for such introduced Geraghty type-I contractive mapping using C-triangular property. Theorem 1. Suppose (Ȳ,Mz, ⋄) is a G-complete b-fuzzy metric space with C-triangular fuzzy metric and a self map L defined on Ȳ is a Geraghty type-I contractive map. Then L has a unique fixed point in Ȳ. V. Chandra, U. D. Patel, S. Radenović / Eur. J. Pure Appl. Math, 17 (4) (2024), 2384-2404 2387 Proof. Consider a Picard sequence {δn} such that δn+1 = Lδn. We assume that δn ̸= δn+1 for all n ∈ N ∪ {0}, otherwise we will get fixed point. Now 1−Mz(δn+1, δn+2, l) ≤ β(1−Mz(δn, δn+1, l))(1−Mz(δn, δn+1, l)) < 1−Mz(δn, δn+1, l) (3) Thus, we conclude that Mz(δn+1, δn+2, l) ≥ Mz(δn, δn+1, l) for all n ∈ N. Hence {Mz(δn+1, δn, l)} is an increasing sequence of positive real numbers in (0, 1]. So, there exists s(l) ∈ (0, 1] such that lim n→+∞ Mz(δn, δn+1, l) = s(l) for all l > 0. Now, we need to prove s(l) = 1. Suppose s(l0) < 1, for any l0 > 0. By (3), we obtain lim n→+∞ β(1−Mz(δn, δn+1, l)) = 1. Since β ∈ B. This implies that lim n→+∞ Mz(δn, δn+1, l) = s(l) = 1, a contradiction to our assumption. Hence, we conclude that lim n→+∞ Mz(δn, δn+1, l) = 1, (4) for all l > 0. Next, we need to show {δn} is a Cauchy sequence. Consider a contrary, λ = lim n,m→+∞ Mz(δn, δm, l) < 1. (5) By (2), 1−Mz(δn+1, δm+1, l) ≤ β(1−Mz(δn, δm, l))(1−Mz(δn, δm, l)). Taking the limit as n,m → +∞ in the above inequality where n > m, then we get lim n,m→+∞ (1−Mz(δn+1, δm+1, l)) ≤ lim n,m→+∞ β(1−Mz(δn, δm, l))(1−Mz(δn, δm, l)). By using (5), we get lim n,m→+∞ (1−Mz(δn+1, δm+1, l)) ≤ lim n,m→+∞ β(1−Mz(δn, δm, l))(1− λ). (6) On the flip side, using C-traingular property 1−Mz(δn, δm, l) ≤ 1−Mz(δn, δn+1, l) + 1−Mz(δn+1, δm, l) ≤ 1−Mz(δn, δn+1, l) + 1−Mz(δn+1, δm+1, l) + 1−Mz(δm+1, δm, l). Putting limit as n,m → +∞ and using (4) and (6), we get (1− λ) ≤ lim n,m→+∞ (1−Mz(δn+1, δm+1, l)) ≤ lim n,m→+∞ β(1−Mz(δn, δm, l))(1− λ) V. Chandra, U. D. Patel, S. Radenović / Eur. J. Pure Appl. Math, 17 (4) (2024), 2384-2404 2388 this implies lim n,m→+∞ β(1−Mz(δn, δm, l)) = 1 which implies lim n,m→+∞ (1−Mz(δn, δm, l)) = 0. This yields that lim n,m→+∞ Mz(δn, δm, l) = λ = 1, a contradiction with the assumption (5). Therefore, sequence {δn} is a G-Cauchy in Ȳ. Since the space Ȳ is complete then there exists u ∈ Ȳ such that sequence {δn} converges to u, lim n→+∞ Mz(δn, u, l) = 1, (7) for all l > 0. Next we need to show u is a fixed point of L. 1−Mz(δn+1,Lu, l) ≤ (1−Mz(δn, u, l))β(1−Mz(δn, u, l)), consider limit as n → +∞, this implies lim n→+∞ Mz(δn+1,Lu, l) = 1, (8) for all l > 0. Using triangle inequality, we write Mz(u,Lu, l) ≥ Mz(u, δn+1, l) ⋄Mz(δn+1,Lu, l). considering limit as n → +∞ and with (4) and (8), we obtain Mz(u,Lu, l) = 1 for all l > 0. Consider v is another fixed point of L such that u ̸= v, Mz(u, v, l) < 1. Thus 1−Mz(u, v, l) = 1−Mz(Lu,Lv, l) ≤ (1−Mz(u, v, l))β(1−Mz(u, v, l)) < 1−Mz(u, v, l). We get a contrary, thus fixed point is unique. Example 2. Consider Ȳ = [0, 1] and let Mz : Ȳ × Ȳ → [0, 1] defined by Mz(δ, γ, l) = e− |δ−γ|2 l+0.5 and Mz be a C-triangular for all δ, γ ∈ Ȳ and l > 0. Then (Ȳ,Mz, ⋄) is a G-complete b-fuzzy metric space. Consider the mapping L : Ȳ → Ȳ defined by L(δ) = { 1 3δ 2, if δ ∈ [0, 1) 1 4 , if δ = 1, for all δ, γ ∈ Ȳ and l > 0. Now, in the following three cases will be formed for which Geraghty type-I contraction is to be verified for β(t1) = 1− t1. Case 1. If δ, γ ∈ [0, 1) then Mz(Lδ,Lγ, l) = e− |Lδ−Lγ|2 l+0.5 V. Chandra, U. D. Patel, S. Radenović / Eur. J. Pure Appl. Math, 17 (4) (2024), 2384-2404 2389 = e− 1 9 |δ2−γ2|2 l+0.5 1−Mz(Lδ,Lγ, l) = (1− e− |Lδ−Lγ|2 l+0.5 ) = (1− e− 1 9 |δ2−γ2|2 l+0.5 ) ≤ e− |δ−γ|2 l+0.5 (1− e− |δ−γ|2 l+0.5 ) (9) = β(1−Mz(δ, γ, l))(1−Mz(δ, γ, l)), Case 2. If δ ∈ [0, 1), γ = 1 then Mz(Lδ,Lγ, l) = e− |Lδ−Lγ|2 l+0.5 = e− | δ 2 3 − 1 4 |2 l+0.5 1−Mz(Lδ,Lγ, l) = 1− e− | δ 2 3 − 1 4 |2 l+0.5 ≤ e− |δ−1|2 l+0.5 (1− e− |δ−1|2 l+0.5 ) (10) = β(1−Mz(δ, γ, l))(1−Mz(δ, γ, l)), Case 3. If δ = γ = 0 then the Geragthy type-I contraction trivially holds. Now, the graphical representation of cases 1 and 2; Graphs of two functions: Yellow - 1-Exp@ -1 9 Abs@∆^2-Γ^2D^2�lD, Red - Exp@ -Abs@∆ - ΓD^2 l D H1-Exp@-Abs@∆-ΓD^2�lDLL 0.0 0.2 0.4 0.6 0.8 ∆ 0.0 0.2 0.4 0.6 0.8 Γ 0.0 0.1 0.2 z Figure 1: (1− e− 1 9 |δ2−γ2|2 l+0.5 ) ≤ e− |δ−γ|2 l+0.5 (1− e− |δ−γ|2 l+0.5 ) Graphs of two functions: Yellow - 1-Exp@-Abs@ ∆^2 3 - 1 4 D^2�lD, Red - Exp@ -Abs@∆ - 1D^2 l D H1-Exp@-Abs@∆-1D^2�lDLL 0.0 0.2 0.4 0.6 0.8 ∆ 0.6 0.8 1.0 Γ 0.0 0.1 0.2 z Figure 2: 1− e− | δ 2 3 − 1 4 |2 l+0.5 ≤ e− |δ−1|2 l+0.5 (1− e− |δ−1|2 l+0.5 ) V. Chandra, U. D. Patel, S. Radenović / Eur. J. Pure Appl. Math, 17 (4) (2024), 2384-2404 2390 In Figure 1, the yellow colour represents the L.H.S. and the red colour represents the R.H.S. of equation (9), and in Figure 2, the yellow colour represents the L.H.S. The red colour represents the R.H.S. of equation (10). From the graphical representations it is clearly visible that the contraction (2) is satis- fied. Hence, the inequality holds in these cases for l > 0 and β(t1) = 1 − t1. Now, for all δ, γ, η ∈ [0, 1], then it is easy to check that Mz is C-triangular. Hence, all assumptions of theorem (1) are satisfied for l > 0 and β(t1) = 1− t1, and 0 is a unique fixed point of L. Now we recall the following definitions from [2]. Definition 6. [2] A self map L is said to be a triangular α-admissible if there exists α : Ȳ2 × (0,+∞) → R such that (i) α(δ, γ, l) ≥ 1 ⇒ α(Lδ,Lγ, l) ≥ 1 (ii) α(δ, η, l) ≥ 1 and α(η, γ, l) ≥ 1 ⇒ α(δ, γ, l) ≥ 1 for all δ, γ, η ∈ Ȳ and any l > 0. Lemma 1. [2] Consider a fuzzy metric space denoted by (Ȳ,Mz, ⋄) and let L : Ȳ → Ȳ be a triangular α-admissible mapping. Suppose there exists an element δ0 ∈ Ȳ such that α(δ0,Lδ0, l) ≥ 1. Let us define a {δn} recursively by setting δn+1 = Lδn. Then α(δm, δn, l) ≥ 1, for all m,n ∈ N with m < n and l > 0. Next we define a new contractive inequality in Suzuki view. Definition 7. A triangular α-admissible self mapping L defined on a b-fuzzy metric space (Ȳ,Mz, ⋄) is called a α-Suzuki-Geraghty type-I if there exists a β ∈ B such that Mz(δ,Lδ, l) > q · Mz(δ, γ, l) ⇒ α(δ, γ, l)(1−Mz(Lδ,Lγ, l)) ≤ β(1−Mz(δ, γ, l))(1−Mz(δ, γ, l)) (11) where q ∈ (0, 1) and δ, γ ∈ Ȳ. Now we write a few definitions which are essential for our next result. Definition 8. A triangular α-admissible self mapping L is defined on a b-fuzzy metric space (Ȳ,Mz, ⋄) is said to have property G1 if for any two sequences {δn}, {δm} in Ȳ such that lim n,m→+∞ Mz(δn, δm, l) = a(l) ∈ (0, 1], where n > m and n,m ∈ N, q ∈ (0, 1), then Mz(δn, δn+1, l) > q · Mz(δn, δm, l), for all l > 0. V. Chandra, U. D. Patel, S. Radenović / Eur. J. Pure Appl. Math, 17 (4) (2024), 2384-2404 2391 Definition 9. A triangular α-admissible self mapping L is defined on a b-fuzzy metric space (Ȳ,Mz, ⋄) is said to have property G2 if for any convergent sequence {δn} in Ȳ converging to u, Mz(δn,Lδn, l) > q · Mz(δn, u, l), where n ∈ N and q ∈ (0, 1). Example 3. Let Ȳ = [0, 1] and define Mz(δ, γ, l) = l l+|δ−γ|2 . Let (Ȳ,Mz, ⋄) is a G- complete b-fuzzy metric space. Let a self-map L : Ȳ → Ȳ defined by L(δ) = { 1, if δ ∈ (0, 1] 0 otherwise. Let δn = 1− 1 n and δm = 1− 1 m , with n,m ∈ N. Since lim n→+∞ Mz(δn, δn+1, l) = lim n→+∞ Mz ( 1− 1 n , 1− 1 n+ 1 , l ) ∈ (0, 1], Then, Mz(δn, δn+1, l) > q · Mz(δn, δm, l) where q = 1 2 , Definition (8) satisfied. We have Mz(δn,Lδn, l) = Mz(1− 1 n , 1, l) > q ·Mz(1− 1 n , 1, l) for all n ∈ N and q ∈ (0, 1). Hence, Definition 9 holds. Theorem 2. Consider a self map L defined on a G-complete b-fuzzy metric space (Ȳ,Mz, ⋄) where fuzzy metric is C- triangular satisfying: (i) map L is b-fuzzy α-Suzuki-Geraghty type-I; (ii) L has property G1 and G2; (iii) there exists δ0 ∈ Ȳ such that α(δ0,Lδ0, l) ≥ 1 for all l > 0; (iv) if α(δn, δn+1, l) ≥ 1 and δn → u as n → +∞, then α(δn, u, l) ≥ 1 for all n ∈ N. Then L has a fixed point. Proof. By assumption (3), there exists δ0 ∈ Ȳ such that α(δ0, δ1, l) ≥ 1 for all l > 0 and define a sequence {δn} in Ȳ by δn+1 = Lδn for all n ∈ N. Suppose that δn = δn+1 for some n ∈ N∪{0} no need to prove anything automatically completed. Suppose δn ̸= δn+1 for all n ∈ N. By lemma 1, we have α(δn, δn+1, l) ≥ 1, (12) for all n ∈ N and l > 0. By (11), Mz(δn,Lδn, l) > q · Mz(δn, δn+1, l) implies V. Chandra, U. D. Patel, S. Radenović / Eur. J. Pure Appl. Math, 17 (4) (2024), 2384-2404 2392 α(δn, δn+1, l)(1−Mz(Lδn,Lδn+1, l)) ≤ β(1−Mz(δn, δn+1, l))(1−Mz(δn, δn+1, l)). Now (1−Mz(δn+1, δn+2, l)) = (1−Mz(Lδn,Lδn+1, l)) ≤ α(δn, δn+1, l)(1−Mz(Lδn,Lδn+1, l)) ≤ β(1−Mz(δn, δn+1, l))(1−Mz(δn, δn+1, l)) < (1−Mz(δn, δn+1, l)). (13) This concludes that {Mz(δn, δn+1, l)} is non-decreasing sequence of positive real number in (0, 1]. So there exists s(l) ∈ (0, 1] such that lim n→+∞ Mz(δn, δn+1, l) = s(l). Suppose to the contrary, s(l0) < 1 for any l0 > 0. Now put limit as n → +∞ lim n→+∞ β(1−Mz(δn, δn+1, l)) = 1. By the characteristic of B, we have lim n→+∞ Mz(δn, δn+1, l) = 1, (14) a contradiction. Hence, we need to show {δn} is a G-Cauchy sequence. Suppose λ = Mz(δn, δm, l) < 1, By using property (G1), Mz(δn, δn+1, l) > q · Mz(δn, δm, l) implies α(δn, δm, l)(1−Mz(Lδn,Lδm, l)) ≤ β(1−Mz(δn, δm, l))(1−Mz(δn, δm, l)). We have (1−Mz(δn+1, δm+1, l)) = (1−Mz(Lδn,Lδm, l)) ≤ α(δn, δm, l)(1−Mz(Lδn,Lδm, l)) ≤ β(1−Mz(δn, δm, l))(1−Mz(δn, δm, l)). Taking the limit as n,m → +∞ and lemma 1, lim n,m→+∞ (1−Mz(δn+1, δm+1, l)) ≤ lim n,m→+∞ α(δn, δm, l)(1−Mz(δn+1, δm+1, l)) ≤ lim n,m→+∞ β(1−Mz(δn, δm, l))(1− λ). (15) On the flip side, (1−Mz(δn, δm, l)) ≤ (1−Mz(δn, δn+1, l)) + (1−Mz(δn+1, δm, l)) ≤ (1−Mz(δn, δn+1, l)) + (1−Mz(δn+1, δm+1, l)) + (1−Mz(δm+1, δm, l)). V. Chandra, U. D. Patel, S. Radenović / Eur. J. Pure Appl. Math, 17 (4) (2024), 2384-2404 2393 Putting limit as n,m → +∞ and using (14) and (15), (1− λ) ≤ lim n,m→+∞ (1−Mz(δn+1, δm+1, l)) ≤ lim n,m→+∞ β(1−Mz(δn, δm, l))(1− λ). which gives lim n,m→+∞ β(1−Mz(δm, δn, l)) = 1, lim n,m→+∞ Mz(δm, δn, l) = 1. Which is a contradiction with λ. Thus, {δn} is a G-Cauchy sequence. Since Ȳ is a G-complete, there exists u ∈ Ȳ such that lim n→+∞ Mz(δn, u, l) = 1. (16) By the property of (G2), Mz(δn−1,Lδn−1, l) > q · Mz(δn−1, u, l) ⇒ α(δn−1, u, l)(1−Mz(Lδn−1,Lu, l)) ≤ β(1−Mz(δn−1, u, l))(1−Mz(δn−1, u, l)) (1−Mz(Lδn−1,Lu, l)) ≤ α(δn−1, u, l)(1−Mz(Lδn−1,Lu, l)) ≤ β(1−Mz(δn−1, u, l))(1−M(δn−1, u, l)) < (1−M(δn−1, u, l)) 1−Mz(δn,Lu, l) < 1−Mz(δn−1, u, l), Put limit as n → +∞, we get Mz(u,Lu, l) = 1. that is, Lu = u. Next, assume v is another fixed point of L such that u ̸= v that is Mz(u, v, l) < 1. By the property of G2, we know that Mz(u, u, l) = Mz(u,Lu, l) > q · Mz(u, v, l) ⇒ (1−Mz(u, v, l)) = (1−Mz(Lu,Lv, l)) ≤ α(u, v, l)(1−Mz(Lu,Lv, l)) ≤ β(1−Mz(u, v, l))(1−Mz(u, v, l)) < (1−Mz(u, v, l)), a contradiction with the assumption, so fixed point u is unique. Now we write an example which is α-Suzuki Geraghty type-I contractive mapping but not a Geraghty type-I mapping. Example 4. Define a fuzzy metric Mz(δ, γ, l) = e− |δ−γ|2 l for all l > 0 on [0, 1] with standard triangular norm and space is G-complete and Define a self map like L(δ) = { δ 2 , if δ ∈ [0, 1) 0, δ = 1. V. Chandra, U. D. Patel, S. Radenović / Eur. J. Pure Appl. Math, 17 (4) (2024), 2384-2404 2394 Also define function α by α(δ, γ, l) = { 1, if δ, γ ∈ [0, 1) 0, otherwise. for all l > 0 and β(t1) = e−t1. Suppose q = 1 2 . Put the following cases to verify the α-Suzuki-Geraghty type-I contraction mapping: Case 1. If δ, γ ∈ [0, 1) then α(δ, γ, l) = 1, Mz(δ, δ 2 , l) > q ·Mz(δ, γ, l), that is, e − |δ− δ 2 |2 l > q · e− |δ−γ|2 l implies α(δ, γ, l)[1− e− | δ2− γ 2 |2 l ] ≤ e−(1−e− |δ−γ|2 l )[1− e− |δ−γ|2 l ]. Graphs of two functions: Yellow - Exp@ -AbsA∆ - ∆ 2 E^2 l D, Red - 1 2 HExp@ -Abs@∆ - ΓD^2 l DLL 0.0 0.5 1.0 ∆ 0.0 0.5 1.0 Γ 0.0 0.5 1.0 z Figure 3: e− |δ− δ 2 |2 l > q · e− |δ−γ|2 l Graphs of two functions: Yellow - 1 - Exp@-Abs@∆�2 - Γ�2D^2�lD, Red - Exp@-1D H1 - Exp@-Abs@∆ - ΓD^2�lDL 0.0 0.5 1.0 ∆ 0.0 0.5 1.0 Γ 0.0 0.2 0.4 0.6 0.8 z Figure 4: α(δ, γ, l)[1− e− | δ 2 − γ 2 |2 l ] ≤ e−(1−e − |δ−1|2 l )[1− e− |δ−γ|2 l ] Figure 3. The yellow colour represents the value of e− |δ− δ 2 |2 l , and the red colour rep- resents the value of q · e− |δ−γ|2 l and Figure 4. The yellow colour represents the value of α(δ, γ, l)[1 − e− | δ2− γ 2 |2 l ], and the red colour represents the value of e−(1−e− |δ−γ|2 l )(1 − V. Chandra, U. D. Patel, S. Radenović / Eur. J. Pure Appl. Math, 17 (4) (2024), 2384-2404 2395 e− |δ−γ|2 l ). Hence, it is clear that the hypothesis of inequality does not hold, and also the conclusion part does not hold. So, the inequality holds for this particular case. Case 2. If δ ∈ [0, 1) and γ = 1, then α(δ, γ, l) = 0, Mz(δ, 1, l) > q · Mz(δ, 1, l), that is, e− |δ− δ 2 |2 l > q · e− |δ−1|2 l . Since the hypothesis inequality is not supportive, there is no need to continue for further calculations, but even then, 0 ≤ e−(1−e− |δ−1|2 l )[1− e− |δ−1|2 l ]. Hence, the inequality holds for this case. Case 3. If δ = γ = 1, then α(δ, γ, l) = 0. Mz(1, 1 2 , l) > q ·Mz(1, 1, l), that is, e − 1 4l > q · 1 implies 0 ≤ 0. Case 4. If δ = 1 and γ ∈ [0, 1), then α(δ, γ, l) = 0, Mz(1, 0, l) > q · Mz(1, γ, l), that is, e− 1 l > q · e− |1−γ|2 l . the hypothesis is not supportive, there is no need to continue further calculations, but even then, we get conclusion part will be zero from both sides. Hence, the inequality holds in this case. This example demonstrate that the self-mapping L is α-Suzuki Geraghty type-I contractive mapping but not a Geraghty type-I mapping. Let δ, γ ∈ Ȳ for all l > 0 such that α(δ, γ, l) ≥ 1 this implies that δ, γ ∈ [0, 1), then Lδ,Lγ ∈ [0, 1). Thus, α(Lδ,Lγ, l) = 1 for all l > 0. Let δ, γ, η ∈ [0, 1] such that α(δ, η, l) ≥ 1 and α(η, γ, l) ≥ 1 for all l > 0. This implies that δ, γ, η ∈ [0, 1). So, α(δ, γ, l) ≥ 1 for all l > 0. Therefore, L is triangular α-admissible. Hence all the assump- tions of the above theorem are gratified and also hold property G1, G2 and condition (3) for q = 0.5, β(t1) = e−t1. So, δ = 0 is a fixed point of L. This is another supportive example of our results. Example 5. Let Ȳ = {0, 12 , 1, 2} with Mz(δ, γ, l) = l l+|δ−γ|2 for l > 0 is a G-complete b-fuzzy metric space. Define L(0) = L(12) = L(1) = 1 2 and L(2) = 1. we can calculate L satisfies each assumptions Theorem 2 with unique fixed point δ = 1 2 for function α like α(δ, γ, l) = { 1, if δ, γ ∈ {0, 12 , 1} 0 otherwise. for all l > 0, β(t1) = 1− t1 and q ∈ (0, 1). Now we introduce one more inequality. Definition 10. A triangular α-admissible self mapping L defined on a b-fuzzy metric space (Ȳ,Mz, ⋄) is called a α-Suzuki-Geraghty type-II if there exists a β ∈ B such that Mz(δ,Lδ, l) > q · Mz(δ, γ, l) implies α(δ, γ, l) ( 1 Mz(Lδ,Lγ, l) − 1 ) ≤ β ( 1 Mz(δ, γ, l) − 1 )( 1 Mz(δ, γ, l) − 1 ) , (17) for all δ, γ ∈ Ȳ and l > 0, q ∈ (0, 1). V. Chandra, U. D. Patel, S. Radenović / Eur. J. Pure Appl. Math, 17 (4) (2024), 2384-2404 2396 Theorem 3. Consider a self map L defined on a G-complete b-fuzzy metric space (Ȳ,Mz, ⋄) where fuzzy metric is triangular satisfying: (i) map L is b-fuzzy α-Suzuki-Geraghty type-II; (ii) L has property G1 and G2; (iii) there exists δ0 ∈ Ȳ such that α(δ0,Lδ0, l) ≥ 1 for all l > 0; (iv) if α(δn, δn+1, l) ≥ 1 and δn → u as n → +∞, then α(δn, u, l) ≥ 1 for all n ∈ N. Then L has a fixed point. Proof. Constructing of Picard sequence such that δn ̸= δn+1 for all n ∈ N. By lemma 1, we have α(δn, δn+1, l) ≥ 1 for all n ∈ N and l > 0. By the α-Suzuki-Geraghty type -II contraction, we have Mz(δn, δn+1, l) > q · Mz(δn, δn+1, l) ⇒α(δn, δn+1, l) ( 1 Mz(Lδn,Lδn+1, l) − 1 ) ≤ β ( 1 Mz(δn, δn+1, l) − 1 )( 1 Mz(δn, δn+1, l) − 1 ) for all l > 0. ( 1 Mz(Lδn,Lδn+1, l) − 1 ) ≤ α(δn, δn+1, l) ( 1 Mz(Lδn,Lδn+1, l) − 1 ) ≤ β ( 1 Mz(δn, δn+1, l) − 1 )( 1 Mz(δn, δn+1, l) − 1 ) < ( 1 Mz(δn, δn+1, l) − 1 ) . (18) we conclude that Mz(δn+1, δn+2, l) > Mz(δn, δn+1, l) for all n ∈ N it means, it is non- decreasing sequence of positive real numbers. So there exists s(l) ∈ (0, 1] such that lim n→+∞ Mz(δn, δn+1, l) = s(l), for all l > 0. Next, we require to prove s(l) = 1. Taken a contrary, s(l0) < 1 for all l0 > 0. Now taking limit as n → +∞ lim n→+∞ β ( 1 Mz(δn, δn+1, l) − 1 ) = 1 ⇒ lim n→+∞ Mz(δn, δn+1, l) = 1, (19) a contradiction. Next we must prove that sequence is a G-Cauchy sequence. Suppose λ = Mz(δn, δm, l) < 1. By (G1) property, Mz(δn, δn+1, l) > q · Mz(δn, δm, l) implies α(δn, δm, l) ( 1 Mz(Lδn,Lδm, l) − 1 ) ≤ β ( 1 Mz(δn, δm, l) − 1 )( 1 Mz(δn, δm, l) − 1 ) . V. Chandra, U. D. Patel, S. Radenović / Eur. J. Pure Appl. Math, 17 (4) (2024), 2384-2404 2397 By (17) and lemma (1), we get( 1 Mz(δn+1, δm+1, l) − 1 ) = ( 1 Mz(Lδn,Lδm, l) − 1 ) ≤ α(δn, δm, l) ( 1 Mz(Lδn,Lδm, l) − 1 ) ≤ β ( 1 Mz(δn, δm, l) − 1 )( 1 Mz(δn, δm, l) − 1 ) < ( 1 Mz(δn, δm, l) − 1 ) taking limit on both sides, lim n,m→+∞ ( 1 Mz(δn+1, δm+1, l) − 1 ) ≤ lim n,m→+∞ α(δn, δm, l) ( 1 Mz(Lδn,Lδm, l) − 1 ) ≤ lim n,m→+∞ β ( 1 Mz(δn, δm, l) − 1 )( 1 λ − 1 ) < ( 1 λ − 1 ) . (20) On the flip side,( 1 Mz(δn, δm, l) − 1 ) ≤ ( 1 Mz(δn, δn+1, l) − 1 ) + ( 1 Mz(δn+1, δm, l) − 1 ) ≤ ( 1 Mz(δn, δn+1, l) − 1 ) + ( 1 Mz(δn+1, δm+1, l) − 1 ) + ( 1 Mz(δm+1, δm, l) − 1 ) . taking limit as n,m → +∞ and using (19, 20), lim n,m→+∞ ( 1 Mz(δn, δm, l) − 1 ) ≤ lim n,m→+∞ α(δn, δm, l) ( 1 Mz(Lδn,Lδm, l) − 1 ) ≤ lim n,m→+∞ β ( 1 Mz(δn, δm, l) − 1 ) lim n,m→+∞ ( 1 Mz(δn, δm, l) − 1 ) < lim n,m→+∞ ( 1 Mz(δn, δm, l) − 1 ) 1 λ − 1 ≤ lim n,m→+∞ ( 1 Mz(Lδn,Lδm, l) − 1 ) ≤ lim n,m→+∞ β ( 1 Mz(δn, δm, l) − 1 ) ( 1 λ − 1) < 1 λ − 1. which suggest that lim n,m→+∞ β ( 1 Mz(δn, δm, l) − 1 ) = 1 ⇒ lim n,m→+∞ 1 Mz(δn, δm, l) = 1, V. Chandra, U. D. Patel, S. Radenović / Eur. J. Pure Appl. Math, 17 (4) (2024), 2384-2404 2398 a contradiction. Thus, {δn} is a G-Cauchy sequence. Since Ȳ is a G-complete then there exists u ∈ Ȳ such that lim n→+∞ Mz(δn, u, l) = 1. (21) Next to prove fixed point of map L, we require property (G2), Mz(δn,Lδn, l) > q · Mz(δn−1, u, l) implies α(δn−1, u, l) ( 1 Mz(δn,Lu, l) − 1 ) ≤ β ( 1 Mz(δn−1, u, l) − 1 )( 1 Mz(δn−1, u, l) − 1 ) ( 1 Mz(δn,Lu, l) − 1 ) ≤ α(δn−1, u, l) ( 1 Mz(δn,Lu, l) − 1 ) ≤ β ( 1 Mz(δn−1, u, l) − 1 )( 1 Mz(δn−1, u, l) − 1 ) . Put limit as n → +∞, lim n→+∞ ( 1 Mz(δn−1,Lu,l) − 1 ) ≤ 0. So, Mz(u,Lu, l) = 1 that is Lu = u. Finally, we require to show uniqueness of the fixed point. Consider another fixed point v such that u ̸= v, it means Mz(u, v, l) < 1. Using (G2) property, Mz(u, u, l) = Mz(u,Lu, l) > q · Mz(u, v, l) implies α(u, v, l) ( 1 Mz(u, v, l) − 1 ) ≤ β ( 1 Mz(u, v, l) − 1 )( 1 Mz(u, v, l) − 1 ) 1 Mz(u, v, l) − 1 ≤ α(u, v, l) ( 1 Mz(u, v, l) − 1 ) ≤ β ( 1 Mz(u, v, l) − 1 )( 1 Mz(u, v, l) − 1 ) < 1 Mz(u, v, l) − 1, a contradiction. Hence, u is a unique fixed point for self map L. Remark 1. If α(δ, γ, l) = 1 in definition (10), then mapping L becomes a Suzuki Geraghty type-II contractive map. Corollary 1. Suppose (Ȳ,Mz, ⋄) is a G-complete b-fuzzy metric space with triangular fuzzy metric and a self map L defined on Ȳ is a Suzuki Geraghty type-II contractive map with properties (G1) and (G2). Then L has a unique fixed point. Now we present another definition which is not in view of Suzuki type. Definition 11. A triangular α-admissible self mapping L defined on a b-fuzzy metric space (Ȳ,Mz, ⋄) is called a α-Geraghty type-II if there exists a β ∈ B such that α(δ, γ, l) ( 1 Mz(Lδ,Lγ, l) − 1 ) ≤ β ( 1 Mz(δ, γ, l) − 1 )( 1 Mz(δ, γ, l) − 1 ) , (22) q ∈ (0, 1), for all δ, γ ∈ Ȳ and l > 0. V. Chandra, U. D. Patel, S. Radenović / Eur. J. Pure Appl. Math, 17 (4) (2024), 2384-2404 2399 Theorem 4. Consider a self map L defined on a G-complete b-fuzzy metric space (Ȳ,Mz, ⋄) where fuzzy metric is triangular satisfying: (i) map L is b-fuzzy α-Geraghty type-II; (ii) there exists δ0 ∈ Ȳ such that α(δ0,Lδ0, l) ≥ 1 for all l > 0; (iii) if α(δn, δn+1, l) ≥ 1 and δn → u as n → +∞, then α(δn, u, l) ≥ 1 for all n ∈ N. Then L has a fixed point. Remark 2. α(δ, γ, l) = 1 in definition (11) results in the following: Corollary 2. Consider a self map L which is Geraghty type-II contractive defined on a G-complete b-fuzzy metric space (Ȳ,Mz, ⋄) where fuzzy metric is triangular then the self-mapping L has a unique fixed point. In the support of Corollary 2, we have an example. Example 6. Consider a fuzzy metric Mz(δ, γ, l) = l+0.3 l+0.3+|δ−γ|2 for all δ, γ ∈ Ȳ = [0, 1] and l > 0. We can check it is a G-complete b-fuzzy metric space with respect to standard triangular norm. Define a self map L such as L(δ) = { δ 2 , if δ, γ ∈ (0, 1] 0, if δ = 0. To show mapping L is a Geraghty type-II contractive with β(t1) = 1 1+t1 . Case 1. If δ, γ ∈ (0, 1] then 1 Mz(Lδ,Lγ, l) − 1 = 1 l+0.3 l+0.3+|Lδ−Lγ|2 − 1 = 1 l+0.3 l+0.3+ 1 4 |δ−γ|2 − 1 = 1− l+0.3 l+0.3+ 1 4 |δ−γ|2 l+0.3 l+0.3+ 1 4 |δ−γ|2 = 1 4 |δ − γ|2 l + 0.3 + 1 4 |δ − γ|2 · l + 0.3 + 1 4 |δ − γ|2 l + 0.3 = 1 4 |δ − γ|2 l + 0.3 ≤ |δ − γ|2 l + 0.3 + |δ − γ|2 = 1− l + 0.3 l + 0.3 + |δ − γ|2 V. Chandra, U. D. Patel, S. Radenović / Eur. J. Pure Appl. Math, 17 (4) (2024), 2384-2404 2400 = β ( 1 Mz(δ, γ, l) − 1 )( 1 Mz(δ, γ, l) − 1 ) , Case2. If δ = 0, γ ∈ (0, 1] then 1 Mz(Lδ,Lγ, l) − 1 = 1 l+0.3 l+0.3+|Lδ−Lγ|2 − 1 = 1 l+0.3 l+0.3+ 1 4 |γ|2 − 1 = 1 4 |γ| 2 l + 0.3 ≤ |γ|2 l + 0.3 + |γ|2 = 1− l + 0.3 l + 0.3 + |γ|2 = β ( 1 Mz(δ, γ, l) − 1 )( 1 Mz(δ, γ, l) − 1 ) , Case 3. If δ = γ = 0 then it is trivial. It is easy to check that Mz is triangular. Hence, δ = 0 is a unique fixed point of L. 3. Application In this section, we discuss the existence of a unique solution of a non-linear integral equation and need some specific conditions for the solution. A b-fuzzy metric space that resembles C ([a, b],R) is the space Ȳ of all continuous real valued functions defined on the interval [a, b] with the b-fuzzy metric Mz(δ, γ, l) = l l + max a≤s1≤b |δ(s1)− γ(s1)|2 . Consider an integral equation δ(l1) = f(l1) + ∫ b a h(l1, s1)F (l1, s1, δ(s1))ds1, (23) where f : [a, b] → R, h : [a, b] × [a, b] → R and F : [a, b] × [a, b] × R → R are continuous functions. Theorem 5. Suppose (i) for all l1, s1 ∈ [a, b], δ, γ ∈ Ȳ Mz(δ(s1),L(δ(s1)), l) > q · Mz(δ(s1), γ(s1), l) ⇒ |F (l1, s1, δ(s1))− F (l1, s1, γ(s1))|2 ≤ e− maxa≤s1≤b |δ(s1)−γ(s1)| 2 l |δ(s1)− γ(s1)|2, V. Chandra, U. D. Patel, S. Radenović / Eur. J. Pure Appl. Math, 17 (4) (2024), 2384-2404 2401 (ii) for all l1, s1 ∈ [a, b] (∫ b a h(l1, s1)ds1 )2 ≤ 1 b− a . (iii) if {δn(l1)} and {δm(l1)} are the two sequences in Ȳ such that lim n,m→+∞ max a≤l1≤b |δn(l1)− δm(l1)|2 → r(k) ⇒ q · (l + max a≤l1≤b |δn(l1)− δn+1(l1)|)2 < (l + max a≤l1≤b |δn(l1)− δm(l1)|2), for all n,m ∈ N such that n > m, q ∈ (0, 1). (iv) if {δn(l1)} is a sequence in C ([a, b],R) such that δn(l1) → δ(l1) ⇒ q · (l + max a≤l1≤b |δn(l1)− δn+1(l1)|2) < (l + max a≤l1≤b |δn(l1)− δ(l1)|2), for all n ∈ N and l > 0, q ∈ (0, 1). Then the integral equation (23) has a solution in Ȳ. Proof. Suppose L : Ȳ → Ȳ is an integral operator Lδ(l1) = f(l1) + ∫ b a h(l1, s1)F (l1, s1, δ(s1))ds1, for δ ∈ Ȳ. Now 1 Mz(Lδ,Lγ, l) − 1 = 1 l l+ max a≤s1≤b |Lδ(s1)−Lγ(s1)|2 − 1 = 1− l l+ max a≤s1≤b |Lδ(s1)−Lγ(s1)|2 l l+ max a≤s1≤b |Lδ(s1)−Lγ(s1)|2 = max a≤s1≤b |Lδ(s1)− Lγ(s1)|2 l = max a≤s1≤b | ∫ b a h(l1, s1)F (l1, s1, δ(s1))ds1 − ∫ b a h(l1, s1)F (l1, s1, y(s1))ds1|2 l ≤ max a≤s1≤b e− |δ(s1)−γ(s1)| 2 l · |δ(s1)− γ(s1)|2 l ≤ β ( 1 Mz(δ, γ, l) − 1 )( 1 Mz(δ, γ, l) − 1 ) . V. Chandra, U. D. Patel, S. Radenović / Eur. J. Pure Appl. Math, 17 (4) (2024), 2384-2404 2402 Therefore, L is a Suzuki Geraghty type-II contractive mapping for β(t1) = e−t1 and l1 > 0. For two sequences in Ȳ such that n > m and n,m ∈ N, by using the assumption (3) Mz (δn(l1), δm(l1), l) = l l +maxa≤l1≤b |δn(l1)− δm(l1)|2 > q · l l + r(k) = r(l) ∈ (0, 1] implies Mz (δn(l1), δn+1(l1), l) = l l +maxa≤l1≤b |δn(l1)− δn+1(l1)|2 > q · l l +maxa≤l1≤b |δn(l1)− δm(l1)|2 = q · Mz (δn(l1), δm(l1), l) . Hence, property-(G1) holds. If a sequence {δn(l1)} in Ȳ such that δn(l1) → δ(l1) in Ȳ by using assumption (4), Mz (δn(l1), δn+1(l1), l) = l l +maxa≤l1≤b |δn(l1)− δn+1(l1)|2 > q · l l +maxa≤l1≤b |δn(l1)− δ(l1)|2 = q · Mz (δn(l1), δ(l1), l) . Therefore, property (G2) holds. Therefore every requirements of corollary (2) are gratified with the consideration of the function β(t1) = e−t1 . Hence, we conclude that there exists δ(l1) ∈ C ([a, b],R) such that Lδ(l1) = δ(l1) and the integral equations (23) has a solution. This way we complete the proof. 4. Conclusion and Future Work In this study, we explore the concept of fuzzy α-Geraghty type mappings and α-Suzuki- Geraghty type mappings within the framework of b-fuzzy metric space. We extended the theory of fixed-point theorems by employing these mappings and demonstrated their utility through several illustrative examples, including a graphical approach for better visualization. Our findings provide a foundation for solving fixed-point problems in more generalized settings, such as G-complete b-fuzzy metric space. The application to non- linear integral equations highlights the practical significance of these results. Future research could focus on expanding these concepts to more complex metric spaces or hybrid structures, exploring their applications in various fields such as optimization, dynamic systems, and network theory. 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