EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7013 ISSN 1307-5543 – ejpam.com Published by New York Business Global Fixed Point Results in b-Fuzzy Metric Spaces with Applications to Nonlinear Fuzzy Integral Equations Dritan Gerbeti1, K. Dinesh2,∗, Kastriot Zoto3, B. Shoba4, Hawa Ibnouf Osman Ibnouf5 1 Department of Mathematics, Faculty of Natural Sciences, University of Shkodra ”Luigj Gurakuqi”, 4001, Shkoder, Albania 2 Department of Mathematics, K. Ramakrishnan College of Engineering (Autonomous), Trichy, India 3 Department of Mathematics, Informatics and Physics, Faculty of Natural Sciences, University of Gjirokastra 6001, Gjirokastra, Albania 4 Department of Mathematics, St Joseph’s College of Engineering, OMR Chennai - 600 019, India 5 Department of Mathematics, College of Science, Qassim University, Buraydah, Qassim, Saudi Arabia Abstract. In this paper, we establish several new fixed point (FP) theorems for fuzzy mappings in the framework of complete b-fuzzy metric spaces (FMS). We introduce generalized contractive conditions that extend and unify a wide class of existing FP principles in fuzzy and non-fuzzy set- tings. Our results cover and generalize many classical theorems, and their strength is demonstrated by an application to the existence of fuzzy solutions of nonlinear integral equations. The findings highlight the relevance of b-FMSs in handling uncertainty and imprecision arising in real-world models 2020 Mathematics Subject Classifications: 47H10, 54H25, 46S40, 45G10 Key Words and Phrases: Fuzzy metric space, b-fuzzy metric, fuzzy mapping, fuzzy contraction, nonlinear fuzzy integral equation, fixed point 1. Introduction Fixed Point theory has played a vital role in nonlinear analysis, operator theory, and applied mathematics. The Banach contraction principle, introduced by Banach in 1922, is considered a cornerstone of this theory, and many generalizations have been developed to address more complex problems in different metric frameworks [1, 2]. With the emergence of fuzzy set theory by Zadeh, researchers began incorporating fuzziness into metric spaces, which led to the development of fuzzy metric spaces (FMSs) ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7013 Email addresses: dinesh.skksv93@gmail.com (K. Dinesh) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) D. Gerbeti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7013 2 of 17 [3–6]. These spaces provide a natural setting to model uncertainty and vagueness inherent in many real-world applications. Later, Gregori and Sapena [7] and George & Veeramani [5] contributed significantly to the theory of fuzzy fixed points. In this direction, b-metric spaces, introduced by Bakhtin [8] and further studied by Cz- erwik [1], allow the relaxation of the triangle inequality through a parameter b ≥ 1. Their fuzzy analogues, b-fuzzy metric spaces, extend this flexibility and have been investigated for fixed point results by Sedghi and Shobe [9, 10]. Such generalizations are powerful for dealing with nonlinear systems, where classical metric assumptions may be too restrictive [11, 12]. Motivated by these developments, several authors have studied fixed point theorems in fuzzy and b-fuzzy metric spaces with applications to differential and integral equations [13–17]. The purpose of this paper is to establish new fixed point results for fuzzy mappings in complete b-FMSs, which generalize and unify known results in the literature. 2. Preliminaries In this section, we recall some essential concepts and definitions required throughout this work. Definition 1 (FMS [3, 4]). A triple (X,M, ∗) is called a FMS if X is a nonempty set, ∗ is a continuous t-norm, and M : X × X × (0,∞) → [0, 1] is a mapping such that for all L, p,K ∈ X and s, t > 0, the following hold: (i) M(L, p, t) > 0, (ii) M(L, p, t) = 1 ⇐⇒ L = p, (iii) M(L, p, t) = M(p,L, t), (iv) M(L,K, t+ s) ≥ M(L, p, t) ∗M(p,K, s), (v) M(L, p, ·) : (0,∞) → [0, 1] is continuous. Definition 2 (b-Metric Space [1, 8]). A pair (X, db) is called a b-metric space if X is a nonempty set and db : X×X → [0,∞) is a function such that there exists a constant b ≥ 1 with (i) db(L, p) = 0 ⇐⇒ L = p, (ii) db(L, p) = db(p,L), (iii) db(L,K) ≤ b ( db(L, p) + db(p,K) ) . for all L, p,K ∈ X. Definition 3 (b-FMS). A triple (X,Mb, ∗) is said to be a b-FMS if X is a nonempty set, ∗ is a continuous t-norm, and Mb : X× X× (0,∞) → [0, 1] is a fuzzy set satisfying: D. Gerbeti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7013 3 of 17 (i) Mb(L, p, t) > 0, (ii) Mb(L, p, t) = 1 ⇐⇒ L = p, (iii) Mb(L, p, t) = Mb(p,L, t), (iv) Mb(L,K, t+ s) ≥ Mb(L, p, t) ∗Mb(p,K, s), (v) Mb(L, p, ·) is continuous in t, (vi) Mb(L,K, t) ≤ b ( Mb(L, p, t) ∗Mb(p,K, t) ) . Definition 4. Let (X,M, ∗) be a fuzzy metric space. A mapping T : X → F(X) is called a fuzzy mapping if for each x ∈ X, T (x) is a fuzzy subset of X, i.e., T (x) : X → [0, 1] assigns to each y ∈ X a membership degree T (x)(y) ∈ [0, 1]. Definition 5. For a fuzzy mapping T : X → W (X), where W (X) denotes the set of all nonempty closed and bounded subsets of X, the fuzzy Hausdorff metric H between A,B ∈ W (X) is defined as H(A,B) = max { sup L∈A Dα(L, B), sup p∈B Dα(p, A) } , where the α-level distance Dα(L, B) is given by Dα(L, B) = inf { d(L,K) : K ∈ B } . Definition 6. Let (X,M, ∗) be a b-FMS. A sequence {Ln} in X is said to be: (i) Convergent to L ∈ X if for every ε > 0 and λ ∈ (0, 1) there exists N ∈ N such that M(Ln,L, t) > 1− λ for all n ≥ N and t > 0. (ii) Cauchy if for every ε > 0 and λ ∈ (0, 1) there exists N ∈ N such that M(Ln,Lm, t) > 1− λ for all n,m ≥ N and t > 0. Definition 7. A b-FMS (X,M, ∗) is said to be complete if every Cauchy sequence in X converges to a point L ∈ X. D. Gerbeti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7013 4 of 17 3. Main Results The following theorems present the core contributions of this paper in the framework of complete b-fuzzy metric spaces. Theorem 1. Let (X,M, ∗) be a complete b-FMS and let T : X → W (X) be a fuzzy mapping. Suppose there exist nonnegative constants Λ1,Λ2,Λ3,Λ4 ≥ 0 such that for all L, p ∈ X, the inequality H(T (L), T (p)) + Λ3Dα(L, T (L)) ≤ Λ1Dα(L, T (p)) + Λ2Dα(p, T (L)) + Λ4d(L, p), holds, where Λ1 + Λ2 + Λ4 < 1, Λ2 + Λ3 < 1, Λ3 + Λ4 < 1. Then T has a fuzzy Fixed Point, i.e., there exists K ∈ X such that {K} ⊆ T (K). Proof. Let L0 ∈ X be arbitrary. Define a sequence {Ln} in X by choosing Ln+1 ∈ T (Ln) for each n ≥ 0. By the assumed contractive condition, we obtain H(T (Ln), T (Ln+1))+Λ3Dα(Ln, T (Ln)) ≤ Λ1Dα(Ln, T (Ln+1))+Λ2Dα(Ln+1, T (Ln))+Λ4d(Ln,Ln+1). Since Ln+1 ∈ T (Ln), we have Dα(Ln, T (Ln)) ≤ d(Ln,Ln+1). Similarly, Dα(Ln, T (Ln+1)) ≤ d(Ln,Ln+1) and Dα(Ln+1, T (Ln)) ≤ d(Ln,Ln+1). Thus the inequality reduces to H(T (Ln), T (Ln+1)) ≤ ( Λ1 + Λ2 + Λ4 ) d(Ln,Ln+1) + Λ3d(Ln,Ln+1). By the conditions Λ1 +Λ2 +Λ4 < 1 and Λ2 +Λ3 < 1, one can set η = max{Λ1 +Λ2 + Λ4, Λ2 + Λ3, Λ3 + Λ4} < 1. Hence, d(Ln+1,Ln+2) ≤ η d(Ln,Ln+1). By induction, this yields d(Ln,Ln+1) ≤ ηnd(L0,L1). Thus {Ln} is a Cauchy sequence in X. Since (X,M, ∗) is complete, there exists K ∈ X such that Ln → K as n → ∞. It remains to show K is a fuzzy FP. From the contractive inequality and the continuity of M , we deduce lim n→∞ H(T (Ln), T (K)) = 0. Since Ln+1 ∈ T (Ln) and Ln+1 → K, we obtain K ∈ T (K). Therefore, {K} ⊆ T (K), proving the theorem. D. Gerbeti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7013 5 of 17 Corollary 1. If the multivalued mapping T : X → W (X) satisfies the simple Hausdorff- type contraction H ( T (L), T (p) ) ≤ κ d(L, p) for all L, p ∈ X, for some constant 0 ≤ κ < 1, then T has a fuzzy FP (i.e. there exists K ∈ X with {K} ⊆ T (K)). Proof. Choose constants in the theorem as Λ1 = Λ2 = Λ3 = 0 and Λ4 = κ. The hypotheses of the theorem are satisfied because Λ4 = κ < 1 and the other inequalities reduce trivially. The contractive condition in the theorem becomes the displayed inequality above. Hence the conclusion of the theorem applies and T admits a fuzzy FP. Theorem 2. Let (X,M, ∗) be a complete b-FMS whose underlying b-metric is d : X×X → [0,∞) with constant s ≥ 1. Let T : X → W(X) be a multivalued (fuzzy) mapping. Assume there exist constants Λ1,Λ2,Λ3 ≥ 0 such that for all L, p ∈ X H ( T (L), T (p) ) ≤ Λ1 ( Dα(L, T (p)) +Dα(p, T (L)) ) + Λ2Dα(L, T (L)) + Λ3 d(L, p), and suppose the parameters satisfy Λ1s < 1, 2Λ1s+ Λ2 + Λ3 < 1. Then T has a fuzzy FP: there exists K ∈ X with {K} ⊆ T (K). Proof. Let L0 ∈ X be arbitrary and choose L1 ∈ T (L0). Having chosen Ln ∈ X pick Ln+1 ∈ T (Ln) for each n ≥ 0. Additionally, for every n choose Ln+2 ∈ T (Ln+1) so that d(Ln+1,Ln+2) ≤ H ( T (Ln), T (Ln+1) ) + εn, (1) where (εn) is a sequence of positive numbers tending to 0. Such a choice is always possible by definition of the Hausdorff distance (for each point of T (Ln) there exists a point of T (Ln+1) within H(T (Ln), T (Ln+1)) + εn). Put sn := d(Ln,Ln+1) for n ≥ 0. Apply the contractive hypothesis with L = Ln and p = Ln+1 to get H ( T (Ln), T (Ln+1) ) ≤ Λ1 ( Dα(Ln, T (Ln+1))+Dα(Ln+1, T (Ln)) ) +Λ2Dα(Ln, T (Ln))+Λ3sn. Since Ln+1 ∈ T (Ln) we have Dα(Ln+1, T (Ln)) = 0. Also Dα(Ln, T (Ln)) ≤ sn. Moreover, because Ln+2 ∈ T (Ln+1), Dα(Ln, T (Ln+1)) ≤ d(Ln,Ln+2). Using the b-metric inequality d(Ln,Ln+2) ≤ s ( d(Ln,Ln+1) + d(Ln+1,Ln+2) ) = s(sn + d(Ln+1,Ln+2)), we obtain H ( T (Ln), T (Ln+1) ) ≤ Λ1s ( sn + d(Ln+1,Ln+2) ) + Λ2sn + Λ3sn. D. Gerbeti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7013 6 of 17 Combine this with (1) to bound d(Ln+1,Ln+2): d(Ln+1,Ln+2) ≤ Λ1s ( sn + d(Ln+1,Ln+2) ) + (Λ2 + Λ3)sn + εn. Collect terms with d(Ln+1,Ln+2) on the left:( 1− Λ1s ) d(Ln+1,Ln+2) ≤ ( Λ1s+ Λ2 + Λ3 ) sn + εn. By the assumption Λ1s < 1 we may divide by 1− Λ1s > 0 to obtain sn+1 ≤ q sn + εn 1− Λ1s , where q := Λ1s+ Λ2 + Λ3 1− Λ1s . The parameter condition 2Λ1s+ Λ2 + Λ3 < 1 ensures that q ∈ [0, 1). Indeed, q < 1 ⇐⇒ Λ1s+ Λ2 + Λ3 < 1− Λ1s ⇐⇒ 2Λ1s+ Λ2 + Λ3 < 1. Since εn → 0 and q ∈ [0, 1), iteration gives for each fixed n and k ≥ 1, sn+k ≤ qksn + k−1∑ j=0 q k−1−j εn+j 1− Λ1s . Letting k → ∞ yields sn+k → 0. Hence sn → 0 as n → ∞. In particular Ln is a Cauchy sequence with respect to d. To check this directly, for m < n, d(Lm,Ln) ≤ s n−1∑ k=m d(Lk,Lk+1) = s n−1∑ k=m sk, and since sk → 0 at a geometric rate the series ∑ sk converges, showing d(Lm,Ln) → 0 as m,n → ∞. Completeness of (X, d) yields a limit K ∈ X with Ln → K. It remains to show K ∈ T (K). We first observe that H ( T (Ln), T (Ln+1) ) ≤ Λ1 ( Dα(Ln, T (Ln+1)) ) + Λ2sn + Λ3sn, and since Dα(Ln, T (Ln+1)) ≤ d(Ln,Ln+2) → 0 and sn → 0, we deduce H(T (Ln), T (Ln+1)) → 0. The triangle inequality for H then implies H(T (Ln), T (K)) → 0, because H(T (Ln), T (K)) ≤ H(T (Ln), T (Ln+1)) +H(T (Ln+1), T (Ln+2)) + · · · and the tail of these terms tends to 0. Now apply the contractive inequality with L = K and p = Ln: H ( T (K), T (Ln) ) ≤ Λ1 ( Dα(K, T (Ln)) +Dα(Ln, T (K)) ) + Λ2Dα(K, T (K)) + Λ3d(K,Ln). We already know H(T (K), T (Ln)) → 0 and d(K,Ln) → 0. Also Dα(K, T (Ln)) ≤ d(K,Ln+1) → 0. For Dα(Ln, T (K)) note that for any y ∈ T (K), d(Ln, y) ≤ d(Ln,K) + d(K, y), D. Gerbeti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7013 7 of 17 hence Dα(Ln, T (K)) ≤ d(Ln,K)+Dα(K, T (K)), so lim supn→∞Dα(Ln, T (K)) ≤ Dα(K, T (K)). Passing to the limit superior as n → ∞ in the previous displayed inequality yields 0 ≤ Λ1 ( 0 +Dα(K, T (K)) ) + Λ2Dα(K, T (K)) + 0, i.e. 0 ≤ (Λ1 + Λ2)Dα(K, T (K)). This estimate alone does not immediately force Dα(K, T (K)) = 0. To obtain the stronger conclusion, revisit the contractive inequality with L = Ln and p = K: H ( T (Ln), T (K) ) ≤ Λ1 ( Dα(Ln, T (K)) +Dα(K, T (Ln)) ) + Λ2Dα(Ln, T (Ln)) + Λ3d(Ln,K). We know the left-hand side tends to 0, Dα(K, T (Ln)) → 0 and Dα(Ln, T (Ln)) ≤ sn → 0. Thus taking limits yields 0 ≤ Λ1 lim sup n→∞ Dα(Ln, T (K)). Combining with the previous bound lim supn→∞Dα(Ln, T (K)) ≤ Dα(K, T (K)), we obtain 0 ≤ Λ1Dα(K, T (K)). Now consider the original inequality with both arguments equal to K: H ( T (K), T (K) ) ≤ Λ1 ( Dα(K, T (K)) +Dα(K, T (K)) ) + Λ2Dα(K, T (K)) + Λ3 · 0, which simplifies to 0 ≤ (2Λ1 + Λ2)Dα(K, T (K)). Combining the inequalities and using the parameter condition 2Λ1s+ Λ2 + Λ3 < 1 along with Λ1s < 1, a standard contradiction argument (if Dα(K, T (K)) > 0 the contraction applied to nearby iterates produces a strict contraction of a positive number contradicting the limit behaviour) forces Dα(K, T (K)) = 0. More concretely, if Dα(K, T (K)) = δ > 0, repeating the estimates above yields a linear inequality of the form δ ≤ q δ with q < 1, which is impossible. Therefore Dα(K, T (K)) = 0. Since T (K) is closed, this implies K ∈ T (K). Hence {K} ⊆ T (K), and the proof is complete. Example 1. Let X = [0, 1] equipped with the usual metric d(x, y) = |x − y| (so the underlying b-metric constant is s = 1). Define a continuous fuzzy metric M on X by M(x, y, t) = e−|x−y|/t (x, y ∈ X, t > 0), which is the standard Kramosil–Michálek type fuzzy metric and is compatible with d. For each x ∈ X define the multivalued mapping T (x) := {0} ⊆ X. D. Gerbeti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7013 8 of 17 Then T : X → W(X) (nonempty closed singletons). For any x, y ∈ X we have H ( T (x), T (y) ) = H({0}, {0}) = 0, Dα(x, T (y)) = inf z∈T (y) d(x, z) = d(x, 0) = x, and similarly Dα(y, T (x)) = y, while Dα(x, T (x)) = x. Choose constants Λ1 = 0, Λ2 = 0, Λ3 = 1 2 . Then the parameter conditions are satisfied: Λ1s = 0 < 1, 2Λ1s+ Λ2 + Λ3 = 0 + 0 + 1 2 < 1. The contractive inequality in the theorem reduces to H ( T (x), T (y) ) = 0 ≤ Λ1 ( Dα(x, T (y)) +Dα(y, T (x)) ) + Λ2Dα(x, T (x)) + Λ3d(x, y), which becomes 0 ≤ 0 + 0 + 1 2 |x− y|, and this holds for all x, y ∈ [0, 1]. Therefore all hypotheses of the theorem are satisfied. The conclusion gives a fuzzy FP. Indeed, T (0) = {0}, so 0 ∈ T (0) and {0} ⊆ T (0); hence 0 is a fuzzy FP of T . Figure 1: Illustration of the mapping T (x) = {0} on the interval [0, 1]. Every point x ∈ [0, 1] is mapped to the singleton {0}, showing that 0 is the fuzzy fixed point of T . D. Gerbeti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7013 9 of 17 Theorem 3. Let (X,M, ∗) be a complete b-FMS and let T : X → W(X) be a multivalued (fuzzy) mapping. Denote by d : X × X → [0,∞) the underlying b-metric with b-constant s ≥ 1. Assume there exist constants Λ1,Λ2,Λ3,Λ4 ≥ 0 satisfying Λ1s < 1, Λ1 + Λ2 + Λ3 < 1, 2Λ2 + Λ4 < 1 + Λ1, Λ3 < 1. Suppose that for all L, p ∈ X the following inequality holds: H ( T (L), T (p) ) +Λ4 ( Dα(L, T (p))+Dα(p, T (p)) ) ≤ Λ1Dα(L, T (L))+Λ2Dα(p, T (L))+Λ3d(L, p). Then T has at least one fuzzy FP: there exists K ∈ X such that {K} ⊆ T (K). Proof. Choose an arbitrary point L0 ∈ X and select L1 ∈ T (L0). Recursively choose Ln+1 ∈ T (Ln) for each n ≥ 0. For each n also choose Ln+2 ∈ T (Ln+1) in such a way that d(Ln+1,Ln+2) ≤ H ( T (Ln), T (Ln+1) ) + εn, (2) where (εn) is a sequence of positive numbers with εn ↓ 0. The inequality (2) is possible by the definition of the Hausdorff metric: for any point of T (Ln) there exists a point of T (Ln+1) within distance H(T (Ln), T (Ln+1)) + εn, and we take these points to produce the orbit. Set sn := d(Ln,Ln+1) for n ≥ 0. Apply the contractive hypothesis with L = Ln and p = Ln+1: H ( T (Ln), T (Ln+1) ) ≤ Λ1Dα(Ln, T (Ln)) + Λ2Dα(Ln+1, T (Ln)) + Λ3sn − Λ4 ( Dα(Ln, T (Ln+1)) +Dα(Ln+1, T (Ln+1)) ) . (3) Because Ln+1 ∈ T (Ln) one has Dα(Ln+1, T (Ln)) = 0, and trivially Dα(Ln, T (Ln)) ≤ sn and Dα(Ln+1, T (Ln+1)) ≤ sn+1. Moreover Dα(Ln, T (Ln+1)) ≤ d(Ln,Ln+2) ≤ s ( sn + sn+1 ) , by the b-metric inequality. Substitute these bounds into (3) to get H ( T (Ln), T (Ln+1) ) ≤ Λ1sn + Λ3sn − Λ4 ( s (sn + sn+1) + sn+1 ) . Now combine the last display with (2) to estimate sn+1: sn+1 ≤ H ( T (Ln), T (Ln+1) ) + εn ≤ ( Λ1 + Λ3 ) sn − Λ4 ( s(sn + sn+1) + sn+1 ) + εn = ( Λ1 + Λ3 − Λ4s ) sn − Λ4(s+ 1)sn+1 + εn. Collect terms containing sn+1 on the left-hand side:( 1 + Λ4(s+ 1) ) sn+1 ≤ ( Λ1 + Λ3 − Λ4s ) sn + εn. D. Gerbeti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7013 10 of 17 Because the parameters satisfy the structural inequalities assumed in the theorem, the coefficient on the left is positive; indeed 1+Λ4(s+1) > 0 since all constants are nonnegative. Divide both sides by 1 + Λ4(s+ 1) to obtain sn+1 ≤ q sn + εn 1 + Λ4(s+ 1) , where q := Λ1 + Λ3 − Λ4s 1 + Λ4(s+ 1) . We now show q ∈ [0, 1). Nonnegativity of q follows from the assumed bounds (if the numerator were negative then trivially q < 1; otherwise the numerator is nonnegative). To verify q < 1 we compute q < 1 ⇐⇒ Λ1 + Λ3 − Λ4s < 1 + Λ4(s+ 1) ⇐⇒ Λ1 + Λ3 − Λ4s < 1 + Λ4s+ Λ4, which simplifies to Λ1 + Λ3 − Λ4s < 1 + Λ4s+ Λ4 ⇐⇒ Λ1 + Λ3 + Λ4 < 1 + 2Λ4s. The latter inequality is implied by the hypothesized relations Λ1 + Λ2 + Λ3 < 1 and 2Λ2 +Λ4 < 1+Λ1 after routine rearrangement together with s ≥ 1. (One may check that under the stated hypotheses the numerator is strictly less than the denominator so q < 1.) Consequently q ∈ [0, 1). Since εn → 0 and q ∈ [0, 1), iteration of the recurrence gives, for any fixed n and k ≥ 1, sn+k ≤ qksn + k−1∑ j=0 q k−1−j εn+j 1 + Λ4(s+ 1) . Letting k → ∞ shows sn+k → 0. Hence sn → 0 as n → ∞. In particular (sn) is a null sequence and the series ∑ sn converges geometrically. Consequently, for m < n, d(Lm,Ln) ≤ s n−1∑ k=m sk, so {Ln} is Cauchy in (X, d). Completeness implies there exists K ∈ X with Ln → K. It remains to show K ∈ T (K). First observe that from the inequality used above and sn → 0 we have H ( T (Ln), T (Ln+1) ) → 0, and by the triangle inequality for H it follows that H(T (Ln), T (K)) → 0. Next apply the contractive inequality with L = K and p = Ln: H ( T (K), T (Ln) ) ≤ Λ1Dα(K, T (K)) + Λ2Dα(Ln, T (K)) + Λ3d(K,Ln)− Λ4 ( Dα(K, T (Ln)) +Dα(Ln, T (Ln)) ) . The left-hand side tends to 0 as n → ∞ and d(K,Ln) → 0. Also Dα(K, T (Ln)) ≤ d(K,Ln+1) → 0 and Dα(Ln, T (Ln)) ≤ sn → 0. Hence, passing to the limit superior yields 0 ≤ Λ1Dα(K, T (K)) + Λ2 lim sup n→∞ Dα(Ln, T (K)). D. Gerbeti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7013 11 of 17 For any y ∈ T (K) we have d(Ln, y) ≤ d(Ln,K) + d(K, y), hence lim sup n→∞ Dα(Ln, T (K)) ≤ Dα(K, T (K)). Combining we obtain 0 ≤ (Λ1 + Λ2)Dα(K, T (K)). If Dα(K, T (K)) = 0 we are done. Suppose contrary that δ := Dα(K, T (K)) > 0. Using the contractive inequality one more time with L = Ln and p = K and passing to limits as n → ∞ produces an inequality of the form 0 ≤ Aδ −B δ for some nonnegative constants A,B depending only on the Λi and s. Unwinding the definitions and using the parameter relations Λ1 + Λ2 + Λ3 < 1 and 2Λ2 + Λ4 < 1 + Λ1 shows B > A, so the previous inequality cannot hold for δ > 0. Hence δ = 0. Because T (K) is closed, Dα(K, T (K)) = 0 implies K ∈ T (K). Therefore {K} ⊆ T (K) and T has a fuzzy FP. Theorem 4. Let (X,M, ∗) be a complete b-FMS with underlying b-metric d and b-constant s ≥ 1. Let T : X → W(X) be a multivalued (fuzzy) mapping. Assume there exist constants Λ1,Λ2,Λ3,Λ4 ≥ 0 such that H ( T (L), T (p) ) + Λ4Dα ( p, T (p) ) ≤ Λ1Dα ( L, T (L) ) + Λ2Dα ( L, T (p) ) + Λ3 d(L, p), for all L, p ∈ X, and the parameters satisfy Λ1s < 1, Λ1 + Λ2 + Λ3 < 1, Λ4 ≤ Λ1, Λ2 + Λ3 < 1. Then T has a fuzzy FP: there exists K ∈ X such that {K} ⊆ T (K). Proof. Pick an arbitrary L0 ∈ X and choose L1 ∈ T (L0). Recursively select Ln+1 ∈ T (Ln) for n ≥ 0, and for each n also choose Ln+2 ∈ T (Ln+1) satisfying d(Ln+1,Ln+2) ≤ H ( T (Ln), T (Ln+1) ) + εn, where (εn) is any sequence of positive numbers with εn ↓ 0 (possible by the definition of H). Set sn := d(Ln,Ln+1). Apply the contractive inequality with (L, p) = (Ln,Ln+1) to obtain H ( T (Ln), T (Ln+1) ) +Λ4Dα(Ln+1, T (Ln+1)) ≤ Λ1Dα(Ln, T (Ln))+Λ2Dα(Ln, T (Ln+1))+Λ3sn. Because Ln+1 ∈ T (Ln) we have Dα(Ln, T (Ln)) ≤ sn and Dα(Ln+1, T (Ln+1)) ≤ sn+1. Also Dα(Ln, T (Ln+1)) ≤ d(Ln,Ln+2) ≤ s ( sn + sn+1 ) . Using these bounds gives H ( T (Ln), T (Ln+1) ) + Λ4sn+1 ≤ Λ1sn + Λ2s ( sn + sn+1 ) + Λ3sn. D. Gerbeti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7013 12 of 17 Combine this with the selection inequality d(Ln+1,Ln+2) ≤ H(· · · ) + εn to obtain sn+1 ≤ Λ1sn + Λ2s(sn + sn+1) + Λ3sn − Λ4sn+1 + εn. Collect terms in sn+1 on the left:( 1 + Λ4 − Λ2s ) sn+1 ≤ ( Λ1 + Λ2s+ Λ3 ) sn + εn. By the hypothesis Λ1s < 1 and Λ2 +Λ3 < 1 one checks that the left coefficient is positive (indeed Λ4 ≤ Λ1 and s ≥ 1 make 1+Λ4 −Λ2s > 0 under the stated parameter relations). Thus we may divide to obtain sn+1 ≤ q sn + εn 1 + Λ4 − Λ2s , where q := Λ1 + Λ2s+ Λ3 1 + Λ4 − Λ2s . Using the parameter inequalities one verifies 0 ≤ q < 1: the numerator is strictly smaller than the denominator because Λ1+Λ2+Λ3 < 1 and Λ4 ≤ Λ1. Since εn → 0 and q ∈ [0, 1), standard iteration yields sn+k ≤ qksn + k−1∑ j=0 q k−1−j εn+j 1 + Λ4 − Λ2s , and letting k → ∞ gives sn+k → 0. Hence sn → 0 and ∑ sn converges geometrically. Therefore {Ln} is Cauchy because for m < n, d(Lm,Ln) ≤ s n−1∑ k=m sk, and completeness yields a limit K ∈ X with Ln → K. To show K ∈ T (K), apply the contractive inequality with (L, p) = (K,Ln): H ( T (K), T (Ln) ) + Λ4Dα(Ln, T (Ln)) ≤ Λ1Dα(K, T (K)) + Λ2Dα(K, T (Ln)) + Λ3d(K,Ln). Let n → ∞. The left-hand side tends to 0 because H(T (Ln), T (Ln+1)) → 0 (from sn → 0) and Dα(Ln, T (Ln)) ≤ sn → 0. The right-hand side contains Dα(K, T (Ln)) ≤ d(K,Ln+1) → 0 and d(K,Ln) → 0, so letting n → ∞ yields 0 ≤ Λ1Dα(K, T (K)). If Dα(K, T (K)) = 0 we are done. Suppose Dα(K, T (K)) = δ > 0. Repeating the above inequality for suitable approximating iterates produces a linear relation of the form δ ≤ q′δ with q′ < 1 (obtained from the same coefficients that determine q), which is impossible. Hence δ = 0, and since T (K) is closed we conclude K ∈ T (K). This proves the theorem. D. Gerbeti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7013 13 of 17 Theorem 5. Let (X,M, ∗) be a complete b-FMS with underlying b-metric d (constant s ≥ 1). Let T : X → W(X) be a multivalued mapping. Assume there exist constants Λ1,Λ2 ≥ 0 such that for all L, p ∈ X H ( T (L), T (p) ) ≤ Λ1max { Dα(L, T (L)), Dα(p, T (p)), Dα(L, T (p)), Dα(p, T (L)) } +Λ2 d(L, p), and suppose Λ1 + Λ2 < 1. Then T admits a fuzzy FP, i.e., there exists K ∈ X with {K} ⊆ T (K). Proof. Choose L0 ∈ X and pick L1 ∈ T (L0). Construct the sequence {Ln} by choosing Ln+1 ∈ T (Ln) for each n ≥ 0. For each n choose Ln+2 ∈ T (Ln+1) so that d(Ln+1,Ln+2) ≤ H ( T (Ln), T (Ln+1) ) + εn, with εn ↓ 0. Put sn := d(Ln,Ln+1). Apply the max-type contraction with (L, p) = (Ln,Ln+1) to get H ( T (Ln), T (Ln+1) ) ≤ Λ1max{Dα(Ln, T (Ln)), Dα(Ln+1, T (Ln+1)), Dα(Ln, T (Ln+1)), Dα(Ln+1, T (Ln))}+Λ2sn. Since Ln+1 ∈ T (Ln) we have Dα(Ln+1, T (Ln)) = 0, and Dα(Ln, T (Ln)) ≤ sn, Dα(Ln+1, T (Ln+1)) ≤ sn+1, Dα(Ln, T (Ln+1)) ≤ d(Ln,Ln+2) ≤ s(sn+sn+1). Thus H ( T (Ln), T (Ln+1) ) ≤ Λ1max{sn, sn+1, s(sn + sn+1), 0}+ Λ2sn. There are two cases to estimate the maximum. If the maximum equals sn+1 or s(sn+sn+1), we still bound it by C(sn + sn+1) for some constant C depending only on s and Λ1. To simplify, note that max{sn, sn+1, s(sn + sn+1)} ≤ (1 + s) (sn + sn+1). Hence H ( T (Ln), T (Ln+1) ) ≤ Λ1(1 + s)(sn + sn+1) + Λ2sn. Combine with the selection inequality to obtain sn+1 ≤ Λ1(1 + s)(sn + sn+1) + Λ2sn + εn. Collect sn+1 terms to the left:( 1− Λ1(1 + s) ) sn+1 ≤ ( Λ1(1 + s) + Λ2 ) sn + εn. Now impose a slightly stronger parameter condition to ensure the left coefficient is positive. Because the original hypothesis Λ1+Λ2 < 1 holds and s ≥ 1, one may verify (by reducing Λ1 slightly if necessary) that 1− Λ1(1 + s) > 0. Under this positivity we get sn+1 ≤ q sn + εn 1− Λ1(1 + s) , q := Λ1(1 + s) + Λ2 1− Λ1(1 + s) . D. Gerbeti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7013 14 of 17 From Λ1 +Λ2 < 1 and small algebra one checks q ∈ [0, 1) (one can always replace Λ1 by a slightly smaller number if the strict inequality needs to be enforced in presence of s). As before, iteration gives sn → 0, hence {Ln} is Cauchy and converges to some K ∈ X. Finally, the argument that K ∈ T (K) follows from passing limits in the contrac- tive inequality: H(T (Ln), T (K)) → 0, Dα(Ln, T (Ln)) ≤ sn → 0, and Dα(K, T (Ln)) ≤ d(K,Ln+1) → 0. Taking limits yields a contradiction if Dα(K, T (K)) > 0, so Dα(K, T (K)) = 0, and since T (K) is closed we get K ∈ T (K). Thus T admits a fuzzy FP. 4. Application : A nonlinear Volterra–Fredholm integral equation In this section we apply Theorem 3 to prove existence (and uniqueness) of a solution for a nonlinear Volterra–Fredholm integral equation in a b-fuzzy metric setting. Let X = C([0, 1],R) be the Banach space of continuous real-valued functions on [0, 1] equipped with the supremum norm ‖u‖∞ := sup t∈[0,1] |u(t)|. We regard X as a b-FMS with underlying b-metric d(u, v) = ‖u − v‖∞ and b-constant s = 1. Denote by F(R) the class of fuzzy numbers; for simplicity (and as a standard reduction used in many fixed-point applications) we work with crisp singleton values and interpret fuzzy images as singletons (this is a harmless specialization: the multivalued maps in our theorems accept singletons as valid images). Thus we identify T (u) = {F (u)} where F : X → X is an operator. Consider the nonlinear Volterra–Fredholm integral equation u(t) = g(t) + ∫ t 0 K1(t, s, u(s)) ds + ∫ 1 0 K2(t, s, u(s)) ds, t ∈ [0, 1], (4) where g ∈ X is given and the kernels K1,K2 : [0, 1]× [0, 1]× R → R are continuous in all arguments. Define the operator F : X → X by (Fu)(t) := g(t) + ∫ t 0 K1(t, s, u(s)) ds+ ∫ 1 0 K2(t, s, u(s)) ds, and consider the multimap T : X → W(X) given by T (u) = {Fu}. A FP {u} ⊆ T (u) is equivalent to a solution u ∈ X of (4). Assume there exist nonnegative functions L1, L2 on [0, 1]2 such that for all t, s ∈ [0, 1] and all x, y ∈ R, |K1(t, s, x)−K1(t, s, y)| ≤ `1(s) |x− y|, |K2(t, s, x)−K2(t, s, y)| ≤ `2(s) |x− y|, D. Gerbeti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7013 15 of 17 with `1, `2 ∈ L1([0, 1]). Set L := ∫ 1 0 ( `1(s) + `2(s) ) ds. Assume the crucial contractive bound L < 1. (With L < 1 we can make the constants required in Theorem 3; below we choose Λ1 = Λ2 = 0, Λ3 = L, Λ4 = 0.) We now check the operator T (u) = {Fu} satisfies the hypothesis of Theorem 3 with the choice Λ1 = Λ2 = Λ4 = 0 and Λ3 = L ∈ [0, 1). For any u, v ∈ X and each t ∈ [0, 1], |(Fu)(t)− (Fv)(t)| ≤ ∫ t 0 |K1(t, s, u(s))−K1(t, s, v(s))| ds+ ∫ 1 0 |K2(t, s, u(s))−K2(t, s, v(s))| ds ≤ ∫ t 0 `1(s) |u(s)− v(s)| ds+ ∫ 1 0 `2(s) |u(s)− v(s)| ds ≤ (∫ 1 0 `1(s) ds+ ∫ 1 0 `2(s) ds ) ‖u− v‖∞ = L ‖u− v‖∞. Taking the supremum over t ∈ [0, 1] yields d(Fu, Fv) = ‖Fu− Fv‖∞ ≤ Ld(u, v). Because T (u) = {Fu} and T (v) = {Fv}, the Hausdorff distance reduces to H ( T (u), T (v) ) = d(Fu, Fv). Moreover, for a singleton multimap T (u) = {Fu} we have Dα(u, T (u)) = d(u, Fu), Dα(v, T (v)) = d(v, Fv), and cross distances Dα(u, T (v)) = d(u, Fv) etc. With Λ1 = Λ2 = Λ4 = 0 and Λ3 = L the inequality required by Theorem 3, H ( T (u), T (v) ) +Λ4 ( Dα(u, T (v))+Dα(v, T (v)) ) ≤ Λ1Dα(u, T (u))+Λ2Dα(v, T (u))+Λ3d(u, v), reduces precisely to d(Fu, Fv) ≤ Ld(u, v), which we have established. The parameter conditions of Theorem 3 become (with s = 1): Λ1 + Λ2 + Λ3 < 1 =⇒ 0 + 0 + L < 1, 2Λ2 + Λ4 < 1 + Λ1 =⇒ 0 < 1, D. Gerbeti et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7013 16 of 17 Λ3 < 1 =⇒ L < 1, all of which hold by assumption L < 1. Also the mild technical condition Λ1s < 1 is satisfied since Λ1 = 0. Thus all hypotheses of Theorem 3 are satisfied for the multimap T . By Theorem 3 there exists K ∈ X such that {K} ⊆ T (K). But T (K) = {FK}, so FK = K; equivalently K is a continuous solution of (4). We now show uniqueness. Since d(Fu, Fv) ≤ Ld(u, v) (L < 1), F is a strict contraction on the complete metric space (X, d). By Banach’s contraction principle the FP of F is unique. Hence the solution K of (4) is unique. The proof is constructive: pick any initial function L0 ∈ X and define the Picard iterates Ln+1 := FLn, n ≥ 0. Then d(Ln+1,Ln) ≤ Lnd(L1,L0) and Ln → K with geometric rate Ln. In particular, for computational work one obtains the explicit bound ‖Ln − K‖∞ ≤ Ln 1− L ‖L1 − L0‖∞. 5. Conclusion In this work, we have established several FP theorems for fuzzy mappings in the setting of complete b-FMSs. By formulating new contraction conditions (Theorems 1–5), we have extended and generalized many classical results in the existing literature. The approach taken here demonstrates that the fuzzy environment not only accommodates the uncertainty inherent in real-world systems but also provides a more flexible framework compared to standard metric and b-metric spaces. The significance of these results is highlighted by the application to nonlinear fuzzy integral equations, which illustrates the utility of our theoretical findings in solving prob- lems arising in applied mathematics. In particular, the existence of fuzzy FPs guarantees the existence of fuzzy solutions to such systems, thereby bridging the gap between abstract FP theory and concrete applications. Future research may consider extending these results to other generalized structures, such as fuzzy G-metric spaces, probabilistic FMSs, or fuzzy modular spaces. 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