$\mathfrak{F}$-Bipolar Metric Spaces: Fixed Point Results and Their Applications

Abstract

The aim of this research article is to broaden the scope of fixed point theory in $\mathfrak{F}$-bipolar metric spaces by introducing the concept of rational ($⋏,⋎,\psi$)-contractions. These new contractions allow for the formulation of fixed point theorems specifically designed for contravariant mappings. The validity of our approach is substantiated by a meticulously crafted example. Moreover, we explore the practical implications of these theorems beyond the realm of fixed point theory. Notably, we demonstrate their effectiveness in establishing the existence and uniqueness of solutions to integral equations. Additionally, we investigate homotopy problems, focusing on the conditions for the existence of a unique solution within this framework.

1. Introduction

Fixed point (FP) theory, a fundamental branch of mathematical analysis, explores self-referential functions. It aims to locate members of a set that are unaffected by the application of these functions. This seemingly basic concept has far-reaching consequences in diverse areas of mathematics, such as differential equations, geometry and optimization, and Antón–Sancho [1,2] studied the FPs of automorphisms on the moduli space of vector bundles over a compact Riemann surface, as well as the FPs of involutions on the moduli space of G-Higgs bundles over a compact Riemann surface with a classical complex structure group. The Banach Contraction Principle (BCP) [3] is widely recognized as a cornerstone in the development of this theory. This powerful tool tackles existence problems and has been extensively generalized by mathematicians over time. Samet et al. [4] introduced the concept of ⋏-$\psi$-contractions in the background of metric spaces (MSs), proving FP results for such mappings. Building on this, Salimi et al. [5] refined these notions and established new FP theorems in the same framework. The classical MS introduced by Fréchet [6] in 1906 has been significantly extended by weakening its axioms or modifying its domain and range. Bakhtin [7] gave the conception of b-metric spaces (b-MSs) and made notable extensions to the concept of MSs, which were formally defined by Czerwik [8] in 1993 with the aim of generalizing the BCP. Later on, Branciari [9] introduced a variant called generalized metric spaces (g-MSs). These spaces relax the triangle inequality in favor of a weaker property known as the rectangular metric inequality. Jleli et al. [10] proposed a captivating extension encompassing b-MSs and g-MSs, referred to as the $\mathfrak{F}$-metric space ($\mathfrak{F}$-MS). Subsequently, Hussain et al. [11] leveraged $\mathfrak{F}$-MS to establish FP results for $\left(⋎,\psi \right)$-contractions. While existing MS generalizations focus on distances within a single set, questions arise regarding measuring distances between elements of distinct sets. To address this challenge in various disciplines, Mutlu et al. [12] awarded bipolar metric spaces (bip MS). In a subsequent studies, Gürdal et al. [13] and Gaba et al. [14] investigated FP results for different generalized contractions in the situation of bip MSs, respectively. This concept has furthered the development of FP theory. However, significant work remains on the existence of FPs within bip MS settings (see [15,16,17,18,19]). Recognizing this need, Rawat et al. [20] ingeniously combined the novel ideas of $\mathfrak{F}$-MSs and bip MSs, introducing the concept of $\mathfrak{F}$-bipolar metric spaces ($\mathfrak{F}$-bip MSs) and establishing an initial FP result. Subsequently, Albargi [21] presented some new FP theorems in this new and generalized MS and solved an integral equation by the implementations of the leading result. They also obtained some coupled FP results by applying the main theorems. Recently, Alamri [22] exploited the idea of $\mathfrak{F}$-bip MS and proved some new result in FP theory.

This research introduces the concept of rational ($⋏,⋎,\psi$)-contractions in $\mathfrak{F}$-bip MS and establishes FP theorems for contravariant mappings. We then demonstrate the applicability of our main result by investigating the existence of solutions for a specific integral equation. Our results extend the leading results of Al-Mazrooei et al. [23], Albargi [21], and Alamri [22] in the generalized sense of the contractive conditions.

2. Preliminaries

The well-known BCP [3] is given in the following way:

Theorem 1

([3]). Consider a complete metric space (CMS) represented by $\left(\mho ,\mathfrak{d}\right)$ and $\mathcal{Z}:\mho$$\to \mho$. Suppose there exists a constant λ belonging to the closed interval $[0,1)$, such that

$$\mathfrak{d}\left(\mathcal{Z}\varsigma ,\mathcal{Z}\mathfrak{y}\right)\le \lambda \mathfrak{d}\left(\varsigma ,\mathfrak{y}\right),$$

for all $\varsigma ,\mathfrak{y}\in \mho$; then, $\mathcal{Z}$ attains a unique fixed point (UFP).

Samet et al. [4] pioneered these concepts.

Definition 1.

The set Ψ of functions $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ fulfills the subsequent requirements:

$\left({\psi }_1\right)$

for any $a,b\in {\mathbb{R}}^{+}$ with $a\le b,$ it holds that $\psi \left(a\right)\le \psi \left(b\right),$

$\left({\psi }_2\right)$

${\displaystyle \sum _{𝚤=1}^{\infty }}{\psi }^{𝚤}\left(a\right)<+\infty ,$ for all $a>0.$

Lemma 1.

Elements within the set Ψ satisfy the property that for any positive value a, and the function evaluated at a ($\psi \left(a\right))$ is strictly less than a. Additionally, the function always evaluates to zero at $a=0$ ($\psi \left(0\right)=0).$

Definition 2

([4]). Considering an arbitrary function $⋏:\mho \times \mho \to [0,+\infty ),$ a mapping $\mathcal{Z}:\mho$$\to \mho$ is classified as ⋏-admissible if

$$⋏\left(\varsigma ,\mathfrak{y}\right)\ge 1⟹⋏\left(\mathcal{Z}\varsigma ,\mathcal{Z}\mathfrak{y}\right)\ge 1,$$

for all $\varsigma ,\mathfrak{y}\in \mho .$

Definition 3

([4]). In a metric space $\left(\mho ,\mathfrak{d}\right),$ a mapping $\mathcal{Z}:\mho$$\to \mho$ is considered an ($⋏,\psi$)-contraction if there exist some $⋏:\mho \times \mho \to [0,+\infty )$ and $\psi \in \Psi$, such that

$$⋏\left(\varsigma ,\mathfrak{y}\right)\mathfrak{d}\left(\mathcal{Z}\varsigma ,\mathcal{Z}\mathfrak{y}\right)\le \psi \left(\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right)\right),$$

for all $\varsigma ,\mathfrak{y}\in \mho .$

Jleli et al. [10] introduced a captivating generalization of MS, known as $\mathfrak{F}$-MS.

Let $\mathfrak{F}$ be the family of functions $g:(0,+\infty )\to \mathbb{R}$ satisfying the following claims:

(${\mathcal{F}}_1$)

$g\left(a\right)<g\left(b\right),$ for $a<b,$

(${\mathcal{F}}_2$)

for every sequence $\left\{a_{𝚤}\right\}\subseteq {\mathbb{R}}^{+}$, ${lim}_{𝚤\to \infty }{a}_{𝚤}=0$ if and only if ${lim}_{𝚤\to \infty }g\left(a_{𝚤}\right)=-\infty .$

Definition 4

([10]). Considering a non-empty set ℧ and a distance function $\mathfrak{d}:\mho \times \mho \to [0,+\infty )$, such that

(i) 

$\mathfrak{d}(\varsigma ,\mathfrak{y})=0⟺\varsigma =\mathfrak{y}$,

(ii) 

$\mathfrak{d}(\varsigma ,\mathfrak{y})=\mathfrak{d}(\mathfrak{y},\varsigma ),$

(iii) 

For each ${\left(u_{𝚤}\right)}_{𝚤=1}^p\subset \mho$ with $(u_1,u_p)=(\varsigma ,\mathfrak{y})$, we have

$$\mathfrak{d}(\varsigma ,\mathfrak{y})>0\Rightarrow g\left(\mathfrak{d}(\varsigma ,\mathfrak{y})\right)\le g\left(\sum _{𝚤=1}^{p-1}\mathfrak{d}\left(u_{𝚤},u_{𝚤+1}\right)\right)+\alpha ,$$

for all $(\varsigma ,\mathfrak{y})\in \mho \times \mho$ and for $p\in \mathbb{N}$ with $p\ge 2$. If there exists a pair $(g,\alpha )\in \mathfrak{F}\times [0,+\infty )$ satisfying the aforementioned properties, then $\mathfrak{d}$ is called an $\mathfrak{F}$-metric on ℧, and the combination $(\mho ,\mathfrak{d})$ is referred to as an $\mathfrak{F}$-MS.

Example 1

([10]). Let $\mho =\mathbb{N},$ $f\left(a\right)=ln\left(a\right)$ and $\varrho =ln\left(3\right).$ Define $\mathfrak{d}:\mho \times \mho \to [0,+\infty )$ by

$$\mathfrak{d}(\varsigma ,\mathfrak{y})=\left\{\begin{array}{c}{(\varsigma -\mathfrak{y})}^2\mathrm{if}(\varsigma ,\mathfrak{y})\in [0,3]\times [0,3]\\ |\varsigma -\mathfrak{y}|\mathrm{if}(\varsigma ,\mathfrak{y})\notin [0,3]\times [0,3]\end{array}\right.$$

then ($\mho ,\mathfrak{d}$) is an $\mathfrak{F}$-MS.

The concept of bip MS was introduced by Mutlu et al. [12] in this manner.

Definition 5

([12]). Let $\mho \ne \varnothing$ and $\Omega \ne \varnothing$ and let $\mathfrak{d}:\mho \times \Omega \to [0,+\infty )$; this satisfies

($b{i}_1)$

$\mathfrak{d}(\varsigma ,\mathfrak{y})=0⟺\varsigma =\mathfrak{y}$,

($b{i}_2)$

$\mathfrak{d}(\varsigma ,\mathfrak{y})=\mathfrak{d}(\mathfrak{y},\varsigma )$, if $\varsigma ,\mathfrak{y}\in \mho \cap \Omega ,$

($b{i}_3)$

$\mathfrak{d}(\varsigma ,\mathfrak{y})\le \mathfrak{d}(\varsigma ,{\mathfrak{y}}^{/})+\mathfrak{d}({\varsigma }^{/},{\mathfrak{y}}^{/})+\mathfrak{d}({\varsigma }^{/},\mathfrak{y})$,

for all $(\varsigma ,\mathfrak{y}),({\varsigma }^{/},{\mathfrak{y}}^{/})\in \mho \times \Omega .$

In this case, $\left(\mho ,\Omega ,\mathfrak{d}\right)$ can be classified as a bip MS.

Example 2

([12]). Considering two sets ℧ and Ω containing singleton and all compact subsets of the real numbers $\mathbb{R},$ respectively, a distance function $\mathfrak{d}:\mho \times \Omega \to [0,+\infty )$ is defined as

$$\mathfrak{d}(\varsigma ,\Xi )=\left|\varsigma -inf\left(\Xi \right)\right|+\left|\varsigma -sup\left(\Xi \right)\right|$$

for any singleton set $\left\{\varsigma \right\}$ in ℧ and any set Ξ in Ω. With this distance function, the triplet $\left(\mho ,\Omega ,\mathfrak{d}\right)$ forms a complete bip MS.

Definition 6.

Consider two bip MSs $\left({\mho }_1,{\Omega }_1,{\mathfrak{d}}_1\right)$ and $\left({\mho }_2,{\Omega }_2,{\mathfrak{d}}_2\right)$. A mapping $\mathcal{Z}:{\mho }_1\cup {\Omega }_1$$\rightrightarrows {\mho }_2\cup {\Omega }_2$ is classified as covariant if $\mathcal{Z}\left({\mho }_1\right)\subseteq {\mho }_2$ and $\mathcal{Z}\left({\Omega }_1\right)\subseteq {\Omega }_2$, and a mapping $\mathcal{Z}:{\mho }_1\cup {\Omega }_1$ $\rightrightarrows {\mho }_2\cup {\Omega }_2$ is classified as a contravariant mapping if $\mathcal{Z}\left({\mho }_1\right)\subseteq {\Omega }_2$ and $\mathcal{Z}\left({\mho }_2\right)\subseteq {\Omega }_1.$

According to Mutlu et al. [12], the covariant mapping is denoted by $\mathcal{Z}:\left({\mho }_1,{\Omega }_1\right)$$\rightrightarrows \left({\mho }_2,{\Omega }_2\right),$ while the contravariant mapping is represented as $\mathcal{Z}:\left({\mho }_1,{\Omega }_1\right)$$\rightleftarrows \left({\mho }_2,{\Omega }_2\right)$.

Rawat et al. [20] integrated the concepts of bip MS and $\mathfrak{F}$-MS, thereby introducing the idea of an $\mathfrak{F}$-bip MS.

Definition 7

([20]). Considering non-empty sets ℧ and Ω and a distance function $\mathfrak{d}:\mho \times \Omega \to [0,+\infty )$, such that

$\left(D_1\right)$

$\mathfrak{d}(\varsigma ,\mathfrak{y})=0⟺\varsigma =\mathfrak{y}$,

$\left(D_2\right)$

$\mathfrak{d}(\varsigma ,\mathfrak{y})=\mathfrak{d}(\mathfrak{y},\varsigma )$, if $\varsigma ,\mathfrak{y}\in \mho \cap \Omega ,$

$\left(D_3\right)$

for every ${\left(u_{𝚤}\right)}_{𝚤=1}^p\subset \mho$ and ${\left(v_{𝚤}\right)}_{𝚤=1}^p\subset \Omega$ with $(u_1,v_p)=(\varsigma ,\mathfrak{y})$, we have

$$\mathfrak{d}(\varsigma ,\mathfrak{y})>0\Rightarrow g\left(\mathfrak{d}(\varsigma ,\mathfrak{y})\right)\le g\left(\sum _{𝚤=1}^{p-1}\mathfrak{d}\left(u_{𝚤+1},v_{𝚤}\right)+\sum _{𝚤=1}^p\mathfrak{d}\left(u_{𝚤},v_{𝚤}\right)\right)+\varrho ,$$

for $p\in \mathbb{N}$ with $p\ge 2$. If there exists a pair $(g,\varrho )\in \mathfrak{F}\times [0,+\infty )$ satisfying the aforementioned properties, then $(\mho ,\Omega ,\mathfrak{d})$ is called an $\mathfrak{F}$-bip MS.

Example 3.

Let $\mho =\left\{1,2\right\}$ and $\Omega =\left\{2,7\right\}.$ Introduce a distance function $\mathfrak{d}:\mho \times \Omega \to [0,+\infty )$ by

$$\mathfrak{d}(1,2)=6,\mathfrak{d}(1,7)=10,\mathfrak{d}(2,7)=2,\mathfrak{d}(2,2)=0.$$

However, under these conditions, $\mathfrak{d}$ fulfills all the requirements for an $\mathfrak{F}$-bip metric with a specific value of $\varrho (\varrho =0)$ and a particular function $g\left(a\right)=lna,$ for positive values of a. Consequently, while $(\mho ,\Omega ,\mathfrak{d})$ qualifies as an $\mathfrak{F}$-bip MS, it does not satisfy the standard definition of a bip MS.

Remark 1

([20]). By setting $\Omega =\mho$, $p=2𝚤,$ $u_j=u_{2j-1}$ and $v_j=u_{2j}$ in the aforementioned Definition 7, we derive a sequence ${\left(u_j\right)}_{j=1}^{2𝚤}\in \mho$ with $\left(u_1,u_{2𝚤}\right)=(\varsigma ,\mathfrak{y})$ that satisfies the condition (iii) of Definition 4. Consequently, every $\mathfrak{F}$-MS is an $\mathfrak{F}$-bip MS, though the reverse does not generally hold true.

Definition 8

([20]). In the context of $\mathfrak{F}$-bip MS, a point $\varsigma \in \mho \cup \Omega$ is said to be

• Left point if it belongs to the set ℧ but not necessarily to Ω.
• Right point if it belongs to the set Ω but not necessarily to ℧.
• Central point if it belongs to both sets ℧ and Ω.

Definition 9

([20]). In the context of $\mathfrak{F}$-bip MS,

• Left sequence: An element-by-element progression within the set ℧, denoted as (${\varsigma }_{𝚤}$), where each term ${\varsigma }_{𝚤}$ belongs to Ω.
• Right sequence: An element-by-element progression within the set Ω, denoted as (${\mathfrak{y}}_{𝚤})$, where each term ${\mathfrak{y}}_{𝚤}$ belongs to Ω.

Definition 10

([20]). In the context of $\mathfrak{F}$-bip MS,

• A sequence (${\varsigma }_{𝚤}$) in ℧ converges to a point ς if it satisfies specific conditions related to right points and the limit of the distance function $\mathfrak{d}$(${\varsigma }_{𝚤}$,ς) as ı approaches positive infinity. Similar conditions apply for sequences in Ω converging to left points.
• A bisequence (${\varsigma }_{𝚤}$, ${\mathfrak{y}}_{𝚤}$) on (℧, Ω, $\mathfrak{d}$) represents a sequence of element pairs, where the first element comes from ℧ and the second from Ω. Convergence of a bisequence typically involves the convergence of both individual sequences (${\varsigma }_{𝚤}$) and (${\mathfrak{y}}_{𝚤})$. Additionally, a bisequence is considered biconvergent if both sequences converge to the same element.

Definition 11

([20]). In the context of $\mathfrak{F}$-bip MS, a bisequence $\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)$ is termed a Cauchy bisequence if, for every $ϵ>0,$ there exists ${𝚤}_0\in \mathbb{N},$ such that $\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_p\right)<ϵ,$∀$𝚤,p\ge {𝚤}_0.$

Definition 12

([20]). If every Cauchy bisequence in $\mathfrak{F}$-bip MS $\left(\mho ,\Omega ,\mathfrak{d}\right)$ is convergent, then $\left(\mho ,\Omega ,\mathfrak{d}\right)$ is complete.

3. Fixed Point Results for Contravariant Mappings

Throughout this section, we consider $\left(\mho ,\Omega ,\mathfrak{d}\right)$ as $\mathfrak{F}$-bip MS and $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ as contravariant mapping.

Definition 13.

A mapping $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ is classified as ⋏-admissible if a function $⋏:\mho \times \Omega \to [0,+\infty )$ exists and satisfies the following condition:

$$⋏\left(\varsigma ,\mathfrak{y}\right)\ge 1⟹⋏\left(\mathcal{Z}\mathfrak{y},\mathcal{Z}\varsigma \right)\ge 1,$$

(1)

for all $\left(\varsigma ,\mathfrak{y}\right)\in \mho \times \Omega .$

Example 4.

Let $\mho =\left[0,+\infty \right)$ and $\Omega =\left(-\infty ,0\right]$, and $⋏:\mho \times \Omega \to [0,+\infty )$ is defined as

$$⋏\left(\varsigma ,\mathfrak{y}\right)=\left\{\begin{array}{c}1,\mathrm{if}\varsigma \ne \mathfrak{y},\\ 0,\mathrm{if}\varsigma =\mathfrak{y}.\end{array}\right.$$

A mapping $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ defined by $\mathcal{Z}\left(\varsigma \right)=-\varsigma$ is ⋏-admissible.

Definition 14.

A mapping $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ is defined as ⋏-admissible with respect to ⋎ if $⋏,⋎:\mho \times \Omega \to [0,+\infty )$ exists and satisfies

$$⋏\left(\varsigma ,\mathfrak{y}\right)\ge ⋎\left(\varsigma ,\mathfrak{y}\right)⟹⋏\left(\mathcal{Z}\mathfrak{y},\mathcal{Z}\varsigma \right)\ge ⋎\left(\mathcal{Z}\mathfrak{y},\mathcal{Z}\varsigma \right),$$

(2)

for all $\left(\varsigma ,\mathfrak{y}\right)\in \mho \times \Omega .$

Definition 15.

A mapping $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ is rational ($⋏,⋎,\psi$)-contraction if $⋏,⋎:\mho \times \Omega \to [0,+\infty )$ and $\psi \in \Psi$ exist and satisfy

$$⋏\left(\varsigma ,\mathfrak{y}\right)\ge ⋎\left(\varsigma ,\mathfrak{y}\right)⟹\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathcal{Z}\varsigma \right)\le \psi \left(\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)\right),$$

(3)

where

$$\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)=max\left\{\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right),{\displaystyle \frac{\mathfrak{d}\left(\varsigma ,\mathcal{Z}\varsigma \right)\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathfrak{y}\right)}{1+\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right)}}\right\}$$

for all $\left(\varsigma ,\mathfrak{y}\right)\in \mho \times \Omega .$

Theorem 2.

Let $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ be a rational ($⋏,⋎,\psi$)-contraction. Suppose the following assumptions are satisfied:

(i) 

$\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ is ⋏-admissible with respect to $⋎,$

(ii) 

$\exists {\varsigma }_0\in \mho$ such that $⋏\left({\varsigma }_0,\mathcal{Z}{\varsigma }_0\right)\ge ⋎\left({\varsigma }_0,\mathcal{Z}{\varsigma }_0\right),$

(iii) 

$\mathcal{Z}$ is continuous or, if $\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\subseteq$$\left(\mho ,\Omega ,\mathfrak{d}\right)$ is bisequence, provided that $⋏\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\ge ⋎\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right),$ $\forall 𝚤\in \mathbb{N}$ with ${\varsigma }_{𝚤}\to \mu$ and ${\mathfrak{y}}_{𝚤}\to \mu ,$ as $𝚤\to \infty$ for $\mu \in \mho \cap \Omega ,$ then $⋏\left({\varsigma }_{𝚤},\mu \right)\ge ⋎\left({\varsigma }_{𝚤},\mu \right)$, $\forall 𝚤\in \mathbb{N}.$

Then, $\mathcal{Z}:\mho \cup \Omega \to$$\mho \cup \Omega$ has a FP. Moreover, if property (P) is satisfied, then the FP of the mapping is unique.

Proof. 

Within the F-bip MS, let us begin by selecting arbitrary elements ${\varsigma }_0\in \mho$ and ${\mathfrak{y}}_0\in \Omega$. We then assume an initial condition where $⋏\left({\varsigma }_0,\mathcal{Z}{\varsigma }_0\right)\ge ⋎\left({\varsigma }_0,\mathcal{Z}{\varsigma }_0\right).$ Define the bisequence $\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)$ in $\left(\mho ,\Omega ,\mathfrak{d}\right)$ by

$${\mathfrak{y}}_{𝚤}=\mathcal{Z}{\varsigma }_{𝚤}\mathrm{and}{\varsigma }_{𝚤+1}=\mathcal{Z}{\mathfrak{y}}_{𝚤}$$

for all $𝚤\in \mathbb{N}.$ Since

$$⋏\left({\varsigma }_0,{\mathfrak{y}}_0\right)=⋏\left({\varsigma }_0,\mathcal{Z}{\varsigma }_0\right)\ge ⋎\left({\varsigma }_0,\mathcal{Z}{\varsigma }_0\right)=⋎\left({\varsigma }_0,{\mathfrak{y}}_0\right).$$

So by the assumption (i), we have

$$⋏\left({\varsigma }_1,{\mathfrak{y}}_0\right)=⋏\left(\mathcal{Z}{\mathfrak{y}}_0,\mathcal{Z}{\varsigma }_0\right)\ge ⋎\left(\mathcal{Z}{\mathfrak{y}}_0,\mathcal{Z}{\varsigma }_0\right)=⋎\left({\varsigma }_1,{\mathfrak{y}}_0\right)$$

and $⋏\left({\varsigma }_1,{\mathfrak{y}}_0\right)\ge ⋎\left({\varsigma }_1,{\mathfrak{y}}_0\right)$ implies

$$⋏\left({\varsigma }_1,{\mathfrak{y}}_1\right)=⋏\left(\mathcal{Z}{\mathfrak{y}}_0,\mathcal{Z}{\varsigma }_1\right)\ge ⋎\left(\mathcal{Z}{\mathfrak{y}}_0,\mathcal{Z}{\varsigma }_1\right)=⋎\left({\varsigma }_1,{\mathfrak{y}}_1\right).$$

Similarly, $⋏\left({\varsigma }_1,{\mathfrak{y}}_1\right)\ge ⋎\left({\varsigma }_1,{\mathfrak{y}}_1\right)$ implies

$$⋏\left({\varsigma }_2,{\mathfrak{y}}_1\right)=⋏\left(\mathcal{Z}{\mathfrak{y}}_1,\mathcal{Z}{\varsigma }_1\right)\ge ⋎\left(\mathcal{Z}{\mathfrak{y}}_1,\mathcal{Z}{\varsigma }_1\right)=⋎\left({\varsigma }_2,{\mathfrak{y}}_1\right),$$

and $⋏\left({\varsigma }_2,{\mathfrak{y}}_1\right)\ge ⋎\left({\varsigma }_2,{\mathfrak{y}}_1\right)$ implies

$$⋏\left({\varsigma }_2,{\mathfrak{y}}_2\right)=⋏\left(\mathcal{Z}{\mathfrak{y}}_1,\mathcal{Z}{\varsigma }_2\right)\ge ⋎\left(\mathcal{Z}{\mathfrak{y}}_1,\mathcal{Z}{\varsigma }_2\right)=⋎\left({\varsigma }_2,{\mathfrak{y}}_2\right).$$

Continuing in this way, we have

$$⋏\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\ge ⋎\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\mathrm{and}⋏\left({\varsigma }_{𝚤+1},{\mathfrak{y}}_{𝚤}\right)\ge ⋎\left({\varsigma }_{𝚤+1},{\mathfrak{y}}_{𝚤}\right)$$

(4)

for every $𝚤\in \mathbb{N}.$ Using inequalities (3) and (4), we can conclude that

$$\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)=\mathfrak{d}\left(\mathcal{Z}{\mathfrak{y}}_{𝚤-1},\mathcal{Z}{\varsigma }_{𝚤}\right)\le \psi \left(\mathfrak{M}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)\right)$$

(5)

where

$$\begin{array}{ccc}\hfill \mathfrak{M}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)& =& max\left\{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right),{\displaystyle \frac{\mathfrak{d}\left({\varsigma }_{𝚤},\mathcal{Z}{\varsigma }_{𝚤}\right)\mathfrak{d}\left(\mathcal{Z}{\mathfrak{y}}_{𝚤-1},{\mathfrak{y}}_{𝚤-1}\right)}{1+\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)}}\right\}\hfill \\ & =& max\left\{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right),{\displaystyle \frac{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)}{1+\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)}}\right\}\hfill \end{array}$$

for all $𝚤\in \mathbb{N}.$ If $max\left\{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right),{\displaystyle \frac{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)}{1+\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)}}\right\}=\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right),$ then from (5), we have

$$\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\le \psi \left(\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)\right).$$

(6)

If $max\left\{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right),{\displaystyle \frac{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)}{1+\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)}}\right\}={\displaystyle \frac{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)}{1+\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)}},$ then from (5), we have

$$\begin{array}{ccc}\hfill \mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)& \le & \psi \left({\displaystyle \frac{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)}{1+\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)}}\right)\hfill \\ & <& {\displaystyle \frac{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)}{1+\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)}}\le \mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\hfill \end{array}$$

which is a contradiction. Similarly,

$$\mathfrak{d}\left({\varsigma }_{𝚤+1},{\mathfrak{y}}_{𝚤}\right)=\mathfrak{d}\left(\mathcal{Z}{\mathfrak{y}}_{𝚤},\mathcal{Z}{\varsigma }_{𝚤}\right)\le \psi \left(\mathfrak{M}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\right)$$

(7)

where

$$\begin{array}{ccc}\hfill \mathfrak{M}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)& =& max\left\{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right),{\displaystyle \frac{\mathfrak{d}\left({\varsigma }_{𝚤},\mathcal{Z}{\varsigma }_{𝚤}\right)\mathfrak{d}\left(\mathcal{Z}{\mathfrak{y}}_{𝚤},{\mathfrak{y}}_{𝚤}\right)}{1+\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)}}\right\}\hfill \\ & =& max\left\{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right),{\displaystyle \frac{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\mathfrak{d}\left({\varsigma }_{𝚤+1},{\mathfrak{y}}_{𝚤}\right)}{1+\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)}}\right\}\hfill \end{array}$$

for all $𝚤\in \mathbb{N}.$ If $max\left\{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right),{\displaystyle \frac{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\mathfrak{d}\left({\varsigma }_{𝚤+1},{\mathfrak{y}}_{𝚤}\right)}{1+\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)}}\right\}=$$\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right),$ then from (7), we have

$$\mathfrak{d}\left({\varsigma }_{𝚤+1},{\mathfrak{y}}_{𝚤}\right)\le \psi \left(\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\right).$$

(8)

If $max\left\{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right),{\displaystyle \frac{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\mathfrak{d}\left({\varsigma }_{𝚤+1},{\mathfrak{y}}_{𝚤}\right)}{1+\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)}}\right\}=$${\displaystyle \frac{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\mathfrak{d}\left({\varsigma }_{𝚤+1},{\mathfrak{y}}_{𝚤}\right)}{1+\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)}},$ then from (7), we have

$$\begin{array}{ccc}\hfill \mathfrak{d}\left({\varsigma }_{𝚤+1},{\mathfrak{y}}_{𝚤}\right)& \le & \psi \left({\displaystyle \frac{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\mathfrak{d}\left({\varsigma }_{𝚤+1},{\mathfrak{y}}_{𝚤}\right)}{1+\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)}}\right)\hfill \\ & <& {\displaystyle \frac{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\mathfrak{d}\left({\varsigma }_{𝚤+1},{\mathfrak{y}}_{𝚤}\right)}{1+\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)}}\le \mathfrak{d}\left({\varsigma }_{𝚤+1},{\mathfrak{y}}_{𝚤}\right)\hfill \end{array}$$

which is a contradiction. By mathematical induction for (6) and (8), we get

$$\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\le {\psi }^{𝚤}\left(\mathfrak{d}\left({\varsigma }_1,{\mathfrak{y}}_0\right)\right)$$

(9)

and

$$\mathfrak{d}\left({\varsigma }_{𝚤+1},{\mathfrak{y}}_{𝚤}\right)\le {\psi }^{𝚤+1}\left(\mathfrak{d}\left({\varsigma }_0,{\mathfrak{y}}_0\right)\right)$$

(10)

for all $𝚤\in \mathbb{N}.$ Let $\left(g,\varrho \right)\in \mathfrak{F}\times [0,\infty )$ be such that ($D_3$) is satisfied. Let $ϵ>0$ be fixed. By (${\mathcal{F}}_2$), there exists $\delta >0$, such that

$$0<a<\delta ⟹g\left(a\right)<g\left(ϵ\right)-\varrho .$$

(11)

Let there exists $ϵ>0$ and $𝚤\left(ϵ\right)\in \mathbb{N}$ such that

$$\sum _{𝚤\ge 𝚤\left(ϵ\right)}{\psi }^{𝚤}\left(\mathfrak{d}\left({\varsigma }_1,{\mathfrak{y}}_0\right)\right)<{\displaystyle \frac{ϵ}{2}}$$

and

$$\sum _{𝚤\ge 𝚤\left(ϵ\right)}{\psi }^{𝚤+1}\left(\mathfrak{d}\left({\varsigma }_0,{\mathfrak{y}}_0\right)\right)<{\displaystyle \frac{ϵ}{2}}.$$

Proceeding under the assumption that $p>𝚤\ge 𝚤\left(ϵ\right)$ (where $𝚤\left(ϵ\right)$ is a specific value dependent on $ϵ$), and considering condition ($D_3$), we can infer that $\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_p\right)$$>0$ implies

$$\begin{array}{ccc}\hfill g\left(\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_p\right)\right)& \le & g\left(\begin{array}{c}\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)+\mathfrak{d}\left({\varsigma }_{𝚤+1},{\mathfrak{y}}_{𝚤}\right)+\mathfrak{d}\left({\varsigma }_{𝚤+1},{\mathfrak{y}}_{𝚤+1}\right)+\\ ...+\mathfrak{d}\left({\varsigma }_p,{\mathfrak{y}}_{p-1}\right)+\mathfrak{d}\left({\varsigma }_p,{\mathfrak{y}}_p\right)\end{array}\right)\hfill \\ & \le & g\left(\sum _{j=𝚤}^p\mathfrak{d}\left({\varsigma }_j,{\mathfrak{y}}_j\right)+\sum _{j=𝚤}^{p-1}\mathfrak{d}\left({\varsigma }_{j+1},{\mathfrak{y}}_j\right)\right)+\varrho \hfill \\ & \le & g\left(\sum _{j=𝚤}^p{\psi }^j\left(\mathfrak{d}\left({\varsigma }_1,{\mathfrak{y}}_0\right)\right)+\sum _{j=𝚤}^{p-1}{\psi }^{𝚤+1}\left(\mathfrak{d}\left({\varsigma }_0,{\mathfrak{y}}_0\right)\right)\right)+\varrho \hfill \\ & \le & g\left(\sum _{𝚤\ge 𝚤\left(ϵ\right)}{\psi }^{𝚤}\left(\mathfrak{d}\left({\varsigma }_1,{\mathfrak{y}}_0\right)\right)+\sum _{𝚤\ge 𝚤\left(ϵ\right)}{\psi }^{𝚤+1}\left(\mathfrak{d}\left({\varsigma }_0,{\mathfrak{y}}_0\right)\right)\right)+\varrho \hfill \\ & <& g\left(ϵ\right).\hfill \end{array}$$

∀$j\in \mathbb{N}.$ Likewise, if $𝚤>p\ge 𝚤\left(ϵ\right),$ applying condition ($D_3$) yields that a positive distance between elements $\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_p\right)$$>0$, which implies the following:

$$\begin{array}{ccc}\hfill g\left(\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_p\right)\right)& \le & g\left(\begin{array}{c}\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)+\mathfrak{d}\left({\varsigma }_{𝚤-1},{\mathfrak{y}}_{𝚤-1}\right)+\mathfrak{d}\left({\varsigma }_{𝚤-1},{\mathfrak{y}}_{𝚤-2}\right)+\\ ...+\mathfrak{d}\left({\varsigma }_p,{\mathfrak{y}}_{p-1}\right)+\mathfrak{d}\left({\varsigma }_p,{\mathfrak{y}}_p\right)\end{array}\right)\hfill \\ & \le & g\left(\sum _{j=p}^{𝚤-1}\mathfrak{d}\left({\varsigma }_j,{\mathfrak{y}}_j\right)+\sum _{j=𝚤}^{𝚤}\mathfrak{d}\left({\varsigma }_j,{\mathfrak{y}}_{j-1}\right)\right)+\varrho \hfill \\ & \le & g\left(\sum _{j=p}^{𝚤-1}{\psi }^j\left(\mathfrak{d}\left({\varsigma }_1,{\mathfrak{y}}_0\right)\right)+\sum _{j=p}^{𝚤}{\psi }^{𝚤+1}\left(\mathfrak{d}\left({\varsigma }_0,{\mathfrak{y}}_0\right)\right)\right)+\varrho \hfill \\ & \le & g\left(\sum _{𝚤\ge 𝚤\left(ϵ\right)}{\psi }^{j+1}\left(\mathfrak{d}\left({\varsigma }_0,{\mathfrak{y}}_0\right)\right)+\sum _{𝚤\ge 𝚤\left(ϵ\right)}{\psi }^{𝚤}\left(\mathfrak{d}\left({\varsigma }_1,{\mathfrak{y}}_0\right)\right)\right)+\varrho \hfill \\ & <& g\left(ϵ\right)\hfill \end{array}$$

for every $j\in \mathbb{N}.$ As a result (${\mathcal{F}}_1$), $\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_p\right)<ϵ,$ for all $p,𝚤\ge {𝚤}_0.$ Therefore, $\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)$ is a Cauchy bisequence in $\left(\mho ,\Omega ,\mathfrak{d}\right).$ Since $\left(\mho ,\Omega ,\mathfrak{d}\right)$ is complete, thus $\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)$ biconverges to an element $\mu \in \mho \cap \Omega .$ Consequently, $\left({\varsigma }_{𝚤}\right)\to \mu ,\left({\mathfrak{y}}_{𝚤}\right)\to \mu .$ Also as $\mathcal{Z}$ is continuous, so

$$\left({\varsigma }_{𝚤}\right)\to \mu ⟹\left({\mathfrak{y}}_{𝚤}\right)=\left(\mathcal{Z}{\varsigma }_{𝚤}\right)\to \mathcal{Z}\mu .$$

Also, since the sequence $\left({\mathfrak{y}}_{𝚤}\right)$ converges to a unique limit $\mu$ in $\mho \cap \Omega$. Thus, $\mathcal{Z}\mu =\mu$. So, $\mathcal{Z}$ has a FP. As $⋏\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\ge ⋎\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right),$$\forall 𝚤\in \mathbb{N}$ with ${\varsigma }_{𝚤}\to \mu$ and ${\mathfrak{y}}_{𝚤}\to \mu ,$ as $𝚤\to \infty$ for $\mu \in \mho \cap \Omega ,$ then via Proposition (iii), we have $⋏\left({\varsigma }_{𝚤},\mu \right)\ge ⋎\left({\varsigma }_{𝚤},\mu \right)$$\forall 𝚤\in \mathbb{N}.$ By (3), we have

$$\begin{array}{ccc}\hfill g\left(\mathfrak{d}\left(\mathcal{Z}\mu ,\mu \right)\right)& \le & g\left(\mathfrak{d}\left(\mathcal{Z}\mu ,\mathcal{Z}{\varsigma }_{𝚤}\right)+\mathfrak{d}\left(\mathcal{Z}{\mathfrak{y}}_{𝚤},\mathcal{Z}{\varsigma }_{𝚤}\right)+\mathfrak{d}\left(\mathcal{Z}{\mathfrak{y}}_{𝚤},\mu \right)\right)+\varrho \hfill \\ & \le & g\left(\psi \left(\mathfrak{M}\left({\varsigma }_{𝚤},\mu \right)\right)+\psi \left(\mathfrak{M}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\right)+\mathfrak{d}\left({\varsigma }_{𝚤+1},\mu \right)\right)+\varrho \hfill \end{array}$$

where

$$\mathfrak{M}\left({\varsigma }_{𝚤},\mu \right)=max\left\{\mathfrak{d}\left({\varsigma }_{𝚤},\mu \right),{\displaystyle \frac{\mathfrak{d}\left({\varsigma }_{𝚤},\mathcal{Z}{\varsigma }_{𝚤}\right)\mathfrak{d}\left(\mathcal{Z}\mu ,\mu \right)}{1+\mathfrak{d}\left({\varsigma }_{𝚤},\mu \right)}}\right\}\to 0$$

as $𝚤\to \infty .$ Also,

$$\mathfrak{M}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)=max\left\{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right),{\displaystyle \frac{\mathfrak{d}\left({\varsigma }_{𝚤},\mathcal{Z}{\varsigma }_{𝚤}\right)\mathfrak{d}\left(\mathcal{Z}{\mathfrak{y}}_{𝚤},{\mathfrak{y}}_{𝚤}\right)}{1+\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)}}\right\}\to 0$$

as $𝚤\to \infty .$ Taking the limit as ı approaches positive infinity ($𝚤\to \infty )$, and by leveraging the continuity of both g and $\psi$ at $a=0,$ we obtain the equation $\mathfrak{d}\left(\mathcal{Z}\mu ,\mu \right)=0.$ Hence, $\mathcal{Z}\mu =\mu .$ Therefore, the existence of a FP for $\mathcal{Z}$ is established. Now, suppose another FP $\nu$ exists for $\mathcal{Z}$, such that $\mathcal{Z}\nu =\nu$. This implies $\nu$ belongs to both sets ($\nu \in \mho \cap \Omega$). However, due to property (P), there must exist another element $\sigma$ within the intersection ($\sigma \in \mho \cap \Omega$) that satisfies a specific condition:

$$⋏\left(\mu ,\sigma \right)\ge ⋎\left(\mu ,\sigma \right)\mathrm{and}⋏\left(\sigma ,\nu \right)\ge ⋎\left(\sigma ,\nu \right).$$

(12)

By supposition (i) and (12), we have

$$⋏\left(\mu ,{\mathcal{Z}}^{𝚤}\sigma \right)\ge ⋎\left(\mu ,{\mathcal{Z}}^{𝚤}\sigma \right)\mathrm{and}⋏\left({\mathcal{Z}}^{𝚤}\sigma ,\nu \right)\ge ⋎\left({\mathcal{Z}}^{𝚤}\sigma ,\nu \right)$$

(13)

for every $𝚤\in \mathbb{N}.$ Using (${\mathcal{F}}_1$) and (3), we can deduce that

$$\begin{array}{ccc}\hfill g\left(\mathfrak{d}\left(\mu ,{\mathcal{Z}}^{𝚤}\sigma \right)\right)& \le & g\left(\mathfrak{d}\left(\mathcal{Z}\mu ,\mathcal{Z}\left({\mathcal{Z}}^{𝚤-1}\sigma \right)\right)\right)\hfill \\ & \le & g\left(\mathfrak{d}\left(\mathcal{Z}\mu ,{\mathcal{ZZ}}^{𝚤-1}\sigma \right)\right)\hfill \\ & \le & g\left(\psi \left(\mathfrak{M}\left(\mu ,{\mathcal{Z}}^{𝚤-1}\sigma \right)\right)\right)\hfill \end{array}$$

(14)

where

$$\begin{array}{ccc}\hfill \mathfrak{M}\left(\mu ,{\mathcal{Z}}^{𝚤-1}\sigma \right)& =& max\left\{\mathfrak{d}\left(\mu ,{\mathcal{Z}}^{𝚤-1}\sigma \right),{\displaystyle \frac{\mathfrak{d}\left({\mathcal{Z}}^{𝚤-1}\sigma ,{\mathcal{ZZ}}^{𝚤-1}\sigma \right)\mathfrak{d}\left(\mathcal{Z}\mu ,\mu \right)}{1+\mathfrak{d}\left(\mu ,{\mathcal{Z}}^{𝚤-1}\sigma \right)}}\right\}\hfill \\ & =& \mathfrak{d}\left(\mu ,{\mathcal{Z}}^{𝚤-1}\sigma \right).\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill g\left(\mathfrak{d}\left(\mu ,{\mathcal{Z}}^{𝚤}\sigma \right)\right)& \le & g\left(\psi \left(\mathfrak{d}\left(\mu ,{\mathcal{Z}}^{𝚤-1}\sigma \right)\right)\right)\hfill \\ & \le & ...\le g\left({\psi }^{𝚤}\left(\mathfrak{d}\left(\mu ,\sigma \right)\right)\right).\hfill \end{array}$$

Likewise, we have

$$\begin{array}{ccc}\hfill g\left(\mathfrak{d}\left({\mathcal{Z}}^{𝚤}\sigma ,\nu \right)\right)& \le & g\left(\mathfrak{d}\left(\mathcal{Z}\left({\mathcal{Z}}^{𝚤-1}\sigma \right),\mathcal{Z}\nu \right)\right)\hfill \\ & \le & g\left(\mathfrak{d}\left(\mathcal{Z}\left({\mathcal{Z}}^{𝚤-1}\sigma \right),\mathcal{Z}\nu \right)\right)\hfill \\ & \le & g\left(\psi \left(\mathfrak{M}\left({\mathcal{Z}}^{𝚤-1}\sigma ,\nu \right)\right)\right)\hfill \end{array}$$

(15)

where

$$\begin{array}{ccc}\hfill \mathfrak{M}\left({\mathcal{Z}}^{𝚤-1}\sigma ,\nu \right)& =& max\left\{\mathfrak{d}\left({\mathcal{Z}}^{𝚤-1}\sigma ,\nu \right),{\displaystyle \frac{\mathfrak{d}\left(\nu ,\mathcal{Z}\nu \right)\mathfrak{d}\left(\mathcal{Z}\left({\mathcal{Z}}^{𝚤-1}\sigma \right),{\mathcal{Z}}^{𝚤-1}\sigma \right)}{1+\mathfrak{d}\left({\mathcal{Z}}^{𝚤-1}\sigma ,\nu \right)}}\right\}\hfill \\ & =& \mathfrak{d}\left({\mathcal{Z}}^{𝚤-1}\sigma ,\nu \right).\hfill \end{array}$$

$$\begin{array}{ccc}\hfill g\left(\mathfrak{d}\left({\mathcal{Z}}^{𝚤}\sigma ,\nu \right)\right)& \le & g\left(\psi \left(\mathfrak{d}\left({\mathcal{Z}}^{𝚤-1}\sigma ,\nu \right)\right)\right)\hfill \\ & \le & ...\le g\left({\psi }^{𝚤}\left(\mathfrak{d}\left({\mathcal{Z}}^{𝚤-1}\sigma ,\nu \right)\right)\right).\hfill \end{array}$$

Taking the limit as ı approaches positive infinity ($𝚤\to +\infty )$ in the inequalities established earlier (14) and (15), and by leveraging the continuity of both g and $\psi$, we arrive at the equation

$$\underset{𝚤\to \infty }{lim}g\left(\mathfrak{d}\left(\mu ,{\mathcal{Z}}^{𝚤}\sigma \right)\right)=-\infty$$

(16)

and

$$\underset{𝚤\to \infty }{lim}g\left(\mathfrak{d}\left({\mathcal{Z}}^{𝚤}\sigma ,\nu \right)\right)=-\infty .$$

(17)

Thus, from (16) and (17) by (${\mathcal{F}}_2$), we have

$${\mathcal{Z}}^{𝚤}\sigma \to \mu \mathrm{and}{\mathcal{Z}}^{𝚤}\sigma \to \nu$$

leads to a contradiction. Convergent sequences in complete $\mathfrak{F}$-bip MS, like the one we are dealing with, have a unique limit. Therefore, $\mu$ and $\nu$ must be the same element ($\mu =$$\nu$) and reside within the intersection of sets ℧ and $\Omega$ ($\mu =\nu \in \mho \cap \Omega ).$ □

Example 5.

Let $\mho =\left\{7,8,17,19\right\}$ and $\Omega =\left\{2,4,9,17\right\}$. Define $\mathfrak{d}:\mho \times \Omega \to [0,\infty )$ by

$$\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right)=\left|\varsigma -\mathfrak{y}\right|.$$

Then, $\left(\mho ,\Omega ,\mathfrak{d}\right)$ is complete $\mathfrak{F}$-bip MS. Define $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ by

$$\mathcal{Z}\left(\varsigma \right)=\left\{\begin{array}{c}17,\mathrm{if}\varsigma \in \mho \cup \left\{9\right\}\\ 19,\mathit{otherwise}.\end{array}\right.$$

Then, the mapping $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ is a contravariant mapping. Now we introduce the functions $⋏,⋎:\mho \times \Omega \to [0,+\infty )$ by

$$⋏\left(\varsigma ,\mathfrak{y}\right)=⋏\left(\varsigma ,\mathfrak{y}\right)=\left\{\begin{array}{c}1,\mathrm{if}\varsigma \in \mho \mathrm{and}\mathfrak{y}=9,\\ 0,\mathit{otherwise}.\end{array}\right.$$

Then, all the hypotheses of Theorem 2 are satisfied and the mapping $\mathcal{Z}$ have a unique FP $17.$

In a special case where $⋎\left(\varsigma ,\mathfrak{y}\right)=1$ in Theorem 2, we obtain the following consequence:

Corollary 1.

Let $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ be a contravariant mapping and let the following inequality hold:

$$⋏\left(\varsigma ,\mathfrak{y}\right)\ge 1⟹\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathcal{Z}\varsigma \right)\le \psi \left(\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)\right),$$

where

$$\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)=max\left\{\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right),{\displaystyle \frac{\mathfrak{d}\left(\varsigma ,\mathcal{Z}\varsigma \right)\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathfrak{y}\right)}{1+\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right)}}\right\}$$

for all $\left(\varsigma ,\mathfrak{y}\right)\in \mho \times \Omega$ and $\psi \in \Psi$ and $⋏:\mho \times \Omega \to [0,+\infty ).$

Additionally, let us assume the following conditions:

(i) 

$\mathcal{Z}$ is ⋏-admissible;

(ii) 

$\exists {\varsigma }_0\in \mho ,$${\mathfrak{y}}_0\in \Omega$ such that $⋏\left({\varsigma }_0,\mathcal{Z}{\mathfrak{y}}_0\right)\ge 1$;

(iii) 

$\mathcal{Z}$ is continuous or, if $\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\subseteq$$\left(\mho ,\Omega ,\mathfrak{d}\right)$ is such that $⋏\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\ge 1,$$\forall 𝚤\in \mathbb{N}$ with ${\varsigma }_{𝚤}\to \mu$ and ${\mathfrak{y}}_{𝚤}\to \mu ,$ as $𝚤\to \infty$ for $\mu \in \mho \cap \Omega ,$ then $⋏\left(\mu ,{\mathfrak{y}}_{𝚤}\right)\ge 1$$\forall 𝚤\in \mathbb{N}.$

Then, $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ attains a FP.

Remark 2.

If $max\left\{\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right),{\displaystyle \frac{\mathfrak{d}\left(\varsigma ,\mathcal{Z}\varsigma \right)\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathfrak{y}\right)}{1+\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right)}}\right\}=\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right),$ then we can derive the leading result of Alamri [22] as a direct consequence.

Remark 3.

If we define $⋏:\mho \times \Omega \to [0,+\infty )$ by $⋏\left(\varsigma ,\mathfrak{y}\right)=1$ for all $\varsigma \in \mho$ and $\mathfrak{y}\in \Omega$ and $\psi :[0,+\infty )\to [0,+\infty )$ by $\psi \left(a\right)=\lambda a$ for $0<\lambda <1,$ then Corollary 1 is an extension of Alamri [24].

Remark 4.

Assuming that $\mho =\Omega$ in the previous Corollary, then $\mathfrak{F}$-bip MS limited to $\mathfrak{F}$-MS and Corollary 1 is an extension of Al-Mazrooei et al. [23].

If we consider Theorem 2 and restrict our attention to the scenario where the function $⋏\left(\varsigma ,\mathfrak{y}\right)=1,$ then the theorem yields the following result:

Corollary 2.

Let $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ be a contravariant mapping and let the following inequality hold:

$$⋎\left(\varsigma ,\mathfrak{y}\right)\le 1⟹\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathcal{Z}\varsigma \right)\le \psi \left(\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)\right),$$

where

$$\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)=max\left\{\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right),{\displaystyle \frac{\mathfrak{d}\left(\varsigma ,\mathcal{Z}\varsigma \right)\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathfrak{y}\right)}{1+\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right)}}\right\}$$

for all $\left(\varsigma ,\mathfrak{y}\right)\in \mho \times \Omega$ and for some specific functions $\psi \in \Psi$ and $⋎:\mho \times \Omega \to [0,+\infty ).$

Furthermore, suppose the following assumptions are satisfied:

(i) 

$\mathcal{Z}$ is ⋎-subadmissible;

(ii) 

$\exists {\varsigma }_0\in \mho ,$ ${\mathfrak{y}}_0\in \Omega$ such that $⋎\left({\varsigma }_0,{\mathfrak{y}}_0\right)\le 1$ and $⋎\left({\varsigma }_0,\mathcal{Z}{\mathfrak{y}}_0\right)\le 1$;

(iii) 

$\mathcal{Z}$ is continuous or, if $\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)$ in $\left(\mho ,\Omega ,\mathfrak{d}\right)$ is such that $⋎\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\le 1,$$\forall 𝚤\in \mathbb{N}$ with ${\varsigma }_{𝚤}\to \mu$ and ${\mathfrak{y}}_{𝚤}\to \mu ,$ as $𝚤\to \infty$ for $\mu \in \mho \cap \Omega ,$ then $⋎\left(\mu ,{\mathfrak{y}}_{𝚤}\right)\le 1$$\forall 𝚤\in \mathbb{N}.$

Then, $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ has a FP.

The next outcome is a natural implication of the conditions outlined in Corollary 1.

Corollary 3.

Suppose $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ be a contravariant mapping and suppose that the following inequality holds:

$$⋏\left(\varsigma ,\mathfrak{y}\right)\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathcal{Z}\varsigma \right)\le \psi \left(\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)\right),$$

where

$$\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)=max\left\{\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right),{\displaystyle \frac{\mathfrak{d}\left(\varsigma ,\mathcal{Z}\varsigma \right)\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathfrak{y}\right)}{1+\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right)}}\right\}$$

for all $\left(\varsigma ,\mathfrak{y}\right)\in \mho \times \Omega$ and for some specific functions $\psi \in \Psi$ and $⋏:\mho \times \Omega \to [0,+\infty ).$

Additionally, assume that the following assertions hold:

(i) 

$\mathcal{Z}$ is ⋏-admissible;

(ii) 

$\exists {\varsigma }_0\in \mho ,$ ${\mathfrak{y}}_0\in \Omega$ such that $⋏\left({\varsigma }_0,\mathcal{Z}{\mathfrak{y}}_0\right)\ge 1$;

(iii) 

$\mathcal{Z}$ is continuous or, if $\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)$ in $\left(\mho ,\Omega ,\mathfrak{d}\right)$ is such that $⋏\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\ge 1,$ $\forall 𝚤\in \mathbb{N}$ with ${\varsigma }_{𝚤}\to \mu$ and ${\mathfrak{y}}_{𝚤}\to \mu ,$ as $𝚤\to \infty$ for $\mu \in \mho \cap \Omega ,$ then $⋏\left(\mu ,{\mathfrak{y}}_{𝚤}\right)\ge 1$$\forall 𝚤\in \mathbb{N}.$

Then, $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ attains a FP.

Corollary 4.

Suppose $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ be a contravariant mapping and that the following inequality holds:

$${\left(⋏\left(\varsigma ,\mathfrak{y}\right)+\ell \right)}^{\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathcal{Z}\varsigma \right)}\le {\left(1+\ell \right)}^{\psi \left(\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)\right)},$$

(18)

where

$$\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)=max\left\{\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right),{\displaystyle \frac{\mathfrak{d}\left(\varsigma ,\mathcal{Z}\varsigma \right)\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathfrak{y}\right)}{1+\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right)}}\right\}$$

for all $\left(\varsigma ,\mathfrak{y}\right)\in \mho \times \Omega$ and for some specific functions $\psi \in \Psi$ and $⋏:\mho \times \Omega \to [0,+\infty )$ and $\ell >0.$

Additionally, assume that the following conditions are met:

(i) 

$\mathcal{Z}$ is ⋏-admissible,

(ii) 

$\exists {\varsigma }_0\in \mho ,$${\mathfrak{y}}_0\in \Omega$ such that $⋏\left({\varsigma }_0,\mathcal{Z}{\mathfrak{y}}_0\right)\ge 1,$

(iii) 

$\mathcal{Z}$ is continuous or, if $\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)$ is a bisequence in $\left(\mho ,\Omega ,\mathfrak{d}\right)$ under the condition that $⋏\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\ge 1,$$\forall 𝚤\in \mathbb{N}$ with ${\varsigma }_{𝚤}\to \mu$ and ${\mathfrak{y}}_{𝚤}\to \mu ,$ as $𝚤\to \infty$ for $\mu \in \mho \cap \Omega ,$ then $⋏\left(\mu ,{\mathfrak{y}}_{𝚤}\right)\ge 1$$\forall 𝚤\in \mathbb{N}.$

Then, $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ attains a FP.

Proof. 

Let $⋏\left(\varsigma ,\mathfrak{y}\right)\ge 1.$ Then, by (18), we have

$${\left(1+\ell \right)}^{\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathcal{Z}\varsigma \right)}\le {\left(⋏\left(\varsigma ,\mathfrak{y}\right)+\ell \right)}^{\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathcal{Z}\varsigma \right)}\le {\left(1+\ell \right)}^{\psi \left(\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)\right)},$$

which implies $\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathcal{Z}\varsigma \right)\le \psi \left(\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)\right),$ where

$$\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)=max\left\{\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right),{\displaystyle \frac{\mathfrak{d}\left(\varsigma ,\mathcal{Z}\varsigma \right)\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathfrak{y}\right)}{1+\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right)}}\right\}.$$

Hence, all the axioms of Corollary 1 are met, and $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ attains a FP. □

Likewise, we derive the subsequent corollary.

Corollary 5.

Suppose $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ be a contravariant mapping and that the following inequality holds:

$${\left(\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathcal{Z}\varsigma \right)+\ell \right)}^{⋏\left(\varsigma ,\mathfrak{y}\right)}\le \psi \left(\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)\right)+\ell ,$$

(19)

where

$$\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)=max\left\{\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right),{\displaystyle \frac{\mathfrak{d}\left(\varsigma ,\mathcal{Z}\varsigma \right)\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathfrak{y}\right)}{1+\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right)}}\right\}$$

for all $\left(\varsigma ,\mathfrak{y}\right)\in \mho \times \Omega$ and for some specific functions $\psi \in \Psi$ and $⋏:\mho \times \Omega \to [0,+\infty )$ and $\ell >0.$

Additionally, assume that the following conditions are met:

(i) 

$\mathcal{Z}$ is ⋏-admissible;

(ii) 

$\exists {\varsigma }_0\in \mho ,$ ${\mathfrak{y}}_0\in \Omega$ such that $⋏\left({\varsigma }_0,\mathcal{Z}{\mathfrak{y}}_0\right)\ge 1$;

(iii) 

$\mathcal{Z}$ is continuous or, if $\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)$ is a bisequence in $\left(\mho ,\Omega ,\mathfrak{d}\right)$ such that $⋏\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\ge 1,$ $\forall 𝚤\in \mathbb{N}$ with ${\varsigma }_{𝚤}\to \mu$ and ${\mathfrak{y}}_{𝚤}\to \mu ,$ as $𝚤\to \infty$ for $\mu \in \mho \cap \Omega ,$ then $⋏\left(\mu ,{\mathfrak{y}}_{𝚤}\right)\ge 1$$\forall 𝚤\in \mathbb{N}.$

Then, $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ has a FP.

Proof. 

Let $⋏\left(\varsigma ,\mathfrak{y}\right)\ge 1.$ Then, by (19), we have

$$\left(\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathcal{Z}\varsigma \right)+\ell \right)\le {\left(\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathcal{Z}\varsigma \right)+\ell \right)}^{⋏\left(\varsigma ,\mathfrak{y}\right)}\le \psi \left(\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)\right)+\ell ,$$

which implies $\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathcal{Z}\varsigma \right)\le \psi \left(\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)\right),$ where

$$\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)=max\left\{\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right),{\displaystyle \frac{\mathfrak{d}\left(\varsigma ,\mathcal{Z}\varsigma \right)\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathfrak{y}\right)}{1+\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right)}}\right\}.$$

Hence, all the hypotheses of Corollary 1 are met, and $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ attains a FP. □

Corollary 6.

Suppose $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ be a contravariant and continuous mapping and that the following inequality holds:

$$\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathcal{Z}\varsigma \right)\le \psi \left(\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)\right),$$

where

$$\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)=max\left\{\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right),{\displaystyle \frac{\mathfrak{d}\left(\varsigma ,\mathcal{Z}\varsigma \right)\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathfrak{y}\right)}{1+\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right)}}\right\}$$

for all $\left(\varsigma ,\mathfrak{y}\right)\in \mho \times \Omega$ and for some specific function $\psi \in \Psi .$

Then, $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ has a unique FP.

Proof. 

Set $⋏:\mho \times \Omega \to [0,+\infty )$ by $⋏\left(\varsigma ,\mathfrak{y}\right)=1,$ for $\varsigma \in \mho$ and $\mathfrak{y}\in \Omega$ in Theorem 2. □

Corollary 7.

Let $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ be a continuous contravariant mapping. Let us consider the scenario where a constant λ exists and satisfies the following property $0<\lambda <1$. Under this assumption, we proceed with

$$\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathcal{Z}\varsigma \right)\le \lambda \mathfrak{M}\left(\varsigma ,\mathfrak{y}\right),$$

where

$$\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)=max\left\{\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right),{\displaystyle \frac{\mathfrak{d}\left(\varsigma ,\mathcal{Z}\varsigma \right)\mathfrak{d}\left(\mathcal{Z}\mathfrak{y},\mathfrak{y}\right)}{1+\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right)}}\right\}$$

for all $\left(\varsigma ,\mathfrak{y}\right)\in \mho \times \Omega .$

Then, $\mathcal{Z}:\left(\mho ,\Omega ,\mathfrak{d}\right)$$\rightleftarrows \left(\mho ,\Omega ,\mathfrak{d}\right)$ attains a unique FP.

Proof. 

Within Theorem 2, a function $\psi$ is introduced that maps non-negative real numbers to themselves ($\psi :[0,+\infty )\to [0,+\infty )$). This function is defined such that $\psi \left(a\right)=\lambda a$ and $0<\lambda <1$. □

4. Application

Fixed point theorems provide a powerful tool to guarantee the existence and uniqueness of solutions to certain types of integral equations by transforming them into fixed point problems.

4.1. Integral Equations

This section investigates the existence and uniqueness of solutions for the following integral equation:

$$\phi \left(\varsigma \right)=h\left(\varsigma \right)+{\int }_{\mho \cup \Omega }K(\varsigma ,\mathfrak{y},\phi \left(\varsigma \right))\mathfrak{dy}$$

(20)

where $\mho \cup \Omega$ represents a Lebesgue measurable set and h is a continuous function.

Theorem 3.

Assume that these axioms are satisfied, as follows:

(i) 

$K:\left({\mho }^2\cup {\Omega }^2\right)\times [0,\infty )\to [0,\infty )$ and $g\in {\mathcal{X}}^{\infty }\left(\mho \right)\cup {\mathcal{X}}^{\infty }\left(\Omega \right)$;

(ii) 

There exists a continuous function ${\rm Y}:{\mho }^2\cup {\Omega }^2\to [0,\infty )$, such that

$$\left|K(\varsigma ,\mathfrak{y},\phi (\mathfrak{y}\left)\right)-K(\varsigma ,\mathfrak{y},\varphi (\mathfrak{y}\left)\right)\right|\le {\displaystyle \frac{1}{2}}{\rm Y}\left(\varsigma ,\mathfrak{y}\right)\mathfrak{M}\left(\phi ,\varphi \right)$$

where

$$\mathfrak{M}\left(\phi ,\varphi \right)=max\left\{\begin{array}{c}\left|\varphi \left(\mathfrak{y}\right)-\phi \left(\mathfrak{y}\right)\right|,\\ {\displaystyle \frac{\left|\varphi \left(\mathfrak{y}\right)-K(\varsigma ,\mathfrak{y},\varphi (\mathfrak{y}\left)\right)\right|\left|K(\varsigma ,\mathfrak{y},\phi (\mathfrak{y}\left)\right)-\phi \left(\mathfrak{y}\right)\right|}{1+\left|\varphi \left(\mathfrak{y}\right)-\phi \left(\mathfrak{y}\right)\right|}}\end{array}\right\}$$

for all $\varsigma ,\mathfrak{y}\in \left({\mho }^2\cup {\Omega }^2\right)$;

(iii) 

$∥{\int }_{\mho \cup \Omega }{\rm Y}\left(\varsigma ,\mathfrak{y}\right)\mathfrak{dy}∥\le 1,$ that is, ${sup}_{\varsigma \in \mho \cup \Omega }{\int }_{\mho \cup \Omega }\left|{\rm Y}\left(\varsigma ,\mathfrak{y}\right)\right|\mathfrak{dy}\le 1.$

Then, the integral Equation (20) has a unique solution in ${\mathcal{X}}^{\infty }\left(\mho \right)\cup {\mathcal{X}}^{\infty }\left(\Omega \right).$

Proof. 

We consider two normed linear spaces, $A={\mathcal{X}}^{\infty }\left(\mho \right)$ and $C={\mathcal{X}}^{\infty }\left(\Omega \right)$, where ℧ and $\Omega$ represent Lebesgue measurable sets with a finite total measure $m\left(\mho \cup \Omega \right)<\infty .$ A distance function $\mathfrak{d}:A\times C\to \left[0,\infty \right)$ is defined as follows:

$$\mathfrak{d}\left(\xi ,\zeta \right)={∥\xi -\zeta ∥}_{\infty }$$

for every $\xi ,\zeta \in A\times C.$ Under this assumption, $\left(A,C,\mathfrak{d}\right)$ forms a complete $\mathfrak{F}$-bip MS. We then introduce a mapping, denoted by $I:A\cup C\to A\cup C$, which is defined as

$$I\left(\phi \left(\varsigma \right)\right)=h\left(\varsigma \right)+{\int }_{\mho \cup \Omega }K(\varsigma ,\mathfrak{y},\phi \left(\varsigma \right))\mathfrak{dy}$$

for $\varsigma \in \mho \cup \Omega$ and $⋏,⋎:A\times C\to [0,+\infty )$ by

$$⋏\left(\phi \left(\varsigma \right),\varphi \left(\varsigma \right)\right)=⋎\left(\phi \left(\varsigma \right),\varphi \left(\varsigma \right)\right)=1.$$

Therefore, we can establish

$$\begin{array}{ccc}\hfill \mathfrak{d}\left(I\left(\phi \left(\varsigma \right)\right),I\left(\varphi \left(\varsigma \right)\right)\right)& =& ∥I\left(\phi \left(\varsigma \right)\right)-I\left(\varphi \left(\varsigma \right)\right)∥\hfill \\ & =& \left|{\int }_{\mho \cup \Omega }K(\varsigma ,\mathfrak{y},\phi \left(\varsigma \right))\mathfrak{dy}-{\int }_{\mho \cup \Omega }K(\varsigma ,\mathfrak{y},\varphi \left(\varsigma \right))\mathfrak{dy}\right|\hfill \\ & \le & {\int }_{\mho \cup \Omega }\left|K(\varsigma ,\mathfrak{y},\phi (\varsigma \left)\right)-K(\varsigma ,\mathfrak{y},\varphi (\varsigma \left)\right)\right|\mathfrak{dy}\hfill \\ & \le & {\int }_{\mho \cup \Omega }{\displaystyle \frac{1}{2}}{\rm Y}\left(\varsigma ,\mathfrak{y}\right)\mathfrak{M}\left(\phi ,\varphi \right)\mathfrak{dy}\hfill \\ & \le & {\displaystyle \frac{1}{2}}\mathfrak{M}\left(\phi \left(\mathfrak{y}\right),\varphi \left(\mathfrak{y}\right)\right){\int }_{\mho \cup \Omega }\left|{\rm Y}\left(\varsigma ,\mathfrak{y}\right)\right|\mathfrak{dy}\hfill \\ & \le & {\displaystyle \frac{1}{2}}\mathfrak{M}\left(\phi \left(\mathfrak{y}\right),\varphi \left(\mathfrak{y}\right)\right)\underset{\varsigma \in \mho \cup \Omega }{sup}{\int }_{\mho \cup \Omega }\left|{\rm Y}\left(\varsigma ,\mathfrak{y}\right)\right|\mathfrak{dy}\hfill \\ & \le & {\displaystyle \frac{1}{2}}\mathfrak{M}\left(\phi ,\varphi \right)\hfill \\ & =& \psi \left(\mathfrak{M}\left(\phi ,\varphi \right)\right)\hfill \end{array}$$

where

$$\mathfrak{M}\left(\phi ,\varphi \right)=max\left\{\begin{array}{c}\left|\varphi \left(\mathfrak{y}\right)-\phi \left(\mathfrak{y}\right)\right|,\\ {\displaystyle \frac{\left|\varphi \left(\mathfrak{y}\right)-K(\varsigma ,\mathfrak{y},\varphi (\mathfrak{y}\left)\right)\right|\left|K(\varsigma ,\mathfrak{y},\phi (\mathfrak{y}\left)\right)-\phi \left(\mathfrak{y}\right)\right|}{1+\left|\varphi \left(\mathfrak{y}\right)-\phi \left(\mathfrak{y}\right)\right|}}\end{array}\right\}$$

We define a function $\psi$ such that $\psi \left(a\right)={\displaystyle \frac{1}{2}}a,$ for $a>0$. Leveraging a result established earlier 2, this implies that the mapping I possesses a unique FP within the combined set $A\cup C.$ □

4.2. Homotopy

Theorem 4.

Consider a complete $\mathfrak{F}$-bip MS $\left(\mho ,\Omega ,\mathfrak{d}\right)$ and let $\left(A,C\right)$ be open and $\left(\overline{A},\overline{C}\right)$ be the closed subsets of $\left(\mho ,\Omega \right)$, respectively, with $\left(A,C\right)\subseteq \left(\overline{A},\overline{C}\right).$ Assume that $\mathcal{X}:\left(\overline{A}\cup \overline{C}\right)\times \left[0,1\right]\to \mho \cup \Omega$ meets the following conditions:

(j₁)

$\varsigma \ne \mathcal{X}\left(\varsigma ,q\right)$ for every $\varsigma \in \partial A\cup \partial C$ and $q\in \left[0,1\right]$,

(j₂)

for every $\varsigma \in \overline{A}$, $\mathfrak{y}\in \overline{C}$ and $q\in \left[0,1\right]$

$$\mathfrak{d}\left(\mathcal{X}\left(\mathfrak{y},q\right),\mathcal{X}\left(\varsigma ,q\right)\right)\le \psi \left(\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)\right),$$

where

$$\mathfrak{M}\left(\varsigma ,\mathfrak{y}\right)=max\left\{\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right),{\displaystyle \frac{\mathfrak{d}\left(\varsigma ,\mathcal{X}\left(\varsigma ,q\right)\right)\mathfrak{d}\left(\mathcal{X}\left(\mathfrak{y},q\right),\mathfrak{y}\right)}{1+\mathfrak{d}\left(\varsigma ,\mathfrak{y}\right)}}\right\}$$

$\psi \in \Psi ,$

(j₃)

∃$M\ge 0$ such that

$$\mathfrak{d}\left(\mathcal{X}\left(\varsigma ,r\right),\mathcal{X}\left(\mathfrak{y},\omega \right)\right)\le M\left|r-\omega \right|$$

for all $\varsigma \in \overline{A}$, $\mathfrak{y}\in \overline{C}$ and $r,\omega \in \left[0,1\right].$

Then, $\mathcal{X}\left(·,0\right)$ has a FP if $\mathcal{X}\left(·,1\right)$ has a FP.

Proof. 

Let ${\mathcal{W}}_1=\left\{\kappa \in \left[0,1\right]:\varsigma =\mathcal{X}\left(\varsigma ,\kappa \right),\varsigma \in A\right\}$ and ${\mathcal{W}}_1=\left\{\omega \in \left[0,1\right]:\mathfrak{y}=\mathcal{X}\left(\mathfrak{y},\omega \right)\right.$, $\left.\mathfrak{y}\in C\right\}.$ Since $\mathcal{X}\left(·,0\right)$ has a FP in $A\cup C,$ then we obtain $0\in {\mathcal{W}}_1\cap {\mathcal{W}}_2.$ Thus, ${\mathcal{W}}_1\cap {\mathcal{W}}_2\ne \varnothing$. We will now demonstrate that ${.}_1\cap {\mathcal{W}}_2$ is both closed and open in $\left[0,1\right]$. Consequently, given that it is connected, therefore ${\mathcal{W}}_1={\mathcal{W}}_2=[0,1].$ Let $\left({\left\{{\kappa }_{𝚤}\right\}}_{𝚤=1}^{\infty }\right),\left({\left\{{\omega }_{𝚤}\right\}}_{𝚤=1}^{\infty }\right)\subseteq \left({\mathcal{W}}_1,{\mathcal{W}}_2\right)$ with $\left({\kappa }_{𝚤},{\omega }_{𝚤}\right)\to \left(\varpi ,\varpi \right)\in [0,1]$ as $𝚤\to \infty .$ It is our contention that $\varpi \in {\mathcal{W}}_1\cap {\mathcal{W}}_2.$ Since $\left({\kappa }_{𝚤},{\omega }_{𝚤}\right)\in {\mathcal{W}}_1\cap {\mathcal{W}}_2,$ for $𝚤\in \mathbb{N}\cup \left\{0\right\}.$ Under these conditions, there exists a sequence pair $\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)$ within the combined set $\left(A,C\right).$ This bisequence satisfies the properties that ${\mathfrak{y}}_{𝚤}=\mathcal{X}\left({\varsigma }_{𝚤},{\kappa }_{𝚤}\right)$ and ${\varsigma }_{𝚤+1}=\mathcal{X}\left({\mathfrak{y}}_{𝚤},{\omega }_{𝚤}\right).$ This leads to

$$\begin{array}{ccc}\hfill \mathfrak{d}\left({\varsigma }_{𝚤+1},{\mathfrak{y}}_{𝚤}\right)& =& \mathfrak{d}\left(\mathcal{X}\left({\mathfrak{y}}_{𝚤},{\omega }_{𝚤}\right),\mathcal{X}\left({\varsigma }_{𝚤},{\kappa }_{𝚤}\right)\right)\hfill \\ & \le & \psi \left(\mathfrak{M}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)\right)\hfill \end{array}$$

where

$$\mathfrak{M}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)=max\left\{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right),{\displaystyle \frac{\mathfrak{d}\left({\varsigma }_{𝚤},\mathcal{X}\left({\varsigma }_{𝚤},{\kappa }_{𝚤}\right)\right)\mathfrak{d}\left(\mathcal{X}\left({\mathfrak{y}}_{𝚤},{\omega }_{𝚤}\right),{\mathfrak{y}}_{𝚤}\right)}{1+\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)}}\right\}.$$

and

$$\begin{array}{ccc}\hfill \mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)& =& \mathfrak{d}\left(\mathcal{X}\left({\mathfrak{y}}_{𝚤-1},{\omega }_{𝚤-1}\right),\mathcal{X}\left({\varsigma }_{𝚤},{\kappa }_{𝚤}\right)\right)\hfill \\ & \le & \psi \left(\mathfrak{M}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)\right),\hfill \end{array}$$

where

$$\mathfrak{M}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)=max\left\{\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right),{\displaystyle \frac{\mathfrak{d}\left({\varsigma }_{𝚤},\mathcal{X}\left({\varsigma }_{𝚤},{\kappa }_{𝚤}\right)\right)\mathfrak{d}\left(\mathcal{X}\left({\mathfrak{y}}_{𝚤-1},{\omega }_{𝚤-1}\right),{\mathfrak{y}}_{𝚤-1}\right)}{1+\mathfrak{d}\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤-1}\right)}}\right\}$$

Following the proof of Theorem 2, it is straightforward to demonstrate that $\left({\varsigma }_{𝚤},{\mathfrak{y}}_{𝚤}\right)$ forms a Cauchy bisequence in $(A,C)$. Since $(A,C)$ is complete, thus $\exists {\varpi }_1\in A\cap C$ such that ${lim}_{𝚤\to \infty }\left({\varsigma }_{𝚤}\right)={lim}_{𝚤\to \infty }\left({\mathfrak{y}}_{𝚤}\right)={\varpi }_1.$ Therefore, we have

$$\begin{array}{ccc}\hfill g\left(\mathfrak{d}\left(\mathcal{X}\left({\varpi }_1,\omega \right),{\mathfrak{y}}_{𝚤}\right)\right)& =& g\left(\mathfrak{d}\left(\mathcal{X}\left({\varpi }_1,\omega \right),\mathcal{X}\left({\varsigma }_{𝚤},{\kappa }_{𝚤}\right)\right)\right)\hfill \\ & \le & g\left(\psi \left(max\left\{\begin{array}{c}\mathfrak{d}\left({\varsigma }_{𝚤},{\varpi }_1\right),\\ {\displaystyle \frac{\mathfrak{d}\left({\varsigma }_{𝚤},\mathcal{X}\left({\varsigma }_{𝚤},{\kappa }_{𝚤}\right)\right)\mathfrak{d}\left(\mathcal{X}\left({\varpi }_1,\omega \right),{\varpi }_1\right)}{1+\mathfrak{d}\left({\varsigma }_{𝚤},{\varpi }_1\right)}}\end{array}\right\}\right)\right)=-\infty \hfill \end{array}$$

as $𝚤\to \infty .$ Therefore, by (${\mathfrak{F}}_2$), we obtain $\mathfrak{d}\left(\mathcal{X}\left({\varpi }_1,\omega \right),{\varpi }_1\right)=0,$ which implies that $\mathcal{X}\left({\varpi }_1,\omega \right)={\varpi }_1.$ Likewise, $\mathcal{X}\left({\varpi }_1,\kappa \right)=v_1.$ Thus, $\kappa =\omega \in {\mathcal{W}}_1\cap {\mathcal{W}}_2,$ and evidently ${\mathcal{W}}_1\cap {\mathcal{W}}_2$ is closed set in $[0,1].$

Next, we need to demonstrate that ${\mathcal{W}}_1\cap {\mathcal{W}}_2$ is open in $[0,1].$ Suppose $\left({\kappa }_0,{\omega }_0\right)\in \left({\mathcal{W}}_1,{\mathcal{W}}_2\right)$; then, there is a bisequence $\left({\varsigma }_0,{\mathfrak{y}}_0\right)$ so that ${\varsigma }_0=\mathcal{X}\left({\varsigma }_0,{\kappa }_0\right),$${\mathfrak{y}}_0=\mathcal{X}\left({\mathfrak{y}}_0,{\omega }_0\right).$ Since $A\cup C$ is open, there exists $r>0$ such that $B_{\mathfrak{d}}\left({\varsigma }_0,r\right)\subseteq A\cup C$ and $B_{\mathfrak{d}}\left(r,{\mathfrak{y}}_0\right)\subseteq A\cup C.$ Select $\kappa \in \left({\omega }_0-ϵ,{\omega }_0+ϵ\right)$ and $\omega \in \left({\kappa }_0-ϵ,{\kappa }_0+ϵ\right)$ such that

$$\left|\kappa -{\omega }_0\right|\le {\displaystyle \frac{1}{M^{𝚤}}}<{\displaystyle \frac{ϵ}{2}}$$

$$\left|\omega -{\kappa }_0\right|\le {\displaystyle \frac{1}{M^{𝚤}}}<{\displaystyle \frac{ϵ}{2}}$$

and

$$\left|{\kappa }_0-{\omega }_0\right|\le {\displaystyle \frac{1}{M^{𝚤}}}<{\displaystyle \frac{ϵ}{2}}.$$

Hence, we have $\mathfrak{y}\in \overline{B_{{\mathcal{W}}_1\cup {\mathcal{W}}_2}\left({\varsigma }_0,r\right)}=\left\{\mathfrak{y}:{\mathfrak{y}}_0\in C|\mathfrak{d}\left({\varsigma }_0,\mathfrak{y}\right)\le r+\mathfrak{d}\left({\varsigma }_0,{\mathfrak{y}}_0\right)\right\}$ and $\varsigma \in \overline{B_{{\mathcal{W}}_1\cup {\mathcal{W}}_2}\left(r,{\mathfrak{y}}_0\right)}=\left\{\varsigma :{\varsigma }_0\in A|\mathfrak{d}\left(\varsigma ,{\mathfrak{y}}_0\right)\le r+\mathfrak{d}\left({\varsigma }_0,{\mathfrak{y}}_0\right)\right\}.$ Moreover, we have

$$\begin{array}{ccc}\hfill \mathfrak{d}\left(\mathcal{X}\left(\varsigma ,\kappa \right),{\mathfrak{y}}_0\right)& =& \mathfrak{d}\left(\mathcal{X}\left(\varsigma ,\kappa \right),\mathcal{X}\left({\mathfrak{y}}_0,{\omega }_0\right)\right)\hfill \\ & \le & \mathfrak{d}\left(\mathcal{X}\left(\varsigma ,\kappa \right),\mathcal{X}\left(\mathfrak{y},{\omega }_0\right)\right)+\mathfrak{d}\left(\mathcal{X}\left({\varsigma }_0,\kappa \right),\mathcal{X}\left(\mathfrak{y},{\omega }_0\right)\right)+\mathfrak{d}\left(\mathcal{X}\left({\varsigma }_0,\kappa \right),\mathcal{X}\left({\mathfrak{y}}_0,{\omega }_0\right)\right)\hfill \\ & \le & 2M\left|\kappa -{\omega }_0\right|+\mathfrak{d}\left(\mathcal{X}\left({\varsigma }_0,\kappa \right),\mathcal{X}\left(\mathfrak{y},{\omega }_0\right)\right)\hfill \\ & \le & {\displaystyle \frac{2}{M^{𝚤}-1}}+\psi \left(\mathfrak{M}\left({\varsigma }_0,\mathfrak{y}\right)\right)\hfill \\ & \le & {\displaystyle \frac{2}{M^{𝚤}-1}}+\mathfrak{M}\left({\varsigma }_0,\mathfrak{y}\right)\hfill \end{array}$$

where

$$\mathfrak{M}\left({\varsigma }_0,\mathfrak{y}\right)=max\left\{\mathfrak{d}\left({\varsigma }_0,\mathfrak{y}\right),{\displaystyle \frac{\mathfrak{d}\left({\varsigma }_0,\mathcal{X}\left({\varsigma }_0,\kappa \right)\right)\mathfrak{d}\left(\mathcal{X}\left(\mathfrak{y},{\omega }_0\right),\mathfrak{y}\right)}{1+\mathfrak{d}\left({\varsigma }_0,\mathfrak{y}\right)}}\right\}$$

Letting $𝚤\to \infty ,$ we get

$$\mathfrak{d}\left(\mathcal{X}\left(\varsigma ,\kappa \right),{\mathfrak{y}}_0\right)\le \mathfrak{d}\left({\varsigma }_0,\mathfrak{y}\right)\le r+\mathfrak{d}\left({\varsigma }_0,{\mathfrak{y}}_0\right).$$

In a corresponding fashion, we get

$$\mathfrak{d}\left({\varsigma }_0,\mathcal{X}\left(\mathfrak{y},\omega \right)\right)\le \mathfrak{d}\left(\varsigma ,{\mathfrak{y}}_0\right)\le r+\mathfrak{d}\left({\varsigma }_0,{\mathfrak{y}}_0\right).$$

But

$$\mathfrak{d}\left({\varsigma }_0,{\mathfrak{y}}_0\right)=\mathfrak{d}\left(\mathcal{X}\left({\varsigma }_0,{\kappa }_0\right),\mathcal{X}\left({\mathfrak{y}}_0,{\omega }_0\right)\right)\le M\left|{\kappa }_0-{\omega }_0\right|\le {\displaystyle \frac{1}{M^{𝚤-1}}}\to 0$$

as $𝚤\to \infty ,$ which yields ${\varsigma }_0={\mathfrak{y}}_0.$ As a result, $\omega =\kappa \in \left({\omega }_0-ϵ,{\omega }_0+ϵ\right)$ for each fixed $\omega$ and $\mathcal{X}\left(·,\kappa \right):\overline{B_{{\mathcal{W}}_1\cup {\mathcal{W}}_2}\left({\varsigma }_0,r\right)}\to \overline{B_{{\mathcal{W}}_1\cup {\mathcal{W}}_2}\left({\varsigma }_0,r\right)}.$ Given that all the conditions of Corollary 1 are satisfied, $\mathcal{X}\left(·,\kappa \right)$ has a FP in $\overline{A}\cap \overline{C},$ which indeed exists in $A\cap C.$ Thus, $\kappa =\omega \in {\mathcal{W}}_1\cap {\mathcal{W}}_2$ for each $\omega \in \left({\omega }_0-ϵ,{\omega }_0+ϵ\right).$ Therefore, $\left({\omega }_0-ϵ,{\omega }_0+ϵ\right)\in {\mathcal{W}}_1\cap {\mathcal{W}}_2$, indicating that ${\mathcal{W}}_1\cap {\mathcal{W}}_2$ is open in $[0,1].$ Likewise, the converse can be demonstrated. □

5. Conclusions and Future Works

In the present manuscript, we have defined the idea of rational ($⋏,⋎,\psi$)-contractions in the context of $\mathfrak{F}$-bip MSs and proved some theorems of FPs. To confirm the genuineness of the given results, a meaningful example was also furnished. This study investigated the solutions for a specific integral equation by leveraging our key theoretical outcomes. Furthermore, we discussed the existence of a unique solution for an associated homotopy problem.

In the background of $\mathfrak{F}$-bip MS, one can obtain fixed points and common fixed points for multi-valued (set-valued) mappings and fuzzy mappings in a future work. Furthermore, one can investigate the solution of differential and integral inclusions as applications of these results.