Applications of Common Fixed-Point Results in Complex-Valued Metric Spaces to Homotopy Theory

Abstract

The aim of this research article is to introduce generalized rational contractions in the context of complex-valued metric spaces and to establish novel common fixed-point theorems. Our findings generalize several well-known fixed-point theorems and contribute to the advancement of fixed-point theory. A non-trivial example is provided to demonstrate the efficacy of the obtained results. Additionally, we demonstrate the practical significance of our findings through a homotopy result.

1. Introduction

Fixed-point theory is a fundamental area of mathematical analysis that deals with the existence and properties of points that remain invariant under certain mappings. The concept of a fixed point (FP), where a function maps a point to itself, has profound implications in various mathematical fields and real-world applications. In this theory, the study of metric spaces (MSs) has been fundamental, providing crucial applications across various scientific disciplines such as mathematics, computer science, physics, chemistry, and biology. The widespread utility of MSs has led to numerous extensions and generalizations by researchers. Azam et al. [1] broadened the scope of metric spaces by replacing the real-valued range with complex numbers, thereby introducing the concept of complex-valued metric spaces (CVMSs). They demonstrated common FPs for two single-valued mappings under generalized contractions involving specific rational expressions. Subsequently, Rouzkard et al. [2] further developed Azam et al.’s [1] results by incorporating a rational term into the contraction conditions. Klin-eam et al. [3] gave generalized contractive-type conditions and presented some theorems that extend previous results in this area. Subsequently, Bhatt et al. [4] presented another extension of Banach’s fixed-point theorem and established common FPs of mappings satisfying the rational inequality in the context of CVMSs. Thereafter, Ahmad et al. [5] introduced the generalized Housdorff distance function in the background of CVMSs and proved common FP theorems for set-valued mappings. They established FP results for Banach and Kannan’s contractive conditions involving rational expressions given by Fisher. Afterwards, Azam et al. [6] proved common FP theorems for Chatterjea’s contractive conditions along with the rational inequality. Humaira et al. [7] obtained FP results for $\alpha$-fuzzy mappings in the scope of CVMSs and investigated the homotopy problem as applications of their leading theorem. Wangwe et al. [8] established FP theorems for extended interpolative Kanann–Ćirić–Reich–Rus non-self mappings within the framework of hyperbolic CVMSs. Later on, FP results involving $\varpi$-distances were presented by Khuangsatung et al. [9] in the setting of CVMSs. The work of Sengar [10] focused on common FP theorems for implicit relations in the context of CVMSs. Additional details and related work can be found in [11,12,13,14].

In this research article, we introduce generalized rational contractions in the setting of CVMSs and obtain some new common FP results. By our main theorem, we derive some famous results of the literature including the leading results of Azam et al. [1], Rouzkard et al. [2], Klin-eam et al. [3], Bhatt et al. [4], Ahmad et al. [5], and Azam et al. [6]. To illustrate the novelty of our primary theorem, we provide a concrete example. Moreover, we demonstrate the significance of our findings by presenting a homotopy result.

2. Preliminaries

In this section, we establish the foundational concepts and notation that will be utilized throughout this paper. Consistent with Azam et al. [1], $R\left(\omega \right)$ and $I\left(\omega \right)$ denote the real and imaginary components of the complex number $\omega$, respectively. Let us define a partial order $\precsim$ on the set $\mathbb{C}$ as described below:

$${\omega }_1\precsim {\omega }_2⟺R\left({\omega }_1\right)⩽R\left({\omega }_2\right),I\left({\omega }_1\right)⩽I\left({\omega }_2\right)$$

for ${\omega }_1,{\omega }_2\in \mathbb{C}$. Consequently,

$${\omega }_1\precsim {\omega }_2,$$

provided that at least one of the following criteria is met:

$$\begin{array}{ccc}\hfill \left(\mathrm{i}\right)R\left({\omega }_1\right)& =& R\left({\omega }_2\right),I\left({\omega }_1\right)<I\left({\omega }_2\right),\hfill \\ \hfill \left(\mathrm{ii}\right)R\left({\omega }_1\right)& <& R\left({\omega }_2\right),I\left({\omega }_1\right)=I\left({\omega }_2\right),\hfill \\ \hfill \left(\mathrm{iii}\right)R\left({\omega }_1\right)& <& R\left({\omega }_2\right),I\left({\omega }_1\right)<I\left({\omega }_2\right),\hfill \\ \hfill \left(\mathrm{iv}\right)R\left({\omega }_1\right)& =& R\left({\omega }_2\right),I\left({\omega }_1\right)=I\left({\omega }_2\right).\hfill \end{array}$$

To be precise, we will use ${\omega }_1⋨{\omega }_2$ when ${\omega }_1\ne {\omega }_2$ and any of the conditions (i), (ii), and (iii) hold, and we will use ${\omega }_1\prec {\omega }_2$ when only condition (iii) is met.

Definition 1.

Let $\Theta \ne \varnothing$ and $\phi :\Theta \times \Theta \to \mathbb{C}$ be a function such that

1. 

$0\precsim \phi (\mathfrak{p},\tau ),$ and $\phi (\mathfrak{p},\tau )=0$⇔$\mathfrak{p}=\tau$;

2. 

$\phi (\mathfrak{p},\tau )=\phi (\tau ,\mathfrak{p})$;

3. 

$\phi (\mathfrak{p},\tau )\precsim \phi (\mathfrak{p},\nu )+\phi (\nu ,\tau )$;

for all $\mathfrak{p},\tau ,\nu \in \Theta$. Then, the couple $(\Theta ,\phi )$ is identified as a CVMS.

A point $\mathfrak{p}\in$$\Theta$ is an interior point of a set $E\subseteq \Theta$ if there exists a positive complex number r such that

$$B\left(\mathfrak{p},r\right)=\left\{\varpi \in \Theta :\phi \left(\mathfrak{p},\varpi \right)\prec r\right\}\subseteq E.$$

Similarly, a point $\mathfrak{p}\in$$\Theta$ is said to be the limit point of a set $E\subseteq \Theta$ if for every positive complex number r, the open ball $B\left(\mathfrak{p},r\right)$ contains at least one point of E other than $\mathfrak{p}$ itself. A set E is open provided that each of its points is an interior point. A subset M of $\Theta$ is closed whenever all of its limit points are elements of $M.$ Also, a set E is bounded if it is contained within a ball of finite radius. The collection of open balls $\left\{B\left(\mathfrak{p},r\right):\mathfrak{p}\in \Theta ,0\prec r\right\}$ forms a sub-basis for a topology on $\Theta .$ We represent this complex topology as ${\tau }_c.$ Importantly, ${\tau }_c$ is a Hausdorff topology on $\Theta .$

According to Ahmad et al. [5], the generalized Hausdorff distance function in a CVMS is defined as follows.

Let $⋏\left({\omega }_1\right)=\{{\omega }_2\in$$\mathbb{C}:$${\omega }_1⪯$${\omega }_2\}$ for ${\omega }_1\in$$\mathbb{C},$ and

$$⋏\left(\mu ,M\right)=\underset{\varpi \in M}{\cup }⋏\left(\phi \left(\mu ,\varpi \right)\right)=\underset{\varpi \in M}{\cup }\{\omega \in \mathbb{C}:\phi \left(\mu ,\varpi \right)⪯\omega \}$$

for $\mu \in$$\Theta$ and $M\in N\left(\Theta \right).$

For $E,M\in B\left(\Theta \right)$, we denote

$$⋏\left(E,M\right)=\left(\underset{\mu \in E}{\cap }⋏\left(\mu ,M\right)\right)\cap \left(\underset{\varpi \in M}{\cap }⋏\left(\varpi ,E\right)\right),$$

where $N\left(\Theta \right)$, $C(\Theta )$, and $B\left(\Theta \right)$ are non-empty, closed, and bounded subsets of the CVMS $\Theta$, respectively.

Remark 1.

If in Definition 1, we take $I\left(\omega \right)=0,$ that is, $\mathbb{C}=\mathbb{R},$ then the CVMS $(\Theta ,\phi )$ is reduced to an MS. Additionally, for $E,M\in C(\Theta ),H(E,M)=inf⋏(E,M)$ represents the Hausdorff distance derived from $\phi .$

Let $\beth :\Theta \to C(\Theta )$ be a set-valued mapping. For $\mathfrak{p}\in \Theta$ and $E\in C(\Theta )$, we define

$$W_{\mathfrak{p}}\left(E\right)=\{\phi (\mathfrak{p},\mu ):\mu \in E\}.$$

Thus, for $\mathfrak{p},\tau \in \Theta$,

$$W_{\mathfrak{p}}(\beth \tau )=\{\phi (\mathfrak{p},u):u\in \beth \tau \}.$$

Definition 2

([5]). A set-valued function $\beth :\Theta \to C(\Theta )$ is said to possess the greatest lower bound (glb) property on Θ if the greatest lower bound of $W_{\mathfrak{p}}(\beth \tau )$ exists within the complex numbers for all $\mathfrak{p},\tau \in \Theta$. We symbolize this glb by $\phi (\mathfrak{p},\beth \tau ),$ i.e.,

$$\phi (\mathfrak{p},\beth \tau )=inf\left\{\phi \right(\mathfrak{p},u):u\in \beth \tau \}.$$

In the upcoming results, the set-valued mappings $\mathcal{K}$ and ℶ satisfy the glb property.

Ahmad et al. [5] proved the following common FP theorem for set-valued mappings satisfying a Banach-type contraction, incorporating a rational expression in the setting of CVMSs.

Theorem 1.

Let $\mathcal{K},\beth :\Theta ⟶$$CB\left(\Theta \right)$. Assume that there exist non-negative real numbers ${\varsigma }_i$ for $i=1,2,3$ with ${\varsigma }_1+2\left({\varsigma }_2+{\varsigma }_3\right)<1$ such that

$${\varsigma }_1\phi \left(\mathfrak{p},\tau \right)+{\varsigma }_2{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\tau ,\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}+{\varsigma }_3{\displaystyle \frac{\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\phi \left(\mathfrak{p},\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\in ⋏\left(\mathcal{K}\mathfrak{p},\beth \tau \right)$$

for all $\mathfrak{p},\tau \in$$\Theta .$ Then, $\mathcal{K}$ and ℶ have a common FP.

They [5] also established a common FP result for a Kannan-type contraction.

Theorem 2.

Let $\mathcal{K},\beth :\Theta ⟶$$CB\left(\Theta \right)$. Assume that there exist non-negative real numbers ${\varsigma }_i$ for $i=1,2,3$ with ${\varsigma }_1+{\varsigma }_2+{\varsigma }_3<{\displaystyle \frac{1}{2}}$ such that

$${\varsigma }_1\left(\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)+\phi \left(\tau ,\beth \tau \right)\right)+{\varsigma }_2{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\tau ,\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}+{\varsigma }_3{\displaystyle \frac{\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\phi \left(\mathfrak{p},\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\in ⋏\left(\mathcal{K}\mathfrak{p},\beth \tau \right)$$

for all $\mathfrak{p},\tau \in$$\Theta .$ Then, $\mathcal{K}$ and ℶ have a common FP.

Theorem 3.

Let $\mathcal{K},\beth :\Theta ⟶$$CB\left(\Theta \right)$. Assume that there exist non-negative real numbers ${\varsigma }_i$ for $i=1,2,3$ with ${\varsigma }_1+{\varsigma }_2+{\varsigma }_3<{\displaystyle \frac{1}{2}}$ such that

$${\varsigma }_1\left(\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)+\phi \left(\mathfrak{p},\beth \tau \right)\right)+{\varsigma }_2{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\tau ,\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}+{\varsigma }_3{\displaystyle \frac{\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\phi \left(\mathfrak{p},\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\in ⋏\left(\mathcal{K}\mathfrak{p},\beth \tau \right)$$

for all $\mathfrak{p},\tau \in$$\Theta .$ Then, $\mathcal{K}$ and ℶ have a common FP.

The following lemmas are necessary:

Lemma 1

([1]). A sequence $\left\{{\mathfrak{p}}_n\right\}$ in the CVMS Θ converges to a point $\mathfrak{p}$ if and only if $\left|\phi ({\mathfrak{p}}_n,\mathfrak{p})\right|\to 0$ as $n\to \infty .$

Lemma 2

([1]). A sequence $\left\{{\mathfrak{p}}_n\right\}$ in the CVMS Θ is Cauchy if and only if $\left|\phi ({\mathfrak{p}}_n,{\mathfrak{p}}_{n+m})\right|\to 0$ as $n\to \infty .$

3. Main Results

In this section, we prove novel FP theorems for set-valued mappings that fulfill generalized contractive conditions within the framework of a complete CVMS.

Theorem 4.

Let $\left(\Theta ,\phi \right)$ be a complete CVMS and let $\mathcal{K},\beth :\Theta ⟶$$CB\left(\Theta \right)$ be set-valued mappings satisfying the glb property. Assume the existence of non-negative real numbers ${\varsigma }_i$ for $i=1,2,\dots ,7$ such that

$${\varsigma }_1+2\left({\varsigma }_2+{\varsigma }_3+{\varsigma }_4+{\varsigma }_5+{\varsigma }_6+{\varsigma }_7\right)<1,$$

and

$$\begin{array}{ccc}\hfill & \hfill & {\varsigma }_1\phi \left(\mathfrak{p},\tau \right)+{\varsigma }_2\left(\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)+\phi \left(\tau ,\beth \tau \right)\right)\hfill \\ \hfill & \hfill & +{\varsigma }_3\left(\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)+\phi \left(\mathfrak{p},\beth \tau \right)\right)\hfill \\ \hfill & \hfill & +{\varsigma }_4{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\tau ,\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\phi \left(\mathfrak{p},\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\hfill \\ \hfill & \hfill & +{\varsigma }_6{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\mathfrak{p},\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\phi \left(\tau ,\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\hfill \\ \hfill & \in \hfill & ⋏\left(\mathcal{K}\mathfrak{p},\beth \tau \right),\hfill \end{array}$$

(1)

for all $\mathfrak{p},\tau \in$$\Theta .$ Then, $\mathcal{K}$ and ℶ have a common FP.

Proof. 

Choose an arbitrary point ${\mathfrak{p}}_0$ in the set $\Theta .$ Subsequently, select a point ${\mathfrak{p}}_1$ that is an element of $\mathcal{K}{\mathfrak{p}}_0,$ i.e., ${\mathfrak{p}}_1\in \mathcal{K}{\mathfrak{p}}_0.$ From (1), we have

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)+\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)+\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}\hfill \\ & \in & ⋏\left(\mathcal{K}{\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right).\hfill \end{array}$$

By definition of “⋏”, we have

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)+\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)\right)+{\varsigma }_3\left(\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)+\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}\hfill \\ & \in & \left(\underset{\mathfrak{p}\in \mathcal{K}{\mathfrak{p}}_0}{\cap }⋏\left(\mathfrak{p},\beth {\mathfrak{p}}_1\right)\right)\cap \left(\underset{{\mathfrak{p}}^{\prime }\in \beth {\mathfrak{p}}_1}{\cap }⋏\left({\mathfrak{p}}^{\prime },\mathcal{K}{\mathfrak{p}}_0\right)\right),\hfill \end{array}$$

which implies

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)+\phi \left({\mathfrak{p}}_{1,}\beth {\mathfrak{p}}_1\right)\right)+{\varsigma }_3\left(\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)+\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_{1,}\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_{0,}\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}\hfill \\ & \in & \left(\underset{\mathfrak{p}\in \mathcal{K}{\mathfrak{p}}_0}{\cap }⋏\left(\mathfrak{p},\beth {\mathfrak{p}}_1\right)\right).\hfill \end{array}$$

This yields

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)+\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)\right)+{\varsigma }_3\left(\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)+\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}\hfill \\ & \in & ⋏\left(\mathfrak{p},\beth {\mathfrak{p}}_1\right),\hfill \end{array}$$

for all $\mathfrak{p}\in \mathcal{K}{\mathfrak{p}}_0.$ Now, since ${\mathfrak{p}}_1\in \mathcal{K}{\mathfrak{p}}_0,$ we have

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)+\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)\right)+{\varsigma }_3\left(\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)+\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}\hfill \\ & \in & ⋏\left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right),\hfill \end{array}$$

that is,

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)+\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)\right)+{\varsigma }_3\left(\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)+\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}\hfill \\ & \in & ⋏\left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)=\underset{\mathfrak{p}\in \beth {\mathfrak{p}}_1}{\cup }⋏\left(\phi \left({\mathfrak{p}}_1,\mathfrak{p}\right)\right).\hfill \end{array}$$

So, there exists some ${\mathfrak{p}}_2\in \beth {\mathfrak{p}}_1$ such that

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)+\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)\right)+{\varsigma }_3\left(\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)+\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}\hfill \\ & \in & ⋏\left(\phi ({\mathfrak{p}}_1,{\mathfrak{p}}_2)\right).\hfill \end{array}$$

That is,

$$\begin{array}{ccc}\hfill \phi ({\mathfrak{p}}_1,{\mathfrak{p}}_2)& ⪯& {\varsigma }_1\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)+\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)\right)+{\varsigma }_3\left(\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)+\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}.\hfill \end{array}$$

Leveraging the glb property inherent to $\mathcal{K}$ and $\beth ,$ we obtain

$$\begin{array}{ccc}\hfill \phi ({\mathfrak{p}}_1,{\mathfrak{p}}_2)& ⪯& {\varsigma }_1\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)+\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right)+{\varsigma }_3\left(\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_1\right)+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_2\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_0\right)\phi \left({\mathfrak{p}}_0,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_2\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_1\right)\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)}{1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)}},\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill \left|\phi ({\mathfrak{p}}_1,{\mathfrak{p}}_2)\right|& \le & {\varsigma }_1\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|+{\varsigma }_2\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|+{\varsigma }_2\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|+{\varsigma }_3\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_2\right)\right|\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|}{\left|1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|}}\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|+{\varsigma }_6{\displaystyle \frac{\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|}{\left|1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|}}\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_2\right)\right|.\hfill \end{array}$$

Now, since ${\displaystyle \frac{\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|}{\left|1+\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|}}<1,$ we have

$$\begin{array}{ccc}\hfill \left|\phi ({\mathfrak{p}}_1,{\mathfrak{p}}_2)\right|& \le & {\varsigma }_1\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|+{\varsigma }_2\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|+{\varsigma }_2\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|+{\varsigma }_3\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_2\right)\right|\hfill \\ & & +{\varsigma }_4\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|+{\varsigma }_6\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_2\right)\right|.\hfill \end{array}$$

According to the triangle inequality of the CVMS, we have

$$\begin{array}{ccc}\hfill \left|\phi ({\mathfrak{p}}_1,{\mathfrak{p}}_2)\right|& \le & {\varsigma }_1\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|+{\varsigma }_2\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|+{\varsigma }_2\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|+{\varsigma }_3\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|\hfill \\ & & +{\varsigma }_3\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|+{\varsigma }_4\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|+{\varsigma }_6\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|+{\varsigma }_6\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|,\hfill \end{array}$$

which implies

$$\left|\phi ({\mathfrak{p}}_1,{\mathfrak{p}}_2)\right|\le {\displaystyle \frac{{\varsigma }_1+{\varsigma }_2+{\varsigma }_3+{\varsigma }_6}{1-{\varsigma }_2-{\varsigma }_3-{\varsigma }_4-{\varsigma }_6}}\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|.$$

(2)

Similarly, according to (1), we have

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)+\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_2\right)+\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}_{1,}\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}\hfill \\ & \in & ⋏\left(\mathcal{K}{\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)=⋏\left(\beth {\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_2\right).\hfill \end{array}$$

By definition of “⋏”, we have

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)+\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_2\right)+\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}_{1,}\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}\hfill \\ & \in & \left(\underset{\mathfrak{p}\in \mathcal{K}{\mathfrak{p}}_2}{\cap }⋏\left(\mathfrak{p},\beth {\mathfrak{p}}_1\right)\right)\cap \left(\underset{{\mathfrak{p}}^{\prime }\in \beth {\mathfrak{p}}_1}{\cap }⋏\left({\mathfrak{p}}^{\prime },\mathcal{K}{\mathfrak{p}}_2\right)\right),\hfill \end{array}$$

which implies

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)+\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_2\right)+\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}_{1,}\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}\hfill \\ & \in & \left(\underset{{\mathfrak{p}}^{\prime }\in \beth {\mathfrak{p}}_1}{\cap }⋏\left({\mathfrak{p}}^{\prime },\mathcal{K}{\mathfrak{p}}_2\right)\right).\hfill \end{array}$$

This yields

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)+\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_2\right)+\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}_{1,}\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}\hfill \\ & \in & ⋏\left({\mathfrak{p}}^{\prime },\mathcal{K}{\mathfrak{p}}_2\right),\hfill \end{array}$$

for all ${\mathfrak{p}}^{\prime }\in \beth {\mathfrak{p}}_1.$ Now, since ${\mathfrak{p}}_2\in \beth {\mathfrak{p}}_1,$ we have

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)+\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_2\right)+\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}_{1,}\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}\hfill \\ & \in & ⋏\left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right),\hfill \end{array}$$

that is,

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)+\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_2\right)+\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}_{1,}\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}\hfill \\ & \in & ⋏\left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)=\underset{\mathfrak{p}\in \mathcal{K}{\mathfrak{p}}_2}{\cup }⋏\left(\phi \left({\mathfrak{p}}_2,\mathfrak{p}\right)\right).\hfill \end{array}$$

So, there exists some ${\mathfrak{p}}_3\in \mathcal{K}{\mathfrak{p}}_2$ such that

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)+\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_2\right)+\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}_{1,}\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}\hfill \\ & \in & ⋏\left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)=⋏\left(\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\right),\hfill \end{array}$$

that is,

$$\begin{array}{ccc}\hfill \phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)& ⪯& {\varsigma }_1\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)+\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_2\right)+\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_2,\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_2,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}_{1,}\mathcal{K}{\mathfrak{p}}_2\right)\phi \left({\mathfrak{p}}_1,\beth {\mathfrak{p}}_1\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}.\hfill \end{array}$$

Leveraging the glb property inherent to both $\mathcal{K}$ and ℶ, we obtain

$$\begin{array}{ccc}\hfill \phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)& ⪯& {\varsigma }_1\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)+\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_3\right)+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_2\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_3\right)\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_2\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_2\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}_{1,}{\mathfrak{p}}_3\right)\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)}{1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)}},\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill \left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\right|& \le & {\varsigma }_1\left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)\right|+{\varsigma }_2\left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\right|\hfill \\ & & +{\varsigma }_2\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|+{\varsigma }_3\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_3\right)\right|\hfill \\ & & +{\varsigma }_4\left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\right|{\displaystyle \frac{\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|}{\left|1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)\right|}}\hfill \\ & & +{\varsigma }_7\left|\phi \left({\mathfrak{p}}_{1,}{\mathfrak{p}}_3\right)\right|{\displaystyle \frac{\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|}{\left|1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)\right|}}.\hfill \end{array}$$

Now, since ${\displaystyle \frac{\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|}{\left|1+\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)\right|}}<1,$ we have

$$\begin{array}{ccc}\hfill \left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\right|& \le & {\varsigma }_1\left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)\right|+{\varsigma }_2\left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\right|\hfill \\ & & +{\varsigma }_2\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|+{\varsigma }_3\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_3\right)\right|\hfill \\ & & +{\varsigma }_4\left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\right|+{\varsigma }_7\left|\phi \left({\mathfrak{p}}_{1,}{\mathfrak{p}}_3\right)\right|.\hfill \end{array}$$

According to the triangle inequality of the CVMS, we have

$$\begin{array}{ccc}\hfill \left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\right|& \le & {\varsigma }_1\left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_1\right)\right|+{\varsigma }_2\left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\right|\hfill \\ & & +{\varsigma }_2\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|+{\varsigma }_3\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|\hfill \\ & & +{\varsigma }_3\left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\right|+{\varsigma }_4\left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\right|\hfill \\ & & +{\varsigma }_7\left|\phi \left({\mathfrak{p}}_{1,}{\mathfrak{p}}_2\right)\right|+{\varsigma }_7\left|\phi \left({\mathfrak{p}}_{2,}{\mathfrak{p}}_3\right)\right|\hfill \\ & =& {\varsigma }_1\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|+{\varsigma }_2\left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\right|\hfill \\ & & +{\varsigma }_2\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|+{\varsigma }_3\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|\hfill \\ & & +{\varsigma }_3\left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\right|+{\varsigma }_4\left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\right|\hfill \\ & & +{\varsigma }_7\left|\phi \left({\mathfrak{p}}_{1,}{\mathfrak{p}}_2\right)\right|+{\varsigma }_7\left|\phi \left({\mathfrak{p}}_{2,}{\mathfrak{p}}_3\right)\right|,\hfill \end{array}$$

which implies

$$\left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\right|\le {\displaystyle \frac{{\varsigma }_1+{\varsigma }_2+{\varsigma }_3+{\varsigma }_7}{1-{\varsigma }_2-{\varsigma }_3-{\varsigma }_4-{\varsigma }_7}}\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|.$$

(3)

Let $h=max\left\{{\displaystyle \frac{{\varsigma }_1+{\varsigma }_2+{\varsigma }_3+{\varsigma }_6}{1-{\varsigma }_2-{\varsigma }_3-{\varsigma }_4-{\varsigma }_6}},{\displaystyle \frac{{\varsigma }_1+{\varsigma }_2+{\varsigma }_3+{\varsigma }_7}{1-{\varsigma }_2-{\varsigma }_3-{\varsigma }_4-{\varsigma }_7}}\right\}<1$; then, from (2) and (3), we have

$$\left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\right|\le h\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|\le h^2\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|.$$

Employing an inductive approach, a sequence $\left\{{\mathfrak{p}}_n\right\}$ can be constructed within $\Theta ,$ where ${\mathfrak{p}}_{2n+1}\in \mathcal{K}{\mathfrak{p}}_{2n}$, ${\mathfrak{p}}_{2n+2}\in \beth {\mathfrak{p}}_{2n+1}$, and

$$|\phi \left({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1}\right)|\le h^n\left|\phi \left({\mathfrak{p}}_{0,}{\mathfrak{p}}_1\right)\right|$$

(4)

for all $n\in \mathbb{N}.$ Now, for $m>n,$ we obtain

$$\begin{array}{ccc}\hfill \left|\phi \right({\mathfrak{p}}_n,{\mathfrak{p}}_m\left)\right|& \le & |\phi ({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1})|+|\phi ({\mathfrak{p}}_{n+1},{\mathfrak{p}}_{n+2})|+\cdots +|\phi ({\mathfrak{p}}_{m-1},{\mathfrak{p}}_m)|\hfill \\ & \le & \left(h^n+h^{n+1}+\cdots +h^{m-1}\right)\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|\hfill \\ & =& h^n\left(1+h+h^2\cdots +h^{m-n-1}\right)\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|\hfill \\ & =& h^n\left({\displaystyle \frac{1-h^{m-n}}{1-h}}\right)\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|.\hfill \end{array}$$

Since $0<h<1,$ in the numerator, we have $1-h^{m-n}<1.$ Consequently,

$$|\phi ({\mathfrak{p}}_n,{\mathfrak{p}}_m)|\le \left({\displaystyle \frac{h^n}{1-h}}\right)\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|,(m>n).$$

Accordingly,

$$|\phi ({\mathfrak{p}}_n,{\mathfrak{p}}_m)|\le \left({\displaystyle \frac{h^n}{1-h}}\right)\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|⟶0$$

as $n⟶\infty ,$ because $0<h<1.$ Consequently, $\left\{{\mathfrak{p}}_n\right\}$ is Cauchy within $\Theta .$ Given the completeness of $\Theta$, there exists ${\mathfrak{p}}^{*}$ in $\Theta$ such that ${\mathfrak{p}}_n⟶{\mathfrak{p}}^{*}$ as $n⟶\infty .$ To demonstrate that ${\mathfrak{p}}^{*}\in \beth {\mathfrak{p}}^{*}$ and ${\mathfrak{p}}^{*}\in \mathcal{K}{\mathfrak{p}}^{*},$ we refer to Equation (1) and observe

$$\begin{array}{ccc}\hfill & \hfill & {\varsigma }_1\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)+\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)\right)\hfill \\ \hfill & \hfill & +{\varsigma }_3\left(\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)+\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)\right)\hfill \\ \hfill & \hfill & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}\hfill \\ \hfill & \hfill & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}\hfill \\ \hfill & \in \hfill & ⋏\left(\mathcal{K}{\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right).\hfill \end{array}$$

(5)

By definition of “⋏”, we have

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)+\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)+\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}\hfill \\ & \in & \left(\underset{\mathfrak{p}\in \mathcal{K}{\mathfrak{p}}_{2n}}{\cap }⋏\left(\mathfrak{p},\beth {\mathfrak{p}}^{*}\right)\right)\cap \left(\underset{{\mathfrak{p}}^{\prime }\in \beth {\mathfrak{p}}^{*}}{\cap }⋏\left({\mathfrak{p}}^{\prime },\mathcal{K}{\mathfrak{p}}_{2n}\right)\right),\hfill \end{array}$$

that is,

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)+\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)+\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}\hfill \\ & \in & \left(\underset{\mathfrak{p}\in \mathcal{K}{\mathfrak{p}}_{2n}}{\cap }⋏\left(\mathfrak{p},\beth {\mathfrak{p}}^{*}\right)\right).\hfill \end{array}$$

This implies that

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)+\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)+\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}\hfill \\ & \in & ⋏\left(\mathfrak{p},\beth {\mathfrak{p}}^{*}\right)\hfill \end{array}$$

for all $\mathfrak{p}\in \mathcal{K}{\mathfrak{p}}_{2n}.$ Since ${\mathfrak{p}}_{2n+1}\in \mathcal{K}{\mathfrak{p}}_{2n},$ we have

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)+\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)+\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}\hfill \\ & \in & ⋏\left({\mathfrak{p}}_{2n+1},\beth {\mathfrak{p}}^{*}\right).\hfill \end{array}$$

By definition of “⋏”, we have

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)+\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)+\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}\hfill \\ & \in & ⋏\left({\mathfrak{p}}_{2n+1},\beth {\mathfrak{p}}^{*}\right)=\underset{{\mathfrak{p}}^{\prime }\in \beth {\mathfrak{p}}^{*}}{\cup }⋏\left(\phi \left({\mathfrak{p}}_{2n+1},{\mathfrak{p}}^{\prime }\right)\right).\hfill \end{array}$$

There exists some ${\mathfrak{p}}_n\in \beth {\mathfrak{p}}^{*}$ such that

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)+\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)+\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}\hfill \\ & \in & ⋏\left(\phi ({\mathfrak{p}}_{2n+1},{\mathfrak{p}}_n)\right),\hfill \end{array}$$

that is,

$$\begin{array}{ccc}\hfill \phi ({\mathfrak{p}}_{2n+1},{\mathfrak{p}}_n)& ⪯& {\varsigma }_1\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)+\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)+\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}_{2n},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},\mathcal{K}{\mathfrak{p}}_{2n}\right)\phi \left({\mathfrak{p}}^{*},\beth {\mathfrak{p}}^{*}\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}.\hfill \end{array}$$

Leveraging the glb property inherent to both $\mathcal{K}$ and ℶ, we obtain

$$\begin{array}{ccc}\hfill \phi ({\mathfrak{p}}_{2n+1},{\mathfrak{p}}_n)& ⪯& {\varsigma }_1\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}_{2n+1}\right)+\phi \left({\mathfrak{p}}^{*},{\mathfrak{p}}_n\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left({\mathfrak{p}}^{*},{\mathfrak{p}}_{2n+1}\right)+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}_n\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}_{2n+1}\right)\phi \left({\mathfrak{p}}^{*},{\mathfrak{p}}_n\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},{\mathfrak{p}}_{2n+1}\right)\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}_n\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}_{2n+1}\right)\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}_n\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},{\mathfrak{p}}_{2n+1}\right)\phi \left({\mathfrak{p}}^{*},{\mathfrak{p}}_n\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}.\hfill \end{array}$$

(6)

Since

$$\phi ({\mathfrak{p}}^{*},{\mathfrak{p}}_{n)}⪯\phi ({\mathfrak{p}}^{*},{\mathfrak{p}}_{2n+1})+\phi ({\mathfrak{p}}_{2n+1},{\mathfrak{p}}_n),$$

using (6), we obtain

$$\phi ({\mathfrak{p}}^{*},{\mathfrak{p}}_{n)}⪯{\varsigma }_1\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)+{\varsigma }_2\left(\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}_{2n+1}\right)+\phi \left({\mathfrak{p}}^{*},{\mathfrak{p}}_n\right)\right)+{\varsigma }_3\left(\phi \left({\mathfrak{p}}^{*},{\mathfrak{p}}_{2n+1}\right)+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}_n\right)\right)$$

$$+{\varsigma }_4{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}_{2n+1}\right)\phi \left({\mathfrak{p}}^{*},{\mathfrak{p}}_n\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},{\mathfrak{p}}_{2n+1}\right)\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}_n\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}$$

$$+{\varsigma }_6{\displaystyle \frac{\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}_{2n+1}\right)\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}_n\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left({\mathfrak{p}}^{*},{\mathfrak{p}}_{2n+1}\right)\phi \left({\mathfrak{p}}^{*},{\mathfrak{p}}_n\right)}{1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)}},$$

that is,

$$\begin{array}{ccc}\hfill \phi ({\mathfrak{p}}^{*},{\mathfrak{p}}_{n)}& \le & {\varsigma }_1\left|\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)\right|+{\varsigma }_2\left|\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}_{2n+1}\right)\right|\hfill \\ & & +{\varsigma }_2\left|\phi \left({\mathfrak{p}}^{*},{\mathfrak{p}}_n\right)\right|+{\varsigma }_3\left|\phi \left({\mathfrak{p}}^{*},{\mathfrak{p}}_{2n+1}\right)\right|+{\varsigma }_3\left|\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}_n\right)\right|\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\left|\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}_{2n+1}\right)\right|}{\left|1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)\right|}}\left|\phi \left({\mathfrak{p}}^{*},{\mathfrak{p}}_n\right)\right|+{\varsigma }_5\left|\phi \left({\mathfrak{p}}^{*},{\mathfrak{p}}_{2n+1}\right)\right|{\displaystyle \frac{\left|\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}_n\right)\right|}{\left|1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)\right|}}\hfill \\ & & +{\varsigma }_6\left|\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}_{2n+1}\right)\right|{\displaystyle \frac{\left|\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}_n\right)\right|}{\left|1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)\right|}}+{\varsigma }_7{\displaystyle \frac{\left|\phi \left({\mathfrak{p}}^{*},{\mathfrak{p}}_{2n+1}\right)\right|}{\left|1+\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}^{*}\right)\right|}}\left|\phi \left({\mathfrak{p}}^{*},{\mathfrak{p}}_n\right)\right|.\hfill \end{array}$$

Taking the limit as $n⟶\infty ,$ we obtain $\left(1-{\varsigma }_2-{\varsigma }_3\right)\left|\phi \right({\mathfrak{p}}^{*},{\mathfrak{p}}_{n)}|\le 0.$ Since $\left(1-{\varsigma }_2-{\varsigma }_3\right)\ne 0,$ $\left|\phi \right({\mathfrak{p}}^{*},{\mathfrak{p}}_{n)}|=0.$ Thus, by Lemma 2, we have ${\mathfrak{p}}_n⟶{\mathfrak{p}}^{*}$ as $n⟶\infty .$ Since $\beth {\mathfrak{p}}^{*}$ is closed, ${\mathfrak{p}}^{*}\in \beth {\mathfrak{p}}^{*}.$ Likewise, it can be deduced that ${\mathfrak{p}}^{*}\in \mathcal{K}{\mathfrak{p}}^{*}.$ Consequently, $\mathcal{K}$ and $\beth$ have a common FP. □

Notation 1.

For the remainder of this work, we will consider $\left(\Theta ,\phi \right)$ a complete CVMS and assume that the set-valued mappings $\mathcal{K},\beth :\Theta ⟶$$CB\left(\Theta \right)$ possess the glb property.

Corollary 1.

Let $\mathcal{K}:\Theta ⟶$$CB\left(\Theta \right)$. Assume that there exist non-negative real numbers ${\varsigma }_i$ for $i=1,2,\dots ,7$ such that

$${\varsigma }_1+2\left({\varsigma }_2+{\varsigma }_3+{\varsigma }_4+{\varsigma }_5+{\varsigma }_6+{\varsigma }_7\right)<1,$$

and

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left(\mathfrak{p},\tau \right)+{\varsigma }_2\left(\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)+\phi \left(\tau ,\mathcal{K}\tau \right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)+\phi \left(\mathfrak{p},\mathcal{K}\tau \right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\tau ,\mathcal{K}\tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\phi \left(\mathfrak{p},\mathcal{K}\tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\mathfrak{p},\mathcal{K}\tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\phi \left(\tau ,\mathcal{K}\tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\hfill \\ & \in & ⋏\left(\mathcal{K}\mathfrak{p},\mathcal{K}\tau \right)\hfill \end{array}$$

for all $\mathfrak{p},\tau \in$$\Theta .$ Then, $\mathcal{K}$ has an FP.

Proof. 

Set $\mathcal{K}=\beth$ in Theorem 4. □

Corollary 2.

Let $\mathcal{K},\beth :\Theta ⟶$$CB\left(\Theta \right)$. Assume the existence of non-negative real numbers ${\varsigma }_i$ for $i=1,2,\dots ,6$ such that

$${\varsigma }_1+2\left({\varsigma }_2+{\varsigma }_3+{\varsigma }_4+{\varsigma }_5+{\varsigma }_6\right)<1,$$

and

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left(\mathfrak{p},\tau \right)+{\varsigma }_2\left(\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)+\phi \left(\tau ,\beth \tau \right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)+\phi \left(\mathfrak{p},\beth \tau \right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\tau ,\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\phi \left(\mathfrak{p},\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\mathfrak{p},\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\hfill \\ & \in & ⋏\left(\mathcal{K}\mathfrak{p},\beth \tau \right)\hfill \end{array}$$

for all $\mathfrak{p},\tau \in$$\Theta .$ Then, $\mathcal{K}$ and ℶ have a common FP.

Proof. 

Take ${\varsigma }_7=0$ in Theorem 4. □

Corollary 3.

Let $\mathcal{K},\beth :\Theta ⟶$$CB\left(\Theta \right)$. Assume the existence of non-negative real numbers ${\varsigma }_i$ for $i=1,2,\dots ,5$ such that

$${\varsigma }_1+2\left({\varsigma }_2+{\varsigma }_3+{\varsigma }_4+{\varsigma }_5\right)<1,$$

and

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left(\mathfrak{p},\tau \right)+{\varsigma }_2\left(\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)+\phi \left(\tau ,\beth \tau \right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)+\phi \left(\mathfrak{p},\beth \tau \right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\tau ,\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\phi \left(\mathfrak{p},\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\hfill \\ & \in & ⋏\left(\mathcal{K}\mathfrak{p},\beth \tau \right)\hfill \end{array}$$

for all $\mathfrak{p},\tau \in$$\Theta .$ Then, $\mathcal{K}$ and ℶ have a common FP.

Proof. 

Take ${\varsigma }_6={\varsigma }_7=0$ in Theorem 4. □

Corollary 4.

Let $\mathcal{K},\beth :\Theta ⟶$$CB\left(\Theta \right)$. Assume the existence of non-negative real numbers ${\varsigma }_i$ for $i=1,2,\dots ,4$ such that

$${\varsigma }_1+2\left({\varsigma }_2+{\varsigma }_3+{\varsigma }_4\right)<1,$$

and

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left(\mathfrak{p},\tau \right)+{\varsigma }_2\left(\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)+\phi \left(\tau ,\beth \tau \right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)+\phi \left(\mathfrak{p},\beth \tau \right)\right)+{\varsigma }_4{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\tau ,\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\hfill \\ & \in & ⋏\left(\mathcal{K}\mathfrak{p},\beth \tau \right)\hfill \end{array}$$

for all $\mathfrak{p},\tau \in$$\Theta .$ Then, $\mathcal{K}$ and ℶ have a common FP.

Proof. 

Set ${\varsigma }_5={\varsigma }_6={\varsigma }_7=0$ in Theorem 4. □

A Hardy–Roger-type theorem for set-valued mappings can be deduced by our prime theorem in such manner.

Corollary 5.

Let $\mathcal{K},\beth :\Theta ⟶$$CB\left(\Theta \right)$. Assume the existence of non-negative real numbers ${\varsigma }_i$ for $i=1,2,\dots ,3$ such that ${\varsigma }_1+2\left({\varsigma }_2+{\varsigma }_3\right)<1$ and

$${\varsigma }_1\phi \left(\mathfrak{p},\tau \right)+{\varsigma }_2\left(\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)+\phi \left(\tau ,\beth \tau \right)\right)+{\varsigma }_3\left(\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)+\phi \left(\mathfrak{p},\beth \tau \right)\right)\in ⋏\left(\mathcal{K}\mathfrak{p},\beth \tau \right)$$

for all $\mathfrak{p},\tau \in$$\Theta .$ Then, $\mathcal{K}$ and ℶ have a common FP.

Proof. 

Put ${\varsigma }_4={\varsigma }_5={\varsigma }_6={\varsigma }_7=0$ in Theorem 4. □

We now present a Reich-type FP theorem for set-valued mappings in the context of CVMSs.

Corollary 6.

Let $\mathcal{K},\beth :\Theta ⟶$$CB\left(\Theta \right)$. Assume the existence of non-negative real numbers ${\varsigma }_1$ and ${\varsigma }_2$ such that ${\varsigma }_1+2{\varsigma }_2<1$ and

$${\varsigma }_1\phi \left(\mathfrak{p},\tau \right)+{\varsigma }_2\left(\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)+\phi \left(\tau ,\beth \tau \right)\right)\in ⋏\left(\mathcal{K}\mathfrak{p},\beth \tau \right)$$

for all $\mathfrak{p},\tau \in$$\Theta .$ Then, $\mathcal{K}$ and ℶ have a common FP.

Proof. 

Substitute ${\varsigma }_3={\varsigma }_4={\varsigma }_5={\varsigma }_6={\varsigma }_7=0$ in Theorem 4. □

Here, we state a Banach-type FP theorem for set-valued mappings in the CVMS as the corollary of our prime result.

Corollary 7.

Let $\mathcal{K},\beth :\Theta ⟶$$CB\left(\Theta \right)$. Assume the existence of non-negative real number ς$\in [0,1)$ such that

$$\varsigma \phi \left(\mathfrak{p},\tau \right)\in ⋏\left(\mathcal{K}\mathfrak{p},\beth \tau \right)$$

for all $\mathfrak{p},\tau \in$$\Theta .$ Then, $\mathcal{K}$ and ℶ have a common FP.

Proof. 

Take ${\varsigma }_1=\varsigma$ and ${\varsigma }_2={\varsigma }_3={\varsigma }_4={\varsigma }_5={\varsigma }_6={\varsigma }_7=0$ in Theorem 4. □

A Kannan-type theorem for set-valued mappings in the setting of a CVMS can be deduced in such a way.

Corollary 8.

Let $\mathcal{K},\beth :\Theta ⟶$$CB\left(\Theta \right)$. Assume the existence of non-negative real number ς$\in [0,{\displaystyle \frac{1}{2}})$ such that

$$\varsigma \left(\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)+\phi \left(\tau ,\beth \tau \right)\right)\in ⋏\left(\mathcal{K}\mathfrak{p},\beth \tau \right)$$

for all $\mathfrak{p},\tau \in$$\Theta .$ Then, $\mathcal{K}$ and ℶ have a common FP.

Proof. 

Set ${\varsigma }_2=\varsigma$ and ${\varsigma }_1={\varsigma }_3={\varsigma }_4={\varsigma }_5={\varsigma }_6={\varsigma }_7=0$ in Theorem 4. □

Now, we derive a result which is a Chaterrjea-type theorem for set-valued mappings in the background of a CVMS.

Corollary 9.

Let $\mathcal{K},\beth :\Theta ⟶$$CB\left(\Theta \right)$. Assume the existence of non-negative real number ς$\in [0,{\displaystyle \frac{1}{2}})$ such that

$$\varsigma \left(\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)+\phi \left(\mathfrak{p},\beth \tau \right)\right)\in ⋏\left(\mathcal{K}\mathfrak{p},\beth \tau \right)$$

for all $\mathfrak{p},\tau \in$$\Theta .$ Then, $\mathcal{K},\beth$ have a common FP.

Proof. 

Set ${\varsigma }_3=\varsigma$ and ${\varsigma }_1={\varsigma }_2={\varsigma }_4={\varsigma }_5={\varsigma }_6={\varsigma }_7=0$ in Theorem 4. □

Our main Theorem 4 can be specialized to yield a result that improves upon the main theorem of Ahmad et al. [5].

Corollary 10.

Let $\mathcal{K},\beth :\Theta ⟶$$CB\left(\Theta \right)$. Assume the existence of non-negative real numbers ${\varsigma }_i$ for $i=1,2,3$ such that ${\varsigma }_1+2\left({\varsigma }_2+{\varsigma }_3\right)<1$ and

$${\varsigma }_1\phi \left(\mathfrak{p},\tau \right)+{\varsigma }_2{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\tau ,\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}+{\varsigma }_3{\displaystyle \frac{\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\phi \left(\mathfrak{p},\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\in ⋏\left(\mathcal{K}\mathfrak{p},\beth \tau \right)$$

for all $\mathfrak{p},\tau \in$$\Theta .$ Then, $\mathcal{K},\beth$ have a common FP.

Proof. 

Take ${\varsigma }_2={\varsigma }_3={\varsigma }_6={\varsigma }_7=0$ in Theorem 4. □

Building upon our primary theorem, Theorem 4, we also establish a result that generalizes the key finding of Ahmad et al. [5].

Corollary 11.

Let $\mathcal{K},\beth :\Theta ⟶$$CB\left(\Theta \right)$. Assume the existence of non-negative real numbers ${\varsigma }_i$ for $i=1,2,3$ such that ${\varsigma }_1+{\varsigma }_2+{\varsigma }_3<{\displaystyle \frac{1}{2}}$ and

$${\varsigma }_1\left(\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)+\phi \left(\tau ,\beth \tau \right)\right)+{\varsigma }_2{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\tau ,\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}+{\varsigma }_3{\displaystyle \frac{\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\phi \left(\mathfrak{p},\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\in ⋏\left(\mathcal{K}\mathfrak{p},\beth \tau \right)$$

for all $\mathfrak{p},\tau \in$$\Theta .$ Then, $\mathcal{K},\beth$ have a common FP.

Proof. 

Take ${\varsigma }_1={\varsigma }_3={\varsigma }_6={\varsigma }_7=0$ in Theorem 4. □

As a corollary to our main theorem, Theorem 4, we present a result that encompasses the primary theorem of Azam et al. [6].

Corollary 12.

Let $\mathcal{K},\beth :\Theta ⟶$$CB\left(\Theta \right)$. Assume the existence of non-negative real numbers ${\varsigma }_i$ for $i=1,2,3$ such that ${\varsigma }_1+{\varsigma }_2+{\varsigma }_3<{\displaystyle \frac{1}{2}}$ and

$${\varsigma }_1\left(\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)+\phi \left(\mathfrak{p},\beth \tau \right)\right)+{\varsigma }_2{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\tau ,\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}+{\varsigma }_3{\displaystyle \frac{\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\phi \left(\mathfrak{p},\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\in ⋏\left(\mathcal{K}\mathfrak{p},\beth \tau \right)$$

for all $\mathfrak{p},\tau \in$$\Theta .$ Then, $\mathcal{K},\beth$ have a common FP.

Proof. 

Take ${\varsigma }_1={\varsigma }_2={\varsigma }_6={\varsigma }_7=0$ in Theorem 4. □

Example 1.

Let $\Theta =[0,1]$, and let us define $\phi :\Theta \times \Theta \to \mathbb{C}$ by

$$\phi (\mathfrak{p},\tau )=\left|\mathfrak{p}-\tau \right|e^{i\theta },\theta ={tan}^{-1}\left|{\displaystyle \frac{\tau }{\mathfrak{p}}}\right|.$$

Then, $(\Theta ,\phi )$ is a CVMS. Consider the mappings $\mathcal{K}$,$\beth :\Theta \to C(\Theta )$ defined by

$$\mathcal{K}\mathfrak{p}=[0,{\displaystyle \frac{1}{5}}\mathfrak{p}]\mathrm{and}\beth \mathfrak{p}=[0,{\displaystyle \frac{1}{10}}\mathfrak{p}].$$

The contractive condition in the main theorem becomes vacuous when both $\mathfrak{p}$ and τ equal zero. Henceforth, we assume without a loss of generality that $\mathfrak{p}$ and τ are nonzero and $\mathfrak{p}$ is strictly less than τ. Consequently,

$$\begin{array}{ccc}\hfill \phi \left(\mathfrak{p},\tau \right)& =& \left|\tau -\mathfrak{p}\right|e^{i\theta },\hfill \\ \hfill \phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)& =& \left|\mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{5}}\right|e^{i\theta },\hfill \\ \hfill \phi \left(\tau ,\beth \tau \right)& =& \left|\tau -{\displaystyle \frac{\tau }{10}}\right|e^{i\theta },\hfill \\ \hfill \phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)& =& \left|\tau -{\displaystyle \frac{\mathfrak{p}}{5}}\right|e^{i\theta },\hfill \\ \hfill \phi \left(\mathfrak{p},\beth \tau \right)& =& \left|\mathfrak{p}-{\displaystyle \frac{\tau }{10}}\right|e^{i\theta },\hfill \end{array}$$

and

$$⋏\left(\mathcal{K}\mathfrak{p},\beth \tau \right)=⋏\left(\left|{\displaystyle \frac{\mathfrak{p}}{5}}-{\displaystyle \frac{\tau }{10}}\right|e^{i\theta }\right).$$

Let us consider

$$\begin{array}{ccc}& & {\varsigma }_1\left|\phi \left(\mathfrak{p},\tau \right)\right|+{\varsigma }_2\left(|\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)|+|\phi \left(\tau ,\beth \tau \right)|\right)+{\varsigma }_3\left(|\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)|+\left|\phi \left(\mathfrak{p},\beth \tau \right)\right|\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{|\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\left|\right|\phi \left(\tau ,\beth \tau \right)|}{|1+\phi \left(\mathfrak{p},\tau \right)|}}+{\varsigma }_5{\displaystyle \frac{\left|\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\right|\left|\phi \left(\mathfrak{p},\beth \tau \right)\right|}{|1+\phi \left(\mathfrak{p},\tau \right)|}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{|\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\left|\right|\phi \left(\mathfrak{p},\beth \tau \right)|}{|1+\phi \left(\mathfrak{p},\tau \right)|}}+{\varsigma }_7{\displaystyle \frac{\left|\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\right||\phi \left(\tau ,\beth \tau \right)}{|1+\phi \left(\mathfrak{p},\tau \right)|}}\hfill \\ & =& {\varsigma }_1\left|\tau -\mathfrak{p}\right|e^{i\theta }+{\varsigma }_2\left(\left|\mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{5}}\right|+\left|\tau -{\displaystyle \frac{\tau }{10}}\right|\right)e^{i\theta }\hfill \\ & & +{\varsigma }_3\left(\left|\tau -{\displaystyle \frac{\mathfrak{p}}{5}}\right|+\left|\mathfrak{p}-{\displaystyle \frac{\tau }{10}}\right|\right)e^{i\theta }\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\left(\left|\mathfrak{p}-\frac{\mathfrak{p}}{5}\right|\left|\tau -\frac{\tau }{10}\right|\right)e^{2i\theta }}{\left|1+\left|\tau -\mathfrak{p}\right|e^{i\theta }\right|}}+{\varsigma }_5{\displaystyle \frac{\left(\left|\tau -\frac{\mathfrak{p}}{5}\right|\left|\mathfrak{p}-\frac{\tau }{10}\right|\right)e^{2i\theta }}{\left|1+\left|\tau -\mathfrak{p}\right|e^{i\theta }\right|}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\left(\left|\mathfrak{p}-\frac{\mathfrak{p}}{5}\right|\left|\mathfrak{p}-\frac{\tau }{10}\right|\right)e^{2i\theta }}{\left|1+\left|\tau -\mathfrak{p}\right|e^{i\theta }\right|}}+{\varsigma }_7{\displaystyle \frac{\left(\left|\tau -\frac{\mathfrak{p}}{5}\right|\left|\tau -\frac{\tau }{10}\right|\right)e^{2i\theta }}{\left|1+\left|\tau -\mathfrak{p}\right|e^{i\theta }\right|}}.\hfill \end{array}$$

It is apparent that for any selection of ${\varsigma }_i$ (where $i=2,3,\dots ,7$) and ${\varsigma }_1={\displaystyle \frac{1}{5}}$, we observe

$$\begin{array}{ccc}\hfill \left|{\displaystyle \frac{\mathfrak{p}}{5}}-{\displaystyle \frac{\tau }{10}}\right|& \le & {\displaystyle \frac{1}{5}}\left|\tau -\mathfrak{p}\right|+{\varsigma }_2\left(\left|\mathfrak{p}-{\displaystyle \frac{\mathfrak{p}}{5}}\right|+\left|\tau -{\displaystyle \frac{\tau }{10}}\right|\right)e^{i\theta }\hfill \\ & & +{\varsigma }_3\left(\left|\tau -{\displaystyle \frac{\mathfrak{p}}{5}}\right|+\left|\mathfrak{p}-{\displaystyle \frac{\tau }{10}}\right|\right)e^{i\theta }\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\left(\left|\mathfrak{p}-\frac{\mathfrak{p}}{5}\right|\left|\tau -\frac{\tau }{10}\right|\right)e^{2i\theta }}{\left|1+\left|\tau -\mathfrak{p}\right|e^{i\theta }\right|}}+{\varsigma }_5{\displaystyle \frac{\left(\left|\tau -\frac{\mathfrak{p}}{5}\right|\left|\mathfrak{p}-\frac{\tau }{10}\right|\right)e^{2i\theta }}{\left|1+\left|\tau -\mathfrak{p}\right|e^{i\theta }\right|}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\left(\left|\mathfrak{p}-\frac{\mathfrak{p}}{5}\right|\left|\mathfrak{p}-\frac{\tau }{10}\right|\right)e^{2i\theta }}{\left|1+\left|\tau -\mathfrak{p}\right|e^{i\theta }\right|}}+{\varsigma }_7{\displaystyle \frac{\left(\left|\tau -\frac{\mathfrak{p}}{5}\right|\left|\tau -\frac{\tau }{10}\right|\right)e^{2i\theta }}{\left|1+\left|\tau -\mathfrak{p}\right|e^{i\theta }\right|}}.\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left(\mathfrak{p},\tau \right)+{\varsigma }_2\left(\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)+\phi \left(\tau ,\beth \tau \right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)+\phi \left(\mathfrak{p},\beth \tau \right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\tau ,\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\phi \left(\mathfrak{p},\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\mathfrak{p},\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\phi \left(\tau ,\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\hfill \\ & \in & ⋏\left(\mathcal{K}\mathfrak{p},\beth \tau \right).\hfill \end{array}$$

Consequently, all conditions of our primary theorem, Theorem 4, are fulfilled. Thus, 0 is a common FP of mappings $\mathcal{K}$ and ℶ.

Theorem 5.

Let $\mathcal{K},\beth :\Theta ⟶$$CB\left(\Theta \right)$. Assume that there exist ${\mathfrak{p}}_0\in \Theta$, $0\prec r\in \mathbb{C}$ and non-negative real numbers ${\varsigma }_i$ for $i=1,2,\dots ,7$ such that

$${\varsigma }_1+2\left({\varsigma }_2+{\varsigma }_3+{\varsigma }_4+{\varsigma }_5+{\varsigma }_6+{\varsigma }_7\right)<1$$

and

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left(\mathfrak{p},\tau \right)+{\varsigma }_2\left(\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)+\phi \left(\tau ,\beth \tau \right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)+\phi \left(\mathfrak{p},\beth \tau \right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\tau ,\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\phi \left(\mathfrak{p},\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\mathfrak{p},\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\phi \left(\tau ,\beth \tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\hfill \\ & \in & ⋏\left(\mathcal{K}\mathfrak{p},\beth \tau \right)\hfill \end{array}$$

(7)

for all $\mathfrak{p},\tau \in$$\widetilde{B\left({\mathfrak{p}}_0,r\right)}$ and

$$\left(1-h\right)r\in ⋏\left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right),$$

(8)

where $h=max\left\{{\displaystyle \frac{{\varsigma }_1+{\varsigma }_2+{\varsigma }_3+{\varsigma }_6}{1-{\varsigma }_2-{\varsigma }_3-{\varsigma }_4-{\varsigma }_6}},{\displaystyle \frac{{\varsigma }_1+{\varsigma }_2+{\varsigma }_3+{\varsigma }_7}{1-{\varsigma }_2-{\varsigma }_3-{\varsigma }_4-{\varsigma }_7}}\right\}<1$. Then, $\mathcal{K}$ and ℶ have a common FP in $\widetilde{B\left({\mathfrak{p}}_0,r\right)}$.

Proof. 

Consider an arbitrary point ${\mathfrak{p}}_0$ in the set $\Theta$. Based on the results established in (8), it can be readily shown that

$$|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)|\le (1-h)|r|.$$

(9)

Thus, we have ${\mathfrak{p}}_1\in$$\widetilde{B\left({\mathfrak{p}}_0,r\right)}$. The proof of Theorem 4 demonstrates that

$$\left|\phi ({\mathfrak{p}}_1,{\mathfrak{p}}_2)\right|\le {\displaystyle \frac{{\varsigma }_1+{\varsigma }_2+{\varsigma }_3+{\varsigma }_6}{1-{\varsigma }_2-{\varsigma }_3-{\varsigma }_4-{\varsigma }_6}}\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|$$

(10)

and

$$\left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\right|\le {\displaystyle \frac{{\varsigma }_1+{\varsigma }_2+{\varsigma }_3+{\varsigma }_7}{1-{\varsigma }_2-{\varsigma }_3-{\varsigma }_4-{\varsigma }_7}}\left|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)\right|.$$

(11)

Let $h=max\left\{{\displaystyle \frac{{\varsigma }_1+{\varsigma }_2+{\varsigma }_3+{\varsigma }_6}{1-{\varsigma }_2-{\varsigma }_3-{\varsigma }_4-{\varsigma }_6}},{\displaystyle \frac{{\varsigma }_1+{\varsigma }_2+{\varsigma }_3+{\varsigma }_7}{1-{\varsigma }_2-{\varsigma }_3-{\varsigma }_4-{\varsigma }_7}}\right\}<1$; then, from (10) and (11), we have

$$\left|\phi ({\mathfrak{p}}_1,{\mathfrak{p}}_2)\right|\le h\left|\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)\right|$$

(12)

and

$$\left|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)\right|\le h\left|\phi ({\mathfrak{p}}_1,{\mathfrak{p}}_2)\right|.$$

(13)

Now, from (9) and (12), we obtain

$$|\phi \left({\mathfrak{p}}_1,{\mathfrak{p}}_2\right)|\le h(1-h)|r|.$$

(14)

Note that

$$\begin{array}{ccc}\hfill \left|\phi \right({\mathfrak{p}}_0,{\mathfrak{p}}_2\left)\right|& \le & |\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)|+|\phi ({\mathfrak{p}}_1,{\mathfrak{p}}_2)|\hfill \\ & \le & (1-h)\left|r\right|+h(1-h)\left|r\right|\hfill \\ & =& (1-h)(1+h)\left|r\right|\hfill \\ & \le & (1-h^2)\left|r\right|\le \left|r\right|.\hfill \end{array}$$

Thus, we have ${\mathfrak{p}}_2\in \widetilde{B\left({\mathfrak{p}}_0,r\right)}.$ Now, by (8), (13), and (14), we have that

$$\begin{array}{ccc}\hfill \left|\phi \right({\mathfrak{p}}_0,{\mathfrak{p}}_3\left)\right|& \le & |\phi \left({\mathfrak{p}}_0,{\mathfrak{p}}_1\right)|+|\phi ({\mathfrak{p}}_1,{\mathfrak{p}}_2)|+|\phi \left({\mathfrak{p}}_2,{\mathfrak{p}}_3\right)|\hfill \\ & \le & (1-h)\left|r\right|+h(1-h)\left|r\right|+h^2(1-h)\left|r\right|\hfill \\ & =& (1-h^3)\left|r\right|\le \left|r\right|.\hfill \end{array}$$

So, ${\mathfrak{p}}_3\in \widetilde{B\left({\mathfrak{p}}_0,r\right)}.$ By iteratively applying this process, we can generate a sequence $\left\{{\mathfrak{p}}_n\right\}$ within the closed ball $\widetilde{B\left({\mathfrak{p}}_0,r\right)}$ satisfying ${\mathfrak{p}}_{2n+1}\in \mathcal{K}{\mathfrak{p}}_{2n}$, ${\mathfrak{p}}_{2n+2}\in \beth {\mathfrak{p}}_{2n+1}$,

$$|\phi \left({\mathfrak{p}}_{2n},{\mathfrak{p}}_{2n+1}\right)|\le h^{2n}\left|\phi \left({\mathfrak{p}}_{0,}{\mathfrak{p}}_1\right)\right|,$$

(15)

and

$$|\phi \left({\mathfrak{p}}_{2n+1},{\mathfrak{p}}_{2n+2}\right)|\le h^{2n+1}\left|\phi \left({\mathfrak{p}}_{0,}{\mathfrak{p}}_1\right)\right|$$

(16)

for $n\in \mathbb{N}$ and $h<1$. Now, by (15) and (16), we have

$$|\phi \left({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1}\right)|\le h^n\left|\phi \left({\mathfrak{p}}_{0,}{\mathfrak{p}}_1\right)\right|$$

for $n\in \mathbb{N}.$ Employing a technique analogous to the proof of Theorem 4, we establish the existence of a common FP for mappings $\mathcal{K}$ and ℶ in $\widetilde{B\left({\mathfrak{p}}_0,r\right)}$. □

Corollary 13.

Let $\mathcal{K}:\Theta ⟶$$CB\left(\Theta \right)$. Assume that there exist ${\mathfrak{p}}_0\in \Theta$, $0\prec r\in \mathbb{C}$ and non-negative real numbers ${\varsigma }_i$ for $i=1,2,\dots ,7$ such that

$${\varsigma }_1+2\left({\varsigma }_2+{\varsigma }_3+{\varsigma }_4+{\varsigma }_5+{\varsigma }_6+{\varsigma }_7\right)<1$$

and

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left(\mathfrak{p},\tau \right)+{\varsigma }_2\left(\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)+\phi \left(\tau ,\mathcal{K}\tau \right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)+\phi \left(\mathfrak{p},\mathcal{K}\tau \right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\tau ,\mathcal{K}\tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\phi \left(\mathfrak{p},\mathcal{K}\tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left(\mathfrak{p},\mathcal{K}\mathfrak{p}\right)\phi \left(\mathfrak{p},\mathcal{K}\tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left(\tau ,\mathcal{K}\mathfrak{p}\right)\phi \left(\tau ,\mathcal{K}\tau \right)}{1+\phi \left(\mathfrak{p},\tau \right)}}\hfill \\ & \in & ⋏\left(\mathcal{K}\mathfrak{p},\mathcal{K}\tau \right),\hfill \end{array}$$

for all $\mathfrak{p},\tau \in$$\widetilde{B\left({\mathfrak{p}}_0,r\right)}$ and

$$\left(1-h\right)r\in ⋏\left({\mathfrak{p}}_0,\mathcal{K}{\mathfrak{p}}_0\right),$$

where $h=max\left\{{\displaystyle \frac{{\varsigma }_1+{\varsigma }_2+{\varsigma }_3+{\varsigma }_6}{1-{\varsigma }_2-{\varsigma }_3-{\varsigma }_4-{\varsigma }_6}},{\displaystyle \frac{{\varsigma }_1+{\varsigma }_2+{\varsigma }_3+{\varsigma }_7}{1-{\varsigma }_2-{\varsigma }_3-{\varsigma }_4-{\varsigma }_7}}\right\}<1$. Then, $\mathcal{K}$ has an FP in $\widetilde{B\left({\mathfrak{p}}_0,r\right)}$.

4. Application

Homotopy theory, a cornerstone of algebraic topology, delves into the study of topological spaces up to homotopy equivalence. In recent years, this theory has revealed profound connections to various mathematical fields. Consequently, it has given rise to numerous FP theorems for homotopic mappings [15,16]. In this section, we leverage our findings to establish FP theorems for homotopic mappings.

Theorem 6.

Consider a complete CVMS $(\Theta ,\phi )$ and an open subset U of Θ and let $\mathcal{S}:[0,1]\times \overline{U}\to CB\left(\Theta \right)$ be set-valued mapping such that both $\mathcal{K}$ and ℶ possess the glb property. Assume that $\exists \widehat{\sigma }\in \Theta$ and $0\prec r\in \mathbb{C}$ such that

(Hom1): $\sigma \notin \left[\mathcal{S}\right(\iota ,\sigma \left)\right]$ for all $\sigma \in \partial U$ and $\iota \in [0,1]$;

(Hom2): $\mathcal{S}(\iota ,·):\overline{U}\to CB(\Theta )$ fulfilling

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left(\sigma ,\acute{\sigma }\right)+{\varsigma }_2\left(\phi \left(\sigma ,\mathcal{S}(\iota ,\sigma )\right)+\phi \left(\acute{\sigma },\mathcal{S}(\acute{\iota },\acute{\sigma })\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left(\acute{\sigma },\mathcal{S}(\iota ,\sigma )\right)+\phi \left(\sigma ,\mathcal{S}(\acute{\iota },\acute{\sigma })\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left(\sigma ,\mathcal{S}(\iota ,\sigma )\right)\phi \left(\acute{\sigma },\mathcal{S}(\acute{\iota },\acute{\sigma })\right)}{1+\phi \left(\sigma ,\acute{\sigma }\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left(\acute{\sigma },\mathcal{S}(\iota ,\sigma )\right)\phi \left(\sigma ,\mathcal{S}(\acute{\iota },\acute{\sigma })\right)}{1+\phi \left(\sigma ,\acute{\sigma }\right)}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left(\sigma ,\mathcal{S}(\iota ,\sigma )\right)\phi \left(\sigma ,\mathcal{S}(\acute{\iota },\acute{\sigma })\right)}{1+\phi \left(\sigma ,\acute{\sigma }\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left(\acute{\sigma },\mathcal{S}(\iota ,\sigma )\right)\phi \left(\acute{\sigma },\mathcal{S}(\acute{\iota },\acute{\sigma })\right)}{1+\phi \left(\sigma ,\acute{\sigma }\right)}}\hfill \\ & \in & ⋏\left(\mathcal{S}(\iota ,\sigma ),\mathcal{S}(\acute{\iota },\acute{\sigma })\right),\hfill \end{array}$$

and

$$(1-h)r\in ⋏(\widehat{\sigma },\mathcal{S}(\widehat{\iota },\widehat{\sigma })),$$

where $h=max\left\{{\displaystyle \frac{{\varsigma }_1+{\varsigma }_2+{\varsigma }_3+{\varsigma }_6}{1-{\varsigma }_2-{\varsigma }_3-{\varsigma }_4-{\varsigma }_6}},{\displaystyle \frac{{\varsigma }_1+{\varsigma }_2+{\varsigma }_3+{\varsigma }_7}{1-{\varsigma }_2-{\varsigma }_3-{\varsigma }_4-{\varsigma }_7}}\right\}<1,$

(Hom3): there exists a continuous increasing function $\varrho :(0,1]\to \Psi \cup \left\{0\right\}$ such that

$$\varrho \left(s\right)-\varrho \left(\iota \right)\in ⋏(\mathcal{S}(s,\sigma ),\mathcal{S}(\iota ,\acute{\sigma })),\varrho \left(s\right)\in \varrho \left(\iota \right)$$

for all $s,\iota \in [0,1]$ and $\sigma \in \overline{U}$, where $\Psi =\{\omega \in \mathbb{C}:0\prec \omega \}.$ Then, $\mathcal{S}(0,·)$ attains an FP if $\mathcal{S}(1,·)$ attains an FP.

Proof. 

Assume that $\mathcal{S}(0,·)$ attains an FP $\omega ,$ so $\omega \in \mathcal{S}(0,\omega ).$ By the condition (Hom1), $\omega \in U.$ We define the following set:

$$\Omega :=\left\{\right(\iota ,\sigma )\in [0,1]\times U:\sigma \in \mathcal{S}(\iota ,\sigma \left)\right\}.$$

Clearly, $\Omega \ne \varnothing .$ A partial order ≾ on $\Omega$ is defined as follows:

$$(\iota ,\sigma )\precsim (s,\acute{\sigma })⟺\iota \le s,\phi (\sigma ,\acute{\sigma })⪯{\displaystyle \frac{2}{1-h}}(\varrho \left(s\right)-\varrho \left(\iota \right)).$$

Consider a totally ordered subset $\mathcal{M}$ of $\Omega$ and $\widehat{\iota }=sup\{\iota :(\iota ,\sigma )\in \mathcal{M}\}.$ Consider $\left\{({\iota }_n,{\sigma }_n)\right\}$ in $\mathcal{M}$, provided that $({\iota }_n,{\sigma }_n)\precsim ({\iota }_{n+1},{\sigma }_{n+1})$ and ${\iota }_n\to \widehat{\iota }$ as $n\to \infty .$ Then, for any $n\ge 1$ with $m>n,$ we have

$$\phi ({\sigma }_m,{\sigma }_n)⪯{\displaystyle \frac{2}{1-h}}(\varrho \left({\iota }_m\right)-\varrho \left({\iota }_n\right))\to 0$$

as $n,m\to \infty$, which yields that $\left\{{\sigma }_n\right\}$ is Cauchy. Since $(\Theta ,\phi )$ is complete, we can find $\widehat{\sigma }\in \Theta$ such that ${\sigma }_n\to \widehat{\sigma }.$ From (Hom1), we choose $n_0\in \mathbb{N}$ in such a way that

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\sigma }_n,\widehat{\sigma }\right)+{\varsigma }_2\left(\phi \left({\sigma }_n,\mathcal{S}({\iota }_n,{\sigma }_n)\right)+\phi \left(\widehat{\sigma },\mathcal{S}(\widehat{\iota },\widehat{\sigma })\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left(\widehat{\sigma },\mathcal{S}({\iota }_n,{\sigma }_n)\right)+\phi \left({\sigma }_n,\mathcal{S}(\widehat{\iota },\widehat{\sigma })\right)\right)+{\varsigma }_4{\displaystyle \frac{\phi \left({\sigma }_n,\mathcal{S}({\iota }_n,{\sigma }_n)\right)\phi \left(\widehat{\sigma },\mathcal{S}(\widehat{\iota },\widehat{\sigma })\right)}{1+\phi \left({\sigma }_n,\widehat{\sigma }\right)}}\hfill \\ & & +{\varsigma }_5{\displaystyle \frac{\phi \left(\widehat{\sigma },\mathcal{S}({\iota }_n,{\sigma }_n)\right)\phi \left({\sigma }_n,\mathcal{S}(\widehat{\iota },\widehat{\sigma })\right)}{1+\phi ({\sigma }_n,\widehat{\sigma })}}+{\varsigma }_6{\displaystyle \frac{\phi \left({\sigma }_n,\mathcal{S}({\iota }_n,{\sigma }_n)\right)\phi \left({\sigma }_n,\mathcal{S}(\widehat{\iota },\widehat{\sigma })\right)}{1+\phi \left({\sigma }_n,\widehat{\sigma }\right)}}\hfill \\ & & +{\varsigma }_7{\displaystyle \frac{\phi \left(\widehat{\sigma },\mathcal{S}({\iota }_n,{\sigma }_n)\right)\phi \left(\widehat{\sigma },\mathcal{S}(\widehat{\iota },\widehat{\sigma })\right)}{1+\phi \left({\sigma }_n,\widehat{\sigma }\right)}}\hfill \\ & \in & ⋏(\mathcal{S}({\iota }_n,{\sigma }_n),\mathcal{S}(\widehat{\iota },\widehat{\sigma }))\hfill \end{array}$$

for all $n\ge n_0$, which implies

$$\begin{array}{ccc}& & {\varsigma }_1\phi \left({\sigma }_n,\widehat{\sigma }\right)+{\varsigma }_2\left(\phi \left({\sigma }_n,\mathcal{S}({\iota }_n,{\sigma }_n)\right)+\phi \left(\widehat{\sigma },\mathcal{S}(\widehat{\iota },\widehat{\sigma })\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left(\widehat{\sigma },\mathcal{S}({\iota }_n,{\sigma }_n)\right)+\phi \left({\sigma }_n,\mathcal{S}(\widehat{\iota },\widehat{\sigma })\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\sigma }_n,\mathcal{S}({\iota }_n,{\sigma }_n)\right)\phi \left(\widehat{\sigma },\mathcal{S}(\widehat{\iota },\widehat{\sigma })\right)}{1+\phi \left({\sigma }_n,\widehat{\sigma }\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left(\widehat{\sigma },\mathcal{S}({\iota }_n,{\sigma }_n)\right)\phi \left({\sigma }_n,\mathcal{S}(\widehat{\iota },\widehat{\sigma })\right)}{1+\phi ({\sigma }_n,\widehat{\sigma })}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\sigma }_n,\mathcal{S}({\iota }_n,{\sigma }_n)\right)\phi \left({\sigma }_n,\mathcal{S}(\widehat{\iota },\widehat{\sigma })\right)}{1+\phi \left({\sigma }_n,\widehat{\sigma }\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left(\widehat{\sigma },\mathcal{S}({\iota }_n,{\sigma }_n)\right)\phi \left(\widehat{\sigma },\mathcal{S}(\widehat{\iota },\widehat{\sigma })\right)}{1+\phi \left({\sigma }_n,\widehat{\sigma }\right)}}\hfill \\ & \in & ⋏({\sigma }_n,\mathcal{S}(\widehat{\iota },\widehat{\sigma })),\hfill \end{array}$$

since ${\sigma }_n\in \mathcal{S}({\iota }_n,{\sigma }_n).$ Consequently, there exists an element ${\sigma }_k\in \mathcal{S}(\widehat{\iota },\widehat{\sigma })$ such that

$$\begin{array}{ccc}\hfill \phi ({\sigma }_n,{\sigma }_k)& ⪯& {\varsigma }_1\phi \left({\sigma }_n,\widehat{\sigma }\right)+{\varsigma }_2\left(\phi \left({\sigma }_n,\mathcal{S}({\iota }_n,{\sigma }_n)\right)+\phi \left(\widehat{\sigma },\mathcal{S}(\widehat{\iota },\widehat{\sigma })\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left(\widehat{\sigma },\mathcal{S}({\iota }_n,{\sigma }_n)\right)+\phi \left({\sigma }_n,\mathcal{S}(\widehat{\iota },\widehat{\sigma })\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\sigma }_n,\mathcal{S}({\iota }_n,{\sigma }_n)\right)\phi \left(\widehat{\sigma },\mathcal{S}(\widehat{\iota },\widehat{\sigma })\right)}{1+\phi \left({\sigma }_n,\widehat{\sigma }\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left(\widehat{\sigma },\mathcal{S}({\iota }_n,{\sigma }_n)\right)\phi \left({\sigma }_n,\mathcal{S}(\widehat{\iota },\widehat{\sigma })\right)}{1+\phi ({\sigma }_n,\widehat{\sigma })}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\sigma }_n,\mathcal{S}({\iota }_n,{\sigma }_n)\right)\phi \left({\sigma }_n,\mathcal{S}(\widehat{\iota },\widehat{\sigma })\right)}{1+\phi \left({\sigma }_n,\widehat{\sigma }\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left(\widehat{\sigma },\mathcal{S}({\iota }_n,{\sigma }_n)\right)\phi \left(\widehat{\sigma },\mathcal{S}(\widehat{\iota },\widehat{\sigma })\right)}{1+\phi \left({\sigma }_n,\widehat{\sigma }\right)}}.\hfill \end{array}$$

Utilizing the glb characteristic of $\mathcal{S}$, it follows that

$$\begin{array}{ccc}\hfill \phi ({\sigma }_n,{\sigma }_k)& ⪯& {\varsigma }_1\phi \left({\sigma }_n,\widehat{\sigma }\right)+{\varsigma }_2\left(\phi \left({\sigma }_n,{\sigma }_n\right)+\phi \left(\widehat{\sigma },{\sigma }_k\right)\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left(\widehat{\sigma },{\sigma }_n\right)+\phi \left({\sigma }_n,{\sigma }_k\right)\right)\hfill \\ & & +{\varsigma }_4{\displaystyle \frac{\phi \left({\sigma }_n,{\sigma }_n\right)\phi \left(\widehat{\sigma },{\sigma }_k\right)}{1+\phi \left({\sigma }_n,\widehat{\sigma }\right)}}+{\varsigma }_5{\displaystyle \frac{\phi \left(\widehat{\sigma },{\sigma }_n\right)\phi \left({\sigma }_n,{\sigma }_k\right)}{1+\phi ({\sigma }_n,\widehat{\sigma })}}\hfill \\ & & +{\varsigma }_6{\displaystyle \frac{\phi \left({\sigma }_n,{\sigma }_n\right)\phi \left({\sigma }_n,{\sigma }_k\right)}{1+\phi \left({\sigma }_n,\widehat{\sigma }\right)}}+{\varsigma }_7{\displaystyle \frac{\phi \left(\widehat{\sigma },{\sigma }_n\right)\phi \left(\widehat{\sigma },{\sigma }_k\right)}{1+\phi \left({\sigma }_n,\widehat{\sigma }\right)}},\hfill \end{array}$$

that is,

$$\begin{array}{ccc}\hfill \phi ({\sigma }_n,{\sigma }_k)& ⪯& {\varsigma }_1\phi \left({\sigma }_n,\widehat{\sigma }\right)+{\varsigma }_2\phi \left(\widehat{\sigma },{\sigma }_k\right)\hfill \\ & & +{\varsigma }_3\left(\phi \left(\widehat{\sigma },{\sigma }_n\right)+\phi \left({\sigma }_n,{\sigma }_k\right)\right)\hfill \\ & & +{\varsigma }_5{\displaystyle \frac{\phi \left(\widehat{\sigma },{\sigma }_n\right)\phi \left({\sigma }_n,{\sigma }_k\right)}{1+\phi ({\sigma }_n,\widehat{\sigma })}}+{\varsigma }_7{\displaystyle \frac{\phi \left(\widehat{\sigma },{\sigma }_n\right)\phi \left(\widehat{\sigma },{\sigma }_k\right)}{1+\phi \left({\sigma }_n,\widehat{\sigma }\right)}},\hfill \end{array}$$

which yields that

$$\begin{array}{ccc}\hfill \left|\phi ({\sigma }_n,{\sigma }_k)\right|& \le & {\varsigma }_1\left|\phi \left({\sigma }_n,\widehat{\sigma }\right)\right|+{\varsigma }_2\left|\phi \left(\widehat{\sigma },{\sigma }_k\right)\right|\hfill \\ & & +{\varsigma }_3\left|\phi \left(\widehat{\sigma },{\sigma }_n\right)\right|+{\varsigma }_3\left|\phi \left({\sigma }_n,{\sigma }_k\right)\right|\hfill \\ & & +{\varsigma }_5{\displaystyle \frac{\left|\phi \left(\widehat{\sigma },{\sigma }_n\right)\right|}{\left|1+\phi ({\sigma }_n,\widehat{\sigma })\right|}}\left|\phi \left({\sigma }_n,{\sigma }_k\right)\right|\hfill \\ & & +{\varsigma }_7{\displaystyle \frac{\left|\phi \left(\widehat{\sigma },{\sigma }_n\right)\right|}{\left|1+\phi \left({\sigma }_n,\widehat{\sigma }\right)\right|}}\left|\phi \left(\widehat{\sigma },{\sigma }_k\right)\right|.\hfill \end{array}$$

Since $|1+\phi ({\sigma }_n,\widehat{\sigma })|>|\phi ({\sigma }_n,\widehat{\sigma })|$, we have

$$\begin{array}{ccc}\hfill \left|\phi ({\sigma }_n,{\sigma }_k)\right|& \le & {\varsigma }_1\left|\phi \left({\sigma }_n,\widehat{\sigma }\right)\right|+{\varsigma }_2\left|\phi \left(\widehat{\sigma },{\sigma }_k\right)\right|\hfill \\ & & +{\varsigma }_3\left|\phi \left(\widehat{\sigma },{\sigma }_n\right)\right|+{\varsigma }_3\left|\phi \left({\sigma }_n,{\sigma }_k\right)\right|\hfill \\ & & +{\varsigma }_5\left|\phi \left({\sigma }_n,{\sigma }_k\right)\right|+{\varsigma }_7\left|\phi \left(\widehat{\sigma },{\sigma }_k\right)\right|,\hfill \end{array}$$

which implies

$$|\phi ({\sigma }_n,{\sigma }_k)|\le {\displaystyle \frac{{\varsigma }_1+{\varsigma }_3}{1-{\varsigma }_3-{\varsigma }_5}}\left|\phi ({\sigma }_n,\widehat{\sigma })\right|+{\displaystyle \frac{{\varsigma }_2+{\varsigma }_7}{1-{\varsigma }_3-{\varsigma }_5}}\left|\phi \left(\widehat{\sigma },{\sigma }_k\right)\right|.$$

Note that

$$\begin{array}{ccc}\hfill \left|\phi \right(\widehat{\sigma },{\sigma }_k\left)\right|& \le & |\phi (\widehat{\sigma },{\sigma }_n)|+|\phi ({\sigma }_n,{\sigma }_k)|\hfill \\ & \le & |\phi (\widehat{\sigma },{\sigma }_n)|+{\displaystyle \frac{{\varsigma }_1+{\varsigma }_3}{1-{\varsigma }_3-{\varsigma }_5}}\left|\phi ({\sigma }_n,\widehat{\sigma })\right|\hfill \\ & & +{\displaystyle \frac{{\varsigma }_2+{\varsigma }_7}{1-{\varsigma }_3-{\varsigma }_5}}\left|\phi \left(\widehat{\sigma },{\sigma }_k\right)\right|\hfill \\ & \to & 0,\hfill \end{array}$$

for all $n\ge n_0$. Letting $k\to \infty ,$ we obtain

$$|\phi (\widehat{\sigma },{\sigma }_k)|\le {\displaystyle \frac{{\varsigma }_2+{\varsigma }_7}{1-{\varsigma }_3-{\varsigma }_5}}\left|\phi \left(\widehat{\sigma },{\sigma }_k\right)\right|,$$

which is possible only if ${\sigma }_k$ converges to an element $\widehat{\sigma }$ within $\mathcal{S}(\widehat{\iota },\widehat{\sigma }),$ consequently implying that $\widehat{\sigma }\in U,$ which yields $(\widehat{\iota },\widehat{\sigma })\in \Omega .$ Therefore, $(\iota ,\sigma )\precsim (\widehat{\iota },\widehat{\sigma }),$∀ $(\iota ,\sigma )\in \mathcal{M},$ signifying that $(\widehat{\iota },\widehat{\sigma })$ is an upper bound of $\mathcal{M}$. Consequently, by Zorn’s Lemma, $\Omega$ possesses a maximal element $(\widehat{\iota },\widehat{\sigma }).$

Now, we propose $\widehat{\iota }=1$. Suppose that $\widehat{\iota }\le 1$. We choose $0\prec r\in \mathbb{C}$ and $\widehat{\iota }\le \iota$, where $\overline{B}(\widehat{\sigma },r)\subset U$ and $r={\displaystyle \frac{2}{1-h}}(\varrho \left(\iota \right)-\varrho \left(\widehat{\iota }\right))$. Using (Hom3), we have

$$\varrho \left(\iota \right)-\varrho \left(\widehat{\iota }\right)\in ⋏(\mathcal{S}(\iota ,\sigma ),\mathcal{S}(\widehat{\iota },\widehat{\sigma })),\varrho \left(\iota \right)-\varrho \left(\widehat{\iota }\right)\in ⋏\left(\widehat{\sigma },\mathcal{S}(\iota ,\sigma )\right)$$

for all $\widehat{\sigma }\in \mathcal{S}(\widehat{\iota },\widehat{\sigma })$. So, there exists $\sigma \in \mathcal{S}(\iota ,\sigma )$ such that $\varrho \left(\iota \right)-\varrho \left(\widehat{\iota }\right)\in ⋏(\phi \left(\widehat{\sigma },\sigma )\right)$ and so

$$\phi (\sigma ,\widehat{\sigma })⪯\varrho \left(\iota \right)-\varrho \left(\widehat{\iota }\right)⪯{\displaystyle \frac{(1-h)r}{2}}\prec (1-h)r,$$

which implies that

$$|\phi (\sigma ,\widehat{\sigma })|\le (1-h)\left|r\right|.$$

Furthermore, by applying condition (Hom2), it follows that $\mathcal{S}(\iota ,·):\overline{B}(\widehat{\sigma },r)\to CB(\Theta )$ fulfills all hypotheses of Corollary 13 for each $\iota \in [0,1]$. Consequently, for every $\iota$ in $[0,1]$, there exists an element $\sigma$ within $\overline{B}(\widehat{\sigma },r)$ satisfying $\sigma$ belonging to $\mathcal{S}(\iota ,\sigma ).$ This implies that the pair $(\sigma ,\iota )$ is an element of $\Omega$. Given that

$$\phi (\sigma ,\widehat{\sigma })\prec r={\displaystyle \frac{2}{1-h}}(\varrho \left(\iota \right)-\varrho \left(\widehat{\iota }\right)),$$

this yields $(\widehat{\iota },\widehat{\sigma })\precsim (\iota ,\sigma )$, which leads to a contradiction implying that $\widehat{\iota }$ must equal $1.$ Consequently, $\mathcal{S}(·,1)$ attains an FP.

On the other hand, if $\mathcal{S}(1,·)$ attains an FP, a similar argument establishes the existence of an FP for $\mathcal{S}(0,·)$. □

5. Conclusions

In the present research article, we strengthened the idea of CVMSs pioneered by Azam et al. [1] and established the common FP theorems for set-valued mappings. Our findings extended upon prominent outcomes in the literature including the leading findings of Azam et al. [1], Rouzkard et al. [2], Klin-eam et al. [3], Bhatt et al. [4], Ahmad et al. [5], and Azam et al. [6]. To underscore the groundbreaking character of our principal theorem, a concrete example has been provided. Furthermore, the practical utility of our results is demonstrated through the development of a homotopy result.

Potential avenues for future research include extending FP and common FP theorems to fuzzy and L-fuzzy mappings in the background of these generalized contractions. Likewise, the exploration of differential and integral inclusions in CVMSs is warranted. The results presented herein may inspire further investigations and refinements by other researchers to address a wider range of applications.