Introducing Fixed-Point Theorems and Applications in Fuzzy Bipolar b-Metric Spaces with ψ_α- and ϝ_η-Contractive Maps

Abstract

In this study, we introduce novel concepts within the framework of fuzzy bipolar b-metric spaces, focusing on various mappings such as ${\psi }_{\alpha }$-contractive and ${\mathcal{ϝ}}_{\eta }$-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.

1. 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 $(\alpha -\psi )$ Meir–Keeler contractive mappings by defining $\alpha$-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 ${\psi }_{\alpha }$-contractive type covariant mappings, contravariant mappings, and ${\mathcal{ϝ}}_{\eta }$-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 ${\psi }_{\alpha }$-contractive and ${\mathcal{ϝ}}_{\eta }$-contractive mappings introduced in this study can exhibit symmetry properties based on their definitions.

A mapping T is ${\psi }_{\alpha }$-contractive if it satisfies a condition involving a function ${\psi }_{\alpha }$, which can include symmetric or asymmetric terms. Similarly, ${\mathcal{ϝ}}_{\eta }$-contractive mappings involve a function ${\mathcal{ϝ}}_{\eta }$ 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 ${\psi }_{\alpha }$-contractive and ${\mathcal{ϝ}}_{\eta }$-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 ${\psi }_{\alpha }$-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 ${\mathcal{ϝ}}_{\eta }$-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.

2. Preliminaries

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

Definition 1

([16]). A binary operation $\ast :[0,1]\times [0,1]\to [0,1]$ is said to be a continuous τ-norm if $\left(\right[0,1],\ast )$ is a topological monoid with unit 1, such that $\xi \ast \zeta \le \gamma \ast \delta$ whenever $\xi \le \gamma ,\zeta \le \delta$ for all $\xi ,\zeta ,\gamma ,\delta \in [0,1]$.

Definition 2

([14]). Let $\mathcal{A}$ and $\mathcal{B}$ be two nonempty sets. A quadruple $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\ast )$ is called a fuzzy bipolar metric space (FBMS), where ∗ and ${\mathcal{F}}_{\upsilon }$ are a continuous τ-norm and a fuzzy set on $\mathcal{A}\times \mathcal{B}\times (0,\infty )$, respectively, such that for all $\tau ,\rho ,\nu >0$:

(FBMS1) 

${\mathcal{F}}_{\upsilon }(\omega ,\xi ,\tau )>0$ for all $(\omega ,\xi )\in \mathcal{A}\times \mathcal{B}$;

(FBMS2) 

${\mathcal{F}}_{\upsilon }(\omega ,\xi ,\tau )=1$ if and only if $\omega =\xi$ for $\omega \in \mathcal{A}$ and $\xi \in \mathcal{B}$;

(FBMS3) 

${\mathcal{F}}_{\upsilon }(\omega ,\xi ,\tau )={\mathcal{F}}_{\upsilon }(\xi ,\omega ,\tau )$ for all $\omega ,\xi \in \mathcal{A}\cap \mathcal{B}$;

(FBMS4) 

${\mathcal{F}}_{\upsilon }({\omega }_1,{\xi }_2,\tau +\rho +\nu )\ge {\mathcal{F}}_{\upsilon }({\omega }_1,{\xi }_1,\tau )\ast {\mathcal{F}}_{\upsilon }({\omega }_2,{\xi }_1,\rho )\ast {\mathcal{F}}_{\upsilon }({\omega }_2,{\xi }_2,\nu )$ for all ${\omega }_1,{\omega }_2\in \mathcal{A}$ and ${\xi }_1,{\xi }_2\in \mathcal{B}$;

(FBMS5) 

${\mathcal{F}}_{\upsilon }(\omega ,\xi ,·):[0,\infty )\to [0,1]$ is left continuous;

(FBMS6) 

${\mathcal{F}}_{\upsilon }(\omega ,\xi ,·)$ is non-decreasing for all $\omega \in \mathcal{A}$ and $\xi \in \mathcal{B}$.

Definition 3

([17]). Let $\mathcal{A}$ be a non-empty set and let $\theta \ge 1$ be a given real number. A function $\varrho :\mathcal{A}\times \mathcal{A}\to [0,\infty )$ is said to be a b-metric space if for all $x,y,z\in \mathcal{A}$ the following conditions hold:

(BM1) 

$\varrho (x,y)=0$ if and only if $x=y$;

(BM2) 

$\varrho (x,y)=\varrho (y,x)$;

(BM3) 

$\varrho (x,z)\le \theta \left[\varrho (x,y)+\varrho (y,z)\right]$.

The pair $(\mathcal{A},\varrho )$ is a b-metric space.

Remark 1

([18]). It is important to discuss that every b-metric space is not necessarily a metric space. With $\theta =1$, every b-metric space is a metric space.

Definition 4

([10]). Let $\mathcal{A}$ and $\mathcal{B}$ be two non-empty sets, and and let $\theta \ge 1$ be a given real number. Function $\varrho :\mathcal{A}\times \mathcal{A}\to [0,\infty )$ satisfies the following conditions:

(BBM1) 

$\varrho (x,y)=0$ if and only if $x=y$ for all $(x,y)\in \mathcal{A}\times \mathcal{B}$;

(BBM2) 

$\varrho (x,y)=\varrho (y,x)$ for all $x,y\in \mathcal{A}\cap \mathcal{B}$;

(BBM3) 

$\varrho (x_1,y_2)\le \theta \left[\varrho (x_1,y_1)+\varrho (x_2,y_1)+\varrho (x_2,y_2,w)\right]$ for all $x_1,x_2\in \mathcal{A}$ and $y_1,y_2\in \mathcal{B}$.

Then, ϱ is a b-bipolar metric and $(\mathcal{A},\mathcal{B},\varrho )$ is a b-bipolar metric space. If $\mathcal{A}\cap \mathcal{B}=\varnothing$, then the space is called a disjoint; otherwise, it is called a joint. Set $\mathcal{A}$ is the left pole and set $\mathcal{B}$ is the right pole of $(\mathcal{A},\mathcal{B},\varrho )$. The elements of $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{A}\cap \mathcal{B}$ are the left, right, and central elements, respectively.

Definition 5

([10]). Let $\mathcal{A}$ and $\mathcal{B}$ be two non-empty sets and $\theta \ge 1$. A five tuple $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ is called a fuzzy bipolar b-metric space (FBBMS), where ∗ and ${\mathcal{F}}_{\upsilon }$ are the continuous τ-norm and the fuzzy set on $\mathcal{A}\times \mathcal{B}\times (0,\infty )$, respectively, such that for all $\tau ,\rho ,\nu >0$, the following is applicable:

(FBMS1) 

${\mathcal{F}}_{\upsilon }(\omega ,\xi ,\tau )>0$ for all $(\omega ,\xi )\in \mathcal{A}\times \mathcal{B}$;

(FBMS2) 

${\mathcal{F}}_{\upsilon }(\omega ,\xi ,\tau )=1$ if and only if $\omega =\xi$ for $\omega \in \mathcal{A}$ and $\xi \in \mathcal{B}$;

(FBMS3) 

${\mathcal{F}}_{\upsilon }(\omega ,\xi ,\tau )={\mathcal{F}}_{\upsilon }(\xi ,\omega ,\tau )$ for all $\omega ,\xi \in \mathcal{A}\cap \mathcal{B}$;

(FBMS4) 

${\mathcal{F}}_{\upsilon }({\omega }_1,{\xi }_2,\theta (\tau +\rho +\nu ))\ge {\mathcal{F}}_{\upsilon }({\omega }_1,{\xi }_1,\tau )\ast {\mathcal{F}}_{\upsilon }({\omega }_2,{\xi }_1,\rho )\ast {\mathcal{F}}_{\upsilon }({\omega }_2,{\xi }_2,\nu )$ for all ${\omega }_1,{\omega }_2\in \mathcal{A}$ and ${\xi }_1,{\xi }_2\in \mathcal{B}$;

(FBMS5) 

${\mathcal{F}}_{\upsilon }(\omega ,\xi ,·):[0,\infty )\to [0,1]$ is left continuous;

(FBMS6) 

${\mathcal{F}}_{\upsilon }(\omega ,\xi ,·)$ is non-decreasing for all $\omega \in \mathcal{A}$ and $\xi \in \mathcal{B}$.

Definition 6

([10]). Let $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ be a fuzzy bipolar b-metric space.

(S1) 

Point $\omega \in \mathcal{A}\cup \mathcal{B}$ is called the left, right, and central point if $\omega \in \mathcal{A}$, $\omega \in \mathcal{B}$, and both hold. Similarly, sequence $\left\{{\omega }_{\beta }\right\}$, ${\xi }_{\beta }$ on set $\mathcal{A},\mathcal{B}$ is said to be a left and right sequence, respectively.

(S2) 

Sequence $\left\{{\omega }_{\beta }\right\}$ is convergent to point ω if and only if $\left\{{\omega }_{\beta }\right\}$ is a left sequence, ω is a right point, and ${lim}_{\beta \to \infty }{\mathcal{F}}_{\upsilon }({\omega }_{\beta },\theta ,\ast )=1$ for $\tau >0$, or ${\omega }_{\beta }$ is a right sequence, ω is a left point, and ${\mathcal{F}}_{\upsilon }(\omega ,{\omega }_{\beta },\tau )=1$ for $\tau >0$.

Definition 7

([10]). In an FBBMS, sequence $\left(\left\{{\omega }_n\right\},\left\{{\xi }_n\right\}\right)$ is called a bisequence on $\mathcal{A}\times \mathcal{B}$ and it is said to be convergent if both $\left\{{\omega }_n\right\}$ and $\left\{{\xi }_n\right\}$ are convergent. If both sequences converge to a common point q, then $\left(\left\{{\omega }_n\right\},\left\{{\xi }_n\right\}\right)$ is a biconvergent.

The bisequence $\left(\left\{{\omega }_n\right\},\left\{{\xi }_n\right\}\right)$ in a FBBMS ${\mathcal{F}}_{\upsilon }({\omega }_{\beta },\theta ,\ast )$ is called a Cauchy bisequence if, for any $\delta >0$, there exist a number $\gamma \in \mathbb{N}$ such that for all $w,g\ge \delta$ and $w,g\in \mathbb{N}$, we have

$${\mathcal{F}}_{\upsilon }({\omega }_w,{\xi }_g,\tau )>1-\delta ,for all \tau >0.$$

In other words, $\left(\left\{{\omega }_n\right\},\left\{{\xi }_n\right\}\right)$ is a Cauchy bisequence if

$$\underset{w,g\to \infty }{lim}{\mathcal{F}}_{\upsilon }({\omega }_w,{\xi }_g,\tau )=1,for all \tau >0.$$

Lemma 1.

In an FBBMS $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$, the limit $\gamma \in \mathcal{A}\cap \mathcal{B}$ of a bisequence is always unique.

Lemma 2

([19]). In an FBBMS $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$, if a Cauchy bisequence is convergent, it is biconvergent.

Lemma 3

([19]). An FBBMS $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ is considered complete if every Cauchy bisequence within $\mathcal{A}\times \mathcal{B}$ converges within it.

Definition 8

([10]). Let $({\mathcal{A}}_1,{\mathcal{B}}_1,{\mathcal{F}}_{\upsilon },\theta ,\ast )$ and $({\mathcal{A}}_2,{\mathcal{B}}_2,{\mathcal{M}}_{\varsigma },\theta ,\ast )$ be two FBBMSs and a function $\varphi :{\mathcal{A}}_1\cup {\mathcal{B}}_1\to {\mathcal{A}}_2\cup {\mathcal{B}}_2$. Then, the following is applicable:

(i) 

If $\varphi \left({\mathcal{A}}_1\right)\subseteq {\mathcal{B}}_2$ and $\varphi \left({\mathcal{B}}_1\right)\subseteq {\mathcal{A}}_2$, then ϕ is a contravariant from $({\mathcal{A}}_1,{\mathcal{B}}_1,{\mathcal{F}}_{\upsilon },\theta ,\ast )$ to $({\mathcal{A}}_2,{\mathcal{B}}_2,{\mathcal{M}}_{\varsigma },\theta ,\ast )$ and it is denoted by $\varphi :({\mathcal{A}}_1,{\mathcal{B}}_1,{\mathcal{F}}_{\upsilon },\theta ,\ast )\leftrightarrows ({\mathcal{A}}_2,{\mathcal{B}}_2,{\mathcal{M}}_{\varsigma },\theta ,\ast )$.

(ii) 

If $\varphi \left({\mathcal{A}}_1\right)\subseteq {\mathcal{A}}_2$ and $\varphi \left({\mathcal{B}}_1\right)\subseteq {\mathcal{B}}_2$, then ϕ is a covariant from $({\mathcal{A}}_1,{\mathcal{B}}_1,{\mathcal{F}}_{\upsilon },\theta ,\ast )$ to $({\mathcal{A}}_2,{\mathcal{B}}_2,{\mathcal{M}}_{\varsigma },\theta ,\ast )$ and it is denoted by $\varphi :({\mathcal{A}}_1,{\mathcal{B}}_1,{\mathcal{F}}_{\upsilon },\theta ,\ast )\rightrightarrows ({\mathcal{A}}_2,{\mathcal{B}}_2,{\mathcal{M}}_{\varsigma },\theta ,\ast )$.

We establish the continuity of covariant and contravariant mappings within fuzzy bipolar b-metric spaces.

Definition 9.

Let $({\mathcal{A}}_1,{\mathcal{B}}_1,{\mathcal{F}}_{\upsilon },\theta ,\ast )$ and $({\mathcal{A}}_2,{\mathcal{B}}_2,{\mathcal{M}}_{\varsigma },\theta ,\ast )$ be two FBBMSs.

(a) 

Mapping $\varphi :({\mathcal{A}}_1,{\mathcal{B}}_1,{\mathcal{F}}_{\upsilon },\theta ,\ast )\rightrightarrows ({\mathcal{A}}_2,{\mathcal{B}}_2,{\mathcal{M}}_{\varsigma },\theta ,\ast )$ is said to be left-continuous at a particular point $a_0\in {\mathcal{A}}_1$ if for any given $\epsilon >0$ there exists $\delta >0$; such that, for all $b\in {\mathcal{B}}_1$, conditions ${\mathcal{F}}_{\upsilon }(a_0,b,\tau )<\delta$ and ${\mathcal{M}}_{\varsigma }(f\left(a_0\right),f\left(b\right),\tau )<\epsilon$ hold.

(b) 

Mapping $\varphi :({\mathcal{A}}_1,{\mathcal{B}}_1,{\mathcal{F}}_{\upsilon },\theta ,\ast )\rightrightarrows ({\mathcal{A}}_2,{\mathcal{B}}_2,{\mathcal{M}}_{\varsigma },\theta ,\ast )$ is said to be left-continuous at a particular point $b_0\in {\mathcal{B}}_1$ if for any given $\epsilon >0$ there exists $\delta >0$; such that, for all $a\in {\mathcal{A}}_1$, conditions ${\mathcal{F}}_{\upsilon }(a,b_0,\tau )<\delta$ and ${\mathcal{M}}_{\varsigma }(f\left(a\right),f\left(b_0\right),\tau )<\epsilon$ hold.

(c) 

Mapping ϕ is said to be continuous if it is left-continuous at every point $a\in {\mathcal{A}}_1$ and right-continuous at each point $b\in {\mathcal{B}}_1$.

(d) 

Contravariant $\varphi :\varphi :({\mathcal{A}}_1,{\mathcal{B}}_1,{\mathcal{F}}_{\upsilon },\theta ,\ast )\leftrightarrows ({\mathcal{A}}_2,{\mathcal{B}}_2,{\mathcal{M}}_{\varsigma },\theta ,\ast )$ is continuous if and only if it is continuous when considered as a covariant mapping $\varphi :({\mathcal{A}}_1,{\mathcal{B}}_1,{\mathcal{F}}_{\upsilon },\theta ,\ast )\rightrightarrows ({\mathcal{A}}_2,{\mathcal{B}}_2,{\mathcal{M}}_{\varsigma },\theta ,\ast )$.

Definition 10

([10]). Let $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ be a fuzzy bipolar b-metric space. The fuzzy bipolar b-metric space ${\mathcal{F}}_{\upsilon }$ is b-triangular (BT) if the following is applicable:

$$\begin{array}{ccc}\hfill \frac{1}{{\mathcal{F}}_{\upsilon }({\omega }_1,{\xi }_2,\tau )}-1& \le & \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }({\omega }_1,{\xi }_1,\tau )}-1\right)+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }({\omega }_2,{\xi }_1,\tau )}-1\right)\hfill \\ & +& \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }({\omega }_2,{\xi }_2,\tau )}-1\right).\hfill \end{array}$$

Lemma 4.

Let $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ be a fuzzy bipolar b-metric space, where ∗ is a continuous τ-norm and ${\mathcal{F}}_{\upsilon }:\mathcal{A}\times \mathcal{B}\times (0,\infty )\to [0,1]$ is defined as

$${\mathcal{F}}_{\upsilon }(\omega ,\xi ,\tau )=\frac{\tau }{\tau +\varphi (\omega ,\xi )},$$

where $\varphi (\omega ,\xi )$ is a bipolar b-metric space on $\mathcal{A}\times \mathcal{B}$. Then, the FBBMS is b-triangular.

Proof. 

For any ${\omega }_1,{\omega }_2\in \mathcal{A}$ and ${\xi }_1,{\xi }_2\in \mathcal{B}$, we have the following:

$$\begin{array}{ccc}\hfill \frac{1}{{\mathcal{F}}_{\upsilon }({\omega }_1,{\xi }_2,\tau )}-1& =& \frac{\varphi ({\omega }_1,{\xi }_2)}{\tau }\hfill \\ & \le & \theta \frac{\varphi ({\omega }_1,{\xi }_1)}{\tau }+\theta \frac{\varphi ({\omega }_2,{\xi }_1)}{\tau }+\theta \frac{\varphi ({\omega }_2,{\xi }_2)}{\tau }\hfill \\ & \le & \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }({\omega }_1,{\xi }_1,\tau )}-1\right)+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }({\omega }_2,{\xi }_1,\tau )}-1\right)\hfill \\ & +& \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }({\omega }_2,{\xi }_2,\tau )}-1\right).\hfill \end{array}$$

Hence, ${\mathcal{F}}_{\upsilon }$ is b-triangular. □

Example 1.

Let $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ be a fuzzy bipolar b-metric space, where ∗ is a continuous τ-norm defined by $w\ast q=min\left\{w,q\right\}$ and ${\mathcal{F}}_{\upsilon }:\mathcal{A}\times \mathcal{B}\times (0,\infty )\to [0,1]$ defined by

$${\mathcal{F}}_{\upsilon }(\omega ,\xi ,\tau )=\frac{\tau }{\tau +d_b(\omega ,\xi )},$$

where $d_b$ is a b-metric space. Then, the FBBMS is b-triangular.

Proof. 

For any ${\omega }_1,{\omega }_2\in \mathcal{A}$ and ${\xi }_1,{\xi }_2\in \mathcal{B}$, we have the following:

$$\begin{array}{ccc}\hfill \frac{1}{{\mathcal{F}}_{\upsilon }({\omega }_1,{\xi }_2,\tau )}-1& =& \frac{d_p({\omega }_1,{\xi }_2)}{\tau }\hfill \\ & \le & \theta \frac{d_p({\omega }_1,{\xi }_1)}{\tau }+\theta \frac{d_p({\omega }_2,{\xi }_1)}{\tau }+\theta \frac{d_p({\omega }_2,{\xi }_2)}{\tau }\hfill \\ & \le & \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }({\omega }_1,{\xi }_1,\tau )}-1\right)+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }({\omega }_2,{\xi }_1,\tau )}-1\right)\hfill \\ & +& \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }({\omega }_2,{\xi }_2,\tau )}-1\right),\mathrm{for}\tau >0.\hfill \end{array}$$

Consequently, the FBBMS is b-triangular. □

Definition 11.

Let $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ be a complete FBBSM with a constant $\theta \ge 1$, where ∗ is a continuous τ-norm and mapping $\psi :\mathcal{A}\cup \mathcal{B}\to \mathcal{A}\cup \mathcal{B}$ is called a fuzzy b-contraction if there exists $\gamma \in (0,\frac{1}{\theta })$, such that

$$\frac{1}{{\mathcal{F}}_{\upsilon }(\psi \left(\omega \right),\psi \left(\xi \right),\tau )}-1\le \gamma \left(\frac{1}{{\mathcal{F}}_{\upsilon }(\omega ,\xi ,\tau )}-1\right)$$

(1)

for all $\omega \in \mathcal{A},\xi \in \mathcal{B}$, and $\tau >0$, such that $\theta \gamma <1$.

3. ${\mathbf{\psi }}_{\mathbf{\alpha }}$-Contraction Mappings and Fixed-Point Results

Definition 12.

Let ${\Psi }_{\theta }$ be the family of all right-continuous and non-decreasing functions $\psi :[0,\infty )\to [0,\infty )$ such that ${\sum }_{n=1}^{\infty }{\psi }^n\left(t\right)<\infty$ for all $t>0$, where ${\psi }^n$ is the n-th iterate of ψ, satisfying the following conditions:

(Q1) 

$\psi \left(0\right)=0$;

(Q2) 

$\psi \left(\kappa \right)<\kappa$ for all $\kappa >0$;

(Q3) 

${lim}_{n\to \infty }{\psi }^n\left(\kappa \right)=0$ for all $\kappa >0$, where ${\psi }^n\left(\kappa \right)$ is the n-th iteration of ψ at κ.

Remark 2.

For our purpose, for $\theta \ge 1$, we define the following:

$${\Psi }_{\theta }:=\left\{\psi :[0,\infty )\to [0,\infty ),\psi is non-decreasing,right-continuous,and{\sum }_{n=1}^{\infty }{\theta }^n{\psi }^n\left(t\right)<\infty \right\}.$$

It is clear that, with the help of conditions (Q1)–(Q3), if $\psi \in {\Psi }_{\theta }$, then ${lim}_{n\to \infty }{\theta }^n{\psi }^n\left(t\right)=0$ for all $t>0$; hence, $\psi \left(t\right)<t$.

Definition 13.

Let $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ be an FBBMS. Mapping

$$\varphi :(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )\rightrightarrows (\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$$

is purported to be an ${\psi }_{\alpha }$-contractive covariant mapping if for the functions $\alpha :\mathcal{A}\times \mathcal{B}\times (0,\infty )\to [0,\infty )$, $\psi \in {\Psi }_{\theta }$, and $\gamma \in (0,\frac{1}{\theta })$, the below condition holds:

$$\alpha (a,b,\tau )\left(\frac{1}{{\mathcal{F}}_{\upsilon }(\varphi \left(a\right),\varphi \left(b\right),\tau )}-1\right)\le \gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }(a,b,\tau )}-1\right)$$

(2)

for all $a\in \mathcal{A},b\in \mathcal{B}$ and $\tau >0$, such that $\theta \gamma <1$.

Definition 14.

Let $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ be an FBBMS. Mapping

$$\varphi :(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )\rightleftarrows (\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$$

is purported to be an ${\psi }_{\alpha }$-contractive contravariant mapping for the functions $\alpha :\mathcal{A}\times \mathcal{B}\times (0,\infty )\to [0,\infty )$, $\psi \in {\Psi }_{\theta }$, and $\gamma \in (0,\frac{1}{\theta })$, such that the below condition holds:

$$\alpha (a,b,\tau )\left(\frac{1}{{\mathcal{F}}_{\upsilon }(\varphi \left(b\right),\varphi \left(a\right),\tau )}-1\right)\le \gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }(a,b,\tau )}-1\right)$$

(3)

for all $a\in \mathcal{A},b\in \mathcal{B}$ and $\tau >0$, such that $\theta \gamma <1$.

Definition 15.

Let $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ be an FBBMS. Mapping

$$\varphi :(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )\rightrightarrows (\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$$

is purported to be a covariant that is α-admissible if there exists a function $\alpha :\mathcal{A}\times \mathcal{B}\times (0,\infty )\to [0,\infty )$ such that, for all $a\in \mathcal{A},b\in \mathcal{B}$ and $\tau >0$,

$$\alpha (a,b,\tau )\ge 1implies\alpha \left(\varphi \right(a),\varphi (b),\tau )\ge 1.$$

Definition 16.

Let $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ be an FBBMS. Mapping

$$\varphi :(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )\leftrightarrows (\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$$

is purported to be a contravariant that is α-admissible if there exists a function $\alpha :\mathcal{A}\times \mathcal{B}\times (0,\infty )\to [0,\infty )$ such that, for all $a\in \mathcal{A},b\in \mathcal{B}$ and $\tau >0$,

$$\alpha (a,b,\tau )\ge 1implies\alpha \left(\varphi \right(b),\varphi (a),\tau )\ge 1.$$

Theorem 1.

Let $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ be a complete FBBMS. Assume that $\varphi :(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )\rightrightarrows (\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ is an ${\psi }_{\alpha }$-contractive covariant mapping satisfying the following conditions:

(i) 

ϕ is continuous.

(ii) 

ϕ is α-admissible.

(iii) 

There exists $a_0\in \mathcal{A}$, $b_0\in \mathcal{B}$ such that $\alpha (a_0,b_0,\tau )\ge 1$, $\alpha (a_0,\varphi \left(b_0\right),\tau )\ge 1$ for all $\tau >0$.

Under these conditions, ϕ admits a fixed point. That is, $\varphi \left(\mu \right)=\mu$ for some $\mu \in \mathcal{A}\cap \mathcal{B}$.

Proof. 

Fix $a_0\in \mathcal{A}$ and $b_0\in \mathcal{B}$ such that $\alpha (a_0,\varphi \left(b_0\right),\tau )\ge 1$ for all $\tau >0$. Define $\varphi \left(a_n\right)=a_{n+1}$ and $\varphi \left(b_n\right)=b_{n+1}$ for all $n\in \mathbb{N}\cup \left\{0\right\}$. Then, $\left(\left\{a_n\right\},\left\{b_n\right\}\right)$ is a bisequence in $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$.

For any $\tau >0$, from condition (3) and the $\alpha$-admissibility of covariant mapping $\varphi$, we obtain the following:

$$\begin{array}{ccc}\hfill \alpha (a_0,b_0,\tau )\ge 1& \Rightarrow & \alpha (\varphi \left(a_0\right),\varphi \left(b_0\right),\tau )\ge 1,\hfill \\ \hfill \alpha (a_0,b_1,\tau )=\alpha (a_0,\varphi \left(b_0\right),\tau )\ge 1& \Rightarrow & \alpha (\varphi \left(a_0\right),\varphi \left(b_1\right),\tau )=\alpha (a_1,b_2,\tau )\ge 1,\hfill \\ \hfill \alpha (a_1,b_1,\tau )=\alpha (\varphi \left(a_0\right),\varphi \left(b_0\right),\tau )\ge 1& \Rightarrow & \alpha (\varphi \left(a_1\right),\varphi \left(b_1\right),\tau )=\alpha (a_2,b_2,\tau )\ge 1,\hfill \\ \hfill \alpha (a_1,b_2,\tau )=\alpha (a_1,\varphi \left(b_1\right),\tau )\ge 1& \Rightarrow & \alpha (\varphi \left(a_1\right),\varphi \left(b_1\right),\tau )=\alpha (a_2,b_2,\tau )\ge 1,\hfill \\ \hfill \alpha (a_2,b_2,\tau )=\alpha (\varphi \left(a_1\right),\varphi \left(b_1\right),\tau )\ge 1& \Rightarrow & \alpha (\varphi \left(a_2\right),\varphi \left(b_2\right),\tau )=\alpha (a_3,b_3,\tau )\ge 1.\hfill \end{array}$$

By repeating this process, we obtain the following:

$$\alpha (a_{n+1},b_{n+1},\tau )\ge 1\mathrm{and}\alpha (a_n,b_{n+1},\tau )\ge 1,\mathrm{for} \mathrm{all}n\in \mathbb{N}.$$

(4)

Using conditions (2) and (4) for $a=a_n$ and $b=b_n$, we obtain the following:

$$\begin{array}{ccc}\hfill \frac{1}{{\mathcal{F}}_{\upsilon }\left(a_{n+1},b_{n+1},\tau \right)}-1& =& \frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(a_n\right),\varphi \left(b_n\right),\tau \right)}-1\hfill \\ & \le & \alpha \left(a_n,b_n,\tau \right)\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(a_n\right),\varphi \left(b_n\right),\tau \right)}-1\right)\hfill \\ & \le & \gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(a_n\right),\varphi \left(b_n\right),\tau \right)}-1\right),\hfill \end{array}$$

and for $a=a_{n-1}$ and $b=b_n$, we obtain

$$\begin{array}{ccc}\hfill \frac{1}{{\mathcal{F}}_{\upsilon }\left(a_n,b_{n+1},\tau \right)}-1& =& \frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(a_{n-1}\right),\varphi \left(b_n\right),\tau \right)}-1\hfill \\ & \le & \alpha \left(a_{n-1},b_n,\tau \right)\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(a_{n-1}\right),\varphi \left(b_n\right),\tau \right)}-1\right)\hfill \\ & \le & \gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_n,b_n,\tau \right)}-1\right).\hfill \end{array}$$

By the process of induction, we can obtain

$$\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_{n+1},b_{n+1},\tau \right)}-1\le {\gamma }^{n+1}{\psi }^{n+1}\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)$$

and

$$\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_n,b_{n+1},\tau \right)}-1\le {\gamma }^n{\psi }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_1,\tau \right)}-1\right).$$

Now, for $m>n$, $m,n\in \mathbb{N}$, using the properties of $\psi$ and b-triangularity of ${\mathcal{F}}_{\upsilon }$, we obtain the following:

$$\begin{array}{ccc}\hfill \frac{1}{{\mathcal{F}}_{\upsilon }(a_n,b_m,\tau )}-1& \le & \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }(a_n,b_n,\tau )}-1\right)+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }(a_{n+1},b_n,\tau )}-1\right)\hfill \\ & +& \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }(a_{n+1},b_m,\tau )}-1\right)\hfill \\ & & ⋮\hfill \\ & \le & \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }(a_n,b_n,\tau )}-1\right)+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }(a_{n+1},b_m,\tau )}-1\right)\hfill \\ & +& \cdots +{\theta }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }(a_{m-1},b_{m-1},\tau )}-1\right)+{\theta }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }(a_m,b_{m-1},\tau )}-1\right)\hfill \\ & +& {\theta }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }(a_m,b_m,\tau )}-1\right)\hfill \\ & \le & \theta {\gamma }^n{\psi }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)+\theta {\gamma }^n{\psi }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_1,b_0,\tau \right)}-1\right)\hfill \\ & +& \cdots +{\theta }^m{\gamma }^{m-1}{\psi }^{m-1}\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)\hfill \\ & +& {\theta }^m{\gamma }^m{\psi }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_1,b_0,\tau \right)}-1\right)+{\theta }^m{\gamma }^m{\psi }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)\hfill \\ & \le & \theta {\gamma }^n\left(1+\theta \gamma +{\theta }^2{\gamma }^2+\cdots +{\theta }^{m-1}{\gamma }^{m-n}\right){\psi }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)\hfill \\ & +& \theta {\gamma }^n\left(1+\theta \gamma +{\theta }^2{\gamma }^2+\cdots +{\theta }^{m-1}{\gamma }^{m-n}\right){\psi }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_1,b_0,\tau \right)}-1\right)\hfill \\ & \le & \frac{\theta {\gamma }^n}{1-\theta \gamma }{\psi }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)+\frac{\theta {\gamma }^n}{1-\theta \gamma }{\psi }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_1,b_0,\tau \right)}-1\right).\hfill \end{array}$$

Also, for $n>m$, $n,m\in \mathbb{N}$, we obtain the following:

$$\begin{array}{ccc}\hfill \frac{1}{{\mathcal{F}}_{\upsilon }(a_n,b_m,\tau )}-1& \le & \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }(a_m,b_m,\tau )}-1\right)+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }(a_{m+1},b_m,\tau )}-1\right)\hfill \\ & +& \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }(a_{m+1},b_n,\tau )}-1\right)\hfill \\ & & ⋮\hfill \\ & \le & \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }(a_m,b_m,\tau )}-1\right)+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }(a_{m+1},b_n,\tau )}-1\right)\hfill \\ & +& \cdots +{\theta }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }(a_{n-1},b_{n-1},\tau )}-1\right)+{\theta }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }(a_n,b_{n-1},\tau )}-1\right)\hfill \\ & +& {\theta }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }(a_n,b_n,\tau )}-1\right)\hfill \\ & \le & \theta {\gamma }^m{\psi }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)+\theta {\gamma }^m{\psi }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_1,b_0,\tau \right)}-1\right)\hfill \\ & +& \cdots +{\theta }^n{\gamma }^{n-1}{\psi }^{n-1}\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)\hfill \\ & +& {\theta }^n{\gamma }^n{\psi }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_1,b_0,\tau \right)}-1\right)+{\theta }^n{\gamma }^n{\psi }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)\hfill \end{array}$$

$$\begin{array}{ccc}& \le & \theta {\gamma }^m\left(1+\theta \gamma +{\theta }^2{\gamma }^2+\cdots +{\theta }^{n-1}{\gamma }^{n-m}\right){\psi }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)\hfill \\ & +& \theta {\gamma }^m\left(1+\theta \gamma +{\theta }^2{\gamma }^2+\cdots +{\theta }^{n-1}{\gamma }^{n-m}\right){\psi }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_1,b_0,\tau \right)}-1\right)\hfill \\ & \le & \frac{\theta {\gamma }^m}{1-\theta \gamma }{\psi }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)+\frac{\theta {\gamma }^m}{1-\theta \gamma }{\psi }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_1,b_0,\tau \right)}-1\right).\hfill \end{array}$$

Since $\theta \gamma <1$, and letting $m,n\to \infty$ in the above cases, we obtain

$$\underset{n,m\to \infty }{lim}{\mathcal{F}}_{\upsilon }(a_n,b_m,\tau )=1,\tau >0.$$

Thus, we conclude that $\left(\left\{a_n\right\},\left\{b_n\right\}\right)$ is a Cauchy bisequence in $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$. Due to the completeness of FBBMS $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$, $\left(\left\{a_n\right\},\left\{b_n\right\}\right)$ is a convergent bisequence; hence, through Lemma 2 it biconverges to a point $\mu \in \mathcal{A}\cap \mathcal{B}$, i.e., $\left\{a_n\right\}\to \mu$ and $\left\{b_n\right\}\to \mu$.

Now, we show that $\mu$ is a fixed point of $\varphi$. Using the properties of $\varphi$ and b-triangularity of ${\mathcal{F}}_{\upsilon }$, we obtain the following:

$$\begin{array}{ccc}\hfill \frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(\mu \right),\mu ,\tau \right)}-1& \le & \le \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(\mu \right),\varphi \left(a_n\right),\tau \right)}-1\right)+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(a_n\right),\varphi \left(b_n\right),\tau \right)}-1\right)\hfill \\ & +& \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(b_n\right),\mu ,\tau \right)}-1\right)\hfill \\ & \le & \theta \gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\mu ,b_n,\tau \right)}-1\right)+\theta \gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_n,b_n,\tau \right)}-1\right)\hfill \\ & +& \theta \gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\mu ,b_n,\tau \right)}-1\right).\hfill \end{array}$$

By continuity of $\varphi$, $\varphi \left(a_n\right)\to \varphi \left(\mu \right)$ and $\varphi \left(b_n\right)\to \varphi \left(\mu \right)$. Hence, by letting $n\to \infty$, we obtain ${\mathcal{F}}_{\upsilon }(\varphi \left(\mu \right),\mu ,\tau )=1$, $\tau >0$. So, $\varphi \left(\mu \right)=\mu$. □

Example 2.

Let $\mathcal{A}=[-1,1]$ and $\mathcal{B}=\left(\mathbb{N}\cup \left\{0\right\}\right)\setminus \left\{1\right\}$ equipped with a continuous τ-norm. Define ${\mathcal{F}}_{\upsilon }(a,b,\tau )=\frac{\tau }{{\tau +|a-b|}^2}$ for all $a\in \mathcal{A},b\in \mathcal{B}$ and $\tau >0$. Clearly, $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ is a complete FBBMS. Define $\alpha (a,b,\tau )=1$ for all $a\in \mathcal{A},b\in \mathcal{B}$ and $\tau >0$. Suppose $\varphi :\mathcal{A}\cup \mathcal{B}\rightrightarrows \mathcal{A}\cup \mathcal{B}$ can be defined by $\varphi \left(s\right)=sin\left(\frac{s}{5}\right)$. Consider $\psi \left(w\right)=\frac{w}{3}$ for all $w\in [0,\infty )$. Then, it is easy to verify that ψ is right-continuous and non-decreasing and satisfies all conditions stated in Definition 12.

Now, for $a\in \mathcal{A}$ and $b\in \mathcal{B}$, $a\ne b$, we can obtain the following:

$$\begin{array}{ccc}\hfill \alpha (a,b,\tau )\left(\frac{1}{{\mathcal{F}}_{\upsilon }(\varphi \left(a\right),\varphi \left(b\right),\tau )}-1\right)& =& \frac{\left|\varphi \right(a)-\varphi (b\left)\right|}{\tau }\hfill \\ & =& \frac{|sin\left(\frac{a}{5}\right)-sin\left(\frac{b}{5}\right){|}^2}{\tau }\hfill \\ & \le & \frac{1}{25}\frac{{|a-b|}^2}{\tau }\hfill \\ & \le & \frac{1}{9}\frac{{|a-b|}^2}{\tau }\hfill \\ & =& \frac{1}{3}\psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }(a,b,\tau )}-1\right).\hfill \end{array}$$

Thus, $\varphi :\mathcal{A}\cup \mathcal{B}\rightrightarrows \mathcal{A}\cup \mathcal{B}$ is continuous and satisfies the following condition:

$$\alpha (a,b,\tau )\left(\frac{1}{{\mathcal{F}}_{\upsilon }(\varphi \left(a\right),\varphi \left(b\right),\tau )}-1\right)\le \gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }(a,b,\tau )}-1\right)$$

for all $a\in \mathcal{A},b\in \mathcal{B}$ with $a\ne b$ and $\tau >0$.

So, all axioms of Theorem 1 are satisfied with $\gamma =\frac{1}{3}$, and consequently, ϕ has a unique fixed point, i.e., $\mu =1$.

Theorem 2.

Let $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ be a complete FBBMS. Assume that $\varphi :(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )\leftrightarrows (\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ is an ${\psi }_{\alpha }$-contractive contravariant mapping satisfying the following conditions:

(i) 

ϕ is continuous;

(ii) 

ϕ is α-admissible;

(iii) 

There exists $a_0\in \mathcal{A}$, $b_0\in \mathcal{B}$ such that $\alpha (a_0,b_0,\tau )\ge 1$, $\alpha (a_0,\varphi \left(b_0\right),\tau )\ge 1$ for all $\tau >0$.

Under these conditions, ϕ admits a fixed point. That is, $\varphi \left(\mu \right)=\mu$ for $\mu \in \mathcal{A}\cap \mathcal{B}$.

Proof. 

Fix $a_0\in \mathcal{A}$ and $b_0\in \mathcal{B}$ such that $\alpha (a_0,\varphi \left(b_0\right),\tau )\ge 1$ for all $\tau >0$. Define $\varphi \left(a_n\right)=b_n$ and $\varphi \left(b_n\right)=a_{n+1}$ for all $n\in \mathbb{N}\cup \left\{0\right\}$. Then, $\left(\left\{a_n\right\},\left\{b_n\right\}\right)$ is a bisequence in $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$.

For any $\tau >0$, from condition (3) and the $\alpha$-admissibility of covariant mapping $\varphi$, we obtain the following:

$$\begin{array}{ccc}\hfill \alpha (a_0,b_0,\tau )\ge 1& \Rightarrow & \alpha (\varphi \left(b_0\right),\varphi \left(a_0\right),\tau )=\alpha (a_1,b_0,\tau )\ge 1\hfill \\ \hfill \alpha (a_1,b_0,\tau )\ge 1& \Rightarrow & \alpha (\varphi \left(b_0\right),\varphi \left(a_1\right),\tau )=\alpha (a_1,b_1,\tau )\ge 1\hfill \\ \hfill \alpha (a_1,b_1,\tau )\ge 1& \Rightarrow & \alpha (\varphi \left(b_1\right),\varphi \left(a_1\right),\tau )=\alpha (a_2,b_1,\tau )\ge 1\hfill \\ \hfill \alpha (a_2,b_1,\tau )\ge 1& \Rightarrow & \alpha (\varphi \left(b_1\right),\varphi \left(a_2\right),\tau )=\alpha (a_2,b_2,\tau )\ge 1\hfill \\ \hfill \alpha (a_2,b_2,\tau )\ge 1& \Rightarrow & \alpha (\varphi \left(b_2\right),\varphi \left(a_2\right),\tau )=\alpha (a_3,b_2,\tau )\ge 1.\hfill \end{array}$$

By repeating this process, we obtain

$$\alpha (a_n,b_n,\tau )\ge 1\mathrm{and}\alpha (a_{n+1},b_n,\tau )\ge 1,\mathrm{for} \mathrm{all}n\in \mathbb{N}.$$

(5)

Employing the conditions (3) and (5) for $a=a_n$ and $b=b_{n-1}$, we obtain

$$\begin{array}{ccc}\hfill \frac{1}{{\mathcal{F}}_{\upsilon }\left(a_n,b_n,\tau \right)}-1& =& \frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(b_{n-1}\right),\varphi \left(a_n\right),\tau \right)}-1\hfill \\ & \le & \alpha \left(a_n,b_{n-1},\tau \right)\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(b_{n-1}\right),\varphi \left(a_n\right),\tau \right)}-1\right)\hfill \\ & \le & \gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(a_n\right),\varphi \left(b_{n-1}\right),\tau \right)}-1\right),\hfill \end{array}$$

and for $a=a_{n+1}$ and $b=b_n$, we obtain

$$\begin{array}{ccc}\hfill \frac{1}{{\mathcal{F}}_{\upsilon }\left(a_{n+1},b_n,\tau \right)}-1& =& \frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(b_n\right),\varphi \left(a_n\right),\tau \right)}-1\hfill \\ & \le & \alpha \left(a_n,b_n,\tau \right)\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(b_n\right),\varphi \left(a_n\right),\tau \right)}-1\right)\hfill \\ & \le & \gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_n,b_n,\tau \right)}-1\right).\hfill \end{array}$$

By the process of induction, we can obtain

$$\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_n,b_n,\tau \right)}-1\le {\gamma }^{2n-1}{\psi }^{2n-1}\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_1,\tau \right)}-1\right)$$

and

$$\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_{n+1},b_n,\tau \right)}-1\le {\gamma }^{2n+1}{\psi }^{2n+1}\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right).$$

Now, for $m>n$, $m,n\in \mathbb{N}$, using the properties of $\psi$ and b-triangularity of ${\mathcal{F}}_{\upsilon }$, we obtain the following:

$$\begin{array}{ccc}\hfill \frac{1}{{\mathcal{F}}_{\upsilon }\left(a_n,b_m,\tau \right)}-1& \le & \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_n,b_n,\tau \right)}-1\right)+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_n,b_{n+1},\tau \right)}-1\right)\hfill \\ & +& \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_{n+1},b_m,\tau \right)}-1\right)\hfill \\ & & ⋮\hfill \\ & \le & \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_n,b_n,\tau \right)}-1\right)+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_n,b_{n+1},\tau \right)}-1\right)\hfill \\ & +& \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_{n+1},b_{n+1},\tau \right)}-1\right)\hfill \\ & +& \cdots +{\theta }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_{m-1},b_{m-1},\tau \right)}-1\right)+{\theta }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_{m-1},b_m,\tau \right)}-1\right)\hfill \\ & +& {\theta }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_m,b_m,\tau \right)}-1\right)\hfill \\ & \le & \theta {\gamma }^n{\psi }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)+\theta {\gamma }^n{\psi }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_1,\tau \right)}-1\right)\hfill \\ & +& \theta {\gamma }^{n+1}{\psi }^{n+1}\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)\hfill \\ & +& \cdots +{\theta }^m{\gamma }^{m-1}{\psi }^{m-1}\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)\hfill \\ & +& {\theta }^m{\gamma }^{m-1}{\psi }^{m-1}\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_1,\tau \right)}-1\right)\hfill \\ & +& {\theta }^m{\gamma }^m{\psi }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)\hfill \\ & \le & \theta {\gamma }^n\left(1+\theta \gamma +\cdots +{\theta }^{m-1}{\gamma }^{m-1}\right){\psi }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)\hfill \\ & +& \theta {\gamma }^n\left(1+\theta \gamma +\cdots +{\theta }^{m-1}{\gamma }^{m-1}\right){\psi }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_1,\tau \right)}-1\right)\hfill \\ & \le & \frac{\theta {\gamma }^n}{1-\theta \gamma }{\psi }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)+\frac{\theta {\gamma }^n}{1-\theta \gamma }{\psi }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_1,\tau \right)}-1\right).\hfill \end{array}$$

Also, for $n>m$, $n,m\in \mathbb{N}$, we obtain the following:

$$\begin{array}{ccc}\hfill \frac{1}{{\mathcal{F}}_{\upsilon }\left(a_n,b_m,\tau \right)}-1& \le & \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_m,b_m,\tau \right)}-1\right)+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_m,b_{m+1},\tau \right)}-1\right)\hfill \\ & +& \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_{m+1},b_n,\tau \right)}-1\right)\hfill \\ & & ⋮\hfill \\ & \le & \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_m,b_m,\tau \right)}-1\right)+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_m,b_{m+1},\tau \right)}-1\right)\hfill \\ & +& \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_{m+1},b_{m+1},\tau \right)}-1\right)\hfill \\ & +& \cdots +{\theta }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_{n-1},b_{n-1},\tau \right)}-1\right)+{\theta }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_{n-1},b_n,\tau \right)}-1\right)\hfill \\ & +& {\theta }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_n,b_n,\tau \right)}-1\right)\hfill \end{array}$$

$$\begin{array}{ccc}& \le & \theta {\gamma }^m{\psi }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)+\theta {\gamma }^m{\psi }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_1,\tau \right)}-1\right)\hfill \\ & +& \theta {\gamma }^{m+1}{\psi }^{m+1}\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)\hfill \\ & +& \cdots +{\theta }^n{\gamma }^{n-1}{\psi }^{n-1}\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)+{\theta }^n{\gamma }^{n-1}{\psi }^{n-1}\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_1,\tau \right)}-1\right)\hfill \\ & +& {\theta }^n{\gamma }^n{\psi }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)\hfill \\ & \le & \theta {\gamma }^m\left(1+\theta \gamma +\cdots +{\theta }^{n-1}{\gamma }^{n-1}\right){\psi }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)\hfill \\ & +& \theta {\gamma }^n\left(1+\theta \gamma +\cdots +{\theta }^{m-1}{\gamma }^{m-1}\right){\psi }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_1,\tau \right)}-1\right)\hfill \\ & \le & \frac{\theta {\gamma }^m}{1-\theta \gamma }{\psi }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_0,\tau \right)}-1\right)+\frac{\theta {\gamma }^m}{1-\theta \gamma }{\psi }^m\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_0,b_1,\tau \right)}-1\right).\hfill \end{array}$$

Since $\theta \gamma <1$, and letting $m,n\to \infty$ in the above cases, we obtain

$$\underset{n,m\to \infty }{lim}{\mathcal{F}}_{\upsilon }(a_n,b_m,\tau )=1,\tau >0.$$

Thus, we conclude that $\left(\left\{a_n\right\},\left\{b_n\right\}\right)$ is a Cauchy bisequence in $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$. Due to the completeness of FBBMS $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$, $\left(\left\{a_n\right\},\left\{b_n\right\}\right)$ is a convergent bisequence; hence, through Lemma 2, it biconverges to a point $\mu \in \mathcal{A}\cap \mathcal{B}$, i.e., $\left\{a_n\right\}\to \mu$ and $\left\{b_n\right\}\to \mu$.

Now, we show that $\mu$ is a fixed point of $\varphi$. Using the properties of $\varphi$ and b-triangularity of ${\mathcal{F}}_{\upsilon }$, we obtain

$$\begin{array}{ccc}\hfill \frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(\mu \right),\mu ,\tau \right)}-1& \le & \le \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(\mu \right),\varphi \left(a_n\right),\tau \right)}-1\right)+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(a_n\right),\varphi \left(b_n\right),\tau \right)}-1\right)\hfill \\ & +& \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(b_n\right),\mu ,\tau \right)}-1\right).\hfill \end{array}$$

By continuity of $\varphi$, $\varphi \left(a_n\right)\to \varphi \left(\mu \right)$ and $\varphi \left(b_n\right)\to \varphi \left(\mu \right)$. Hence, by letting $n\to \infty$, we obtain ${\mathcal{F}}_{\upsilon }(\varphi \left(\mu \right),\mu ,\tau )=1$, $\tau >0$. So, $\varphi \left(\mu \right)=\mu$. □

Theorem 3.

Let $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ be a complete FBBMS, and let

$$\varphi :(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )\rightrightarrows (\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$$

be an ${\psi }_{\alpha }$-contractive covariant mapping satisfying the following conditions:

1. 

For bisequence $\left(\left\{a_n\right\},\left\{b_n\right\}\right)$, if $\alpha (a_n,b_n,\tau )\ge 1$ for all $n\in \mathbb{N}$, $\left\{a_n\right\}\to \mu$, $\left\{b_n\right\}\to \mu$ as $n\to \infty$ for $\mu \in \mathcal{A}\cap \mathcal{B}$, then $\alpha (\mu ,b_n,\tau )\ge 1$ for all $\tau >0$ and $n\in \mathbb{N}$;

2. 

ϕ is α-admissible;

3. 

There exists $a_0\in \mathcal{A}$, $b_0\in \mathcal{B}$ such that $\alpha (a_0,b_0,\tau )\ge 1$, $\alpha (a_0,\varphi \left(b_0\right),\tau )\ge 1$.

Under these assumptions, ϕ admits a fixed point. That is, $\varphi \left(\mu \right)=\mu$ for some $\mu \in \mathcal{A}\cap \mathcal{B}$.

Proof. 

By proving Theorem 1, we derived a bisequence $\left(\left\{a_n\right\},\left\{b_n\right\}\right)$, which exhibits Cauchy properties within the context of a complete FBBMS $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$. This bisequence, denoted by $\left(\left\{a_n\right\},\left\{b_n\right\}\right)$, biconverges to a point $\mu \in \mathcal{A}\cap \mathcal{B}$, implying that both $\left\{a_n\right\}$ and $\left\{b_n\right\}$ converge to $\mu$ as n tends to infinity.

Now, through condition (1) and (4), we obatin the following:

$$\alpha (\mu ,b_n,\tau )\ge 1\mathrm{for} \mathrm{all}n\in \mathbb{N}\mathrm{and}\tau >0.$$

(6)

Once more, employing conditions (2) and (6), along with the b-triangular property of ${\mathcal{F}}_{\upsilon }$, we achieve the following:

$$\begin{array}{ccc}\hfill & & \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(\mu \right),\mu ,\tau \right)}-1\right)\le \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(\mu \right),\varphi \left(b_n\right),\tau \right)}-1\right)+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(a_n\right),\varphi \left(b_n\right),\tau \right)}-1\right)\hfill \\ & +& \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(a_n\right),\mu ,\tau \right)}-1\right)\hfill \\ & \le & \alpha (\mu ,b_n,\tau )\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(\mu \right),\varphi \left(b_n\right),\tau \right)}-1\right)+\alpha (a_n,b_n,\tau )\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(a_n\right),\varphi \left(b_n\right),\tau \right)}-1\right)\hfill \\ & +& \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(a_{n+1}\right),\varphi \left(\mu \right),\tau \right)}-1\right)\hfill \\ & \le & \theta \gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\mu ,b_n,\tau \right)}-1\right)+\theta \gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_n,b_n,\tau \right)}-1\right)+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_{n+1},\mu ,\tau \right)}-1\right)\hfill \\ & \le & \theta \gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\mu ,b_n,\tau \right)}-1\right)+\theta \gamma \psi \left[\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(a_n\right),\mu ,\tau \right)}-1\right)+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\mu ,\mu ,\tau \right)}-1\right)\right.\hfill \\ & & \left.+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\mu ,b_n,\tau \right)}-1\right)\right]+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_{n+1},\mu ,\tau \right)}-1\right).\hfill \end{array}$$

(7)

Letting $n\to \infty$ in (7), and using the continuity of $\psi$, we obtain

$${\mathcal{F}}_{\upsilon }(\varphi \left(\mu \right),\mu ,\tau )=1\mathrm{for} \mathrm{all}\tau >0,$$

which yields to $\varphi \left(\mu \right)=\mu$. □

Theorem 4.

Let $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ be a complete FBBMS, and let

$$\varphi :(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )\leftrightarrows (\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$$

be an ${\psi }_{\alpha }$-contractive contravariant mapping satisfying the following conditions:

1. 

For bisequence $\left(\left\{a_n\right\},\left\{b_n\right\}\right)$, if $\alpha (a_n,b_n,\tau )\ge 1$ for all $n\in \mathbb{N}$, $\left\{a_n\right\}\to \mu$, $\left\{b_n\right\}\to \mu$ as $n\to \infty$ for $\mu \in \mathcal{A}\cap \mathcal{B}$, then $\alpha (\mu ,b_n,\tau )\ge 1$ for all $\tau >0$ and $n\in \mathbb{N}$;

2. 

ϕ is α-admissible;

3. 

There exists $a_0\in \mathcal{A}$, $b_0\in \mathcal{B}$ such that $\alpha (a_0,b_0,\tau )\ge 1$, $\alpha (a_0,\varphi \left(b_0\right),\tau )\ge 1$.

Under these assumptions, ϕ admits a fixed point. That is, $\varphi \left(\mu \right)=\mu$ for some $\mu \in \mathcal{A}\cap \mathcal{B}$.

Proof. 

By proving Theorem 2, we derived a bisequence $\left(\left\{a_n\right\},\left\{b_n\right\}\right)$, which exhibits Cauchy properties within the context of a complete FBBMS $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$. This bisequence, denoted by $\left(\left\{a_n\right\},\left\{b_n\right\}\right)$, biconverges to point $\mu \in \mathcal{A}\cap \mathcal{B}$, implying that both $\left\{a_n\right\}$ and $\left\{b_n\right\}$ converge to $\mu$ as n tends to infinity.

Now, through condition (1) and (5), we obtain the following:

$$\alpha (a_n,\mu ,\tau )\ge 1\mathrm{for} \mathrm{all}n\in \mathbb{N}\mathrm{and}\tau >0.$$

(8)

Once more, employing conditions (3) and (8), along with the b-triangular property of ${\mathcal{F}}_{\upsilon }$, we achieve the following:

$$\begin{array}{ccc}\hfill & & \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(\mu \right),\mu ,\tau \right)}-1\right)\le \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(\mu \right),\varphi \left(a_n\right),\tau \right)}-1\right)+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(b_n\right),\varphi \left(a_n\right),\tau \right)}-1\right)\hfill \\ & +& \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(b_n\right),\mu ,\tau \right)}-1\right)\hfill \\ & \le & \alpha (a_n,\mu ,\tau )\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(\mu \right),\varphi \left(a_n\right),\tau \right)}-1\right)+\alpha (a_n,b_n,\tau )\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(b_n\right),\varphi \left(a_n\right),\tau \right)}-1\right)\hfill \\ & +& \theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(a_{n+1}\right),\varphi \left(\mu \right),\tau \right)}-1\right)\hfill \\ & \le & \theta \gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_n,\mu ,\tau \right)}-1\right)+\theta \gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_n,b_n,\tau \right)}-1\right)+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_{n+1},\mu ,\tau \right)}-1\right)\hfill \\ & \le & \theta \gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_n,\mu ,\tau \right)}-1\right)+\theta \gamma \psi \left[\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(a_n\right),\mu ,\tau \right)}-1\right)+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\mu ,\mu ,\tau \right)}-1\right)\right.\hfill \\ & & \left.+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\mu ,b_n,\tau \right)}-1\right)\right]+\theta \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a_{n+1},\mu ,\tau \right)}-1\right).\hfill \end{array}$$

(9)

Letting $n\to \infty$ in (9), and using the continuity of $\psi$, we obtain

$${\mathcal{F}}_{\upsilon }(\varphi \left(\mu \right),\mu ,\tau )=1for all \tau >0,$$

which yields to $\varphi \left(\mu \right)=\mu$. □

Theorem 5.

Under the assumption of Theorem 1 or Theorem 3 (or Theorem 2 or Theorem 4), and if there exists a point $\rho \in \mathcal{A}\cap \mathcal{B}$ such that $\alpha (a,\rho ,\tau )\ge 1$ and $\alpha (\rho ,b,\tau )\ge 1$ for all $\tau >0$, $a\in \mathcal{A}$ and $b\in \mathcal{B}$, then the ${\psi }_{\alpha }$-contractive covariant mapping $\varphi :(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )\rightrightarrows (\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ (the ${\psi }_{\alpha }$-contractive contravariant mapping $\varphi :(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )\leftrightarrows (\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ ) has a unique fixed point.

Proof. 

In order to show the uniqueness of a fixed point of the mapping $\varphi :(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )\rightrightarrows (\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ (or,$\varphi :(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )\leftrightarrows (\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ ). Suppose, on the contrary, that $\lambda$ is another fixed point of $\varphi$. Using the assumption, there exists a point $\rho \in \mathcal{A}\cap \mathcal{B}$, such that

$$\alpha (\mu ,\rho ,\tau )\ge 1\mathrm{and}\alpha (\rho ,\lambda ,\tau )\ge 1\mathrm{for} \mathrm{all}\tau >0.$$

(10)

Utilizing the condition (10) and $\alpha$-admissibility of $\varphi$, we obtain

$$\alpha (\mu ,{\varphi }^n\left(\rho \right),\tau )\ge 1\mathrm{and}\alpha ({\varphi }^n\left(\rho \right),\lambda ,\tau )\ge 1\mathrm{for} \mathrm{all}n\in \mathbb{N}\mathrm{and}\tau >0.$$

(11)

Also, using conditions (2) and (11), we obtain the following:

$$\begin{array}{ccc}\hfill \frac{1}{{\mathcal{F}}_{\upsilon }\left(\mu ,{\varphi }^n\left(\rho \right),\tau \right)-1}& =& \frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(\mu \right),\varphi \left({\varphi }^{n-1}\left(\rho \right)\right),\tau \right)-1}\hfill \\ & \le & \alpha \left(\mu ,{\varphi }^{n-1}\left(\rho \right),\tau \right)\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(\mu \right),\varphi \left({\varphi }^{n-1}\left(\rho \right)\right),\tau \right)-1}\right)\hfill \\ & \le & \gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\mu ,{\varphi }^{n-1}\left(\rho \right),\tau \right)-1}\right).\hfill \end{array}$$

Repeating this process, we obtain

$$\frac{1}{{\mathcal{F}}_{\upsilon }\left(\mu ,{\varphi }^n\left(\rho \right),\tau \right)-1}\le {\gamma }^n{\psi }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\mu ,\rho ,\tau \right)-1}\right)\mathrm{for} \mathrm{all}n\in \mathbb{N}\mathrm{and}\tau >0.$$

(12)

In the same way, we can also obtain

$$\frac{1}{{\mathcal{F}}_{\upsilon }\left({\varphi }^n\left(\rho \right),\lambda ,\tau \right)-1}\le {\gamma }^n{\psi }^n\left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(\rho ,\lambda ,\tau \right)-1}\right)\mathrm{for} \mathrm{all}n\in \mathbb{N}\mathrm{and}\tau >0.$$

(13)

Letting $n\to \infty$ in (12) and (13) provides

$${\varphi }^n\left(\rho \right)\to \mu \mathrm{and}{\varphi }^n\left(\rho \right)\to \lambda ,$$

which contradicts the uniqueness of the limit. Hence, $\mu =\lambda \in \mathcal{A}\cap \mathcal{B}$. Therefore, $\varphi$ admits a unique fixed point in $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$. □

Example 3.

Let $\mathcal{A}=(-\infty ,0]$ and $\mathcal{B}=[0,\infty )$ equipped with a continuous τ-norm. Let $d:\mathcal{A}\times \mathcal{B}\to [0,\infty$ be defined as $d(x,y)={|x-y|}^2$. Define ${\mathcal{F}}_{\upsilon }(a,b,\tau )=\frac{\tau }{\tau +d(a,b)}$ for all $a\in \mathcal{A},b\in \mathcal{B}$ and $\tau >0$. Clearly, $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ is a complete FBBMS. Define $\varphi :\mathcal{A}\cup \mathcal{B}\leftrightarrows \mathcal{A}\cup \mathcal{B}$ by $\varphi \left(x\right)=-\frac{x}{3}$ for all $x\in \mathcal{A}\cup \mathcal{B}$ and $\psi \left(t\right)=\frac{1}{2}t$, $\alpha (a,b,\tau )=1$ for all $(a,b)\in \mathcal{A}\times \mathcal{B}$. Then, $\varphi \left([0,\infty )\right)\subset [0,\infty )$ and $\varphi \left([0,\infty )\right)\subset (-\infty ,0]$. It is clear that ϕ is continuous contravariant mapping.

As $x\in (-\infty ,0]$, there exists $a\in [0,\infty )$, such that $x=-a$.

Now,

$$\begin{array}{ccc}\hfill \frac{1}{{\mathcal{F}}_{\upsilon }\left(\varphi \left(a\right),\varphi \left(b\right),\tau \right)}-1& =& \frac{{\left|\varphi \left(a\right)-\varphi \left(b\right)\right|}^2}{\tau }\hfill \\ & =& \frac{|-\frac{a}{3}+\frac{b}{3}|}{\tau }\hfill \\ & =& \frac{1}{9}\frac{{|a-b|}^2}{\tau }\hfill \\ & \le & \frac{1}{3}\times \frac{1}{2}\frac{{|a-b|}^2}{\tau }\hfill \\ & =& \frac{1}{3}\psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a,b,\tau \right)}-1\right)\hfill \end{array}$$

for all $a\in \mathcal{A},b\in \mathcal{B}$ with $a\ne b$ and $\tau >0$.

So, all the axioms of Theorem 5 are satisfied with $\gamma =\frac{1}{3}$, and consequently, ϕ has a unique fixed point, i.e., $\mu =0$.

4. ${\mathcal{ϝ}}_{\mathsf{\eta }}$-Contractive Mappings and Fixed Point Results

In this section, we present the notion of ${\mathcal{ϝ}}_{\eta }$-contractive mappings and $\eta$-admissible mappings within the framework of FBBMS.

Definition 17.

Let ${{\rm Y}}_{\theta }$ be the family of all left-continuous non-decreasing functions $\mathcal{ϝ}:[0,1]\to [0,1]$ satisfying the following conditions:

(F1) 

$\mathcal{ϝ}\left(1\right)=1$;

(F2) 

$\mathcal{ϝ}\left(\nu \right)>\nu$ for all $\nu \in [0,1]$;

(F3) 

${lim}_{n\to \infty }{\mathcal{ϝ}}^n\left(\nu \right)=1$ for all $\nu \in [0,1]$, where ${\mathcal{ϝ}}^n\left(\nu \right)$ is the n-th iteration of ϝ at ν.

Definition 18.

Let $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ be an FBBMS. Mapping

$$\varphi :(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )\rightrightarrows (\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$$

is said to be covariant η-admissible if there exists a function $\eta :\mathcal{A}\times \mathcal{B}\times (0,\infty )\to [0,\infty )$ such that, for all $a\in \mathcal{A}$, $b\in \mathcal{B}$ and $\tau >0$

$$\eta (a,b,\tau )\le 1implies\eta \left(\varphi \right(a),\varphi (b),\tau )\le 1.$$

Definition 19.

Let $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ be an FBBMS. Mapping

$$\varphi :(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )\rightrightarrows (\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$$

is said to be ${\mathcal{ϝ}}_{\eta }$-contractive covariant mapping if for the functions $\eta :\mathcal{A}\times \mathcal{B}\times (0,\infty )\to [0,\infty )$ and $\mathcal{ϝ}\in {\rm Y}$ the following condition holds:

$${\mathcal{F}}_{\upsilon }(a,b,\tau )>0implies\eta (a,b,\tau ){\mathcal{F}}_{\upsilon }(\varphi \left(a\right),\varphi \left(b\right),\tau )\ge \theta \mathcal{ϝ}\left({\mathcal{F}}_{\upsilon }(a,b,\tau )\right)$$

(14)

for all $a\in \mathcal{A},b\in \mathcal{B},a\ne b$ and $\tau >0$.

Theorem 6.

Let $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ be a complete FBBMS. Assume that $\varphi :(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )\rightrightarrows (\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ is ${\mathcal{ϝ}}_{\eta }$-contractive covariant mapping satisfying the following conditions:

(i) 

ϕ is η-admissible;

(ii) 

For bisequence $\left(\left\{a_n\right\},\left\{b_n\right\}\right)$, if $\eta (a_n,b_n,\tau )\le 1$ for all $n\in \mathbb{N}$, $\left\{a_n\right\}\to \mu$, $\left\{b_n\right\}\to \mu$ as $n\to \infty$ for $\mu \in \mathcal{A}\cap \mathcal{B}$, then $\eta (\mu ,a_n,\tau )\le 1$ and $n\in \mathbb{N}$;

(iii) 

There exists $a_0\in \mathcal{A},b_0\in \mathcal{B}$ such that $\eta (a_0,b_0,\tau )\le 1$, $\eta (a_0,\varphi \left(b_0\right),\tau )\le 1$ for $\tau >0$.

Under these axioms, ϕ admits a fixed point. That is, $\varphi \left(\mu \right)=\mu$ for some $\mu \in \mathcal{A}\cap \mathcal{B}$.

Proof. 

Fix $a_0\in \mathcal{A}$ and $b_0\in \mathcal{B}$ such that $\varphi (a_0,\varphi \left(b_0\right),\tau )\le 1$. Define $\varphi \left(a_n\right)=a_{n+1}$ and $\varphi \left(b_n\right)=b_{n+1}$ for all $n\in \mathbb{N}\cup \left\{0\right\}$. Then, $\left(\left\{a_n\right\},\left\{b_n\right\}\right)$ is a bisequence in $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$. For any $\tau >0$, from the axiom (iii) and $\eta$-admissibility of covariant mapping $\varphi$, we obtain the following:

$$\begin{array}{ccc}\hfill \eta (a_0,b_0,\tau )\le 1& \Rightarrow & \eta (\varphi \left(a_0\right),\varphi \left(b_0\right),\tau )\le 1\hfill \\ \hfill \eta (a_0,b_1,\tau )=\eta (a_0,\varphi \left(b_0\right),\tau )\le 1& \Rightarrow & \eta (\varphi \left(a_0\right),\varphi \left(b_1\right),\tau )=\eta (a_1,b_2,\tau )\le 1\hfill \\ \hfill \eta (a_1,b_1,\tau )=\eta (\varphi \left(a_0\right),\varphi \left(b_0\right),\tau )\le 1& \Rightarrow & \eta (\varphi \left(a_1\right),\varphi \left(b_1\right),\tau )=\eta (a_2,b_2,\tau )\le 1\hfill \\ \hfill \eta (a_1,b_2,\tau )=\eta (a_1,\varphi \left(b_1\right),\tau )\le 1& \Rightarrow & \eta (\varphi \left(a_1\right),\varphi \left(b_2\right),\tau )=\eta (a_2,b_3,\tau )\le 1\hfill \\ \hfill \eta (a_2,b_2,\tau )=\eta (\varphi \left(a_1\right),\varphi \left(b_1\right),\tau )\le 1& \Rightarrow & \eta (\varphi \left(a_2\right),\varphi \left(b_2\right),\tau )=\eta (a_3,b_3,\tau )\le 1.\hfill \end{array}$$

By repeating this process, we obtain

$$\eta (a_{n+1},b_{n+1},\tau )\le 1\mathrm{and}\eta (a_n,b_{n+1},\tau )\le 1\mathrm{for} \mathrm{all}n\in \mathbb{N}.$$

(15)

Using conditions (14) and (15), for $a=a_n$ and $b=b_n$, we obtain

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{\upsilon }(a_{n+1},b_{n+1},\tau )& =& {\mathcal{F}}_{\upsilon }(\varphi \left(a_n\right),\varphi \left(b_n\right),\tau )\hfill \\ & \ge & \eta (a_n,b_n,\tau ){\mathcal{F}}_{\upsilon }(\varphi \left(a_n\right),\varphi \left(b_n\right),\tau )\hfill \\ & \ge & \frac{1}{\theta }\mathcal{ϝ}\left(a_n,b_n,\tau \right),\hfill \end{array}$$

and for $a=a_{n-1}$ and $b=b_n$, we obtain

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{\upsilon }(a_n,b_{n+1},\tau )& =& {\mathcal{F}}_{\upsilon }(\varphi \left(a_{n-1}\right),\varphi \left(b_n\right),\tau )\hfill \\ & \ge & \eta (a_{n-1},b_n,\tau ){\mathcal{F}}_{\upsilon }(\varphi \left(a_{n-1}\right),\varphi \left(b_n\right),\tau )\hfill \\ & \ge & \theta \mathcal{ϝ}\left(a_{n-1},b_n,\tau \right).\hfill \end{array}$$

By the process of induction, we can obtain the following:

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{\upsilon }(a_{n+1},b_{n+1},\tau )& \ge & {\theta }^{n+1}{\mathcal{ϝ}}^{n+1}{\mathcal{F}}_{\upsilon }(a_0,b_0,\tau )\mathrm{and}\hfill \\ \hfill {\mathcal{F}}_{\upsilon }(a_n,b_{n+1},\tau )& \ge & {\theta }^n{\mathcal{ϝ}}^n{\mathcal{F}}_{\upsilon }(a_0,b_1,\tau ).\hfill \end{array}$$

Now, for $m>n$, $m\in \mathbb{N}$, using the properties of $\mathcal{ϝ}$ and b-triangularity of ${\mathcal{F}}_{\upsilon }$, we obtain the following:

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{\upsilon }(a_n,b_m,\tau )& \ge & {\mathcal{F}}_{\upsilon }(a_n,b_n,\tau )\ast {\mathcal{F}}_{\upsilon }(a_n,b_{n+1},\tau )\ast {\mathcal{F}}_{\upsilon }(a_{n+1},b_m,\tau )\hfill \\ & & ⋮\hfill \\ & \ge & {\mathcal{F}}_{\upsilon }(a_n,b_n,\tau )\ast {\mathcal{F}}_{\upsilon }(a_{n+1},b_{n+1},\tau )\ast \cdots \ast {\mathcal{F}}_{\upsilon }(a_{m-1},b_{m-1},\tau )\hfill \\ & & \ast {\mathcal{F}}_{\upsilon }(a_{m-1},b_m,\tau )\ast {\mathcal{F}}_{\upsilon }(a_m,b_m,\tau )\hfill \\ & \ge & {\theta }^n{\mathcal{ϝ}}^n\left({\mathcal{F}}_{\upsilon }(a_0,b_0,\tau )\right)\ast {\theta }^n{\mathcal{ϝ}}^n\left({\mathcal{F}}_{\upsilon }(a_0,b_1,\tau )\right)\ast {\theta }^{n+1}{\mathcal{ϝ}}^{n+1}\left({\mathcal{F}}_{\upsilon }(a_0,b_0,\tau )\right)\hfill \\ & & \ast {\theta }^{n+1}{\mathcal{ϝ}}^{n+1}\left({\mathcal{F}}_{\upsilon }(a_0,b_1,\tau )\right)\ast \cdots \ast {\theta }^{m-1}{\mathcal{ϝ}}^{m-1}\left({\mathcal{F}}_{\upsilon }(a_0,b_1,\tau )\right)\ast \hfill \\ & & {\theta }^m\left({\mathcal{ϝ}}^m{\mathcal{F}}_{\upsilon }(a_0,b_0,\tau )\right)\hfill \\ & \ge & {\theta }^m\left({\mathcal{ϝ}}^m\left({\mathcal{F}}_{\upsilon }(a_0,b_0,\tau )\right)\ast {\mathcal{ϝ}}^m\left({\mathcal{F}}_{\upsilon }(a_0,b_1,\tau )\right)\ast \cdots \ast {\mathcal{ϝ}}^m\left({\mathcal{F}}_{\upsilon }(a_0,b_0,\tau )\right)\right)\hfill \\ & \ge & {\mathcal{ϝ}}^m\left({\mathcal{F}}_{\upsilon }(a_0,b_0,\tau )\right)\ast {\mathcal{ϝ}}^m\left({\mathcal{F}}_{\upsilon }(a_0,b_1,\tau )\right)\ast \cdots \ast {\mathcal{ϝ}}^m\left({\mathcal{F}}_{\upsilon }(a_0,b_0,\tau )\right).\hfill \end{array}$$

Letting $n,m\to \infty$ and using the properties of $\mathcal{ϝ}$, we obtain

$$\underset{n,m\to \infty }{lim}{\mathcal{F}}_{\upsilon }(a_n,b_m,\tau )=1\ast 1\ast \cdots \ast 1=1\mathrm{for} \mathrm{all}\tau >0.$$

Thus, we conclude that $\left(\left\{a_n\right\},\left\{b_n\right\}\right)$ is a Cauchy bisequence in $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$. Due to the completeness of FBBMS $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$, $\left(\left\{a_n\right\},\left\{b_n\right\}\right)$ is a convergent bisequence; hence, through Lemma 1, it biconverges to a point $\mu \in \mathcal{A}\cap \mathcal{B}$ i.e., $\left\{a_n\right\}\to \mu$ and $\left\{b_n\right\}\to \mu$.

Finally, we show that $\mu$ is a fixed point of $\varphi$. Using properties of $\mathcal{ϝ}$ and conditions (14) and (15), we obtain the following:

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{\upsilon }(\varphi \left(\mu \right),\mu ,\tau )& \ge & {\mathcal{F}}_{\upsilon }(\varphi \left(\mu \right),\varphi \left(b_n\right),\tau )\ast {\mathcal{F}}_{\upsilon }(\varphi \left(a_n\right),\varphi \left(b_n\right),\tau )\ast {\mathcal{F}}_{\upsilon }(\varphi \left(a_n\right),\mu ,\tau )\hfill \\ & \ge & \eta (\mu ,b_n,\tau ){\mathcal{F}}_{\upsilon }(\varphi \left(\mu \right),\varphi \left(b_n\right),\tau )\ast \eta (a_n,b_n,\tau ){\mathcal{F}}_{\upsilon }(\varphi \left(a_n\right),\varphi \left(b_n\right),\tau )\hfill \\ & \ast & {\mathcal{F}}_{\upsilon }(a_{n+1},\mu ,\tau )\hfill \\ & \ge & \theta \mathcal{ϝ}\left({\mathcal{F}}_{\upsilon }(\mu ,b_n,\tau )\right)\ast \theta \mathcal{ϝ}\left({\mathcal{F}}_{\upsilon }(a_n,b_n,\tau )\right)\ast {\mathcal{F}}_{\upsilon }(a_{n+1},\mu ,\tau )\hfill \\ & \ge & \mathcal{ϝ}\left({\mathcal{F}}_{\upsilon }(\mu ,b_n,\tau )\right)\ast \mathcal{ϝ}\left({\mathcal{F}}_{\upsilon }(a_n,b_n,\tau )\right)\ast {\mathcal{F}}_{\upsilon }(a_{n+1},\mu ,\tau ).\hfill \end{array}$$

As $n\to \infty$, through right-continuity of $\mathcal{ϝ}$, we obtain

$${\mathcal{F}}_{\upsilon }(\varphi \left(\mu \right),\mu ,\tau )=1\mathrm{for}\tau >0.$$

Consequently, $\varphi \left(\mu \right)=\mu$. □

Theorem 7.

Under the conditions stipulated in Theorem 6, and with the additional assumption that

(P) 

there exists a point $\rho \in \mathcal{A}\cap \mathcal{B}$ such that $\eta (a,\rho ,\tau )\le 1$ and $\eta (\rho ,b,\tau )\le 1$ for all $\tau >0$, where $a\in \mathcal{A}$ and $b\in \mathcal{B}$,

then the covariant mapping ϕ, being ${\mathcal{ϝ}}_{\eta }$-contractive, possesses a unique fixed point.

Proof. 

We demonstrate the distinctiveness of the fixed point within the mapping

$$\varphi :(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )\rightrightarrows (\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast ).$$

If we assume otherwise, considering $\lambda$ as another fixed point of $\varphi$ apart from $\mu$, then according to condition (P), there exists a point $\rho \in \mathcal{A}\cap \mathcal{B}$

$$\eta (\mu ,\rho ,\tau )\le 1\mathrm{and}\eta (\rho ,\lambda ,\tau )\le 1\mathrm{for} \mathrm{all}\tau >0.$$

(16)

By employing condition (16) alongside the $\eta$-admissibility of $\varphi$, we obtain

$$\eta (\mu ,{\varphi }^n\left(\rho \right),\tau )\le 1\mathrm{and}\eta ({\varphi }^n\left(\rho \right),\lambda ,\tau )\le 1\mathrm{for} \mathrm{all}n\in \mathbb{N}\mathrm{and}\tau >0.$$

(17)

Also, using conditions (14) and (17), we obtain the following:

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{\upsilon }(\mu ,{\varphi }^n\left(\rho \right),\tau )& =& {\mathcal{F}}_{\upsilon }(\varphi \left(\mu \right),\varphi \left({\varphi }^{n-1}\left(\rho \right)\right),\tau )\hfill \\ & \ge & \eta (\mu ,{\varphi }^{n-1}\left(\rho \right),\tau ){\mathcal{F}}_{\upsilon }(\varphi \left(\mu \right),\varphi \left({\varphi }^{n-1}\left(\rho \right)\right),\tau )\hfill \\ & \ge & \theta \mathcal{ϝ}\left({\mathcal{F}}_{\upsilon }\left(\mu ,{\varphi }^{n-1}\left(\rho \right),\tau \right)\right).\hfill \end{array}$$

Repeating this process, we obtain

$${\mathcal{F}}_{\upsilon }(\mu ,{\varphi }^n\left(\rho \right),\tau )\ge {\theta }^n\mathcal{ϝ}\left({\mathcal{F}}_{\upsilon }\left(\mu ,\rho ,\tau \right)\right)\ge \mathcal{ϝ}\left({\mathcal{F}}_{\upsilon }\left(\mu ,\rho ,\tau \right)\right)\mathrm{for} \mathrm{all}n\in \mathbb{N}\mathrm{and}\tau >0.$$

(18)

In the same way, we can also deduce

$${\mathcal{F}}_{\upsilon }({\varphi }^n\left(\rho \right),\lambda ,\tau )\ge {\theta }^n\mathcal{ϝ}\left({\mathcal{F}}_{\upsilon }\left(\rho ,\lambda ,\tau \right)\right)\ge \mathcal{ϝ}\left({\mathcal{F}}_{\upsilon }\left(\rho ,\lambda ,\tau \right)\right)\mathrm{for} \mathrm{all}n\in \mathbb{N}\mathrm{and}\tau >0.$$

(19)

Letting $n\to \infty$ in (18) and (19) provides

$${\varphi }^n\left(\rho \right)\to \mu \mathrm{and}{\varphi }^n\left(\rho \right)\to \lambda ,$$

which contradicts the uniqueness of the limit. Hence, $\mu =\lambda \in \mathcal{A}\cap \mathcal{B}$. Consequently, $\varphi$ admits a unique fixed point in $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$. □

5. Applications

5.1. 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.

Consider the integral equation in the form:

$$\Phi \left(t\right)=\mathfrak{F}\left(t\right)+\omega {\int }_0^q\Theta (t,s)\Phi \left(s\right)ds,$$

(20)

where $\omega >0$, $\mathfrak{F}\left(t\right)$ is a fuzzy function of $s\in [0,q]$, and $\Theta :[0,q]\times [0,q]\times \mathbb{R}\to \mathbb{R}$ is an integral kernel (see [20]). Our aim is to demonstrate the existence and uniqueness of the solution of Equation (20) by utilizing Theorem 5. We consider $C\left([0,q],\mathbb{R}\right)$ as a collection of all real-valued continuous functions defined on the set $[0,q]$. The induced metric $\mathcal{D}:C\left([0,q],\mathbb{R}\right)\times C\left([0,q],\mathbb{R}\right)\to {\mathbb{R}}^{+}$ is defined as $\mathcal{D}(A,B)=∥A-B∥$, $A,B\in C\left([0,q],\mathbb{R}\right)$.

Now, define a binary relation ∗ as a continuous $\tau$-norm and ${\mathcal{F}}_{\upsilon }:C\left([0,q],\mathbb{R}\right)\times C\left([0,q],\mathbb{R}\right)\times (0,\infty )\to [0,1]$ as

$${\mathcal{F}}_{\upsilon }(A,B,\tau )=\frac{\tau }{\tau +\mathcal{D}(A,B)}$$

for $A,B\in C\left([0,q],\mathbb{R}\right)$ and $\tau >0$. Then, ${\mathcal{F}}_{\upsilon }$ is b-triangular and the quadruple $\left(C\left([0,q],\mathbb{R}\right),C\left([0,q],\mathbb{R}\right),{\mathcal{F}}_{\upsilon },\theta ,\tau \right)$ forms a complete fuzzy bipolar b-metric space.

Theorem 8.

Suppose that for all $A,B\in C\left([0,q],\mathbb{R}\right)$, the following condition holds:

$$∥\varphi \left(A\right)-\varphi \left(B\right)∥\le \frac{{\gamma }^2}{\theta }∥A-B∥,$$

(21)

where $\varphi :C\left([0,q],\mathbb{R}\right)\to C\left([0,q],\mathbb{R}\right)$, $\gamma \in (0,1)$, and $\theta \ge 1$. Then, the integral Equation (20) has a unique solution in $C\left([0,q],\mathbb{R}\right)$.

Proof. 

Define $\varphi :C\left([0,q],\mathbb{R}\right)\to C\left([0,q],\mathbb{R}\right)$ by

$$\left(\varphi B\right)\left(t\right)=\mathfrak{F}\left(t\right)+\omega {\int }_0^q\Theta (t,s)B\left(s\right)ds.$$

(22)

Let $\varphi$ be well defined. It is worth noting that $\varphi$ possesses a unique fixed point in $C\left([0,q],\mathbb{R}\right)$ if and only if the integral Equation (20) admits a unique solution. Let $\alpha (B,A,\tau )=1$ for all $A,B\in C\left([0,q],\mathbb{R}\right)$ and $\tau >0$, and $\psi \left(\nu \right)=\frac{\gamma }{\theta }\nu$ for all $\nu \in [0,\infty )$. It is straightforward to confirm that $\psi$ is right-continuous and fulfills the properties outlined in Definition 12. By employing (21) and (22), for $A,B\in C\left([0,q],\mathbb{R}\right)$, we can establish the following:

$$\begin{array}{ccc}\hfill \alpha (B,A,\tau )\left(\frac{1}{{\mathcal{F}}_{\upsilon }(\varphi \left(B\right),\varphi \left(A\right),\tau )}-1\right)& & =\frac{\mathcal{D}\left(\varphi \right(B),\varphi (A\left)\right)}{\tau }\hfill \\ & & =\frac{∥\varphi \left(B\right)-\varphi \left(A\right)∥}{\tau }\hfill \\ & & \le \frac{{\gamma }^2}{\theta }\frac{∥B-A∥}{\tau }\hfill \\ & & \le \gamma \frac{\gamma }{\theta }\left(\frac{∥B-A∥}{\tau }\right)\hfill \\ & & =\gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }(B,A,\tau )}-1\right).\hfill \end{array}$$

Hence, we obtain

$$\alpha (B,A,\tau )\left(\frac{1}{{\mathcal{F}}_{\upsilon }(\varphi \left(B\right),\varphi \left(A\right),\tau )}-1\right)\le \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }(B,A,\tau )}-1\right)$$

for all $B,A\in C\left([0,q],\mathbb{R}\right)$.

Therefore, the integral operator $\varphi$ satisfies all the conditions specified in Theorem 5. Consequently, according to Theorem 5, there exists a unique fixed point in $C\left([0,q],\mathbb{R}\right)$ for the operator $\varphi$. This implies the existence of a unique solution to Problem (20) in $C\left([0,q],\mathbb{R}\right)$. □

Example 4.

Let $E=C\left([0,1],\mathbb{R}\right)$. Consider the integral equation

$$B\left(t\right)=e^{-3t}+\omega {\int }_0^1{e}^{-(t+s)}B\left(s\right)ds,$$

(23)

where $\left|\omega \right|\le \frac{\gamma }{3}$, $\gamma \in (0,1)$. Then, for $A,B\in E$, we obtain the following:

$$\begin{array}{ccc}& & ∥\left(\varphi \right(B\left)\right)\left(y\right)-\left(\varphi \right(A\left)\right)\left(y\right)∥=∥e^{-3y}+\omega {\int }_0^1{e}^{-(y+s)}B\left(s\right)ds\hfill \\ & & -\left(e^{-3y}+\omega {\int }_0^1{e}^{-(y+s)}A\left(s\right)ds\right)∥\hfill \\ & & =\left|\omega \right|∥{\int }_0^1{e}^{-(y+s)}B\left(s\right)ds-{\int }_0^1{e}^{-(y+s)}A\left(s\right)ds∥\hfill \\ & & =\left|\omega \right|∥{\int }_0^1{e}^{-(y+s)}\left(B\left(s\right)-A\left(s\right)\right)ds∥\hfill \\ & & \le \left|\omega \right|{\int }_0^1{e}^{-(y+s)}∥B\left(s\right)-A\left(s\right)∥ds\hfill \\ & & \le \frac{2}{3}∥B\left(s\right)-A\left(s\right)∥\hfill \\ & & \le \gamma ∥B\left(s\right)-A\left(s\right)∥.\hfill \end{array}$$

All the conditions specified in Theorem 8 are satisfied. Hence, there exists a unique solution to the nonlinear integral problem (23) in the space $C\left([0,1],\mathbb{R}\right)$.

Consider the integral equation as follows.

Theorem 9.

Let us consider the integral equation

$$\Theta \left(\kappa \right)=\mathfrak{F}\left(\kappa \right)+{\int }_{{\mathcal{E}}_1\cup {\mathcal{E}}_2}\mathcal{G}(\kappa ,s,\Theta \left(s\right))ds,\kappa \in {\mathcal{E}}_1\cup {\mathcal{E}}_2,$$

(24)

where ${\mathcal{E}}_1\cup {\mathcal{E}}_2$ is a Lebesgue measurable set and $\mathfrak{F}\left(\kappa \right)$ is a fuzzy function of $\kappa \in [0,p]$. Suppose that

(H1) 

$\mathcal{G}:\left({\mathcal{E}}_1^2\cup {\mathcal{E}}_2^2\right)\times [0,\infty )\to [0,\infty )$ and $\Theta \in L^{\infty }\left({\mathcal{E}}_1\right)\cup L^{\infty }\left({\mathcal{E}}_2\right)$;

(H2) 

There is a continuous function $\varphi :{\mathcal{E}}_1^2\cup {\mathcal{E}}_2^2\to [0,\infty )$ and $\gamma \in (0,1)$ satisfying

$$\left|\mathcal{G}(\kappa ,s,a(s\left)\right)-\mathcal{G}(\kappa ,s,b(s\left)\right)\right|\le \sqrt{\frac{{\gamma }^2}{\theta }\Theta (\kappa ,s)}\left|a\left(\kappa \right)-b\left(\kappa \right)\right|,$$

for $\kappa ,s\in {\mathcal{E}}_1^2\cup {\mathcal{E}}_2^2$, $\gamma \in (0,1)$ and $\theta \ge 1$;

(H3) 

${\int }_{{\mathcal{E}}_1\cup {\mathcal{E}}_2}\Theta (\kappa ,s)ds\le 1$.

Then, the integral Equation (24) has a unique solution in $\Theta \in L^{\infty }\left({\mathcal{E}}_1\right)\cup L^{\infty }\left({\mathcal{E}}_2\right)$.

Proof. 

Let $\mathcal{A}=L^{\infty }\left({\mathcal{E}}_1\right)$ and $\mathcal{B}=L^{\infty }\left({\mathcal{E}}_2\right)$ be two normed linear space, where ${\mathcal{E}}_1,{\mathcal{E}}_2$ are Lebesgue measurable sets and $m({\mathcal{E}}_1\cup {\mathcal{E}}_2)<\infty$.

Let ${\mathcal{F}}_{\upsilon }:\mathcal{A}\times \mathcal{B}\times (0,\infty )\to [0,1]$ given by

$${\mathcal{F}}_{\upsilon }(a,b,\tau )=\frac{\tau }{{\tau +|a-b|}^2}$$

for all $a\in \mathcal{A},b\in \mathcal{B}$ and $\tau >0$. Then, $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ is a complete FBBMS.

Let $\alpha (a,b,\tau )=1$ for all $a\in \mathcal{A},b\in \mathcal{B}$ and $\tau >0$ and $\psi \left(m\right)=\frac{\gamma }{\theta }m$ for all $m\in [0,\infty )$. Then, it is easy to verify that $\psi$ is right-continuous and satisfies the properties stated in Definition 12.

Define $\varphi :L^{\infty }\left({\mathcal{E}}_1\right)\cup L^{\infty }\left({\mathcal{E}}_2\right)\rightrightarrows L^{\infty }\left({\mathcal{E}}_1\right)\cup L^{\infty }\left({\mathcal{E}}_2\right)$ provided by

$$\varphi \left(\Theta \left(\kappa \right)\right)=\mathfrak{F}\left(\kappa \right)+\omega {\int }_{{\mathcal{E}}_1\cup {\mathcal{E}}_2}\mathcal{G}(\kappa ,s,\Theta \left(s\right))ds,\kappa \in {\mathcal{E}}_1\cup {\mathcal{E}}_2.$$

Now,

$$\begin{array}{ccc}& & \frac{1}{{\mathcal{F}}_{\upsilon }(\varphi \left(a\left(\kappa \right)\right),\varphi \left(b\left(\kappa \right)\right),\tau )}-1=\frac{{\left|\varphi \left(a\right(\kappa \left)\right)-\varphi \left(b\right(\kappa \left)\right)\right|}^2}{\tau }\hfill \\ & & =\frac{|\mathfrak{F}\left(\kappa \right)+{\int }_{{\mathcal{E}}_1\cup {\mathcal{E}}_2}\mathcal{G}(\kappa ,s,a\left(s\right))ds-\left(\mathfrak{F}\left(\kappa \right)+{\int }_{{\mathcal{E}}_1\cup {\mathcal{E}}_2}\mathcal{G}(\kappa ,s,b\left(s\right))ds\right){|}^2}{\tau }\hfill \\ & & =\frac{\omega |{\int }_{{\mathcal{E}}_1\cup {\mathcal{E}}_2}\left(\mathcal{G}(\kappa ,s,a(s\left)\right)-\mathcal{G}(\kappa ,s,b(s\left)\right)\right)ds{|}^2}{\tau }\hfill \\ & & \le \frac{{\int }_{{\mathcal{E}}_1\cup {\mathcal{E}}_2}\gamma \Theta (\kappa ,s){\left|a\left(\kappa \right)-b\left(\kappa \right)\right|}^{2}ds}{\tau }\hfill \\ & & \le \frac{{\gamma }^2{\left|a\left(\kappa \right)-b\left(\kappa \right)\right|}^2}{\theta \tau }\hfill \\ & & \le \gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }\left(a\left(\kappa \right),b\left(\kappa \right),\tau \right)}-1\right).\hfill \end{array}$$

Hence, all hypotheses of Theorem 1 are verified, and consequently, the integral Equation (24) has a unique solution. □

5.2. Fractional Differential Equations

We recall many important definitions from fractional calculus theory [21,22]. For a function $\Theta \in C[0,1]$, the order $\nu >0$ of the Riemann—Liouville fractional derivative is

$$\frac{1}{\Gamma \left(\vartheta -\nu \right)}\frac{d^{\vartheta }}{d{\kappa }^{\vartheta }}{\int }_0^{\kappa }\frac{\Theta \left(s\right)}{{(\kappa -s)}^{\nu -\vartheta +1}}ds={\mathcal{D}}^{\nu }\Theta \left(\kappa \right).$$

(25)

From (25), the right-hand side is pointwise defined as $[0,1]$, where $\left[\nu \right]$ and $\Gamma$ are the integer part of the number $\nu$ and the Euler gamma function.

Consider the following fractional differential equation:

$$\begin{array}{ccc}\hfill {}^s\mathcal{D}^{\varsigma }\Theta \left(\kappa \right)& +& \mathcal{G}(\kappa ,\Theta (\kappa \left)\right)=0,0\le \kappa \le 1,\varsigma \in [1,2]\hfill \\ & & \Theta \left(0\right)=\Theta \left(1\right)=0,\hfill \end{array}$$

where $\mathcal{G}:[0,1]\times \mathbb{R}\to \mathbb{R}$ is a continuous function and ${}^s\mathcal{D}^{\varsigma }$ represents the Caputo fractional derivative of order $\varsigma$, which is defined by

$${}^s\mathcal{D}^{\varsigma }=\frac{1}{\Gamma (\vartheta -\varsigma )}{\int }_0^{\delta }\frac{{\Theta }^{\vartheta }\left(s\right)ds}{{(\kappa -s)}^{\varsigma -\vartheta +1}}.$$

Let

$$\begin{array}{ccc}\hfill \mathcal{A}& =& \left(C[0,1],[0,\infty )\right)=\left\{f:[0,1]\to [0,\infty ):f\mathrm{is} \mathrm{continuous} \mathrm{function}\right\}\mathrm{and}\hfill \\ \hfill \mathcal{B}& =& \left(C[0,1],(-\infty ,0]\right)=\left\{f:[0,1]\to (-\infty ,0]:f\mathrm{is} \mathrm{continuous} \mathrm{function}\right\}.\hfill \end{array}$$

Consider ${\mathcal{F}}_{\upsilon }:\mathcal{A}\times \mathcal{B}\times (0,\infty )\to {\mathbb{R}}^{+}$ given by

$${\mathcal{F}}_{\upsilon }(a,b,\tau )=\frac{\tau }{\tau +{sup}_{\kappa \in [0,1]}{\left|a\left(\kappa \right)-b\left(\kappa \right)\right|}^2}$$

for all $(a,b)\in \mathcal{A}\times \mathcal{B}$. Then, $(\mathcal{A},\mathcal{B},{\mathcal{F}}_{\upsilon },\theta ,\ast )$ is a complete FBBMS.

Theorem 10.

Consider the nonlinear fractional differential Equation (25). Suppose that the following hypotheses are held:

(H1) 

We can determine $\gamma \in (0,1),\theta \ge 1$ and $(a,b)\in \mathcal{A}\times \mathcal{B}$ such that

$$\left|\mathcal{G}(\kappa ,a)-\mathcal{G}(\kappa ,b)\right|\le \sqrt{\frac{{\gamma }^2}{\theta }}\left|a\left(\kappa \right)-b\left(\kappa \right)\right|;$$

(26)

(H2) 

${sup}_{\kappa \in (0,1)}{\int }_0^1\left|\mathcal{Q}(\kappa ,s)\right|ds\le 1$.

Then, the FDE (25) has a unique solution in $\mathcal{A}\cup \mathcal{B}$.

Proof. 

The given FDE (25) is equivalent to the following integral equation

$$a\left(\kappa \right)={\int }_0^1\mathcal{Q}(\kappa ,s)\mathcal{G}(a,b\left(s\right))ds,$$

(27)

where

$$\mathcal{Q}(\kappa ,s)=\left\{\begin{array}{cc}\frac{{\left|\kappa (1-s)\right|}^{\varsigma -1}-{(\kappa -s)}^{\varsigma -1}}{\Gamma \left(\varsigma \right)},\hfill & \mathrm{if}0\le s\le \kappa \le 1;\hfill \\ \frac{{\left|\kappa (1-s)\right|}^{\varsigma -1}}{\Gamma \left(\varsigma \right)},\hfill & \mathrm{if}0\le \kappa \le s\le 1.\hfill \end{array}\right.$$

Define $\varphi :\mathcal{A}\cup \mathcal{B}\to \mathcal{A}\cup \mathcal{B}$ by

$$\varphi \left(a\left(\kappa \right)\right)={\int }_0^1\mathcal{Q}(\kappa ,s)\mathcal{G}(\varphi \left(a\right),b\left(s\right))ds,$$

and taking $\psi \left(m\right)=\frac{\gamma }{\theta }m$. Now,

$$\begin{array}{ccc}\hfill \left|\varphi \left(a\right(\kappa \left)\right)-\varphi \left(b\right(\kappa \left)\right)\right|& =& |{\int }_0^1\mathcal{Q}(\kappa ,s)\mathcal{G}(\varphi \left(a\right),a\left(s\right))ds-{\int }_0^1\mathcal{Q}(\kappa ,s)\mathcal{G}(\varphi \left(b\right),b\left(s\right))ds|\hfill \\ & \le & {\int }_0^1{\left|\mathcal{Q}(\kappa ,s)\right|}^{2}ds\times {\int }_0^1{\left|\mathcal{G}\left(\varphi \right(a),a(s\left)\right)-\mathcal{G}\left(\varphi \right(b),b(s\left)\right)\right|}^{2}ds\hfill \\ & \le & \frac{{\gamma }^2}{\theta }{\left|a\left(\kappa \right)-b\left(\kappa \right)\right|}^2.\hfill \end{array}$$

Taking the supremum on both sides, we obtain the following:

$${\mathcal{F}}_{\upsilon }(\varphi \left(a\right),\varphi \left(b\right),\tau )\le \gamma \psi \left({\mathcal{F}}_{\upsilon }(a,b,\tau )\right)\mathrm{for} \mathrm{all}\tau >0.$$

Therefore,

$$\begin{array}{ccc}\hfill \frac{1}{{\mathcal{F}}_{\upsilon }(\varphi \left(a\right),\varphi \left(b\right),\tau )}-1& =& \frac{{sup}_{\kappa \in [0,1]}{\left|\varphi \left(a\right(\kappa \left)\right)-\varphi \left(b\right(\kappa \left)\right)\right|}^2}{\tau }\hfill \\ & \le & \frac{{sup}_{\kappa \in [0,1]}{\left|a\left(\kappa \right)-b\left(\kappa \right)\right|}^2}{\tau }\hfill \\ & =& \gamma \psi \left(\frac{1}{{\mathcal{F}}_{\upsilon }(a\left(\kappa \right),b\left(\kappa \right),\tau )}-1\right).\hfill \end{array}$$

As a result, all the hypotheses of Theorem 5 are fulfilled, and consequently, the fractional differential Equation (25) has a unique solution. □

6. 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 ${\psi }_{\alpha }$-contractive and ${\mathcal{ϝ}}_{\eta }$-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 ${\psi }_{\alpha }$-contractive and ${\mathcal{ϝ}}_{\eta }$-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.