New Studies for Dynamic Programming and Fractional Differential Equations in Partial Modular b-Metric Spaces

Abstract

This study explores innovative insights into the realms of dynamic programming and fractional differential equations, situated explicitly within the framework of partial modular b-metric spaces enriched with a binary relation $\mathcal{R}$, proposing a novel definition for a generalized ${\gimel }_C$-type Suzuki $\mathcal{R}$-contraction specific to these spaces. By doing so, we pave the way for a range of relation-theoretical common fixed-point theorems, highlighting the versatility of our approach. To illustrate the practical relevance of our findings, we present a compelling example. Ultimately, this work aims to enrich the existing academic discourse and stimulate further research and practical applications within the field.

1. Introduction and Preliminaries

Metric fixed-point theory offers a plethora of opportunities for researchers. Among the most notable contributions to this field is the Banach Contraction Principle [1]. This principle has established itself as a fundamental cornerstone within the domain, catalyzing numerous advancements by inspiring an array of generalizations and adaptations related to the concept of contraction mapping.

The concept of metric spaces is crucial in mathematical analysis and serves as a key foundation for various applications. It has seen considerable theoretical development, with researchers working to refine or extend its definition, including the creation of b-metric spaces, a significant advancement in this field. The b-metric structure originated from Bakhtin’s influential study [2], which introduced the b-metric function and significantly contributed to metric space theory. This metric has since been further developed by researchers like Czerwik [3,4]. The main difference between a standard metric and a b-metric is that the latter adapts the triangle inequality into a broader form:

$$b\left(♭,\ell \right)\le \hslash \left[b\left(♭,\mathfrak{z}\right)+b\left(\mathfrak{z},\ell \right)\right].$$

Within this framework, a function $b:\mathcal{U}\times \mathcal{U}\to \left[0,\infty \right)$ is defined as a b-metric with constant ℏ, setting a b-metric space on set $\mathcal{U}$. This essentially generalizes standard metric spaces, especially when $\hslash =1$, but differs by potentially lacking continuity, unlike classical metrics.

Matthews [5] has made significant advancements in denotational semantics by introducing “partial metric spaces”, which deviate from traditional metric spaces by allowing nonzero self-distances. While all metric spaces are a subset of partial metric spaces with self-distances set to zero, the introduction of nonzero self-distances enables the exploration of more complex mathematical relationships. This has made partial metric spaces a crucial development in modern mathematics, significantly expanding the theory and application of metric spaces.

In their notable study of 2014, Mustafa et al. [6] put forth the concept of the partial b-metric, blending elements of partial metrics and b-metrics within metric spaces. Moreover, they developed a vital extension of the Banach contraction principle for these spaces, significantly contributing to the field’s progress.

Definition 1 

([6]). A partial b-metric on a non-empty set $\mathcal{U}$ is a function $\tilde{p}:\mathcal{U}\times \mathcal{U}\to \left[0,\infty \right)$ for all $♭,\ell ,\mathfrak{z}\in \mathcal{U}$, which fulfills the subsequent circumstances:

$\left({\tilde{p}}_1\right)$

$\tilde{p}\left(♭,♭\right)=\tilde{p}\left(\ell ,\ell \right)=\tilde{p}\left(♭,\ell \right)\iff ♭=\ell$;

$\left({\tilde{p}}_2\right)$

$\tilde{p}\left(♭,♭\right)\le \tilde{p}\left(♭,\ell \right)$;

$\left({\tilde{p}}_3\right)$

$\tilde{p}\left(♭,\ell \right)=\tilde{p}\left(\ell ,♭\right)$;

$\left({\tilde{p}}_4\right)$

$\tilde{p}\left(♭,\ell \right)\le \hslash \left[\tilde{p}\left(♭,\mathfrak{z}\right)+\tilde{p}\left(\mathfrak{z},\ell \right)-\tilde{p}\left(\mathfrak{z},\mathfrak{z}\right)\right]+\left({\displaystyle \frac{1-\hslash }{2}}\right)\left(\tilde{p}\left(♭,♭\right)-\tilde{p}\left(\ell ,\ell \right)\right)$.

A partial b-metric space $\left(\mathcal{U},\tilde{p}\right)$ is a pair, such that $\mathcal{U}$ is a non-empty set and $\tilde{p}$ is a partial b-metric on $\mathcal{U}$. The number $\hslash \ge 1$ is also called the coefficient of $\left(\mathcal{U},\tilde{p}\right)$.

Shukla [7] modified the triangle property of partial b-metric to obtain that each partial b-metric generated a b-metric and established a convergence criterion and some working rules in partial b-metric spaces. Definition 1 has been modified in [7] by considering the following condition instead of $\left({\tilde{p}}_4\right)$: $\left({\tilde{p}}_4^{\prime }\right)$ for all $♭,\ell ,\mathfrak{z}\in \mathcal{U}$

$$\tilde{p}\left(♭,\ell \right)\le \hslash \left[\tilde{p}\left(♭,\mathfrak{z}\right)+\tilde{p}\left(\mathfrak{z},\ell \right)\right]-\tilde{p}\left(\mathfrak{z},\mathfrak{z}\right).$$

Since $\hslash \ge 1$, from $\left({\tilde{p}}_4\right)$ we have

$$\tilde{p}\left(♭,\ell \right)\le \hslash \left[\tilde{p}\left(♭,\mathfrak{z}\right)+\tilde{p}\left(\mathfrak{z},\ell \right)-\tilde{p}\left(\mathfrak{z},\mathfrak{z}\right)\right]\le \hslash \left[\tilde{p}\left(♭,\mathfrak{z}\right)+\tilde{p}\left(\mathfrak{z},\ell \right)\right]-\tilde{p}\left(\mathfrak{z},\mathfrak{z}\right).$$

In 2006, Chistyakov [8] introduced the idea of a modular metric for arbitrary sets, paving the way for modular spaces. He expanded on this concept in 2010, introducing the term “modular metric spaces”.

Definition 2 

([8,9]). Let $\mathcal{U}\ne \varnothing$. A function $\omega :\left(0,\infty \right)\times \mathcal{U}\times \mathcal{U}\to \left[0,\infty \right]$, defined by $\omega \left(\lambda ,♭,\ell \right)={\omega }_{\lambda }\left(♭,\ell \right)$, is called a modular metric on $\mathcal{U}$ if it satisfies the following statements for all $♭,\ell ,\mathfrak{z}\in \mathcal{U}$:

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

${\omega }_{\lambda }\left(♭,\ell \right)=0$ for all $\lambda >0$⇔$♭=\ell$,

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

${\omega }_{\lambda }\left(♭,\ell \right)={\omega }_{\lambda }\left(\ell ,♭\right)$ for all $\lambda >0$,

$\left({\omega }_3\right)$

${\omega }_{\lambda +\mu }\left(♭,\ell \right)\le {\omega }_{\lambda }\left(♭,\mathfrak{z}\right)+{\omega }_{\mu }\left(\mathfrak{z},\ell \right)$ for all $\lambda ,\mu >0$.

If in lieu of $\left({\omega }_1\right)$, the axiom

$\left({\omega }_1^{\prime }\right)$

${\omega }_{\lambda }\left(♭,♭\right)=0,\forall \lambda >0$

holds, and thus $\omega$ is called a pseudomodular metric on $\mathcal{U}$.

In 2018, Ege and Alaca [10] advanced metric space theory by introducing the concept of a modular b-metric space. By modifying condition $\left({\omega }_3\right)$, they developed a novel approach to the understanding of modular metric spaces, enhancing the framework and applications of this theory:

$\left({\omega }_3^{\prime }\right)$

There exists $\hslash \ge 1$, such that for all $♭,\ell ,\mathfrak{z}\in \mathcal{U}$ and for all $\lambda ,\mu >0$,

$${\omega }_{\lambda +\mu }\left(♭,\ell \right)\le \hslash \left[{\omega }_{\lambda }\left(♭,\mathfrak{z}\right)+{\omega }_{\mu }\left(\mathfrak{z},\ell \right)\right].$$

In 2010, Hosseinzadeh and Parvaneh [11] introduced the idea of partial modular metric spaces. This idea builds on the existing concept of partial metric spaces. Their work set the stage for Das et al. [12] to later update the definition of the partial modular metric function. They made changes to the rules about non-zero self-distance and the triangle inequality.

In 2023, Kesik et al. [13] introduced a new concept called a partial modular b-metric function. This idea combines aspects of partial, modular, and b-metric spaces. The authors provided results that clarify the topological features of this new space.

Definition 3 

([13]). Let $\mathcal{U}$ be a non-empty set and let $\hslash \ge 1\left(\hslash \in \mathbb{R}\right)$. A function ${\varpi }^{p_b}:\left(0,\infty \right)\times \mathcal{U}\times \mathcal{U}\to \left[0,\infty \right]$ is called a partial modular b-metric (briefly $𝒫{ℳ}_bℳ$) on $\mathcal{U}$ if the following conditions hold for all $♭,\ell ,\mathfrak{z}\in \mathcal{U}$:

$\left({\varpi }_1^{p_b}\right)$

${\varpi }_{\lambda }^{p_b}\left(♭,♭\right)={\varpi }_{\mu }^{p_b}\left(♭,♭\right)$ and ${\varpi }_{\lambda }^{p_b}\left(♭,♭\right)={\varpi }_{\lambda }^{p_b}\left(\ell ,\ell \right)={\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)\iff ♭=\ell$;

$\left({\varpi }_2^{p_b}\right)$

${\varpi }_{\lambda }^{p_b}\left(♭,♭\right)\le {\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)$, for all $\lambda >0$;

$\left({\varpi }_3^{p_b}\right)$

${\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)={\varpi }_{\lambda }^{p_b}\left(\ell ,♭\right)$, for all $\lambda >0$;

$\left({\varpi }_4^{p_b}\right)$

${\varpi }_{\lambda +\mu }^{p_b}\left(♭,\ell \right)\le \hslash \left[{\varpi }_{\lambda }^{p_b}\left(♭,\mathfrak{z}\right)+{\varpi }_{\mu }^{p_b}\left(\mathfrak{z},\ell \right)\right]-{\varpi }_{\lambda }^{p_b}\left(\mathfrak{z},\mathfrak{z}\right)$, for all $\lambda ,\mu >0$.

Then, it is said that $\left(\mathcal{U},{\varpi }_{\lambda }^{p_b}\right)={\mathcal{U}}_{{\varpi }^{p_b}}$ is a partial modular b-metric space which denotes $𝒫{ℳ}_bℳ𝒮$.

An $𝒫{ℳ}_bℳ$ on $\mathcal{U}$ is said to be convex if, in addition to the hypotheses $\left({\varpi }_1^{p_b}\right)$, $\left({\varpi }_2^{p_b}\right)$ and $\left({\varpi }_3^{p_b}\right)$, it satisfies the following: for all $♭,\ell ,\mathfrak{z}\in \mathcal{U}$ and $\lambda ,\mu >0$,

$\left({\varpi }_{4^{\prime }}^{p_b}\right)$

${\varpi }_{\lambda +\mu }^{p_b}\left(♭,\ell \right)\le \hslash \left[{\displaystyle \frac{\lambda }{\lambda +\mu }}{\varpi }_{\lambda }^{p_b}\left(♭,\mathfrak{z}\right)+{\displaystyle \frac{\mu }{\lambda +\mu }}{\varpi }_{\mu }^{p_b}\left(\mathfrak{z},\ell \right)\right]-{\displaystyle \frac{\lambda }{\lambda +\mu }}{\varpi }_{\lambda }^{p_b}\left(\mathfrak{z},\mathfrak{z}\right)$.

Furthermore, a convex modular satisfies the subsequent expression for $0<\mu <\lambda$:

$${\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)\le {\displaystyle \frac{\mu }{\lambda }}{\varpi }_{\mu }^{p_b}\left(♭,\ell \right)\le {\varpi }_{\mu }^{p_b}\left(♭,\ell \right).$$

(1)

For a given ${♭}_0\in \mathcal{U}$, the following two sets ${\mathcal{U}}_{{\varpi }^{p_b}}$ and ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$$𝒫{ℳ}_bℳ𝒮$ are centered at ${♭}_0$:

• ${\mathcal{U}}_{{\varpi }^{p_b}}\left({♭}_0\right)=\left\{♭\in \mathcal{U}:\underset{\lambda \to \infty }{lim}{\varpi }_{\lambda }^{p_b}\left({♭}_0,♭\right)=c\right\}$, for some $c\ge 0$;
• ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}\left({♭}_0\right)=\left\{♭\in \mathcal{U}:\exists \lambda =\lambda \left(♭\right)>0, {\varpi }_{\lambda }^{p_b}\left({♭}_0,♭\right)<\infty \right\}$.

Example 1 

([13]). Let $\mathcal{U}=\mathbb{R}$ and ${\varpi }^{p_b}:\left(0,\infty \right)\times \mathcal{U}\times \mathcal{U}\to \left[0,\infty \right]$ be a mapping defined by for all $♭,\ell \in \mathcal{U}$:

$${\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)=e^{-\lambda }{\left|♭-\ell \right|}^2,\forall \lambda >0.$$

Then, ${\varpi }^{p_b}$ is a $𝒫{ℳ}_bℳ$ on $\mathcal{U}$ with coefficient $\hslash =2$.

In the ensuing discussions, we present some newly introduced examples related to $𝒫{ℳ}_bℳ$.

Example 2. 

Let $\mathcal{U}=\mathbb{R}$, $\chi >1$ be a constant and ${\varpi }^{p_b}:\left(0,\infty \right)\times \mathcal{U}\times \mathcal{U}\to \left[0,\infty \right]$ be a function defined by

$${\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)=e^{-\lambda }{\left|♭-\ell \right|}^{\chi }+\left|♭\right|+\left|\ell \right|,\forall \lambda >0,$$

for all $♭,\ell \in \mathcal{U}$. Then, ${\varpi }^{p_b}$ is a $𝒫{ℳ}_bℳ$ on $\mathcal{U}$ with the coefficient $\hslash =2^{\chi -1}$.

Example 3. 

Let $\mathcal{U}=\mathbb{R}$ and ${\varpi }^{p_b}:\left(0,\infty \right)\times \mathcal{U}\times \mathcal{U}\to \left[0,\infty \right]$ be a function defined by for all $♭,\ell \in \mathcal{U}$

$${\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)={\displaystyle \frac{{\left|♭-\ell \right|}^{\chi }}{\lambda +{\left|♭-\ell \right|}^{\chi }}},\forall \lambda >0,$$

where $\chi >1$ is a constant. Then, ${\varpi }^{p_b}$ is a $𝒫{ℳ}_bℳ$ on $\mathcal{U}$ with the coefficient $\hslash =2^{\chi -1}$.

Remark 1. 

It is a certainty that every partial modular space is a $𝒫{ℳ}_bℳ$ with the parameter $\hslash =1$, and every modular b-metric space is a $𝒫{ℳ}_bℳ$ with the same parameter and zero self-distance. Nevertheless, it is not necessary for the opposite of these facts to be precise. In the subsequent illustration, we will clarify the case.

Example 4. 

Let $\mathcal{U}={\mathbb{R}}^{+}$ and ${\varpi }^{p_b}:\left(0,\infty \right)\times \mathcal{U}\times \mathcal{U}\to \left[0,\infty \right]$ be a function defined by

$${\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)=e^{-\lambda }{\left|♭-\ell \right|}^{\chi }+{\left[max\left\{♭,\ell \right\}\right]}^{\chi },\chi >1,$$

for all $\lambda >0$ and for all $♭,\ell \in \mathcal{U}$. Then, ${\varpi }^{p_b}$ is a $𝒫{ℳ}_bℳ$ on $\mathcal{U}$ with the coefficient $\hslash =2^{q-1}$, but it is neither a modular b-metric nor a partial modular metric space. Indeed, for any $♭>0$, we get ${\varpi }_{\lambda }^{p_b}\left(♭,♭\right)={♭}^{\chi }\ne 0$, which yields that ${\varpi }^{p_b}$ is not a modular b-metric on $\mathcal{U}$. Moreover, for $♭=5$, $\ell =1$ and $\mathfrak{z}=4$, we have ${\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)=e^{-\lambda }{4}^{\chi }+5^{\chi }$ and

$${\varpi }_{\lambda }^{p_b}\left(♭,\mathfrak{z}\right)+{\varpi }_{\lambda }^{p_b}\left(\mathfrak{z},\ell \right)-{\varpi }_{\lambda }^{p_b}\left(\mathfrak{z},\mathfrak{z}\right)=e^{-\lambda }+5^{\chi }+e^{-\lambda }{3}^{\chi }+4^{\chi }-4^{\chi }=e^{-\lambda }\left(1+3^{\chi }\right)+5^{\chi }.$$

Hence, we conclude that $e^{-\lambda }{4}^{\chi }+5^{\chi }>e^{-\lambda }\left(1+3^{\chi }\right)+5^{\chi }$ for all $\chi >1$, which results in ${\varpi }^{p_b}$ is not a partial modular metric on $\mathcal{U}$.

Some additional examples of $𝒫{ℳ}_bℳ$ can be generated by employing the following propositions.

Proposition 1. 

Let $\mathcal{U}$ be a non-empty set, such that ω is a modular b-metric with $\hslash >1$ and ${\varpi }^p$ is a partial modular metric on $\mathcal{U}$. Then, the function ${\varpi }^{p_b}:\left(0,\infty \right)\times \mathcal{U}\times \mathcal{U}\to \left[0,\infty \right]$, defined by

$${\varpi }_{\lambda }^{p_b}\left(x,y\right)={\omega }_{\lambda }\left(x,y\right)+{\varpi }_{\lambda }^p\left(x,y\right)$$

for all $♭,\ell \in \mathcal{U}$ and for all $\lambda >0$, is a $𝒫{ℳ}_bℳ$ on $\mathcal{U}$, i.e., ${\mathcal{U}}_{{\varpi }^{p_b}}$ is a $𝒫{ℳ}_bℳ𝒮$.

Proof. 

Let $\omega$ be a modular b-metric with $\hslash >1$ and ${\varpi }^p$ be a partial modular metric on $\mathcal{U}$. Then, the function ${\varpi }^{p_b}$ is clearly fulfilled the axioms $\left({\varpi }_1^{p_b}\right)$, $\left({\varpi }_2^{p_b}\right)$ and $\left({\varpi }_3^{p_b}\right)$. Hence, for the last axiom of $𝒫{ℳ}_bℳ$, we have

$$\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)={\omega }_{\lambda }\left(♭,\ell \right)+{\varpi }_{\lambda }^p\left(♭,\ell \right)\hfill \\ \le \hslash \left[{\omega }_{\lambda }\left(♭,\mathfrak{z}\right)+{\omega }_{\lambda }\left(\mathfrak{z},\ell \right)\right]+{\varpi }_{\lambda }^p\left(♭,\mathfrak{z}\right)+{\varpi }_{\lambda }^p\left(\mathfrak{z},\ell \right)-{\varpi }_{\lambda }^p\left(\mathfrak{z},\mathfrak{z}\right)\hfill \\ \le \hslash \left[{\omega }_{\lambda }\left(♭,\mathfrak{z}\right)+{\omega }_{\lambda }\left(\mathfrak{z},\ell \right)+{\varpi }_{\lambda }^p\left(♭,\mathfrak{z}\right)+{\varpi }_{\lambda }^p\left(\mathfrak{z},\ell \right)-{\varpi }_{\lambda }^p\left(\mathfrak{z},\mathfrak{z}\right)\right]\hfill \\ =\hslash \left[{\varpi }_{\lambda }^{p_b}\left(♭,\mathfrak{z}\right)+{\varpi }_{\lambda }^{p_b}\left(\mathfrak{z},\ell \right)-{\varpi }_{\lambda }^{p_b}\left(\mathfrak{z},\mathfrak{z}\right)\right]\hfill \\ \le \hslash \left[{\varpi }_{\lambda }^{p_b}\left(♭,\mathfrak{z}\right)+{\varpi }_{\lambda }^{p_b}\left(\mathfrak{z},\ell \right)\right]-{\varpi }_{\lambda }^{p_b}\left(\mathfrak{z},\mathfrak{z}\right),\hfill \end{array}$$

which implies that $\left({\varpi }_4^{p_b}\right)$ holds. Ultimately, we achieve that ${\mathcal{U}}_{{\varpi }^{p_b}}$ is a $𝒫{ℳ}_bℳ𝒮$. □

For a comprehensive understanding of various notions and concepts, such as completeness and convergence, within the framework of partial modular b-$ℳ𝒮$, it is recommended to consult the detailed exposition in [13].

Lemma 1 

([13]). Let ${\varpi }^{p_b}$ be a $𝒫{ℳ}_bℳ$ on a non-empty set $\mathcal{U}$. Define

$${\omega }_{\lambda }\left(♭,\ell \right)=2{\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)-{\varpi }_{\lambda }^{p_b}\left(♭,♭\right)-{\varpi }_{\lambda }^{p_b}\left(\ell ,\ell \right).$$

(2)

Then, ω is a modular b-metric on $\mathcal{U}$.

Lemma 2 

([13]). Let ${\varpi }^{p_b}$ be a $𝒫{ℳ}_bℳ$ on $\mathcal{U}$ and ${\left\{{♭}_{\rho }\right\}}_{\rho \in \mathbb{N}}$ be a sequence in ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$. Then,

$\left(i\right)$

${\left\{{♭}_{\rho }\right\}}_{n\in \mathbb{N}}$ is a ${\varpi }^{p_b}$-Cauchy sequence in $𝒫{ℳ}_bℳ𝒮$${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$⇔ it is a ω-Cauchy sequence in modular b-metric space ${\mathcal{U}}_{\omega }^{*}$.

$\left(ii\right)$

A $𝒫{ℳ}_bℳ𝒮$${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ is ${\varpi }^{p_b}$-complete ⇔ the modular b-metric space ${\mathcal{U}}_{\omega }^{*}$ is ω-complete. Furthermore,

$$\underset{\rho \to \infty }{lim}{\omega }_{\lambda }\left({♭}_{\rho },♭\right)=0\iff \underset{\rho \to \infty }{lim}\left[2{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },♭\right)-{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },{♭}_{\rho }\right)-{\varpi }_{\lambda }^{p_b}\left(♭,♭\right)\right]=0.$$

$\left(iii\right)$

${\left\{{♭}_{\rho }\right\}}_{\rho \in \mathbb{N}}$ is called ω-convergent to ${♭}^{*}\in {\mathcal{U}}_{\omega }^{*}$⇔$\underset{\rho \to \infty }{lim}{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },{♭}^{*}\right)=\underset{\rho ,m\to \infty }{lim}{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },{♭}_m\right)={\varpi }_{\lambda }^{p_b}\left({♭}^{*},{♭}^{*}\right),\forall \lambda >0$.

The lemma mentioned herein is pivotal in establishing the foundational arguments for deriving our principal findings.

Lemma 3 

([13]). Let ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ be a $𝒫{ℳ}_bℳ𝒮$ with the constant $\hslash >1$ and suppose that ${\left\{{♭}_{\rho }\right\}}_{\rho \in \mathbb{N}}$ and ${\left\{{\ell }_{\rho }\right\}}_{\rho \in \mathbb{N}}$ are convergent to ♭ and ℓ, respectively. Then, we have

$$\begin{array}{c}{\displaystyle \frac{1}{{\hslash }^2}}{\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)-{\displaystyle \frac{1}{\hslash }}{\varpi }_{\lambda }^{p_b}\left(♭,♭\right)-{\varpi }_{\lambda }^{p_b}\left(\ell ,\ell \right)\le \underset{\rho \to \infty }{lim}inf{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },{\ell }_{\rho }\right)\le \underset{\rho \to \infty }{lim}sup{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },{\ell }_{\rho }\right)\hfill \\ \\ \le \hslash {\varpi }_{\lambda }^{p_b}\left(♭,♭\right)+{\hslash }^2{\varpi }_{\lambda }^{p_b}\left(\ell ,\ell \right)+{\hslash }^2{\varpi }_{\lambda }^{p_b}\left(♭,\ell \right).\hfill \end{array}$$

In particular, if ${\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)=0$, then we have $\underset{\rho \to \infty }{lim}{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },{\ell }_{\rho }\right)=0$. Moreover, for each $\mathfrak{z}\in {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$, we have

$$\begin{array}{c}{\displaystyle \frac{1}{\hslash }}{\varpi }_{\lambda }^{p_b}\left(♭,\mathfrak{z}\right)-{\varpi }_{\lambda }^{p_b}\left(♭,♭\right)\le \underset{\rho \to \infty }{lim}inf{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },\mathfrak{z}\right)\le \underset{\rho \to \infty }{lim}sup{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },\mathfrak{z}\right)\hfill \\ \\ \le \hslash {\varpi }_{\lambda }^{p_b}\left(♭,♭\right)+\hslash {\varpi }_{\lambda }^{p_b}\left(♭,\mathfrak{z}\right).\hfill \end{array}$$

Also, in case of ${\varpi }_{\lambda }^{p_b}\left(♭,♭\right)=0$, we get

$${\displaystyle \frac{1}{\hslash }}{\varpi }_{\lambda }^{p_b}\left(♭,\mathfrak{z}\right)\le \underset{\rho \to \infty }{lim}inf{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },\mathfrak{z}\right)\le \underset{\rho \to \infty }{lim}sup{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },\mathfrak{z}\right)\le \hslash {\varpi }_{\lambda }^{p_b}\left(♭,\mathfrak{z}\right).$$

Recently, the concept of metric spaces has expanded. Researchers have adapted the Banach contraction principle for these spaces by adding a binary relation, aiming to enhance basic mathematical concepts for more complex structures.

Definition 4 

([14,15]). Let $\mathcal{U}$ be a non-empty set. Thus, now the non-empty binary relation $\mathcal{R}$ is a subset of ${\mathcal{U}}^2$. The set ${\mathcal{U}}^2$ is a universal relation, and the empty set is an empty relation; both are trivial. If any two elements $♭,\ell \in \mathcal{U}$ are related with respect to $\mathcal{R}$, then we shall write $\left(♭,\ell \right)\in \mathcal{R}$. Then, we assert that for all $♭,\ell ,\mathfrak{z}\in \mathcal{U}$, we have the following:

• $\mathcal{R}$ is reflexive if $\left(♭,♭\right)\in \mathcal{R}$;
• $\mathcal{R}$ is symmetric if $\left(♭,\ell \right)\in \mathcal{R}$ implies $\left(\ell ,♭\right)\in \mathcal{R}$;
• $\mathcal{R}$ is asymmetric if $\left(♭,\ell \right)\in \mathcal{R}\mathrm{and}\left(\ell ,♭\right)\in \mathcal{R}$ implies $♭=\ell$;
• $\mathcal{R}$ is transitive if $\left(♭,\ell \right)\in \mathcal{R}\mathrm{and}\left(\ell ,\mathfrak{z}\right)\in \mathcal{R}$ implies $\left(♭,\mathfrak{z}\right)\in \mathcal{R}$.

Alam and Imdad, as referenced in [16], employed a specific definition for non-empty binary relations within their study.

Definition 5 

([16]). Let $\mathcal{U}$ be a non-empty set and $\daleth :\mathcal{U}\to \mathcal{U}$ be a self-mapping. A binary relation $\mathcal{R}$ on $\mathcal{U}$ is said to be ℸ-closed if, for all $♭,\ell \in \mathcal{U}$, we have

• $\left(♭,\ell \right)\in \mathcal{R}\Rightarrow \left(\daleth ♭,\daleth \ell \right)\in \mathcal{R}$.

Zada and Sarvar [17] modified the above definition for two mappings.

Definition 6 

([17]). Let ℸ and $\mathcal{Q}$ be two self-mappings on a non-empty set $\mathcal{U}$. A binary relation $\mathcal{R}$ on $\mathcal{U}$ is said to be $\left(\daleth ,\mathcal{Q}\right)$-closed if, for all $♭,\ell \in \mathcal{U}$, we have

• $\left(♭,\ell \right)\in \mathcal{R}\Rightarrow \left(\daleth ♭,\mathcal{Q}\ell \right)\in \mathcal{R}\mathrm{or}\left(\mathcal{Q}♭,\daleth \ell \right)\in \mathcal{R}$.

The following notion was defined by Kolman et al. [18]:

Definition 7 

([18]). Let $\mathcal{R}$ be a binary relation on $\mathcal{U}$. A path in $\mathcal{R}$ from ♭ to ℓ is a sequence $\left\{{♭}_0,{♭}_1,{♭}_2,{♭}_3,\cdots ,{♭}_{\rho }\right\}\subseteq \mathcal{U}$, such that

$\left(i\right)$

${♭}_0=♭$ and ${♭}_{\rho }=\ell$;

$\left(ii\right)$

$\left({♭}_j,{♭}_{j+1}\right)\in \mathcal{R}$ for all $j=\left\{0,1,2,\cdots \cdots ,\rho -1\right\}$.

The set of all paths from ♭ to ℓ in $\mathcal{R}$ is denoted by $\mathrm{\Gamma }\left(♭,\ell ,\mathcal{R}\right)$. The path of length ρ involves $\rho +1$ elements of $\mathcal{U}$, although they are not necessarily distinct.

Roldan Lopez de Hierro [19] defined a novel notion as follows:

Definition 8 

([19]). A metric space $\left(\mathcal{U},d\right)$ equipped with a binary relation $\mathcal{R}$ is $\mathcal{R}$-regular if for all sequences ${\left\{{♭}_{\rho }\right\}}_{\rho \in \mathbb{N}}\mathrm{in}\mathcal{U}$,

• $\left({♭}_{\rho },{♭}_{\rho +1}\right)\in \mathcal{R}$ and ${♭}_{\rho }\to ♭\in \mathcal{U}$ implies $\left({♭}_{\rho },♭\right)\in \mathcal{R}$ for all $\rho \in \mathbb{N}.$

Furthermore, further findings regarding the aforementioned concept are documented in the references [20,21].

Wardowski [22] introduced F-contractions, a new type of contraction, and proved a related theorem in a complete metric space to enhance results in fixed-point theory.

Theorem 1 

([22]). The self-mapping $\mathrm{\daleth }:\mathcal{U}\to \mathcal{U}$ defined on a complete metric space $\left(\mathcal{U},d\right)$ is called an F-contraction, if there exist $\daleth >0$ and $\mathcal{F}\in \Im$, such that

$$d\left(\daleth ♭,\daleth \ell \right)>0\Rightarrow \daleth +\mathcal{F}\left(d\left(\daleth ♭,\daleth \ell \right)\right)\le \mathcal{F}\left(d\left(♭,\ell \right)\right),\forall ♭,\ell \in \mathcal{U},$$

where the class ℑ consists of the functions $\mathcal{F}:\left(0,\infty \right)\to \left(0,\infty \right)$ that obey the below axioms:

$\left(F_1\right)$

For all $\mathfrak{a},\mathfrak{b}\in \left(0,\infty \right)$, such that $\mathcal{F}\left(\mathfrak{a}\right)<\mathcal{F}\left(\mathfrak{b}\right)$ whenever $\mathfrak{a}<\mathfrak{b}$, that is, $\mathcal{F}$ is strictly increasing;

$\left(F_2\right)$

For each ${\left\{{\mathfrak{a}}_{\rho }\right\}}_{\rho \in \mathbb{N}}$ of positive numbers $\underset{\rho \to \infty }{lim}{\mathfrak{a}}_{\rho }=0$⇔$\underset{\rho \to \infty }{lim}\mathcal{F}\left({\mathfrak{a}}_{\rho }\right)=-\infty$;

$\left(F_3\right)$

A constant $\mathfrak{c}\in \left(0,1\right)$ exits, such that $\underset{\mathfrak{a}\to 0^{+}}{lim}{\mathfrak{a}}^{\mathfrak{c}}\mathcal{F}\left(\mathfrak{a}\right)=0$.

Then, the mapping ℸ, characterized as an F-contraction, possesses a unique fixed point.

Jleli and Samet [23] introduced an advanced generalization of the Banach contraction, termed as $\theta$-contraction, delineated as follows:

Theorem 2 

([23]). On a complete metric space $\left(\mathcal{U},d\right)$, the mapping $\daleth :\mathcal{U}\to \mathcal{U}$ is said to be a θ-contraction, that is, there exists $k\in \left(0,1\right)$, such that the inequality

$$m\left(\daleth ♭,\daleth \ell \right)\ne 0\Rightarrow \theta \left(d\left(\daleth ♭,\daleth \ell \right)\right)\le {\left[\theta \left(d\left(♭,\ell \right)\right)\right]}^k$$

is provided for all $♭,\ell \in \mathcal{U}$, where $\theta :(0,\infty )\to (1,\infty )$ obeys the following circumstances:

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

θ is non-decreasing;

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

For each sequence ${\left\{{\iota }_{\rho }\right\}}_{\rho \in \mathbb{N}}\subset \left(0,\infty \right)$, $\underset{\rho \to \infty }{lim}\theta \left({\iota }_{\rho }\right)=1$⇔$\underset{\rho \to \infty }{lim}{\iota }_{\rho }=0^{+};$

$\left({\Theta }_3\right)$

There exist $r\in \left(0,1\right)$ and $\ell \in \left(0,\infty \right]$, such that $\underset{\iota \to 0^{+}}{lim}{\displaystyle \frac{\theta \left(\iota \right)-1}{{\iota }^r}}=\ell .$

Thereupon, the mapping ℸ owns a unique fixed point.

Piri and Kumam [24] followed Jleli and Samet, but with a minor change in which the condition $\left({\Theta }_3\right)$ was replaced by the continuity of function $\theta$, which this family denoted by ${\Im }^{*}$. Subsequently, a new family of functions $\theta :\left(0,\infty \right)\to \left(1,\infty \right)$ with the following features denoting by $\tilde{\Theta }$ was introduced by Liu et al. [25]:

$\left({\Theta }_{1^{\prime }}\right)$

$\theta$ is non-decreasing and continuous;

$\left({\Theta }_{2^{\prime }}\right)$

$\underset{t\in \left(0,\infty \right)}{inf}\theta \left(t\right)=1.$

Consequently, Liu et al. [25] put forward the following family of functions:

Definition 9 

([25]). Let $\gimel :\left(0,\infty \right)\to \left(0,\infty \right)$ be a mapping satisfying the following conditions:

$\left({\mathrm{\Delta }}_1\right)$

ℷ is non-decreasing, that is for all $t,s\in \left(0,\infty \right)$, $t<s$, one has $\gimel \left(t\right)\le \gimel \left(s\right)$;

$\left({\mathrm{\Delta }}_2\right)$

For each sequence ${\left\{t_{\rho }\right\}}_{\rho \in \mathbb{N}}\subset \left(0,\infty \right)$, such that $\underset{\rho \to \infty }{lim}\gimel \left(t_{\rho }\right)=0\iff \underset{\rho \to \infty }{lim}{t}_{\rho }=0$;

$\left({\mathrm{\Delta }}_3\right)$

ℷ is continuous.

Define the set $\mathrm{\Delta }=\left\{\gimel :\gimel \mathrm{satisfies}\left({\mathrm{\Delta }}_1\right)-\left({\mathrm{\Delta }}_3\right)\right\}$.

Rus [26] introduced comparison functions, marking an important step forward in the field. Berinde [27] built on this idea, expanding and enhancing its significance.

Definition 10. 

$C:\left(0,\infty \right)\to \left(0,\infty \right)$ is a comparison function; which satisfies the following terms:

$\left(i\right)$

C is monotone increasing, that is, $t<s\Rightarrow C\left(t\right)<C\left(s\right)$;

$\left(ii\right)$

$\underset{\rho \to \infty }{lim}{C}^{\rho }\left(t\right)=0$ for all $t>0$, where $C^{\rho }$ stands for the ${\rho }^{th}$ iterate of C;

$\left(iii\right)$

If C is a comparison function, then $C\left(t\right)<t$, for every $t>0$.

In 2016, Liu et al. [25] significantly advanced mathematical analysis by proving a theorem on Suzuki-type contractions, building on Suzuki’s foundational work [28].

Theorem 3 

([25]). Let $\daleth :\mathcal{U}\to \mathcal{U}$ be a mapping on complete metric space $\left(\mathcal{U},d\right)$, where $\mathcal{U}$ is a non-empty set. If there exist a continuous comparison function C and $\gimel \in \mathrm{\Delta }$, such that for all $♭,\ell \in \mathcal{U}$ with $\daleth ♭\ne \daleth \ell$,

$${\displaystyle \frac{1}{2}}d(♭,\daleth ♭)<d(♭,\ell )\Rightarrow \gimel \left(d\left(\daleth ♭,\daleth \ell \right)\right)\le C\left[\gimel \left(𝒫\left(♭,\ell \right)\right)\right],$$

where

$$𝒫\left(♭,\ell \right)=max\left\{d\left(♭,\ell \right),d\left(♭,\daleth ♭\right),d\left(\ell ,\daleth \ell \right),{\displaystyle \frac{1}{2}}d\left(♭,\daleth \ell \right),d\left(\ell ,\daleth ♭\right)\right\}.$$

Then, ℸ owns a unique fixed point in $\mathcal{U}$.

For further elaboration and a more comprehensive understanding, it is recommended to consult the following references: [29,30].

The motivation underlying this study is firmly rooted in the desire to advance the theoretical framework of partial modular b-metric spaces. Specifically, this study introduces a generalized Suzuki $\mathcal{R}$-contraction, which represents a substantial extension of previous contributions made by Liu et al. [25]. This development is pivotal, as it not only enriches the existing literature but also lays the groundwork for new avenues of exploration within the field.

One of the principal strengths of this research lies in its innovative integration of the $\mathcal{R}$ relation. This integration is instrumental in deriving new fixed-point theorems that incorporate quadratic terms, which are often essential in a variety of mathematical applications. The introduction of these theorems broadens the scope of fixed-point theory and provides a deeper understanding of the conditions under which such points can be reliably identified.

Furthermore, the practical relevance of this study is exemplified through an extensive case study that illustrates the application of the proposed theoretical constructs in real-world scenarios. This case study serves to demonstrate the validity and utility of the generalized Suzuki contraction within various practical frameworks, thereby reinforcing the significance of the theoretical advancements.

In addition, this manuscript delves into the investigation of functional equations that emerge from dynamic programming and fractional differential equations. This exploration not only enriches the theoretical context of the research but also significantly broadens its applicability across diverse mathematical disciplines. The findings contribute valuable insights that transcend theoretical boundaries and encourage further empirical investigations, thereby enhancing both the theoretical and practical dimensions of the field.

2. Fixed-Point Results in Partial Modular $\mathit{b}$-Metric Spaces

This section constitutes the core of our discourse, presenting innovative common fixed-point theorems within the framework of partial modular b-metric spaces.

The theorems elaborated herein are predicated upon the conditions stipulated below.

$\left({\mathcal{L}}_1\right)$

${\varpi }_{\lambda }^{p_b}\left(♭,\daleth ♭\right)<\infty$ for all $\lambda >0$ and $♭\in {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$.

$\left({\mathcal{L}}_2\right)$

${\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)<\infty$ for all $\lambda >0$ and $♭,\ell \in {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$.

Definition 11. 

Consider ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ as a partial modular b-metric space, with the mappings $\daleth ,\mathcal{Q}:{\mathcal{U}}_{{\varpi }^{p_b}}^{*}\to {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ and a binary relation $\mathcal{R}$ on ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$. Define $I=\left\{\left(♭,\ell \right)\in \mathcal{R}:{\varpi }_{\lambda }^{p_b}\left(\daleth ♭,\mathcal{Q}\ell \right)>0\right\}.$ The mappings ℸ and $\mathcal{Q}$ form a generalized ${\gimel }_C$-type Suzuki $\mathcal{R}$-contraction if there exist a continuous comparison function C and $\gimel \in \mathrm{\Delta }$, such that for all $♭,\ell \in I$, $\hslash \ge 1$ and $\forall \lambda >0$

$$\begin{array}{c}{\displaystyle \frac{1}{2\hslash }}min\left\{{\varpi }_{\lambda }^{p_b}\left(♭,\daleth ♭\right),{\varpi }_{\lambda }^{p_b}\left(\ell ,\mathcal{Q}\ell \right)\right\}\le {\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)\Rightarrow \hfill \\ \\ \gimel \left({\hslash }^5{\left[{\varpi }_{\lambda }^{p_b}\left(\daleth ♭,\mathcal{Q}\ell \right)\right]}^2\right)\le C\left[\gimel \left(ℳ\left(♭,\ell \right)\right)\right],\hfill \end{array}$$

(3)

where $0\le c<1$ and

$$ℳ\left(♭,\ell \right)=max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left(\ell ,\mathcal{Q}\ell \right)·{\varpi }_{\lambda }^{p_b}\left(♭,\daleth ♭\right),{\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)·{\varpi }_{\lambda }^{p_b}\left(\ell ,\mathcal{Q}\ell \right),\\ {\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)·{\varpi }_{\lambda }^{p_b}\left(♭,\daleth ♭\right),c·{\varpi }_{\lambda }^{p_b}\left(\ell ,\daleth ♭\right)·{\varpi }_{\lambda }^{p_b}\left(♭,\mathcal{Q}\ell \right)\end{array}\right\}.$$

Theorem 4. 

Let ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ be a ${\varpi }^{p_b}$-complete $𝒫{ℳ}_bℳ𝒮$ and $\mathcal{R}$ be a transitive binary relation on ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$. Let ℸ and $\mathcal{Q}$ satisfy the following conditions:

$\left(i\right)$

ℸ and $\mathcal{Q}$ form a generalized ${\gimel }_C$-type Suzuki $\mathcal{R}$-contraction;

$\left(ii\right)$

There exists ${♭}_0\in {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$, such that $\left({♭}_0,\daleth {♭}_0\right)\in \mathcal{R}$;

$\left(iii\right)$

$\mathcal{R}$ is $\left(\daleth ,\mathcal{Q}\right)$-closed;

$\left(vi\right)$

ℸ and $\mathcal{Q}$ are ${\varpi }^{p_b}$-continuous.

If the condition $\left({\mathcal{L}}_1\right)$ is satisfied, then ℸ and $\mathcal{Q}$ have a common fixed point in ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$.

Proof. 

By assumption $\left(ii\right)$, there exists ${♭}_0\in {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$, such that $\left({♭}_0,\daleth {♭}_0\right)\in \mathcal{R}$. Taking ${♭}_0\in {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ as the initial point, we construct the sequence ${\left\{{♭}_{\rho }\right\}}_{\rho \in \mathbb{N}}$ in ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ by ${♭}_1=\daleth {♭}_0$ and ${♭}_2=\mathcal{Q}{♭}_1$. Continuing with the same process, we obtain ${♭}_{2\rho +1}=\daleth {♭}_{2\rho }$ and ${♭}_{2\rho +2}=\mathcal{Q}{♭}_{2\rho +1}$, where $\rho \in \mathbb{N}\cup \{0\}$. Moreover, by the assumptions $\left(ii\right)$ and $\left(iii\right)$, we have

$$\begin{array}{c}\left({♭}_1,{♭}_2\right)=\left(\daleth {♭}_0,\mathcal{Q}{♭}_1\right)\in \mathcal{R},\hfill \\ \left({♭}_2,{♭}_3\right)=\left(\mathcal{Q}{♭}_1,\daleth {♭}_2\right)\in \mathcal{R},\hfill \\ \left({♭}_3,{♭}_4\right)=\left(\daleth {♭}_2,\mathcal{Q}{♭}_3\right)\in \mathcal{R},\hfill \\ \left({♭}_4,{♭}_5\right)=\left(\mathcal{Q}{♭}_3,\daleth {♭}_4\right)\in \mathcal{R}.\hfill \end{array}$$

(4)

In general, we get $\left({♭}_{2\rho },{♭}_{2\rho +1}\right)=\left(\mathcal{Q}{♭}_{2\rho -1},\daleth {♭}_{2\rho }\right)\in \mathcal{R}$ and $\left({♭}_{2\rho +1},{♭}_{2\rho +2}\right)=\left(\daleth {♭}_{2\rho },\mathcal{Q}{♭}_{2\rho +1}\right)\in \mathcal{R}$.

Case 1. If ${♭}_{2{\rho }^{*}}={♭}_{2{\rho }^{*}+1}$, for some ${\rho }^{*}\in \mathbb{N}$, then

$${♭}_{2{\rho }^{*}+1}={♭}_{2{\rho }^{*}+2}.$$

(5)

Indeed, on the contrary, if ${♭}_{2{\rho }^{*}+1}\ne {♭}_{2{\rho }^{*}+2}$, then $\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+2}\right)\in I$ and ${\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+2}\right)>0$, so using (3), we yield that

$${\displaystyle \frac{1}{2\hslash }}min\left\{{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},\daleth {♭}_{2{\rho }^{*}}\right),{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},\mathcal{Q}{♭}_{2{\rho }^{*}+1}\right)\right\}\le {\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+1}\right)\Rightarrow$$

$$\begin{array}{c}\gimel \left({\hslash }^5{\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+2}\right)\right]}^2\right)=\gimel \left({\hslash }^5{\left[{\varpi }_{\lambda }^{p_b}\left(\daleth {♭}_{2{\rho }^{*}},\mathcal{Q}{♭}_{2{\rho }^{*}+1}\right)\right]}^2\right)\hfill \\ \\ \le C\left[\gimel \left(ℳ\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+1}\right)\right)\right],\hfill \end{array}$$

(6)

where $0\le c<1$ and

$$\begin{array}{c}ℳ\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+1}\right)\hfill \\ \\ =max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},\mathcal{Q}{♭}_{2{\rho }^{*}+1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},\daleth {♭}_{2{\rho }^{*}}\right),{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},\mathcal{Q}{♭}_{2{\rho }^{*}+1}\right),\hfill \\ {\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},\daleth {♭}_{2{\rho }^{*}}\right),c·{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},\daleth {♭}_{2{\rho }^{*}}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},\mathcal{Q}{♭}_{2{\rho }^{*}+1}\right)\hfill \end{array}\right\}\hfill \\ \\ =max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+2}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+1}\right),{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+2}\right),\hfill \\ {\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+1}\right),c·{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right)\hfill \end{array}\right\}\hfill \\ \\ =max\left\{{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+2}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right),{\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+1}\right)\right]}^2,0\right\}.\hfill \end{array}$$

Then, it follows from (6) that

$$\begin{array}{c}\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+2}\right)\right]}^2\right)\le \gimel \left({\hslash }^5{\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+2}\right)\right]}^2\right)\hfill \\ \\ \le C\left[\gimel \left({\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+2}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right)\right)\right].\hfill \end{array}$$

Since $C\left(t\right)\le t$ for every $t>0$, we obtain

$$\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+2}\right)\right]}^2\right)<\gimel \left({\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+2}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right)\right).$$

Due to the function ℷ being non-decreasing, we achieve that

$${\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+2}\right)\right]}^2<{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+2}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right),$$

which means

$${\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+2}\right)<{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right).$$

Using the triangle inequality in a $𝒫{ℳ}_bℳ𝒮$ and by (1), we gain

$$\hslash \left[{\varpi }_{{\displaystyle \frac{\lambda }{2}}}^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}}\right)+{\varpi }_{{\displaystyle \frac{\lambda }{2}}}^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right)\right]-{\varpi }_{{\displaystyle \frac{\lambda }{2}}}^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}}\right)<{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right),$$

or

$$\hslash \left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}}\right)+{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right)\right]-{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}}\right)<{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right),$$

which implies

$$\hslash \left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}}\right)+{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right)\right]<{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right)+{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}}\right).$$

Because

$${\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}}\right)+{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right)<\hslash \left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}}\right)+{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right)\right],$$

we have

$${\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}}\right)+{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right)<{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right)+{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}}\right),$$

that is,

$${\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}}\right)<{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}}\right).$$

This observation stands in direct contradiction to the second condition of $𝒫{ℳ}_bℳ$, underscoring a fundamental inconsistency within the framework. If ${\varpi }_{\lambda }^{p_b}\left(♭,♭\right)\le {\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)$, $\forall ♭,\ell \in {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$, then, it follows from (6) that

$$\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+2}\right)\right]}^2\right)\le \gimel \left({\hslash }^5{\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+2}\right)\right]}^2\right)\le C\left[\gimel \left({\left({\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}}\right)\right)}^2\right)\right].$$

Since $C\left(t\right)\le t$ for every $t>0$, we obtain

$$\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+2}\right)\right]}^2\right)<\gimel \left({\left({\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}}\right)\right)}^2\right).$$

In consideration of the property that the function denoted as ℷ exhibits non-decreasing behaviour, it logically follows that

$${\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+2}\right)\right]}^2<{\left({\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}}\right)\right)}^2,$$

which consequently implies that

$${\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}+2}\right)<{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}}\right).$$

This relationship underscores the significance of the ordering in the variables ${♭}_{2{\rho }^{*}+2}$ and ${♭}_{2{\rho }^{*}}$ with respect to the function ${\varpi }^{p_b}$, thereby affirming the foundational assumption regarding the non-decreasing nature of ℷ. Using the triangle inequality in a partial modular b-metric space and by (1), we gain

$$\hslash \left[{\varpi }_{{\displaystyle \frac{\lambda }{2}}}^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}}\right)+{\varpi }_{{\displaystyle \frac{\lambda }{2}}}^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right)\right]-{\varpi }_{{\displaystyle \frac{\lambda }{2}}}^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}}\right)<{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}}\right),$$

such that

$$\hslash \left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}}\right)+{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right)\right]<{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}}\right)+{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}}\right).$$

Since

$${\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}}\right)+{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right)<\hslash \left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}}\right)+{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right)\right],$$

we have

$${\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}}\right)+{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right)<{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}+1},{♭}_{2{\rho }^{*}}\right)+{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}}\right),$$

that is,

$${\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}+2}\right)<{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }^{*}},{♭}_{2{\rho }^{*}}\right)$$

such that this contradicts with the second condition of partial modular b-metric that is ${\varpi }_{\lambda }^{p_b}\left(♭,♭\right)\le {\varpi }_{\lambda }^{p_b}\left(♭,\ell \right),\forall ♭,\ell \in {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$. Hence, ${♭}_{2{\rho }^{*}}={♭}_{2{\rho }^{*}+1}\mathrm{implies}{♭}_{2{\rho }^{*}+1}={♭}_{2{\rho }^{*}+2}$, that is, ${♭}_{2{\rho }^{*}}={♭}_{2{\rho }^{*}+1}={♭}_{2{\rho }^{*}+2}$. Consequently, ${♭}_{2{\rho }^{*}}=\daleth {♭}_{2{\rho }^{*}}={♭}_{2{\rho }^{*}+1}=\mathcal{Q}{♭}_{2{\rho }^{*}}$ and ${♭}_{2{\rho }^{*}}$ is a common fixed point of $\daleth \mathrm{and}\mathcal{Q}$.

Case 2. In light of Case 1, we search for the common fixed point of $\daleth \mathrm{and}\mathcal{Q},$ when ${♭}_{2\rho }\ne {♭}_{2\rho +1}$ for all $\rho \in \mathbb{N}$. We have ${\varpi }_{\lambda }^{p_b}\left(\daleth {♭}_{2\rho },\mathcal{Q}{♭}_{2\rho -1}\right)>0\mathrm{for}\mathrm{all}\rho \in \mathbb{N}$. Since $\left({♭}_{2\rho },{♭}_{2\rho -1}\right)\in \mathcal{R}$, so $\left({♭}_{2\rho },{♭}_{2\rho -1}\right)\in I$. Setting $♭={♭}_{2\rho }$ and $\ell ={♭}_{2\rho -1}$ in (3), we get

$${\displaystyle \frac{1}{2\hslash }}min\left\{{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },\daleth {♭}_{2\rho }\right),{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},\mathcal{Q}{♭}_{2\rho -1}\right)\right\}\le {\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho -1}\right)\Rightarrow$$

$$\begin{array}{c}\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho }\right)\right]}^2\right)\le \gimel \left({\hslash }^5{\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho }\right)\right]}^2\right)=\gimel \left({\hslash }^5{\left[{\varpi }_{\lambda }^{p_b}\left(\daleth {♭}_{2\rho },\mathcal{Q}{♭}_{2\rho -1}\right)\right]}^2\right)\hfill \\ \\ \le C\left[\gimel \left(ℳ\left({♭}_{2\rho },{♭}_{2\rho -1}\right)\right)\right],\hfill \end{array}$$

(7)

where

$$\begin{array}{c}ℳ\left({♭}_{2\rho },{♭}_{2\rho -1}\right)\hfill \\ \\ =max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},\mathcal{Q}{♭}_{2\rho -1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },\daleth {♭}_{2\rho }\right),{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho -1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},\mathcal{Q}{♭}_{2\rho -1}\right),\hfill \\ {\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho -1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },\daleth {♭}_{2\rho }\right),c·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},\daleth {♭}_{2\rho }\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },\mathcal{Q}{♭}_{2\rho -1}\right)\hfill \end{array}\right\}\hfill \\ \\ =max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right),{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho -1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right),\hfill \\ {\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho -1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right),c·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho +1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho }\right)\hfill \end{array}\right\}\hfill \\ \\ =max\left\{{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right),{\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right)\right]}^2,0\right\}.\hfill \end{array}$$

If $ℳ\left({♭}_{2\rho -1},{♭}_{2\rho }\right)={\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right)$, then it follows from (7) that

$$\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho }\right)\right]}^2\right)\le C\left[\gimel \left({\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right)\right)\right].$$

Since $C\left(t\right)\le t$ for every $t>0$, we obtain

$$\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho }\right)\right]}^2\right)<\gimel \left({\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right)\right),$$

and by the non-decreasing property of ℷ, we achieve

$${\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho }\right)\right]}^2<{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right)$$

and, ultimately,

$${\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho }\right)<{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right).$$

Using the triangle inequality in a partial modular b-metric space and by (1), we conclude that

$$\begin{array}{c}\hslash \left[{\varpi }_{{\displaystyle \frac{\lambda }{2}}}^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho -1}\right)+{\varpi }_{{\displaystyle \frac{\lambda }{2}}}^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right)\right]-{\varpi }_{{\displaystyle \frac{\lambda }{2}}}^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho -1}\right)<{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right),\hfill \\ \\ \hslash \left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho -1}\right)+{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right)\right]<{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right)+{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho -1}\right),\hfill \\ \\ {\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho -1}\right)+{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right)<\hslash \left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho -1}\right)+{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right)\right],\hfill \\ \\ {\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho -1}\right)+{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right)<{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right)+{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho -1}\right),\hfill \\ \\ {\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho -1}\right)<{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho -1}\right).\hfill \end{array}$$

This statement embodies a contradiction. Hence, we have $ℳ\left({♭}_{2\rho -1},{♭}_{2\rho }\right)={\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right)\right]}^2$ and so

$$\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho }\right)\right]}^2\right)\le C\left[\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right)\right]}^2\right)\right],$$

which means that

$${\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho }\right)\right]}^2<{\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho -1},{♭}_{2\rho }\right)\right]}^2,$$

and

$${\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho }\right)<{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho -1}\right).$$

(8)

Setting $♭={♭}_{2\rho }$ and $\ell ={♭}_{2\rho +1}$ in (3), we get

$${\displaystyle \frac{1}{2\hslash }}min\left\{{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },\daleth {♭}_{2\rho }\right),{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},\mathcal{Q}{♭}_{2\rho +1}\right)\right\}\le {\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right)\Rightarrow$$

$$\begin{array}{c}\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho +2}\right)\right]}^2\right)\le \gimel \left({\hslash }^5{\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho +2}\right)\right]}^2\right)=\gimel \left({\hslash }^5{\left[{\varpi }_{\lambda }^{p_b}\left(\daleth {♭}_{2\rho },\mathcal{Q}{♭}_{2\rho +1}\right)\right]}^2\right)\hfill \\ \\ \le C\left[\gimel \left(ℳ\left({♭}_{2\rho },{♭}_{2\rho +1}\right)\right)\right],\hfill \end{array}$$

(9)

where

$$\begin{array}{c}ℳ\left({♭}_{2\rho },{♭}_{2\rho +1}\right)\hfill \\ \\ =max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},\mathcal{Q}{♭}_{2\rho +1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },\daleth {♭}_{2\rho }\right),{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},\mathcal{Q}{♭}_{2\rho +1}\right),\hfill \\ {\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },\daleth {♭}_{2\rho }\right),c·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},\daleth {♭}_{2\rho }\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },\mathcal{Q}{♭}_{2\rho +1}\right)\hfill \end{array}\right\}\hfill \\ \\ =max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho +2}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right),{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho +2}\right),\hfill \\ {\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right),c·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho +1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +2}\right)\hfill \end{array}\right\}\hfill \\ \\ =max\left\{{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho +2}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right),{\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right)\right]}^2,0\right\}.\hfill \end{array}$$

If $ℳ\left({♭}_{2\rho +1},{♭}_{2\rho }\right)={\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho +2}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right)$, then we obtain the following by (9):

$$\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho +2}\right)\right]}^2\right)\le C\left[\gimel \left({\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho +2}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right)\right)\right],$$

and, similarly,

$${\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho +2}\right)\right]}^2<{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho +2}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right),$$

and, lastly,

$${\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho +2}\right)<{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right).$$

If $ℳ\left({♭}_{2\rho +1},{♭}_{2\rho }\right)={\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right)\right]}^2$, then the expression (9) turns into

$$\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho +2}\right)\right]}^2\right)\le C\left[\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right)\right]}^2\right)\right],$$

and

$${\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho +2}\right)\right]}^2<{\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right)\right]}^2,$$

which implies

$${\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}_{2\rho +2}\right)<{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho },{♭}_{2\rho +1}\right).$$

(10)

In general, for all $\rho \in \mathbb{N}$, either even or odd, we have

$$\begin{array}{c}\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },{♭}_{\rho +1}\right)\right]}^2\right)\le C\left[\gimel {\left({\varpi }_{\lambda }^{p_b}\left({♭}_{\rho -1},{♭}_{\rho }\right)\right)}^2\right]\hfill \\ \le C^2\left[\gimel {\left({\varpi }_{\lambda }^{p_b}\left({♭}_{\rho -2},{♭}_{\rho -1}\right)\right)}^2\right]\hfill \\ \le ···\hfill \\ \le C^{\rho }\left[\gimel {\left({\varpi }_{\lambda }^{p_b}\left({♭}_0,{♭}_1\right)\right)}^2\right].\hfill \end{array}$$

Letting $\rho \to \infty$ in the above inequality, we get

$$0\le \underset{\rho \to \infty }{lim}\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },{♭}_{\rho +1}\right)\right]}^2\right)\le \underset{\rho \to \infty }{lim}{C}^{\rho }\left[\gimel {\left({\varpi }_{\lambda }^{p_b}\left({♭}_0,{♭}_1\right)\right)}^2\right]=0,$$

(11)

which implies

$$\underset{\rho \to \infty }{lim}\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },{♭}_{\rho +1}\right)\right]}^2\right)=0.$$

This, together with $\left({\mathrm{\Delta }}_{ii}\right)$, allows us to obtain

$$\underset{\rho \to \infty }{lim}{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },{♭}_{\rho +1}\right)=0.$$

(12)

By (2), the above expression implies that

$$\underset{\rho \to \infty }{lim}{\omega }_{\lambda }\left({♭}_{\rho },{♭}_{\rho +1}\right)=0.$$

(13)

Considering the second condition of Definition 3, we derive that

$$\underset{\rho \to \infty }{lim}{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },{♭}_{\rho }\right)\le \underset{\rho \to \infty }{lim}{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },{♭}_{\rho +1}\right)=0.$$

Thus, taking Lemma 1 into account, for all $\rho ,m\ge 1$, we have

$$\begin{array}{c}\underset{\rho \to \infty }{lim}{\omega }_{\lambda }\left({♭}_m,{♭}_{\rho }\right)=2\underset{\rho \to \infty }{lim}{\varpi }_{\lambda }^{p_b}\left({♭}_m,{♭}_{\rho }\right)-\underset{\rho \to \infty }{lim}{\varpi }_{\lambda }^{p_b}\left({♭}_m,{♭}_m\right)-\underset{\rho \to \infty }{lim}{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },{♭}_{\rho }\right),\hfill \end{array}$$

such that

$$\underset{\rho \to \infty }{lim}{\omega }_{\lambda }\left({♭}_m,{♭}_{\rho }\right)=2\underset{\rho \to \infty }{lim}{\varpi }_{\lambda }^{p_b}\left({♭}_m,{♭}_{\rho }\right).$$

(14)

In the subsequent phase of our investigation, we aim to establish that the set ${\left\{{♭}_{\rho }\right\}}_{\rho \in \mathbb{N}}$ constitutes a ${\varpi }^{p_b}$-Cauchy sequence within the space ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$. This endeavor necessitates demonstrating that ${\left\{{♭}_{\rho }\right\}}_{\rho \in \mathbb{N}}$ adheres to the criteria of an $\omega$-Cauchy sequence as delineated in ${\mathcal{U}}_{\omega }^{*}$, in accordance with Lemma 2. Contrary to this assertion, suppose the sequence ${\left\{{♭}_{\rho }\right\}}_{\rho \in \mathbb{N}}$ fails to satisfy the $\omega$-Cauchy condition. Under such a scenario, for certain subsequences ${\left\{{♭}_{2{\rho }_k}\right\}}_{k\in \mathbb{N}}$ and ${\left\{{♭}_{2m_k}\right\}}_{k\in \mathbb{N}}$, there emerges an $\epsilon >0$, accompanied by a positive integer $k\left(\epsilon \right)$, ensuring that for all ${\rho }_k,m_k>k\left(\epsilon \right)$, the condition holds true for any $\lambda >0$ and for every $k\in \mathbb{N}$, as follows:

$${\omega }_{\lambda }\left({♭}_{2m_k},{♭}_{2{\rho }_k}\right)\ge \epsilon \mathrm{and}{\omega }_{\lambda }\left({♭}_{2m_k},{♭}_{2{\rho }_k-2}\right)<\epsilon .$$

(15)

Using (15) and (1), by the triangular inequality, we obtain

$$\begin{array}{c}\epsilon \le {\omega }_{\lambda }\left({♭}_{2m_k},{♭}_{2{\rho }_k}\right)\le \hslash \left({\omega }_{{\displaystyle \frac{\lambda }{2}}}\left({♭}_{2m_k},{♭}_{2m_k+1}\right)+{\omega }_{{\displaystyle \frac{\lambda }{2}}}\left({♭}_{2m_k+1},{♭}_{2{\rho }_k}\right)\right)\hfill \\ \\ \le \hslash \left({\omega }_{\lambda }\left({♭}_{2m_k},{♭}_{2m_k+1}\right)+{\omega }_{\lambda }\left({♭}_{2m_k+1},{♭}_{2{\rho }_k}\right)\right).\hfill \end{array}$$

(16)

Upon taking the limit superior in the aforementioned inequality as $k\to \infty$, we achieve the subsequent result

$${\displaystyle \frac{\epsilon }{\hslash }}\le \underset{k\to \infty }{limsup}{\omega }_{\lambda }\left({♭}_{2m_k+1},{♭}_{2{\rho }_k}\right).$$

(17)

By using the triangular inequality, we get

$${\omega }_{\lambda }\left({♭}_{2m_k},{♭}_{2{\rho }_k-1}\right)\le \hslash \left({\omega }_{{\displaystyle \frac{\lambda }{2}}}\left({♭}_{2m_k},{♭}_{2{\rho }_k-2}\right)+{\omega }_{{\displaystyle \frac{\lambda }{2}}}\left({♭}_{2{\rho }_k-2},{♭}_{2{\rho }_k-1}\right)\right).$$

(18)

Taking the limit as $k\to \infty$ in the above expression, we gain

$$\underset{k\to \infty }{limsup}{\omega }_{\lambda }\left({♭}_{2m_k},{♭}_{2{\rho }_k-1}\right)\le \hslash \epsilon .$$

(19)

Additionally, by (1), we obtain the following expression:

$$\begin{array}{c}\epsilon \le {\omega }_{\lambda }\left({♭}_{2m_k},{♭}_{2{\rho }_k}\right)\le \hslash \left({\omega }_{{\displaystyle \frac{\lambda }{2}}}\left({♭}_{2m_k},{♭}_{2{\rho }_k-1}\right)+{\omega }_{{\displaystyle \frac{\lambda }{2}}}\left({♭}_{2{\rho }_k-1},{♭}_{2{\rho }_k}\right)\right),\hfill \\ \\ \le \hslash {\omega }_{{\displaystyle \frac{\lambda }{2}}}\left({♭}_{2m_k},{♭}_{2{\rho }_k-2}\right)+{\hslash }^2{\omega }_{{\displaystyle \frac{\lambda }{4}}}\left({♭}_{2{\rho }_k-2},{♭}_{2{\rho }_k-1}\right)+{\hslash }^2{\omega }_{{\displaystyle \frac{\lambda }{4}}}\left({♭}_{2{\rho }_k-1},{♭}_{2{\rho }_k}\right)\hfill \\ \\ \le \hslash {\omega }_{\lambda }\left({♭}_{2m_k},{♭}_{2{\rho }_k-2}\right)+{\hslash }^2{\omega }_{\lambda }\left({♭}_{2{\rho }_k-2},{♭}_{2{\rho }_k-1}\right)+{\hslash }^2{\omega }_{\lambda }\left({♭}_{2{\rho }_k-1},{♭}_{2{\rho }_k}\right).\hfill \end{array}$$

(20)

Taking the limit as $k\to \infty$, by using (15) and (1), we conclude that

$$\underset{k\to \infty }{limsup}{\omega }_{\lambda }\left({♭}_{2m_k},{♭}_{2{\rho }_k}\right)\le \hslash \epsilon .$$

(21)

By the triangle inequality of the modular metric, we have

$${\omega }_{\lambda }\left({♭}_{2m_k+1},{♭}_{2{\rho }_k-1}\right)\le \hslash {\omega }_{\lambda }\left({♭}_{2m_k+1},{♭}_{2m_k}\right)+\hslash {\omega }_{\lambda }\left({♭}_{2m_k},{♭}_{2{\rho }_k-1}\right).$$

(22)

Taking the limit as $k\to \infty$ and using (19), the subsequent expression is yielded:

$$\underset{k\to \infty }{limsup}{\omega }_{\lambda }\left({♭}_{2m_k+1},{♭}_{2{\rho }_k-1}\right)\le \hslash 0+\hslash \hslash \epsilon ={\hslash }^2\epsilon .$$

(23)

On the other hand, by using (14), if we use (17), (19), (21) and (23), then we obtain the expressions below:

$${\displaystyle \frac{\epsilon }{2\hslash }}\le \underset{k\to \infty }{limsup}{\varpi }_{\lambda }^{p_b}\left({♭}_{2m_{k+1}},{♭}_{2{\rho }_k}\right),$$

(24)

$$\underset{k\to \infty }{limsup}{\varpi }_{\lambda }^{p_b}\left({♭}_{2m_k},{♭}_{2{\rho }_k-1}\right)\le {\displaystyle \frac{\hslash \epsilon }{2}},$$

(25)

$$\underset{k\to \infty }{limsup}{\varpi }_{\lambda }^{p_b}\left({♭}_{2m_k},{♭}_{2{\rho }_k}\right)\le {\displaystyle \frac{\hslash \epsilon }{2}},$$

(26)

$$\underset{k\to \infty }{limsup}{\varpi }_{\lambda }^{p_b}\left({♭}_{2m_{k+1}},{♭}_{2{\rho }_k-1}\right)\le {\displaystyle \frac{{\hslash }^2\epsilon }{2}}.$$

(27)

Since $\left({♭}_{2m_k},{♭}_{2{\rho }_k-1}\right)\in I$, by (3), we attain

$$\begin{array}{c}ℳ\left({♭}_{2m_k},{♭}_{2{\rho }_k-1}\right)\hfill \\ \\ =max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }_k-1},\mathcal{Q}{♭}_{2{\rho }_k-1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2m_k},\daleth {♭}_{2m_k}\right),{\varpi }_{\lambda }^{p_b}\left({♭}_{2m_k},{♭}_{2{\rho }_k-1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }_k-1},\mathcal{Q}{♭}_{2{\rho }_k-1}\right),\hfill \\ {\varpi }_{\lambda }^{p_b}\left({♭}_{2m_k},{♭}_{2{\rho }_k-1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2m_k},\daleth {♭}_{2m_k}\right),c·{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }_k-1},\daleth {♭}_{2m_k}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2m_k},\mathcal{Q}{♭}_{2{\rho }_k-1}\right)\hfill \end{array}\right\}\hfill \\ \\ =max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }_k-1},{♭}_{2{\rho }_k}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2m_k},{♭}_{2m_k+1}\right),{\varpi }_{\lambda }^{p_b}\left({♭}_{2m_k},{♭}_{2{\rho }_k-1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }_k-1},{♭}_{2{\rho }_k}\right),\hfill \\ {\varpi }_{\lambda }^{p_b}\left({♭}_{2m_k},{♭}_{2{\rho }_k-1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2m_k},{♭}_{2m_k+1}\right),c·{\varpi }_{\lambda }^{p_b}\left({♭}_{2{\rho }_k-1},{♭}_{2m_k+1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2m_k},{♭}_{2{\rho }_k}\right)\hfill \end{array}\right\}.\hfill \end{array}$$

From (24), we achieve

$${\left({\displaystyle \frac{\epsilon }{2\hslash }}\right)}^2\le {\left(\underset{k\to \infty }{limsup}{\varpi }_{\lambda }^{p_b}\left({♭}_{2m_{k+1}},{♭}_{2{\rho }_k}\right)\right)}^2.$$

Thereupon, we conclude that

$$\begin{array}{c}\gimel \left({\hslash }^3{\displaystyle \frac{{\epsilon }^2}{4}}\right)=\gimel \left({\hslash }^5{\left({\displaystyle \frac{\epsilon }{2\hslash }}\right)}^2\right)\le \gimel \left({\hslash }^5{\left[\underset{k\to \infty }{lim}sup{\varpi }_{\lambda }^{p_b}\left({♭}_{2m_{k+1}},{♭}_{2{\rho }_k}\right)\right]}^2\right)\hfill \\ \\ =\underset{k\to \infty }{lim}sup\gimel \left({\hslash }^5·{\varpi }_{\lambda }^{p_b}{\left(\daleth {♭}_{2m_k},\mathcal{Q}{♭}_{2{\rho }_k-1}\right)}^2\right)\hfill \\ \\ \le \underset{k\to \infty }{lim}supC\left[\gimel \left(ℳ\left({♭}_{2m_k},{♭}_{2{\rho }_k-1}\right)\right)\right]\hfill \\ \\ \le C\left[\gimel \left(max\left(0,0,0,c{\displaystyle \frac{{\hslash }^2\epsilon }{2}}.{\displaystyle \frac{\hslash \epsilon }{2}}\right)\right)\right]\hfill \\ \\ <\gimel \left(c{\displaystyle \frac{{\hslash }^3{\epsilon }^2}{4}}\right)\hfill \end{array}$$

which implies that

$$\gimel \left({\hslash }^3{\displaystyle \frac{{\epsilon }^2}{4}}\right)<\gimel \left(c{\displaystyle \frac{{\hslash }^3{\epsilon }^2}{4}}\right)<\gimel \left({\displaystyle \frac{{\hslash }^3{\epsilon }^2}{4}}\right).$$

(28)

The aforementioned expression elucidates a contradiction to the defined properties of the function ℷ. Through rigorous analysis, it has been demonstrated that the sequence ${\left\{{♭}_{\rho }\right\}}_{\rho \in \mathbb{N}}$ constitutes an $\omega$-Cauchy sequence within the context of ${\mathcal{U}}_{\omega }^{*}$. Leveraging Lemma 2$\left(i\right)$, it is further established that this sequence also qualifies as a ${\varpi }^{p_b}$-Cauchy sequence in the space ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$. Given the ${\varpi }^{p_b}$-complete partial modular b-metric space nature of ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$, Lemma 2$\left(ii\right)$ implicates that ${\mathcal{U}}_{\omega }^{*}$ inherently possesses the attribute of being an $\omega$-complete modular b-metric space. Consequently, it is deduced that there exists an element ${♭}^{*}$ within ${\mathcal{U}}_{\varpi }^{*}$ towards which ${♭}_{\rho }$ converges, formally expressed as $\underset{\rho \to \infty }{lim}{\omega }_{\lambda }\left({♭}_{\rho },{♭}^{*}\right)=0$. By Lemma 2$\left(iii\right)$, we get

$$\underset{\rho \to \infty }{lim}{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },{♭}^{*}\right)={\varpi }_{\lambda }^{p_b}\left({♭}^{*},{♭}^{*}\right)=\underset{\rho ,m\to \infty }{lim}{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },{♭}_m\right),\forall \lambda >0.$$

(29)

Since $\underset{\rho ,m\to \infty }{lim}{\varpi }_{\lambda }^{p_b}\left({♭}_{\rho },{♭}_m\right)=0$, so that ${\varpi }_{\lambda }^{p_b}\left({♭}^{*},{♭}^{*}\right)=0$. Thus, the sequence ${\left\{{♭}_{\rho }\right\}}_{\rho \in \mathbb{N}}$ converges to ${♭}^{*}$ in ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$. Now, we claim that $\daleth {♭}^{*}=\mathcal{Q}{♭}^{*}={♭}^{*}$. By (24), for all $\lambda >0$, we obtain

$$\begin{array}{c}\underset{\rho \to \infty }{lim}{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},{♭}^{*}\right)=0\mathrm{and}\underset{\rho \to \infty }{lim}{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +2},{♭}^{*}\right)=0.\hfill \end{array}$$

(30)

Due to ℸ and $\mathcal{Q}$ being continuous, we derive that

$$\underset{\rho \to \infty }{lim}{\varpi }_{\lambda }^{p_b}\left(\daleth {♭}_{2\rho },\daleth {♭}^{*}\right)=0\mathrm{and}\underset{\rho \to \infty }{lim}{\varpi }_{\lambda }^{p_b}\left(\mathcal{Q}{♭}_{2\rho +1},\mathcal{Q}{♭}^{*}\right)=0.$$

(31)

By Lemma 3, the subsequent expression is achieved:

$${\displaystyle \frac{1}{\hslash }}{\varpi }_{\lambda }^{p_b}\left({♭}^{*},\daleth {♭}^{*}\right)\le \underset{\rho \to \infty }{lim}inf{\varpi }_{\lambda }^{p_b}\left({♭}_{2\rho +1},\daleth {♭}^{*}\right)=\underset{\rho \to \infty }{lim}inf{\varpi }_{\lambda }^{p_b}\left(\daleth {♭}_{2\rho },\daleth {♭}^{*}\right)=0.$$

(32)

Thus, ${\varpi }_{\lambda }^{p_b}\left(\daleth {♭}^{*},{♭}^{*}\right)={\varpi }_{\lambda }^{p_b}\left({♭}^{*},{♭}^{*}\right)={\varpi }_{\lambda }^{p_b}\left(\daleth {♭}^{*},\daleth {♭}^{*}\right)=0$. This implies $\daleth {♭}^{*}={♭}^{*}$. Similar arguments lead us to have $\mathcal{Q}{♭}^{*}={♭}^{*}$. Hence, $\daleth {♭}^{*}=\mathcal{Q}{♭}^{*}={♭}^{*},$ that is, $\daleth \mathrm{and}\mathcal{Q}$ have a common fixed point ${♭}^{*}\in {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$. □

The theorem under consideration articulates that, provided $\mathrm{\Gamma }\left(♭,\ell ,\mathcal{R}\right)$ is non-empty and condition $\left({\mathcal{L}}_2\right)$ is satisfied, the common fixed points ℸ and $\mathcal{Q}$ within ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ are unequivocally unique.

Theorem 5. 

Let ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ be a ${\varpi }^{p_b}$-complete $𝒫{ℳ}_bℳ𝒮$, $\daleth ,\mathcal{Q}:{\mathcal{U}}_{{\varpi }^{p_b}}^{*}\to {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ be two mappings on ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ and $\mathcal{R}$ be a transitive binary relation on ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$. If, in addition to assumptions $\left(i\right)$–$\left(iv\right)$ in Theorem 4, $\mathrm{\Gamma }\left(♭,\ell ,\mathcal{R}\right)\ne \varnothing$ and $\left({\mathcal{L}}_2\right)$ are provided, then ℸ and $\mathcal{Q}$ have a unique common fixed point in ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$.

Proof. 

The existence of a common fixed point of ℸ and $\mathcal{Q}$ has been established in Theorem 4. Contrarily, we assume that ♭ and $\widetilde{♭}$ represent two distinct common fixed points of ℸ and $\mathcal{Q}$, it necessitates the consideration of the class of paths of finite length h within $\mathbb{R}$ connecting ♭ and $\widetilde{♭}$. This specific class of paths is denoted as $\mathrm{\Gamma }\left(♭,\widetilde{♭},\mathbb{R}\right)$. Let one of the paths be $\left\{A_0,A_1,A_2,\cdots ,A_h\right\}$ in ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ from ♭ and $\widetilde{♭}$ with

$$A_0=♭,A_h=\widetilde{♭},\left(A_j,A_{j+1}\right)\in \mathcal{R};j=0,1,2,3,\cdots ,\left(h-1\right).$$

By transitivity of $\mathcal{R}$, we have

$$\left(♭,A_1\right)\in \mathcal{R},\left(A_1,A_2\right)\in \mathcal{R},\cdots ,\left(A_{h-1},\widetilde{♭}\right)\in \mathcal{R}\Rightarrow \left(♭,\widetilde{♭}\right)\in \mathcal{R}.$$

Consequently, it can be deduced that the following inequality is maintained:

$$\begin{array}{c}{\displaystyle \frac{1}{2\hslash }}min\left\{{\varpi }_{\lambda }^{p_b}\left(♭,\daleth ♭\right),{\varpi }_{\lambda }^{p_b}\left(\widetilde{♭},\mathcal{Q}\widetilde{♭}\right)\right\}\le {\varpi }_{\lambda }^{p_b}\left(♭,\widetilde{♭}\right)\Rightarrow \hfill \\ \\ \gimel \left({\hslash }^5{\left[{\varpi }_{\lambda }^{p_b}\left(\daleth ♭,\mathcal{Q}\widetilde{♭}\right)\right]}^2\right)\le C\left[\gimel \left(ℳ\left(♭,\widetilde{♭}\right)\right)\right],\hfill \end{array}$$

where $0\le c<1$ and

$$\begin{array}{c}ℳ\left(♭,\widetilde{♭}\right)=max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left(\widetilde{♭},\mathcal{Q}\widetilde{♭}\right)·{\varpi }_{\lambda }^{p_b}\left(♭,\daleth ♭\right),{\varpi }_{\lambda }^{p_b}\left(♭,\widetilde{♭}\right)·{\varpi }_{\lambda }^{p_b}\left(\widetilde{♭},\mathcal{Q}\widetilde{♭}\right),\\ {\varpi }_{\lambda }^{p_b}\left(♭,\widetilde{♭}\right)·{\varpi }_{\lambda }^{p_b}\left(♭,\daleth ♭\right),c·{\varpi }_{\lambda }^{p_b}\left(\widetilde{♭},\daleth ♭\right)·{\varpi }_{\lambda }^{p_b}\left(♭,\mathcal{Q}\widetilde{♭}\right)\end{array}\right\}\hfill \\ \\ =max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left(\widetilde{♭},\widetilde{♭}\right)·{\varpi }_{\lambda }^{p_b}\left(♭,♭\right),{\varpi }_{\lambda }^{p_b}\left(♭,\widetilde{♭}\right)·{\varpi }_{\lambda }^{p_b}\left(\widetilde{♭},\widetilde{♭}\right),\\ {\varpi }_{\lambda }^{p_b}\left(♭,\widetilde{♭}\right)·{\varpi }_{\lambda }^{p_b}\left(♭,♭\right),c·{\varpi }_{\lambda }^{p_b}\left(\widetilde{♭},♭\right)·{\varpi }_{\lambda }^{p_b}\left(♭,\widetilde{♭}\right)\end{array}\right\}\hfill \\ \\ =max\left\{0,0,0,c{\left[{\varpi }_{\lambda }^{p_b}\left(♭,\widetilde{♭}\right)\right]}^2\right\}.\hfill \end{array}$$

Hence, by using $0\le c<1$, we obtain

$$\begin{array}{c}\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left(♭,\widetilde{♭}\right)\right]}^2\right)=\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left(\daleth ♭,\mathcal{Q}\widetilde{♭}\right)\right]}^2\right)\le \gimel \left({\hslash }^5{\left[{\varpi }_{\lambda }^{p_b}\left(\daleth ♭,\mathcal{Q}\widetilde{♭}\right)\right]}^2\right)\hfill \\ \\ \le C\left[\gimel \left(ℳ\left(♭,\widetilde{♭}\right)\right)\right]=C\left[\gimel \left(c{\left[{\varpi }_{\lambda }^{p_b}\left(♭,\widetilde{♭}\right)\right]}^2\right)\right]\hfill \\ \\ <C\left[\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left(♭,\widetilde{♭}\right)\right]}^2\right)\right]\hfill \\ \\ <\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left(♭,\widetilde{♭}\right)\right]}^2\right);\hfill \end{array}$$

however, this contradicts the definition of ℷ, that is, $♭=\widetilde{♭}$, which explains that ♭ is a unique common fixed point of ℸ and $\mathcal{Q}$. □

Definition 12. 

An $𝒫{ℳ}_bℳ𝒮$${\mathcal{U}}_{{\varpi }^{p_b}}$ equipped with a binary relation $\mathcal{R}$ is called $\mathcal{R}$-regular if for each sequence ${\left\{{♭}_{\rho }\right\}}_{\rho \in \mathbb{N}}$ in ${\mathcal{U}}_{{\varpi }^{p_b}}$, whenever $\left({♭}_{\rho },{♭}_{\rho +1}\right)\in \mathcal{R}$ and ${♭}_{\rho }\stackrel{{\varpi }_{\lambda }^{p_b}}{\to }♭$, we have $\left({♭}_{\rho },♭\right)\in \mathcal{R}$ for all $\rho \in \mathbb{N}\cup \left\{0\right\}$.

Remark 2. 

If the self-mappings ℸ and $\mathcal{Q}$ are not ${\varpi }^{p_b}$-continuous, then assuming partial modular b-metric space ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ to be $\mathcal{R}$-regular and that $\left({\mathcal{L}}_1\right)$ is satisfied, the existence of the common fixed points of ℸ and $\mathcal{Q}$ is obtained.

Theorem 6. 

Let ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ be a ${\varpi }^{p_b}$-complete $𝒫{ℳ}_bℳ𝒮$, $\daleth ,\mathcal{Q}:{\mathcal{U}}_{{\varpi }^{p_b}}^{*}\to {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ be two mappings and $\mathcal{R}$ be an anti-symmetric binary relation on ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$. If, in addition to assumptions $\left(i\right)$–$\left(iii\right)$ in Theorem 4, and ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ being $\mathcal{R}$-regular and $\left({\mathcal{L}}_1\right)$ being provided, then there is a common fixed point of ℸ and $\mathcal{Q}$ in ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$.

Proof. 

In accordance with Theorem 4, it is established that the ordered pair $\left({♭}_{\rho },{♭}_{\rho +1}\right)$ is an element of the set $\mathcal{R}$. Furthermore, it is demonstrated that the sequence ${♭}_{\rho }$ converges to ${♭}^{*}$ as $\rho$ takes values within the set of natural numbers, $\mathbb{N}$. It is also specified that for all $\rho$ belonging to $\mathbb{N}$, the entity ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ maintains its characteristic of regularity. This elucidation presents two distinct cases for consideration.

Case 1. Consider that the sequence ${\left\{{♭}_{\rho }\right\}}_{\rho \in \mathbb{N}}$ is constant. Let ${♭}_{\rho }={♭}^{*}$ for each $\rho \in \mathbb{N}$, so that ${♭}_{2\rho }={♭}^{*}$ and $\daleth {♭}^{*}=\daleth {♭}_{2\rho }={♭}_{2\rho +1}$. Since ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ is regular, $\left({♭}_{2\rho +1},{♭}^{*}\right)=\left(\daleth {♭}^{*},{♭}^{*}\right)\in \mathcal{R}$. We know that $\left({♭}_{2\rho },{♭}_{2\rho +1}\right)\in \mathcal{R}$; thus, $\left({♭}^{*},\daleth {♭}^{*}\right)\in \mathcal{R}$. Given that $\mathcal{R}$ is an anti-symmetric relation, ${♭}^{*}=\daleth {♭}^{*}$. Employing analogous reasoning, it can be deduced that ${♭}^{*}=\mathcal{Q}{♭}^{*}$, thereby satisfying the required conditions. This assertion aligns with the foundational principles of the relation theory, further substantiating the anti-symmetric properties of $\mathcal{R}$.

Case 2. If ${\left\{{♭}_{\rho }\right\}}_{\rho \in \mathbb{N}}$ is not constant and is an arbitrary sequence, we claim that ${\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)=0.$ We proceed against this and let ${\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)>0$. It is proved in Theorem 4 that $\underset{i\to \infty }{lim}{♭}_{2i+1}={♭}^{*}$, so there must be an integer ${\rho }_0>0$, such that

$${\varpi }_{\lambda }^{p_b}\left({♭}_{2i+1},\mathcal{Q}{♭}^{*}\right)>0,{\varpi }_{\lambda }^{p_b}\left({♭}_{2i},{♭}^{*}\right)<{\displaystyle \frac{{\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)}{4}},\mathrm{for}\mathrm{all}i\ge {\rho }_0.$$

It is assumed that ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ is $\mathcal{R}$-regular, and by Theorem 4, we know that ${♭}_{2i}\to {♭}^{*}\mathrm{as}i\to \infty$; thus, $\left({♭}_{2i},{♭}^{*}\right)\in \mathcal{R}$. By the contractive condition, the monotonicity of ℷ, and Lemma 3, we obtain

$${\displaystyle \frac{1}{\hslash }}{\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)\le \underset{i\to \infty }{lim}inf{\varpi }_{\lambda }^{p_b}\left({♭}_{2i+1},\mathcal{Q}{♭}^{*}\right)$$

and

$$\begin{array}{c}\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)\right]}^2\right)\le \gimel \left({\hslash }^2\underset{i\to \infty }{lim}inf{\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2i+1},\mathcal{Q}{♭}^{*}\right)\right]}^2\right)\le \gimel \left({\hslash }^5\underset{i\to \infty }{lim}inf{\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2i+1},\mathcal{Q}{♭}^{*}\right)\right]}^2\right)\hfill \\ \\ \le \underset{i\to \infty }{lim}inf\gimel \left({\hslash }^5{\left[{\varpi }_{\lambda }^{p_b}\left(\daleth {♭}_{2i},\mathcal{Q}{♭}^{*}\right)\right]}^2\right).\hfill \end{array}$$

Hence, we have

$$\begin{array}{c}{\displaystyle \frac{1}{2\hslash }}min\left\{{\varpi }_{\lambda }^{p_b}\left({♭}_{2i},\daleth {♭}_{2i}\right),{\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)\right\}\le {\varpi }_{\lambda }^{p_b}\left({♭}_{2i},{♭}^{*}\right)\Rightarrow \hfill \\ \\ \gimel \left({\hslash }^5{\left[{\varpi }_{\lambda }^{p_b}\left(\daleth {♭}_{2i},\mathcal{Q}{♭}^{*}\right)\right]}^2\right)\le C\left[\gimel \left(ℳ\left({♭}_{2i},{♭}^{*}\right)\right)\right],\hfill \end{array}$$

where $0\le c<1$ and

$$\begin{array}{c}ℳ\left({♭}_{2i},{♭}^{*}\right)=max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2i},\daleth {♭}_{2i}\right),{\varpi }_{\lambda }^{p_b}\left({♭}_{2i},{♭}^{*}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right),\hfill \\ {\varpi }_{\lambda }^{p_b}\left({♭}_{2i},{♭}^{*}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2i},\daleth {♭}_{2i}\right),c·{\varpi }_{\lambda }^{p_b}\left({♭}^{*},\daleth {♭}_{2i}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2i},\mathcal{Q}{♭}^{*}\right)\hfill \end{array}\right\}\hfill \\ \\ =max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2i},{♭}_{2i+1}\right),{\varpi }_{\lambda }^{p_b}\left({♭}_{2i},{♭}^{*}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right),\hfill \\ {\varpi }_{\lambda }^{p_b}\left({♭}_{2i},{♭}^{*}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2i},{♭}_{2i+1}\right),c·{\varpi }_{\lambda }^{p_b}\left({♭}^{*},{♭}_{2i+1}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}_{2i},\mathcal{Q}{♭}^{*}\right)\hfill \end{array}\right\}\hfill \\ \\ ={\varpi }_{\lambda }^{p_b}\left({♭}^{*},{♭}_{2i}\right)·{\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right).\hfill \end{array}$$

Thereby, we get

$$\begin{array}{c}\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left(\daleth {♭}_{2i},\mathcal{Q}{♭}^{*}\right)\right]}^2\right)\le \gimel \left({\hslash }^5{\left[{\varpi }_{\lambda }^{p_b}\left(\daleth {♭}_{2i},\mathcal{Q}{♭}^{*}\right)\right]}^2\right)\le C\left[\gimel \left({\varpi }_{\lambda }^{p_b}\left({♭}^{*},{♭}_{2i}\right){\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)\right)\right]\hfill \\ \\ <\gimel \left({\varpi }_{\lambda }^{p_b}\left({♭}^{*},{♭}_{2i}\right){\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)\right)\hfill \end{array}$$

such that

$$\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left(\daleth {♭}_{2i},\mathcal{Q}{♭}^{*}\right)\right]}^2\right)<\gimel \left({\varpi }_{\lambda }^{p_b}\left({♭}^{*},{♭}_{2i}\right){\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)\right)$$

which implies

$${\left[{\varpi }_{\lambda }^{p_b}\left(\daleth {♭}_{2i},\mathcal{Q}{♭}^{*}\right)\right]}^2<{\varpi }_{\lambda }^{p_b}\left({♭}^{*},{♭}_{2i}\right){\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right),$$

that is,

$${\varpi }_{\lambda }^{p_b}\left(\daleth {♭}_{2i},\mathcal{Q}{♭}^{*}\right)<\sqrt{{\varpi }_{\lambda }^{p_b}\left({♭}^{*},{♭}_{2i}\right){\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)}.$$

Then, we obtain

$$\begin{array}{c}\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)\right]}^2\right)\le \gimel \left({\hslash }^2\underset{i\to \infty }{lim}inf{\left[{\varpi }_{\lambda }^{p_b}\left({♭}_{2i+1},\mathcal{Q}{♭}^{*}\right)\right]}^2\right)\hfill \\ \\ \le \underset{i\to \infty }{lim}inf\gimel \left({\hslash }^5{\left[{\varpi }_{\lambda }^{p_b}\left(\daleth {♭}_{2i},\mathcal{Q}{♭}^{*}\right)\right]}^2\right)\hfill \\ \\ \le \underset{i\to \infty }{lim}infC\left[\gimel \left(\sqrt{{\varpi }_{\lambda }^{p_b}\left({♭}^{*},{♭}_{2i}\right){\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)}\right)\right]\hfill \\ \\ <\underset{i\to \infty }{lim}infC\left[\gimel \left(\sqrt{{\displaystyle \frac{{\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)}{4}}{\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)}\right)\right]\hfill \\ \\ <\gimel \left(\sqrt{{\displaystyle \frac{{\left[{\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)\right]}^2}{4}}}\right).\hfill \end{array}$$

Thus,

$$\gimel \left({\left[{\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)\right]}^2\right)<\gimel \left(\sqrt{{\displaystyle \frac{{\left[{\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)\right]}^2}{4}}}\right)$$

implies

$${\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)<{\displaystyle \frac{1}{2}}{\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right).$$

This contradicts the definition of mapping ℷ. Then, ${\varpi }_{\lambda }^{p_b}\left({♭}^{*},\mathcal{Q}{♭}^{*}\right)=0$. By

$${\varpi }_{\lambda }^{p_b}\left(♭,♭\right)\le {\varpi }_{\lambda }^{p_b}\left(♭,\ell \right),\forall ♭,\ell \in {\mathcal{U}}_{{\varpi }_{\lambda }^{p_b}}^{*}$$

we have the following expression:

$${\varpi }_{\lambda }^{p_b}\left(\mathcal{Q}{♭}^{*},\mathcal{Q}{♭}^{*}\right)=0={\varpi }_{\lambda }^{p_b}\left({♭}^{*},{♭}^{*}\right).$$

Therefore, ${♭}^{*}=\mathcal{Q}{♭}^{*}$. By interchanging the roles of $\mathcal{Q}$ and ℸ, we have ${♭}^{*}=\daleth {♭}^{*}$. Hence, $\daleth {♭}^{*}=\mathcal{Q}{♭}^{*}={♭}^{*}$, that is, ${♭}^{*}$ is a common fixed point of ℸ and $\mathcal{Q}$ in ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$. □

Theorem 7. 

Let ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ be a ${\varpi }^{p_b}$-complete $𝒫{ℳ}_bℳ𝒮$, $\daleth ,\mathcal{Q}:{\mathcal{U}}_{{\varpi }^{p_b}}^{*}\to {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ be two mappings and $\mathcal{R}$ be a transitive and anti-symmetric binary relation on ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$, such that $\mathrm{\Gamma }\left(♭,\ell ,\mathcal{R}\right)$ is non-empty for all $♭,\ell \in {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$. If, in addition to assumptions $\left(i\right)$–$\left(iii\right)$ in Theorem 4, ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ is $\mathcal{R}$-regular and $\left({\mathcal{L}}_2\right)$ are provided, then there exists a unique common fixed point of ℸ and $\mathcal{Q}$ in ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$.

Proof. 

The proof of the theorem is obtained directly from Theorems 5 and 6. □

The forthcoming illustration elucidates the primary outcome.

Example 5. 

Let $\mathcal{U}=\left[0,2\right]$ and consider the function ${\varpi }_{\lambda }^{p_b}:\left(0,\infty \right)\times \mathcal{U}\times \mathcal{U}\to \left[0,\infty \right]$ defined by

$${\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)=e^{-\lambda }{\left|♭-\ell \right|}^2$$

for all $♭,\ell \in \mathcal{U}$. Then, ($\mathcal{U},{\varpi }_{\lambda }^{p_b})$ is ${\varpi }^{p_b}$-complete $𝒫{ℳ}_bℳ𝒮$ with the coefficient $\hslash =2$. Consider $\gimel :\left(0,\infty \right)\to \left(0,\infty \right)$ as $\gimel \left(t\right)=t{e}^t$, for every $t>0$. Note that $\gimel \in \mathrm{\Delta }$. Let $C\left(t\right)={\displaystyle \frac{t}{2}}$, which is a continuous comparison function. Define the binary relation by

$$\mathcal{R}=\left\{\left(0,0\right),\left(0,{\displaystyle \frac{1}{5}}\right),\left({\displaystyle \frac{1}{5}},0\right),\left(0,1\right),\left(1,0\right),\left(1,1\right),\left(0,2\right),\left(2,0\right)\right\}.$$

We observe that there exists ${♭}_0\in \mathcal{U}$, such that $\left({♭}_0,\daleth {♭}_0\right)\in \mathcal{R}$ by the definition of $\mathcal{R}$, so the assumption $\left(ii\right)$ is satisfied in Theorem 4. Consider the mappings $\daleth ,\mathcal{Q}:\mathcal{U}\to \mathcal{U}$ as

$$\mathcal{Q}♭=\left\{\begin{array}{c}♭,0\le ♭\le {\displaystyle \frac{1}{5}},\hfill \\ {\displaystyle \frac{1}{5}},{\displaystyle \frac{1}{5}}<♭\le 2,\hfill \end{array}\right.\mathrm{and}\daleth ♭=0$$

for all $♭\in \mathcal{U}$. Clearly, $\mathcal{R}$ is $\left(\daleth ,\mathcal{Q}\right)$-closed and $\daleth ,\mathcal{Q}$ are ${\varpi }^{p_b}$-continuous. These verify the assumptions $\left(iii\right)$ and $\left(iv\right)$ of Theorem 4. Now, we show that ℸ and $\mathcal{Q}$ form a generalized ${\gimel }_C$-type Suzuki $\mathcal{R}$-contraction. Let

$$\begin{array}{c}I=\left\{\left(♭,\ell \right)\in \mathcal{R}:{\varpi }_{\lambda }^{p_b}\left(\daleth ♭,\mathcal{Q}\ell \right)>0\right\},\hfill \\ \\ I=\left\{\left(0,1\right),\left(0,2\right)\right\}.\hfill \end{array}$$

Now, for all $\left(♭,\ell \right)\in I$, we get

$${\displaystyle \frac{1}{2\hslash }}min\left\{{\varpi }_{\lambda }^{p_b}\left(♭,\daleth ♭\right),{\varpi }_{\lambda }^{p_b}\left(\ell ,\mathcal{Q}\ell \right)\right\}\le {\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)\Rightarrow \gimel \left({\hslash }^5{\left[{\varpi }_{\lambda }^{p_b}\left(\daleth ♭,\mathcal{Q}\ell \right)\right]}^2\right)\le C\left[\gimel \left(ℳ\left(♭,\ell \right)\right)\right],$$

where $0\le c<1$, and

$$ℳ\left(♭,\ell \right)=max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left(\ell ,\mathcal{Q}\ell \right)·{\varpi }_{\lambda }^{p_b}\left(♭,\daleth ♭\right),{\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)·{\varpi }_{\lambda }^{p_b}\left(\ell ,\mathcal{Q}\ell \right),\hfill \\ {\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)·{\varpi }_{\lambda }^{p_b}\left(♭,\daleth ♭\right),c·{\varpi }_{\lambda }^{p_b}\left(\ell ,\daleth ♭\right)·{\varpi }_{\lambda }^{p_b}\left(♭,\mathcal{Q}\ell \right)\hfill \end{array}\right\}.$$

Case 1. Since $\left(0,1\right)\in I$, setting $♭=0$ and $\ell =1$ in (3), we obtain

$${\displaystyle \frac{1}{2\hslash }}min\left\{{\varpi }_{\lambda }^{p_b}\left(0,\daleth 0\right),{\varpi }_{\lambda }^{p_b}\left(1,\mathcal{Q}1\right)\right\}={\displaystyle \frac{1}{2\hslash }}min\left\{0,e^{-\lambda }{\displaystyle \frac{16}{5^2}}\right\}=0\le {\varpi }_{\lambda }^{p_b}\left(0,1\right)=e^{-\lambda }.$$

Thereby, we have

$$\begin{array}{c}ℳ\left(0,1\right)=max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left(1,\mathcal{Q}1\right)·{\varpi }_{\lambda }^{p_b}\left(0,\daleth 0\right),{\varpi }_{\lambda }^{p_b}\left(0,1\right)·{\varpi }_{\lambda }^{p_b}\left(1,\mathcal{Q}1\right),\hfill \\ {\varpi }_{\lambda }^{p_b}\left(0,1\right)·{\varpi }_{\lambda }^{p_b}\left(0,\daleth 0\right),c·{\varpi }_{\lambda }^{p_b}\left(1,\daleth 0\right)·{\varpi }_{\lambda }^{p_b}\left(0,\mathcal{Q}1\right)\hfill \end{array}\right\}\hfill \\ \\ =max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left(1,{\displaystyle \frac{1}{5}}\right)·{\varpi }_{\lambda }^{p_b}\left(0,0\right),{\varpi }_{\lambda }^{p_b}\left(0,1\right)·{\varpi }_{\lambda }^{p_b}\left(1,{\displaystyle \frac{1}{5}}\right),\hfill \\ {\varpi }_{\lambda }^{p_b}\left(0,1\right)·{\varpi }_{\lambda }^{p_b}\left(0,0\right),c·{\varpi }_{\lambda }^{p_b}\left(1,0\right)·{\varpi }_{\lambda }^{p_b}\left(0,{\displaystyle \frac{1}{5}}\right)\hfill \end{array}\right\}\hfill \\ \\ =max\left\{0,e^{-\lambda }1e^{-\lambda }{\displaystyle \frac{16}{5^2}},0,c{e}^{-\lambda }{\displaystyle \frac{1}{5^2}}{e}^{-\lambda }1\right\}=e^{-2\lambda }{\displaystyle \frac{16}{5^2}}\hfill \end{array}$$

and, in addition to that,

$$\begin{array}{c}\gimel \left({\hslash }^5{\left[{\varpi }_{\lambda }^{p_b}\left(\daleth 0,\mathcal{Q}1\right)\right]}^2\right)=\gimel \left({\hslash }^5{\left[{\varpi }_{\lambda }^{p_b}\left(0,{\displaystyle \frac{1}{5}}\right)\right]}^2\right)=\gimel \left({\hslash }^5\left(e^{-2\lambda }{\displaystyle \frac{1}{5^4}}\right)\right)\hfill \\ \\ \le C\left[\gimel \left(ℳ\left(0,1\right)\right)\right]={\displaystyle \frac{1}{2}}\gimel \left(e^{-2\lambda }{\displaystyle \frac{16}{5^2}}\right),\hfill \end{array}$$

which yields

$${\hslash }^5\left(e^{-2\lambda }{\displaystyle \frac{1}{5^4}}\right)e^{{\hslash }^5\left(e^{-2\lambda }·{\displaystyle \frac{1}{5^4}}\right)}\le {\displaystyle \frac{1}{2}}\left(e^{-2\lambda }{\displaystyle \frac{16}{5^2}}\right)e^{\left(e^{-2\lambda }{\displaystyle \frac{16}{5^2}}\right)}$$

and

$${\displaystyle \frac{1}{25.8}}{\hslash }^5\le e^{\left(e^{-2\lambda }{\displaystyle \frac{16}{5^2}}\right)-{\hslash }^5\left(e^{-2\lambda }{\displaystyle \frac{1}{5^4}}\right)}.$$

Hence, we obtain

$$ln\left({\displaystyle \frac{1}{25.8}}{\hslash }^5\right)\le \left(e^{-2\lambda }·{\displaystyle \frac{16}{5^2}}\right)-{\hslash }^5\left(e^{-2\lambda }{\displaystyle \frac{1}{5^4}}\right)=e^{-2\lambda }\left({\displaystyle \frac{16}{5^2}}-{\displaystyle \frac{{\hslash }^5}{5^4}}\right),$$

and, for $\hslash =2$, $ln\left(0.16\right)\le \left(e^{-2\lambda }0.5888\right)\Rightarrow {\displaystyle \frac{1.83258}{0.5888}}=-3.1124\le e^{-2\lambda }.$

Case 2.  $\left(0,2\right)\in I$ and similarly setting $♭=0$ and $\ell =2$ in (3), we have

$${\displaystyle \frac{1}{2\hslash }}min\left\{{\varpi }_{\lambda }^{p_b}\left(0,\daleth 0\right),{\varpi }_{\lambda }^{p_b}\left(2,\mathcal{Q}2\right)\right\}={\displaystyle \frac{1}{2\hslash }}min\left\{0,e^{-\lambda }{\displaystyle \frac{9^2}{5^2}}\right\}=0\le e^{-\lambda }4.$$

Thus, by considering the statement

$$\begin{array}{c}ℳ\left(0,2\right)=max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left(2,\mathcal{Q}2\right)·{\varpi }_{\lambda }^{p_b}\left(0,\daleth 0\right),{\varpi }_{\lambda }^{p_b}\left(0,2\right)·{\varpi }_{\lambda }^{p_b}\left(2,\mathcal{Q}2\right),\hfill \\ {\varpi }_{\lambda }^{p_b}\left(0,2\right)·{\varpi }_{\lambda }^{p_b}\left(0,\daleth 0\right),c·{\varpi }_{\lambda }^{p_b}\left(2,\daleth 0\right)·{\varpi }_{\lambda }^{p_b}\left(0,\mathcal{Q}2\right)\hfill \end{array}\right\}\hfill \\ \\ =max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left(2,{\displaystyle \frac{1}{5}}\right)·{\varpi }_{\lambda }^{p_b}\left(0,0\right),{\varpi }_{\lambda }^{p_b}\left(0,2\right)·{\varpi }_{\lambda }^{p_b}\left(2,{\displaystyle \frac{1}{5}}\right),\hfill \\ {\varpi }_{\lambda }^{p_b}\left(0,2\right)·{\varpi }_{\lambda }^{p_b}\left(0,0\right),c·{\varpi }_{\lambda }^{p_b}\left(2,0\right)·{\varpi }_{\lambda }^{p_b}\left(0,{\displaystyle \frac{1}{5}}\right)\hfill \end{array}\right\}\hfill \\ \\ =max\left\{0,e^{-\lambda }4·e^{-\lambda }{\displaystyle \frac{81}{25}},0,c·e^{-\lambda }4·e^{-\lambda }{\displaystyle \frac{1}{25}}\right\}=e^{-2\lambda }{\displaystyle \frac{324}{25}},\hfill \end{array}$$

we get

$$\gimel \left({\hslash }^5{\left[{\varpi }_{\lambda }^{p_b}\left(\daleth 0,\mathcal{Q}2\right)\right]}^2\right)=\gimel \left({\hslash }^5\left(e^{-2\lambda }{\displaystyle \frac{1}{5^4}}\right)\right)\le C\left[\gimel \left(ℳ\left(0,2\right)\right)\right]={\displaystyle \frac{1}{2}}\gimel \left(e^{-2\lambda }{\displaystyle \frac{324}{25}}\right)$$

which implies

$${\hslash }^5\left(e^{-2\lambda }{\displaystyle \frac{1}{5^4}}\right)e^{{\hslash }^5\left(e^{-2\lambda }{\displaystyle \frac{1}{5^4}}\right)}\le {\displaystyle \frac{1}{2}}\left(e^{-2\lambda }{\displaystyle \frac{324}{25}}\right)e^{\left(e^{-2\lambda }{\displaystyle \frac{324}{25}}\right)},$$

and, similarly,

$${\displaystyle \frac{{\hslash }^5}{25.162}}\le e^{\left(e^{-2\lambda }{\displaystyle \frac{324}{25}}\right)-{\hslash }^5\left(e^{-2\lambda }{\displaystyle \frac{1}{5^4}}\right)}.$$

Thereupon, we get

$$ln\left({\displaystyle \frac{{\hslash }^5}{25.162}}\right)\le \left(e^{-2\lambda }{\displaystyle \frac{324}{25}}\right)-{\hslash }^5\left(e^{-2\lambda }{\displaystyle \frac{1}{5^4}}\right)=e^{-2\lambda }\left({\displaystyle \frac{324}{25}}-{\displaystyle \frac{{\hslash }^5}{5^4}}\right).$$

Thereby, for $\hslash =2$, we gain

$$ln\left({\displaystyle \frac{32}{25.162}}\right)\le \left(e^{-2\lambda }{\displaystyle \frac{8068}{625}}\right)\Rightarrow ln\left({\displaystyle \frac{32}{25.162}}\right){\displaystyle \frac{625}{8068}}\le e^{-2\lambda }\Rightarrow -0.374995\le e^{-2\lambda }.$$

Thus, the hypotheses of Theorem 4 are satisfied, and so ℸ and $\mathcal{Q}$ admit a common fixed point in ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$. Therefore, $♭=0$ is the unique common fixed point.

The statement above pertains to a theorem that is derived as a consequence of a single self-mapping.

Corollary 1. 

Let ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ be a ${\varpi }^{p_b}$-complete $𝒫{ℳ}_bℳ𝒮$ and $\mathcal{R}$ be a transitive binary relation on ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$. Let $I=\left\{\left(♭,\ell \right)\in \mathcal{R}:{\varpi }_{\lambda }^{p_b}\left(\daleth ♭,\daleth \ell \right)>0\right\}.$ Presume that the self-mapping ℸ defined on ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ satisfies the following conditions:

$\left(i\right)$

There exist a continuous comparison function C and $\gimel \in \mathrm{\Delta }$, such that for all $♭,\ell \in I$, $\hslash \ge 1$ and $\forall \lambda >0$,

$${\displaystyle \frac{1}{2\hslash }}{\varpi }_{\lambda }^{p_b}\left(♭,\daleth ♭\right)\le {\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)\Rightarrow \gimel \left({\hslash }^4{\left[{\varpi }_{\lambda }^{p_b}\left(\daleth ♭,\daleth \ell \right)\right]}^2\right)\le C\left[\gimel \left(ℳ\left(♭,\ell \right)\right)\right],$$

where $0\le c<1$ and

$$ℳ\left(♭,\ell \right)=max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left(♭,\daleth ♭\right)·{\varpi }_{\lambda }^{p_b}\left(\ell ,\daleth \ell \right),{\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)·{\varpi }_{\lambda }^{p_b}\left(♭,\daleth ♭\right),\hfill \\ {\varpi }_{\lambda }^{p_b}\left(♭,\ell \right)·{\varpi }_{\lambda }^{p_b}\left(\ell ,\daleth \ell \right),c·{\varpi }_{\lambda }^{p_b}\left(♭,\daleth \ell \right)·{\varpi }_{\lambda }^{p_b}\left(\ell ,\daleth ♭\right)\hfill \end{array}\right\};$$

$\left(ii\right)$

$\mathrm{\Gamma }\left(♭,\ell ,\mathcal{R}\right)$ is non-empty for all $♭,\ell \in {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$;

$\left(iii\right)$

There exists ${♭}_0\in {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$, such that $\left({♭}_0,\daleth {♭}_0\right)\in \mathcal{R}$ and $\mathcal{R}$ is ℸ-closed;

$\left(vi\right)$

Either ℸ is ${\varpi }^{p_b}$-continuous or ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ is $\mathcal{R}$-regular.

If $\left({\mathcal{L}}_1\right)$ is satisfied, then ℸ owns a fixed point in ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$. Moreover, if $\left({\mathcal{L}}_2\right)$ is satisfied with $\left({\mathcal{L}}_1\right)$, then ℸ has a unique fixed point in ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$.

Also, we have the following results without the Suzuki restriction.

Corollary 2. 

Let ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ be a ${\varpi }^{p_b}$-complete $𝒫{ℳ}_bℳ𝒮$ and $\mathcal{R}$ be a transitive and anti-symmetric binary relation on ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$. Assume that the mapping $\daleth :{\mathcal{U}}_{{\varpi }^{p_b}}^{*}\to {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ defined on ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ provides the subsequent statements:

$\left(i\right)$

There exist a $\theta \in \tilde{\Theta }$ and $k\in \left(0,1\right)$, such that for all $\left(♭,\ell \right)\in \mathcal{R}$,

$$\theta \left({\hslash }^4{\left[{\varpi }_{\lambda }^{p_b}\left(\daleth ♭,\daleth \ell \right)\right]}^2\right)\le {\left[\theta \left(ℳ\left(♭,\ell \right)\right)\right]}^k;$$

(33)

$\left(ii\right)$

$\mathrm{\Gamma }\left(♭,\ell ,\mathcal{R}\right)$ is non-empty for all $♭,\ell \in {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$;

$\left(iii\right)$

There exists ${♭}_0\in {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$, such that $\left({♭}_0,\daleth {♭}_0\right)\in \mathcal{R}$ and $\mathcal{R}$ is ℸ-closed;

$\left(vi\right)$

Either ℸ is ${\varpi }^{p_b}$-continuous or ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ is $\mathcal{R}$-regular.

Then, the mapping ℸ admits a unique fixed point.

Proof. 

It is adequate to take $C\left(\iota \right)=\left(lnk\right)\iota$ and $\gimel \in \mathrm{\Delta }$ with $\gimel \left(\iota \right):=ln\theta :\left(0,\infty \right)\to \left(0,\infty \right)$ in Corollary 1. Hence, from (33), we get

$$ln\theta \left({\hslash }^4{\left[{\varpi }_{\lambda }^{p_b}\left(\daleth ♭,\daleth \ell \right)\right]}^2\right)\le \left(lnk\right)ln\theta \left(ℳ\left(♭,\ell \right)\right).$$

So the proof is evident from Corollary 1. □

Corollary 3. 

Let ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ be a ${\varpi }^{p_b}$-complete $𝒫{ℳ}_bℳ𝒮$ and $\mathcal{R}$ be a transitive and anti-symmetric binary relation on ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$. Assume that the mapping $\daleth :{\mathcal{U}}_{{\varpi }^{p_b}}^{*}\to {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ defined on ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ provides the subsequent statements:

$\left(i\right)$

There exist a $\mathcal{F}\in {\Im }^{*}$ and $\tau >0$, such that for all $\left(♭,\ell \right)\in \mathcal{R}$,

$$\tau +\mathcal{F}\left({\hslash }^4{\left[{\varpi }_{\lambda }^{p_b}\left(\daleth ♭,\daleth \ell \right)\right]}^2\right)\le \mathcal{F}\left(ℳ\left(♭,\ell \right)\right);$$

(34)

$\left(ii\right)$

$\mathrm{\Gamma }\left(♭,\ell ,\mathcal{R}\right)$ is non-empty for all $♭,\ell \in {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$;

$\left(iii\right)$

There exists ${♭}_0\in {\mathcal{U}}_{{\varpi }^{p_b}}^{*}$, such that $\left({♭}_0,\daleth {♭}_0\right)\in \mathcal{R}$ and $\mathcal{R}$ is ℸ-closed;

$\left(vi\right)$

Either ℸ is ${\varpi }^{p_b}$-continuous or ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ is $\mathcal{R}$-regular.

Then, the mapping ℸ admits a unique fixed point.

Proof. 

We consider $C\left(\iota \right)=e^{-\tau }\iota$ and $\gimel \in \mathrm{\Delta }$ with $\gimel \left(\iota \right):=e^{\mathcal{F}}:\left(0,\infty \right)\to \left(0,\infty \right)$ in Corollary 1. Thus, from (34), we get

$$e^{\mathcal{F}\left({\hslash }^4{\left[{\varpi }_{\lambda }^{p_b}\left(\daleth ♭,\daleth \ell \right)\right]}^2\right)}\le e^{-\tau }{e}^{\mathcal{F}\left(ℳ\left(♭,\ell \right)\right)}.$$

This implies that the proof is attained through Corollary (1). □

3. Applications

3.1. On Fractional Differential Equations

Metric structures are essential to the study of fractional differential equations (FDEs), as they provide a framework for analyzing functions and operators in fractional calculus without relying on traditional Euclidean geometry. This is crucial because solutions to FDEs often lack smoothness and can exhibit non-local behaviour or irregularity, which are effectively modelled using fractional derivatives.

Riemann–Liouville and Caputo fractional derivatives can be defined in metric structures through integral representations or operators that apply to functions in these contexts. The generalization of distance, convergence, and continuity in metric structures is vital for defining fractional derivatives when classical Euclidean structures are not applicable. By incorporating distance functions, metric structures allow for the representation of non-local interactions in FDEs, enabling the capture of long-range effects and memory phenomena.

Operators are central to the definition and solution of FDEs, and metric structures provide a suitable environment for studying these operators, particularly regarding their spectral properties, boundedness, and continuity. The ability to apply fixed-point theorems, like Banach’s fixed-point theorem, is enhanced within metric-type spaces, helping to establish the existence and uniqueness of solutions to fractional differential equations.

In summary, metric structures provide the necessary framework to model and solve FDEs effectively, accommodating non-local effects and irregular behaviours. This field also opens up avenues for studying fractional differential equations with new classes of functions in broader contexts beyond metric spaces. In this section, we will extend our results to Riemann–Liouville derivatives, applicable to fractional differential equations as well. Let us take a moment to revisit and clarify some fundamental definitions that are crucial to our understanding and foundation in this context (see [31,32]).

Definition 13. 

The Riemann–Liouville fractional integral of order $\alpha >0$ for a continuous function $\eta :\left(0,\infty \right)\to \mathbb{R}$ is defined as

$${\mathbb{I}}^{\alpha }\eta \left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}\eta \left(s\right)ds$$

provided the integral exists.

Definition 14. 

For a continuous function $\eta :\left(0,\infty \right)\to \mathbb{R}$, the Riemann–Liouville derivative of fractional order $\alpha >0$, $n=⌊a⌋+1$$(⌊a⌋$ denotes the integer part of the real number $a)$ is defined as

$$D^{\alpha }\eta \left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(n-\alpha \right)}}{\left({\displaystyle \frac{1}{dt}}\right)}^n\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{n-\alpha -1}\eta \left(s\right)ds={\left({\displaystyle \frac{d}{dt}}\right)}^n{\mathbb{I}}^{n-\alpha }\eta \left(t\right)$$

provided it exists.

In this discourse, we delve into the examination of an initial value problem (IVP) delineated by the equations below:

$$D^{\alpha }\eta \left(t\right)=\daleth \left(t,{\eta }_t\right),\forall t\in \gamma =\left[0,b\right],\alpha \in \left[0,1\right],$$

(35)

$$\eta \left(t\right)=\psi \left(t\right),t\in \left(-\infty ,0\right),$$

(36)

wherein $D^{\alpha }$ signifies the conventional Riemann–Liouville fractional derivative, $\daleth :\gamma \times A\to \mathbb{R}$, with $\psi \in A$, satisfying $\psi \left(0\right)=0$, and the set A is identified as phase space or state space. We introduce a $𝒫{ℳ}_bℳ$ function ${\varpi }_{\lambda }^{p_b}$ on $\mathcal{U}$, articulated by

$${\varpi }_{\lambda }^{p_b}\left(\eta ,v\right)=e^{-\lambda }\left|\eta -v\right|+\left|\eta \right|,$$

(37)

$\forall \eta ,v\in \mathcal{U}$. Hence, one can conclude that ${\mathcal{U}}_{{\varpi }^{p_b}}^{*}$ is a ${\varpi }^{p_b}$-complete $𝒫{ℳ}_bℳ𝒮$ with $\hslash =2$. If $\eta :\left(-\infty ,b\right]\to \mathbb{R}$, and ${\eta }_0\in \gamma$, then for every $t\in \left[0,b\right]{\eta }_t$, a $\gamma$-valued continuous function exists on $\left[0,b\right]$. The space $\gamma$ is complete by a solution of problems (35) and (36); we mean a space $\mathrm{\Sigma }=\left\{\left.\eta :\left(-\infty ,b\right]\to \mathbb{R}\right|{\left.\eta \right|}_{\left(-\infty ,0\right)}\mathrm{and}{\left.\eta \right|}_{\left[0,b\right]}\right\}$. Consequently, a function $\eta \in \mathrm{\Sigma }$ is recognized as a solution to Equations (35) and (36) if it adheres to the equation $D^{\alpha }\eta \left(t\right)=\daleth \left(t,{\eta }_t\right)$ across $\gamma$ and fulfills the condition $\eta \left(t\right)=\psi \left(t\right)$ on $\left(-\infty ,0\right]$.

Lemma 4 

([33]). Let $0<ℬ<1$ and $\eta :\left(\infty ,b\right]\to \mathbb{R}$ be continuous and

$$\underset{t\to 0^{+}}{lim}\sigma \left(t\right)=\sigma \left(0^{+}\right)\in \mathbb{R}.$$

Then, η is a solution of the fractional integral equation

$$\eta \left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(ℬ\right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{ℬ-1}\sigma \left(s\right)ds$$

if and only if η is a solution of the initial value problem for the fractional differential equation

$$D^{ℬ}\eta \left(t\right)=\sigma \left(t\right),t\in \left(0,b\right],\eta \left(0\right)=0.$$

Theorem 8. 

Let $\daleth :\gamma \times A\to \mathbb{R}$. Assume that there exists $q\in \left(0,1\right)$, such that

$$\left|\daleth \left(t,\eta \right)-\daleth \left(t,v\right)\right|+\left|\daleth \left(t,\eta \right)\right|\le q\left(\left|\eta -v\right|+\left|\eta \right|\right),$$

(38)

for $t\in \gamma$ and $\forall \eta ,v\in A$. If

$${\displaystyle \frac{b^{ℬ}q{k}_b}{\mathrm{\Gamma }\left(ℬ+1\right)}}={\displaystyle \frac{k_1{e}^{-\lambda }}{4}}<1,$$

where $k_1\in \left(0,{\displaystyle \frac{1}{\hslash }}\right)$, and $k_b=sup\left\{\left|k\left(t\right)\right|:t\in \left[0,b\right]\right\},$ then there exists a unique solution for (IVP) (35) and (36) on the interval $\left(-\infty ,b\right]$.

Proof. 

To commence the analysis of the given initial value problem, it is pivotal to reformulate it as a fixed-point problem. In this context, we introduce an operator denoted as $\mathrm{\Lambda }:\mathrm{\Sigma }\to \mathrm{\Sigma }$, which is defined by the subsequent expression

$$\mathrm{\Lambda }\left(\eta \right)\left(t\right)=\left\{\begin{array}{c}\psi \left(t\right),\mathrm{if}t\in \left(-\infty ,0\right]\hfill \\ \\ {\displaystyle \frac{1}{\mathrm{\Gamma }\left(ℬ\right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{ℬ-1}\daleth \left(s,{\eta }_s\right)ds,\mathrm{if}t\in \left[0,b\right]\hfill \end{array}\right..$$

Let $\rho \left(·\right):\left(-\infty ,0\right]\to \mathbb{R}$ be a function defined by

$$\rho \left(t\right)=\left\{\begin{array}{c}\psi \left(t\right),\mathrm{if}t\in \left(-\infty ,0\right]\hfill \\ \\ 0,\mathrm{if}t\in \left(0,b\right)\hfill \end{array}\right..$$

For each $\vartheta \in C\left(\left[0,b\right],\mathbb{R}\right)$ with $\vartheta \left(0\right)=0$, we denote the function $\overline{\vartheta }$ as

$$\overline{\vartheta }\left(t\right)=\left\{\begin{array}{c}0,\mathrm{if}t\in \left(-\infty ,0\right]\hfill \\ \\ \vartheta \left(t\right),\mathrm{if}t\in \left(0,b\right)\hfill \end{array}\right..$$

If

$$\eta \left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(ℬ\right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{ℬ-1}T\left(s,{\eta }_s\right)ds,$$

for every $0\le t\le b$ and the function $\vartheta \left(·\right)$ provides

$$\vartheta \left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(ℬ\right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{ℬ-1}T\left(s,{\overline{\vartheta }}_s+{\rho }_s\right)ds.$$

Set $C_0=\left\{\vartheta \in C\left(\left[0,b\right],\mathbb{R}\right),\vartheta \left(0\right)=0\right\}$. Let $\daleth :C_0\to C_0$ be a mapping defined by

$$\daleth \vartheta \left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(ℬ\right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{ℬ-1}T\left(s,{\overline{\vartheta }}_s+{\eta }_s\right)ds.$$

The operator $\mathrm{\Lambda }$ has a fixed point equivalent to ℸ; hence, we must prove that ℸ has a fixed point. Indeed, if we consider $\vartheta ,{\vartheta }^{*}\in C_0$ and $t\in \left[0,b\right]$, we achieve

$$\begin{array}{c}\left|T\vartheta \left(t\right)-T{\vartheta }^{*}\left(t\right)\right|+\left|T\vartheta \left(t\right)\right|\hfill \\ =\left|{\displaystyle \frac{1}{\mathrm{\Gamma }\left(ℬ\right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{ℬ-1}T\left(s,{\overline{\vartheta }}_s+{\eta }_s\right)ds-{\displaystyle \frac{1}{\mathrm{\Gamma }\left(ℬ\right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{ℬ-1}T\left(s,\overline{{\vartheta }^{*}}+{\eta }_s\right)ds\right|\hfill \\ \\ +\left|{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}T\left(s,{\overline{\vartheta }}_s+{\eta }_s\right)ds\right|\hfill \end{array}$$

$$\begin{array}{c}<{\displaystyle \frac{1}{\mathrm{\Gamma }\left(ℬ\right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{ℬ-1}\left|T\left(s,{\overline{\vartheta }}_s+{\eta }_s\right)-T\left(s,\overline{{\vartheta }^{*}}+{\rho }_s\right)\right|ds\hfill \\ \\ +{\displaystyle \frac{1}{\mathrm{\Gamma }\left(ℬ\right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{ℬ-1}\left|T\left(s,{\overline{\vartheta }}_s+{\rho }_s\right)\right|ds\hfill \\ \\ <{\displaystyle \frac{1}{\mathrm{\Gamma }\left(ℬ\right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{ℬ-1}\left(\left|T\left(s,{\overline{\rho }}_s+{\rho }_s\right)-T\left(s,\overline{{\vartheta }^{*}}+{\rho }_s\right)\right|+\left|T\left(s,{\overline{\vartheta }}_s+{\eta }_s\right)\right|\right)ds\hfill \\ \\ <{\displaystyle \frac{1}{\mathrm{\Gamma }\left(ℬ\right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{ℬ-1}\left|q\left({\overline{\vartheta }}_s-\overline{{\vartheta }^{*}}\right)+q\left({\overline{\vartheta }}_s\right)\right|ds\hfill \\ \\ <{\displaystyle \frac{1}{\mathrm{\Gamma }\left(ℬ\right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{ℬ-1}q{k}_{b}sup\left(\left|\vartheta \left(s\right)-{\vartheta }^{*}\left(s\right)\right|+\left|\vartheta \left(s\right)\right|\right)ds\hfill \\ \\ <{\displaystyle \frac{k_b}{\mathrm{\Gamma }\left(ℬ\right)}}\underset{0}{\overset{t}{\int }}{\left(t-s\right)}^{ℬ-1}qds\left(\left|\vartheta -{\vartheta }^{*}\right|+\left|\vartheta \right|\right).\hfill \end{array}$$

Therefore, for all $\lambda >0$, we conclude that

$$\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left(T\left(\vartheta \right),T\left({\vartheta }^{*}\right)\right)=e^{-\lambda }\left|\daleth \left(\vartheta \right)-\daleth \left({\vartheta }^{*}\right)\right|+\left|\daleth \left(\vartheta \right)\right|<\left|\daleth \left(\vartheta \right)-\daleth \left({\vartheta }^{*}\right)\right|+\left|\daleth \left(\vartheta \right)\right|\hfill \\ \\ <{\displaystyle \frac{q{b}^{ℬ}{k}_b}{\mathrm{\Gamma }\left(ℬ+1\right)}}\left({\left|\vartheta -{\vartheta }^{*}\right|}_b+\left|\vartheta \right|\right)\hfill \\ \\ <{\displaystyle \frac{k_1}{4}}\left(e^{-\lambda }{\left|\vartheta -{\vartheta }^{*}\right|}_b+\left|\vartheta \right|\right)\hfill \\ \\ ={\displaystyle \frac{k_1}{4}}{\varpi }_{\lambda }^{p_b}\left(\vartheta ,{\vartheta }^{*}\right).\hfill \end{array}$$

(39)

Now, in Corollary 1, without using a Suzuki restriction, we consider

$$\mathcal{R}=\left\{\left(\vartheta ,{\vartheta }^{*}\right)\in \mathcal{R}:\vartheta \le {\vartheta }^{*},\vartheta ,{\vartheta }^{*}\in C_0\right\}.$$

Also, let $I=\left\{\left(\vartheta ,{\vartheta }^{*}\right)\in \mathcal{R}:{\varpi }_{\lambda }^{p_b}\left(\daleth \vartheta \left(♭\right),\daleth {\vartheta }^{*}\left(♭\right)\right)>0\right\}$. Additionally, by considering continuous comparison function C as $C\left(t\right)=\tau t$ with $\tau \in \left(0,1\right)$ and using the non-decreasing property of ℷ, the inequality (39) turns into

$$\begin{array}{c}{\hslash }^4{\varpi }_{\lambda }^{p_b}{\left(\daleth \left(\vartheta \right),\daleth \left({\vartheta }^{*}\right)\right)}^2\le \tau {\varpi }_{\lambda }^{p_b}{\left(\vartheta ,{\vartheta }^{*}\right)}^2\hfill \\ \\ \le \tau max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left(\vartheta ,\daleth \vartheta \right)·{\varpi }_{\lambda }^{p_b}\left({\vartheta }^{*},\daleth {\vartheta }^{*}\right),{\varpi }_{\lambda }^{p_b}\left(\vartheta ,{\vartheta }^{*}\right)·{\varpi }_{\lambda }^{p_b}\left(\vartheta ,\daleth \vartheta \right),\hfill \\ {\varpi }_{\lambda }^{p_b}\left(\vartheta ,{\vartheta }^{*}\right)·{\varpi }_{\lambda }^{p_b}\left({\vartheta }^{*},\daleth {\vartheta }^{*}\right),c·{\varpi }_{\lambda }^{p_b}\left(\vartheta ,\daleth {\vartheta }^{*}\right)·{\varpi }_{\lambda }^{p_b}\left({\vartheta }^{*},\daleth \vartheta \right)\hfill \end{array}\right\}.\hfill \\ \\ =\tau ℳ\left(\vartheta ,{\vartheta }^{*}\right),\hfill \end{array}$$

where $\tau ={\left(k_1\right)}^2\in \left(0,1\right)$. Therefore, it can be deduced that all the prerequisites outlined in Corollary 1 are satisfied. Consequently, we establish that the operator ℸ possesses a unique fixed point. □

3.2. On Dynamic Programming

In the context of mathematical analysis, particularly within the framework of Banach spaces, we direct our attention to the assumption that both $\mathrm{\Lambda }$ and $\mathrm{\Phi }$ represent Banach spaces. Within this framework, we denote $\mathrm{\Sigma }\subseteq \mathrm{\Lambda }$ and $\mathrm{{\rm Y}}\subseteq \mathrm{\Phi }$, where the subsets $\mathrm{\Sigma }$ and $\mathrm{{\rm Y}}$ are, respectively, identified as the state space and the decision space. This configuration lays the foundational groundwork for examining a system of functional equations. The investigation into such a system not only enriches the theoretical landscape but also has implications for practical applications in various domains requiring the resolution of complex functional equations within the confines of Banach spaces:

$$q_1\left(♭\right)=\underset{\ell \in \mathrm{{\rm Y}}}{max}\left\{f\left(♭,\ell \right)+𝒢\left(♭,\ell ,q_1\left(\xi \left(♭,\ell \right)\right)\right)\right\},♭\in \mathrm{\Sigma },$$

$$q_2\left(♭\right)=\underset{\ell \in \mathrm{{\rm Y}}}{max}\left\{g\left(♭,\ell \right)+𝒦\left(♭,\ell ,q_2\left(\xi \left(♭,\ell \right)\right)\right)\right\},♭\in \mathrm{\Sigma },$$

where $f,g:\mathrm{\Sigma }\times \mathrm{{\rm Y}}\to \mathbb{R}$ and $𝒢,𝒦:\mathrm{\Sigma }\times \mathrm{{\rm Y}}\times \mathbb{R}\to \mathbb{R}$ are bounded, $\xi :\mathrm{\Sigma }\times \mathrm{{\rm Y}}\to \mathrm{\Sigma }$. Let ${\mathcal{U}}_{{\varpi }^{p_b}}=B\left(\mathrm{\Sigma }\right)$ denote the space of all bounded real-valued functions on $\mathrm{\Sigma }$. Consider the metric defined by

$${\varpi }_{\lambda }^{p_b}\left(\varsigma ,\sigma \right)=e^{-\lambda }{∥\varsigma -\sigma ∥}^2+∥\varsigma ∥+∥\sigma ∥,$$

for all $\varsigma ,\sigma \in \mathrm{\Lambda }$ and for all $\lambda >0$, where $∥\varsigma ∥=\underset{♭\in \mathrm{\Sigma }}{sup}\left|\varsigma \left(♭\right)\right|$. Note that ${\varpi }^{p_b}$ is a partial modular b-metric on ${\mathcal{U}}_{{\varpi }^{p_b}}$. Also, from (2), we get

$${\omega }_{\lambda }\left(\varsigma ,\sigma \right)=2{\varpi }_{\lambda }^{p_b}\left(\varsigma ,\sigma \right)-{\varpi }_{\lambda }^{p_b}\left(\varsigma ,\varsigma \right)-{\varpi }_{\lambda }^{p_b}\left(\sigma ,\sigma \right)=2e^{-\lambda }∥\varsigma -\sigma ∥.$$

Then, it can be evidently expressed that ${\mathcal{U}}_{{\varpi }^{p_b}}$ is a ${\varpi }^{p_b}$-complete $𝒫{ℳ}_bℳ𝒮$ with $\hslash =2$. Moreover, let the mappings $\daleth ,\mathcal{Q}:{\mathcal{U}}_{{\varpi }^{p_b}}\to {\mathcal{U}}_{{\varpi }^{p_b}}$ be given by

$$\daleth \varsigma \left(♭\right)=\underset{\ell \in \mathrm{{\rm Y}}}{sup}\left\{f\left(♭,\ell \right)+𝒢\left(♭,\ell ,\varsigma \left(\xi \left(♭,\ell \right)\right)\right)\right\},$$

(40)

$$\mathcal{Q}\varsigma \left(♭\right)=\underset{\ell \in \mathrm{{\rm Y}}}{sup}\left\{g\left(♭,\ell \right)+𝒦\left(♭,\ell ,\varsigma \left(\xi \left(♭,\ell \right)\right)\right)\right\},$$

(41)

where $♭\in \mathrm{\Sigma }$ and $\varsigma \in {\mathcal{U}}_{{\varpi }^{p_b}}$. If the functions $f,g$ and $𝒢,𝒦$ are bounded, then $\mathrm{\Lambda }$ and $\mathrm{\Phi }$ are well defined.

Theorem 9. 

Let $\daleth ,\mathcal{Q}:{\mathcal{U}}_{{\varpi }^{p_b}}\to {\mathcal{U}}_{{\varpi }^{p_b}}$ be two operators defined by (40) and (41), respectively, and suppose that the following conditions are fulfilled:

$i.$

$f,g:\mathrm{\Sigma }\times \mathrm{{\rm Y}}\to \mathbb{R}$ and $𝒢,𝒦:\mathrm{\Sigma }\times \mathrm{{\rm Y}}\times \mathbb{R}\to \mathbb{R}$ are bounded;

$ii.$

For $\forall \varsigma ,\sigma \in {\mathcal{U}}_{{\varpi }^{p_b}}$, $\forall ♭\in \mathrm{\Sigma }$, $\forall \ell \in \mathrm{{\rm Y}}$, there exists $\delta \in \left(0,1\right)$, such that

$$\left|𝒢\left(♭,\ell ,\varsigma \left(♭\right)\right)-𝒦\left(♭,\ell ,\sigma \left(♭\right)\right)\right|<{\delta }^{1/2}\left|\varsigma \left(♭\right)-\sigma \left(♭\right)\right|;$$

$iii.$

For every $\left(♭,\ell \right)\in \mathrm{\Sigma }\times \mathrm{{\rm Y}}$, we have

$$𝒢\left(♭,\ell ,\varsigma \left(♭\right)\right)\le \varsigma \left(♭\right)-f\left(♭,\ell \right),$$

$$𝒦\left(♭,\ell ,\varsigma \left(♭\right)\right)\le \sigma \left(♭\right)-g\left(♭,\ell \right),$$

where $f,g:\mathrm{\Sigma }\times \mathrm{{\rm Y}}\to \mathbb{R}$;

$iv.$

For all $\varsigma ,\sigma \in {\mathcal{U}}_{{\varpi }^{p_b}}$, $\left|sup\varsigma \left(♭\right)\right|\le \delta \left|\varsigma \left(♭\right)\right|$ and $\left|sup\sigma \left(♭\right)\right|\le \delta \left|\sigma \left(♭\right)\right|$, where $\delta \in \left(0,1\right)$.

Then, the function Equations (40) and (41) have a unique bounded common solution, that is, ℸ and $\mathcal{Q}$ have a common fixed point.

Proof. 

Let $ϵ\in {\mathbb{R}}^{+}$ be arbitrary, $♭\in \mathrm{\Sigma }$ and $\varsigma \in {\mathcal{U}}_{{\varpi }^{p_b}}$. Then, there exist ${\ell }_1,{\ell }_2\in \mathrm{{\rm Y}}$, such that

$$\daleth \varsigma \left(♭\right)<f\left(♭,{\ell }_1\right)+𝒢\left(♭,{\ell }_1,\varsigma \left(\xi \left(♭,{\ell }_1\right)\right)\right)+ϵ,$$

(42)

$$\mathcal{Q}\sigma \left(♭\right)<g\left(♭,{\ell }_2\right)+𝒦\left(♭,{\ell }_2,\sigma \left(\xi \left(♭,{\ell }_1\right)\right)\right)+ϵ,$$

(43)

$$\daleth \varsigma \left(♭\right)\ge f\left(♭,{\ell }_2\right)+𝒢\left(♭,{\ell }_2,\varsigma \left(\xi \left(♭,{\ell }_2\right)\right)\right),$$

(44)

$$\mathcal{Q}\sigma \left(♭\right)\ge g\left(♭,{\ell }_1\right)+𝒦\left(♭,{\ell }_1,\sigma \left(\xi \left(♭,{\ell }_1\right)\right)\right).$$

(45)

Then, from (42) and (45), we yield that

$$\begin{array}{c}\daleth \varsigma \left(♭\right)-\mathcal{Q}\sigma \left(♭\right)<𝒢\left(♭,{\ell }_1,\varsigma \left(\xi \left(♭,{\ell }_1\right)\right)\right)-𝒦\left(♭,{\ell }_1,\sigma \left(\xi \left(♭,{\ell }_1\right)\right)\right)+ϵ\hfill \\ \\ \le \left|𝒢\left(♭,{\ell }_1,\varsigma \left(\xi \left(♭,{\ell }_1\right)\right)\right)-𝒢\left(♭,{\ell }_1,\sigma \left(\xi \left(♭,{\ell }_1\right)\right)\right)\right|+ϵ\hfill \\ \\ <{\delta }^{1/2}\left|\varsigma \left(♭\right)-\sigma \left(♭\right)\right|+ϵ.\hfill \end{array}$$

Likewise, from (43) and (44), we get

$$\begin{array}{c}\mathcal{Q}\sigma \left(♭\right)-\daleth \varsigma \left(♭\right)<𝒦\left(♭,{\ell }_2,\sigma \left(\xi \left(♭,{\ell }_2\right)\right)\right)-𝒢\left(♭,{\ell }_2,\varsigma \left(\xi \left(♭,{\ell }_2\right)\right)\right)+\epsilon \hfill \\ \\ \le \left|𝒦\left(♭,{\ell }_2,\sigma \left(\xi \left(♭,{\ell }_2\right)\right)\right)-𝒢\left(♭,{\ell }_2,\varsigma \left(\xi \left(♭,{\ell }_2\right)\right)\right)\right|+ϵ\hfill \\ \\ <{\delta }^{1/2}\left|\varsigma \left(♭\right)-\sigma \left(♭\right)\right|+ϵ.\hfill \end{array}$$

Hence, by considering the above inequalities, we conclude that

$$\left|\daleth \varsigma \left(♭\right)-\mathcal{Q}\sigma \left(♭\right)\right|<{\delta }^{1/2}\left|\varsigma \left(♭\right)-\sigma \left(♭\right)\right|+ϵ,$$

and, for an arbitrary $ϵ$

$$\left|\daleth \varsigma \left(♭\right)-\mathcal{Q}\sigma \left(♭\right)\right|\le {\delta }^{1/2}\left|\varsigma \left(♭\right)-\sigma \left(♭\right)\right|,$$

or, equivalently,

$$∥\daleth \varsigma \left(♭\right)-\mathcal{Q}\sigma \left(♭\right)∥\le {\delta }^{1/2}∥\varsigma \left(♭\right)-\sigma \left(♭\right)∥.$$

(46)

Additionally, we have

$$\left|\daleth \varsigma \left(♭\right)\right|=\left|\underset{{\ell }_1\in \Xi }{sup}\left\{f\left(♭,{\ell }_1\right)+𝒢\left(♭,{\ell }_1,\varsigma \left(\xi \left(♭,{\ell }_1\right)\right)\right)\right\}\right|\le \delta \left|\varsigma \left(♭\right)\right|$$

and, hence,

$$∥\daleth \varsigma \left(♭\right)∥\le \delta ∥\varsigma \left(♭\right)∥$$

(47)

and, analogously,

$$∥\mathcal{Q}\sigma \left(♭\right)∥\le \delta ∥\sigma \left(♭\right)∥.$$

(48)

So, from (46), (47) and (48), we derive

$$\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left(\daleth \varsigma \left(♭\right),\mathcal{Q}\sigma \left(♭\right)\right)=e^{-\lambda }{∥\daleth \varsigma \left(♭\right)-\mathcal{Q}\sigma \left(♭\right)∥}^2+∥\daleth \varsigma \left(♭\right)∥+∥\mathcal{Q}\sigma \left(♭\right)∥\hfill \\ \\ \le e^{-\lambda }\delta {∥\varsigma \left(♭\right)-\sigma \left(♭\right)∥}^2+\delta ∥\varsigma \left(♭\right)∥+\delta ∥\sigma \left(♭\right)∥\hfill \\ \\ =\delta \left[e^{-\lambda }{∥\varsigma \left(♭\right)-\sigma \left(♭\right)∥}^2+∥\varsigma \left(♭\right)∥+∥\sigma \left(♭\right)∥\right]\hfill \\ \\ =\delta {\varpi }_{\lambda }^{p_b}\left(\varsigma \left(♭\right),\sigma \left(♭\right)\right).\hfill \end{array}$$

(49)

Now, in Theorem 4, without the Suzuki restriction, we consider the subsequent set

$$\mathcal{R}=\left\{\left(\varsigma ,\sigma \right)\in \mathcal{R}:\varsigma \le \sigma ,\varsigma ,\sigma \in \mathbb{R}\right\}.$$

Let $I=\left\{\left(\varsigma ,\sigma \right)\in \mathcal{R}:{\varpi }_{\lambda }^{p_b}\left(\daleth \varsigma \left(♭\right),\daleth \sigma \left(♭\right)\right)>0\right\}$. Additionally, by considering the continuous comparison function C as $C\left(t\right)={\delta }^{2}t$ with $\delta \in \left(0,1\right)$ and using non-decreasing property of ℷ, for all $\varsigma ,\sigma \in I$, from (49), we have

$$\begin{array}{c}{\varpi }_{\lambda }^{p_b}{\left(\daleth \varsigma \left(♭\right),\mathcal{Q}\sigma \left(♭\right)\right)}^2\le \gamma {\varpi }_{\lambda }^{p_b}{\left(\varsigma \left(♭\right),\sigma \left(♭\right)\right)}^2,\hfill \\ \\ \le \gamma max\left\{\begin{array}{c}{\varpi }_{\lambda }^{p_b}\left(\sigma \left(♭\right),\mathcal{Q}\sigma \left(♭\right)\right)·{\sigma }_{\lambda }^{p_b}\left(\varsigma \left(♭\right),\daleth \varsigma \left(♭\right)\right),\hfill \\ {\varpi }_{\lambda }^{p_b}\left(\varsigma \left(♭\right),\sigma \left(♭\right)\right)·{\varpi }_{\lambda }^{p_b}\left(\sigma \left(♭\right),\mathcal{Q}\sigma \left(♭\right)\right),\hfill \\ {\varpi }_{\lambda }^{p_b}\left(\varsigma \left(♭\right),\sigma \left(♭\right)\right)·{\varpi }_{\lambda }^{p_b}\left(\varsigma \left(♭\right),\daleth \varsigma \left(♭\right)\right),\hfill \\ c{\varpi }_{\lambda }^{p_b}\left(\sigma \left(♭\right),\daleth \varsigma \left(♭\right)\right)·{\varpi }_{\lambda }^{p_b}\left(\varsigma \left(♭\right),\mathcal{Q}\sigma \left(♭\right)\right)\hfill \end{array}\right\}\hfill \\ \\ =\gamma ℳ\left(\varsigma \left(♭\right),\sigma \left(♭\right)\right),\hfill \end{array}$$

where $\gamma ={\delta }^2\in \left(0,1\right)$. Consequently, the last inequality indicates that all the conditions of Theorem 4 are met. Thus, we gain that the functional Equations (40) and (41) have a unique bounded common solution. □

4. Conclusions

In conclusion, this study delineates substantial advancements in the exploration of partial modular b-metric spaces through the introduction of a generalized ${\gimel }_C$-type Suzuki $\mathcal{R}$-contraction. This novel framework not only extends the foundational contributions made by Liu [25] but also integrates the $\mathcal{R}$ relation, thereby enabling the formulation of new common fixed-point theorems that incorporate quadratic terms.

To exemplify the practical relevance of our research, we provided a detailed case study that illustrates how our theoretical advancements can be applied in real-world scenarios. This example serves to underscore the versatility and applicability of our findings within various mathematical contexts.

In addition to our main contributions, we undertook a thorough investigation into the solutions of multiple functional equations arising in the fields of dynamic programming and fractional differential equations. This examination not only highlights the theoretical implications of our work but also emphasizes its potential applications in practical situations, thereby enriching the existing literature and contributing meaningful insights to this evolving area of study. Through these efforts, we aim to pave the way for further research and development in the domain of partial modular b-metric spaces and their applications.