Introducing Fixed-Point Theorems and Applications in Fuzzy Bipolar b -Metric Spaces with ψ α - and 𭟋 η - Contractive Maps

: In this study, we introduce novel concepts within the framework of fuzzy bipolar b -metric spaces, focusing on various mappings such as ψ α -contractive and 𭟋 η -contractive mappings, which are essential for quantifying distances between dissimilar elements. We establish fixed-point theorems for these mappings, demonstrating the existence of invariant points under certain conditions. To enhance the credibility and applicability of our findings, we provide illustrative examples that support these theorems and expand the existing knowledge in this field. Furthermore, we explore practical applications of our research, particularly in solving integral equations and fractional differential equations, showcasing the robustness and utility of our theoretical advancements. Symmetry, both in its traditional sense and within the fuzzy context, is fundamental to our study of fuzzy bipolar b - metric spaces. The introduced contractive mappings and fixed-point theorems expand the theoretical framework and offer robust tools for addressing practical problems where symmetry is significant.


Introduction
Fixed-point theory is very important in many fields, such as engineering, optimization, physics, economics, and mathematics.The Banach fixed-point theorem, introduced by Banach [1], greatly strengthened this theory and sparked extensive research in both mathematics and science.
In 1975, Kramosil and Michalek [2] introduced the innovative idea of fuzzy metric spaces.This concept built on the continuous t-norm introduced by Schweizer and Sklar in 1960 [3] and the foundational fuzzy set theory proposed by L.A. Zadeh in 1965 [4].George and Veeramani [5] expanded this idea by incorporating the Hausdorff topology and adapting classical metric space theorems.This expansion led to significant discoveries in fuzzy metric spaces and their generalizations [6][7][8][9][10][11].In a recent mathematical breakthrough, Mutlu and Gürdal [12] introduced bipolar metric spaces.Unlike traditional metric spaces, which focus on distances within a single set, bipolar metric spaces consider distances between points from two distinct sets.Researchers [7,12,13] have since explored fixed-point theorems in bipolar metric spaces, discovering various applications.Building on this, Bartwal et al. [14] introduced fuzzy bipolar metric spaces, extending the principles of fuzzy metric spaces.They proposed a unique way to measure distances between points in different sets, leading to significant advancements in fixed-point results for fuzzy bipolar metric spaces [12,15].Kumer et al. [9] introduced the concept of contravariant (α − ψ) Meir-Keeler contractive mappings by defining α-orbital admissible mappings and covariant Meir-Keeler contraction in bipolar metric spaces.They proved fixed-point theorems for these contractions and provided some corollaries of their main results.In 2016, Mutlu et al. [12] introduced a new type of metric space called bipolar metric spaces.Since then, researchers have established several fixed-point theorems using various contractive conditions within the context of bipolar metric spaces (see [10]).
This study aims to address a gap in research by introducing new concepts such as ψ α -contractive type covariant mappings, contravariant mappings, and  η -contractive type covariant mappings within fuzzy bipolar metric spaces.We establish fixed-point theorems in this context.Our main goal is to extend the criteria for self-mappings by introducing control functions and admissibility while considering the triangular property of induced fuzzy bipolar metrics.Although existing literature provides valuable insights into fixed-point theory and fuzzy bipolar metric spaces, the study of control functions and admissible self-mappings within fuzzy bipolar metric spaces remains unexplored.Our paper addresses a key research gap by advancing the theoretical foundations of generalized fuzzy metric spaces and enhancing the understanding of fixed-point theory.By integrating a control function and admissible self-mappings with the triangular property, our expanded framework provides a versatile foundation applicable to various fields.
In fuzzy bipolar b-metric spaces, symmetry is essential for defining the structure and properties of the space.A b-metric space generalizes a metric space by relaxing the symmetry requirement, and in the fuzzy context, distances are represented by fuzzy sets instead of exact values, allowing for a more nuanced representation of uncertainty.The ψ α -contractive and  η -contractive mappings introduced in this study can exhibit symmetry properties based on their definitions.
A mapping T is ψ α -contractive if it satisfies a condition involving a function ψ α , which can include symmetric or asymmetric terms.Similarly,  η -contractive mappings involve a function  η that can also reflect symmetry considerations.These mappings ensure the existence of fixed points in fuzzy bipolar b-metric spaces, with symmetry influencing the nature and uniqueness of these fixed points.The fixed-point theorems for ψ α -contractive and  η -contractive mappings often depend on symmetry conditions, which simplify the proofs of existence and uniqueness.
Examples in the study highlight the importance of these symmetry conditions in practical applications, such as solving integral equations and fractional differential equations, where symmetric structures like kernel functions or boundary conditions are involved.In conclusion, symmetry, both in its traditional sense and within the fuzzy context, is fundamental to our study of fuzzy bipolar b-metric spaces.The introduced contractive mappings and fixed-point theorems expand the theoretical framework and provide robust tools for addressing practical problems where symmetry plays a crucial role.
In this study, we thoroughly explore the fundamental concepts of fuzzy bipolar b-metric spaces in Section 2. In Section 3, we establish key results about the existence and uniqueness of fixed points within these spaces by introducing ψ α -contractive mappings.These results leverage a unique property of fuzzy bipolar b-metric spaces, explained with the help of a control function.In Section 4, we introduce another type of mapping called  η -contractive mappings and present additional fixed-point results.Finally, in Section 5, we demonstrate the practical applications of our findings by showing how they can be used to solve nonlinear integral equations.Our work provides valuable insights for both theoretical understanding and real-world applications, enhancing the use of fixed-point theory in fuzzy bipolar b-metric spaces.

Preliminaries
In order to demonstrate our main findings, it is necessary to introduce several fundamental definitions drawn from the existing literature, outlined below:

Definition 2 ([14]
).Let A and B be two nonempty sets.A quadruple (A , B, F υ , * ) is called a fuzzy bipolar metric space (FBMS), where * and F υ are a continuous τ-norm and a fuzzy set on A × B × (0, ∞), respectively, such that for all τ, ρ, ν > 0: Definition 3 ([17]).Let A be a non-empty set and let θ ≥ 1 be a given real number.A function ϱ : A × A → [0, ∞) is said to be a b-metric space if for all x, y, z ∈ A the following conditions hold: Remark 1 ([18]).It is important to discuss that every b-metric space is not necessarily a metric space.With θ = 1, every b-metric space is a metric space.
Then, ϱ is a b-bipolar metric and (A , B, ϱ) is a b-bipolar metric space.If A ∩ B = ∅, then the space is called a disjoint; otherwise, it is called a joint.Set A is the left pole and set B is the right pole of (A , B, ϱ).The elements of A , B, and A ∩ B are the left, right, and central elements, respectively.Definition 5 ([10]).Let A and B be two non-empty sets and θ ≥ 1.A five tuple (A , B, F υ , θ, * ) is called a fuzzy bipolar b-metric space (FBBMS), where * and F υ are the continuous τ-norm and the fuzzy set on A × B × (0, ∞), respectively, such that for all τ, ρ, ν > 0, the following is applicable: Similarly, sequence ω β , ξ β on set A , B is said to be a left and right sequence, respectively.(S2) Sequence ω β is convergent to point ω if and only if ω β is a left sequence, ω is a right point, and lim β→∞ F υ (ω β , θ, * ) = 1 for τ > 0, or ω β is a right sequence, ω is a left point, and F υ (ω, ω β , τ) = 1 for τ > 0.

ψ α -Contraction Mappings and Fixed-Point Results
Definition 12. Let Ψ θ be the family of all right-continuous and non-decreasing functions ψ : < ∞ for all t > 0, where ψ n is the n-th iterate of ψ, satisfying the following conditions: Remark 2. For our purpose, for θ ≥ 1, we define the following: It is clear that, with the help of conditions (Q1)-(Q3), if ψ ∈ Ψ θ , then lim n→∞ θ n ψ n (t) = 0 for all t > 0; hence, ψ(t) < t.
is purported to be an ψ α -contractive contravariant mapping for the functions α : for all a ∈ A , b ∈ B and τ > 0, such that θγ < 1.

Definition 15.
Let (A , B, F υ , θ, * ) be an FBBMS.Mapping is purported to be a covariant that is α-admissible if there exists a function α : is purported to be a contravariant that is α-admissible if there exists a function α : ) is an ψ α -contractive covariant mapping satisfying the following conditions: Under these conditions, ϕ admits a fixed point.That is, ϕ(µ) = µ for some µ ∈ A ∩ B.

Proof. Fix a
For any τ > 0, from condition (3) and the α-admissibility of covariant mapping ϕ, we obtain the following: By repeating this process, we obtain the following: Using conditions ( 2) and ( 4) for a = a n and b = b n , we obtain the following: and for a = a n−1 and b = b n , we obtain By the process of induction, we can obtain Symmetry 2024, 16, 777 of 23 Now, for m > n, m, n ∈ N, using the properties of ψ and b-triangularity of F υ , we obtain the following: Also, for n > m, n, m ∈ N, we obtain the following: Since θγ < 1, and letting m, n → ∞ in the above cases, we obtain lim n,m→∞ Thus, we conclude that ({a n }, {b n }) is a Cauchy bisequence in (A , B, F υ , θ, * ).Due to the completeness of FBBMS (A , B, F υ , θ, * ), ({a n }, {b n }) is a convergent bisequence; hence, through Lemma 2 it biconverges to a point µ ∈ A ∩ B, i.e., {a n } → µ and {b n } → µ.Now, we show that µ is a fixed point of ϕ.Using the properties of ϕ and b-triangularity of F υ , we obtain the following: By continuity of ϕ , ϕ(a n ) → ϕ(µ) and ϕ(b n ) → ϕ(µ).Hence, by letting n → ∞, we obtain F υ (ϕ(µ), µ, τ) = 1, τ > 0. So, ϕ(µ) = µ. 3 for all w ∈ [0, ∞).Then, it is easy to verify that ψ is right-continuous and non-decreasing and satisfies all conditions stated in Definition 12. Now, for a ∈ A and b ∈ B, a ̸ = b, we can obtain the following: Thus, ϕ : A ∪ B ⇒ A ∪ B is continuous and satisfies the following condition: So, all axioms of Theorem 1 are satisfied with γ = 1 3 , and consequently, ϕ has a unique fixed point, i.e., µ = 1.
There exists a Under these assumptions, ϕ admits a fixed point.That is, ϕ(µ) = µ for some µ ∈ A ∩ B.
Proof.By proving Theorem 1, we derived a bisequence ({a n }, {b n }), which exhibits Cauchy properties within the context of a complete FBBMS (A , B, F υ , θ, * ).This bisequence, denoted by ({a n }, {b n }), biconverges to a point µ ∈ A ∩ B, implying that both {a n } and {b n } converge to µ as n tends to infinity.Now, through condition ( 1) and ( 4), we obatin the following: Once more, employing conditions ( 2) and ( 6), along with the b-triangular property of F υ , we achieve the following: Letting n → ∞ in (7), and using the continuity of ψ, we obtain be an ψ α -contractive contravariant mapping satisfying the following conditions: ϕ is α-admissible; 3.
There exists a Under these assumptions, ϕ admits a fixed point.That is, ϕ(µ) = µ for some µ ∈ A ∩ B.
Proof.By proving Theorem 2, we derived a bisequence ({a n }, {b n }), which exhibits Cauchy properties within the context of a complete FBBMS (A , B, F υ , θ, * ).This bisequence, denoted by ({a n }, {b n }), biconverges to point µ ∈ A ∩ B, implying that both {a n } and {b n } converge to µ as n tends to infinity.Now, through condition ( 1) and ( 5), we obtain the following: Once more, employing conditions (3) and ( 8), along with the b-triangular property of F υ , we achieve the following: Letting n → ∞ in (9), and using the continuity of ψ, we obtain which yields to ϕ(µ) = µ.
Also, using conditions ( 2) and ( 11), we obtain the following: Repeating this process, we obtain In the same way, we can also obtain Letting n → ∞ in ( 12) and ( 13) provides which contradicts the uniqueness of the limit.Hence, µ = λ ∈ A ∩ B. Therefore, ϕ admits a unique fixed point in (A , B, F υ , θ, * ).

𭟋 η -Contractive Mappings and Fixed Point Results
In this section, we present the notion of  η -contractive mappings and η-admissible mappings within the framework of FBBMS.Definition 17.Let Υ θ be the family of all left-continuous non-decreasing functions  : [0, 1] → [0, 1] satisfying the following conditions: and  ∈ Υ the following condition holds: ) is  η -contractive covariant mapping satisfying the following conditions: Under these axioms, ϕ admits a fixed point.That is, ϕ(µ) = µ for some µ ∈ A ∩ B.
Using conditions ( 14) and (15), for a = a n and b = b n , we obtain and for a = a n−1 and b = b n , we obtain By the process of induction, we can obtain the following: Letting n, m → ∞ and using the properties of , we obtain lim n,m→∞ Thus, we conclude that ({a n }, {b n }) is a Cauchy bisequence in (A , B, F υ , θ, * ).Due to the completeness of FBBMS (A , B, F υ , θ, * ), ({a n }, {b n }) is a convergent bisequence; hence, through Lemma 1, it biconverges to a point µ ∈ A ∩ B i.e., {a n } → µ and {b n } → µ.
Finally, we show that µ is a fixed point of ϕ.Using properties of  and conditions ( 14) and ( 15), we obtain the following: As n → ∞, through right-continuity of , we obtain Consequently, ϕ(µ) = µ.
Theorem 7.Under the conditions stipulated in Theorem 6, and with the additional assumption that (P) there exists a point ρ ∈ A ∩ B such that η(a, ρ, τ) ≤ 1 and η(ρ, b, τ) ≤ 1 for all τ > 0, where a ∈ A and b ∈ B, then the covariant mapping ϕ, being  η -contractive, possesses a unique fixed point.
Also, using conditions ( 14) and ( 17), we obtain the following: Repeating this process, we obtain )) for all n ∈ N and τ > 0.
In the same way, we can also deduce Letting n → ∞ in ( 18) and ( 19) provides which contradicts the uniqueness of the limit.Hence, µ = λ ∈ A ∩ B. Consequently, ϕ admits a unique fixed point in (A , B, F υ , θ, * ).

Integral Equation
This subsection is devoted to illustrating how the existence and uniqueness of a solution for nonlinear integral equations are demonstrated by employing established findings concerning covariant mappings.
Let ϕ be well defined.It is worth noting that ϕ possesses a unique fixed point in C([0, q], R) if and only if the integral Equation ( 20) admits a unique solution.Let α(B, A, τ) = 1 for all A, B ∈ C([0, q], R) and τ > 0, and ψ(ν) = γ θ ν for all ν ∈ [0, ∞).It is straightforward to confirm that ψ is right-continuous and fulfills the properties outlined in Definition 12.By employing ( 21) and ( 22), for A, B ∈ C([0, q], R), we can establish the following: Hence, we obtain Therefore, the integral operator ϕ satisfies all the conditions specified in Theorem 5. Consequently, according to Theorem 5, there exists a unique fixed point in C([0, q], R) for the operator ϕ.This implies the existence of a unique solution to Problem (20) in C([0, q], R).All the conditions specified in Theorem 8 are satisfied.Hence, there exists a unique solution to the nonlinear integral problem (23) in the space C([0, 1], R).
Consider the integral equation as follows.
Theorem 9. Let us consider the integral equation where Then, the integral Equation ( 24) has a unique solution in Θ ∈ L ∞ (E 1 ) ∪ L ∞ (E 2 ).Taking the supremum on both sides, we obtain the following: As a result, all the hypotheses of Theorem 5 are fulfilled, and consequently, the fractional differential Equation (25) has a unique solution.

Conclusions and Future Work
This study introduces new concepts in the field of fuzzy bipolar b-metric spaces.We investigate various types of mappings, including ψ α -contractive and  η -contractive mappings, which are crucial for measuring distances between different entities.The paper also establishes fixed-point theorems for these mappings, demonstrating the existence of stationary points under certain conditions.We validate these theorems through examples, adding to the existing knowledge in this area.Additionally, we highlight the practical applications of these concepts, particularly in solving integral equations, thereby enhancing the reliability and usefulness of our research findings.
In future research, there is potential to expand upon the innovative concepts presented in this study within fuzzy bipolar b-metric spaces.This could involve a deeper exploration of ψ α -contractive and  η -contractive mappings to gain further insights and applications.Key areas for investigation include broadening the conditions for generalized fixed-point theorems, discovering new types of contractive mappings, and enlarging the categories of fuzzy bipolar b-metric spaces.Furthermore, the development of efficient algorithms for practical fixed-point computation will improve the theoretical results in computational scenarios.These mappings can be applied to complex systems like multi-dimensional fractional differential equations and nonlinear integral equations.Moreover, exploring their utility in diverse fields such as optimization, machine learning, and network theory will enhance their applicability.Finally, empirical validation of these theoretical advancements in real-world problems will ensure their robustness and reliability.By pursuing these avenues, future research has the potential to significantly expand both the theoretical understanding and practical applications of fuzzy bipolar b-metric spaces.