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\begin{document}

\title{Generalized Almost Simulative ${{\hat Z} }_{_{\Psi ^*} }^\Theta  - $Contraction Mappings in Modular $b-$Metric Spaces
}

\author{Mahpeyker \"{O}zt\"{u}rk \and Farhan Golkarmanesh \and
       \newline Abdurrahman B\"{u}y\"{u}kkaya \and Vahid Parvaneh %etc.
}



%\authorrunning{Short form of author list} % if too long for running head

\institute{Mahpeyker \"{O}zt\"{u}rk \at
              Department of Mathematics, Sakarya University, 54187 Sakarya, Turkey. \\
                      \email{mahpeykero@sakarya.edu.tr}
               \and  Farhan Golkarmanesh \at
                 Department of Mathematics, Sanandaj Branch, Islamic
Azad University, Sanandaj, Iran\\
\email{fgolkarmanesh@yahoo.com}
           \and
           Abdurrahman B\"{u}y\"{u}kkaya \at
             Department of mathematics, Karadeniz Technical University, Trabzon, Turkey\\
             \email{abdurrahman.giresun@hotmail.com}
              \and Vahid Parvaneh \at
               Department of Mathematics, Gilan-E-Gharb Branch, Islamic
Azad University, Gilan-E-Gharb, Iran\\
\email{zam.dalahoo@gmail.com}
}

\date{Received: date / Accepted: date}
% The correct dates will be entered by the editor


\maketitle

\begin{abstract}
This article aims to characterize a new class of simulation functions by extending the class of $\mathcal{Z}$ functions demonstrated by Cho in \cite{ze}, and via this novel notion, to establish some common fixed point theorems in modular $b-$metric spaces. Furthermore, some corollaries, which enlarge previously existing literature, have been procured. In conclusion, an example and an application to the nonlinear integral equations have been presented to state the applicability and validity of our outcomes.
\end{abstract}

\section{Introduction and Preliminaries}

Throughout the study, the symbol $\mathbf{N}$ represents the set of all positive natural numbers, and $\mathbf{R_+}$ is used to represent the set of all non-negative real numbers.

Let $\mathcal{G}$ be a non-void set and $J,K:\mathcal{G} \to \mathcal{G}$ be self-mappings. Thereby, the following ones represent the set of fixed points of $J$ and the set of common fixed points of $J$ and $K$, respectively:
$$Fix\left( J \right) = \left\{ {\varsigma  \in {\cal G}:\;J\varsigma  = \varsigma } \right\}\;{\rm{and}}\;{C_{Fix}}\left( {J,K} \right) = \left\{ {\varsigma  \in {\cal G}:\;J\varsigma  = K\varsigma  = \varsigma } \right\}.$$

Metric fixed point theory is a favorite research area at present. The key element in this theory is the Banach contraction principle (BCP), which was published by Banach in 1922 \cite{bnh}. Many studies on BCP are being done to develop this theory. In order to find suitable conditions on mappings that guarantee the existence and uniqueness of fixed points, many researchers have made numerous studies based on BCP, either by revising the contraction conditions with some auxiliary functions, or by generalizing existing spaces, or using both.

Since the metric function, and therefore the metric space structure, is the basis of the metric fixed point theory, this function has been studied intensively by many researchers. Many new distance functions have started to take place in the literature. One of the significant and famous generalizations of the metric function is the $b-$metric function, which first appeared in Bakhtin's work \cite{bah} in 1989, but attracted the attention of researchers  studies with Czerwik (\cite{b1},\cite{b2}) in 1993 and 1998.


\begin{definition}\cite{b1}
Let $\mathcal{G}$ be a non-void set and $\tau  \ge 1,\left( {\tau  \in \mathbf{R}} \right)$. Presume that the function $\sigma:\mathcal{G} \times \mathcal{G}  \to \mathbf{R_+}$ satisfies the following circumstances: for all $\varsigma ,\ell ,\nu \in \mathcal{G}$,
\begin{itemize}
\item[$\left( {{\sigma_1}} \right)$] $\sigma\left( {\varsigma ,\ell } \right) = 0 \Leftrightarrow \varsigma  = \ell,$
\item[$\left( {{\sigma_2}} \right)$] $\sigma\left( {\varsigma ,\ell } \right) = \sigma\left( {\ell,\varsigma} \right),$
\item[$\left( {{\sigma_3}} \right)$] $\sigma\left( {\varsigma,\ell} \right) \le \tau \left[ {\sigma\left( {\varsigma ,\nu } \right) + \sigma\left( {\nu,\ell} \right)} \right].$
\end{itemize}
The function $\sigma$ is entitled as a $b-$metric on $\mathcal{G}$ and  the pair  $\left( {\mathcal{G},\sigma} \right)$ represents a  $b-$metric space.
\end{definition}

If $\tau=1$, then the concepts of $b-$metric and ordinary metric overlap.

Further, unlike the metric space, $b-$metric is not always a continuous function of its variables. Therefore, the following lemma is of considerable importance for the $b-$metric.
\begin{lemma}\emph{\cite{blemma}}
Let $\left( {\mathcal{G},\sigma} \right)$ be a $b-$metric space with $\tau \ge 1$ and $\left\{ {{\varsigma _n}} \right\}$ and $\left\{ {{\ell _n}} \right\}$ be convergent to $\varsigma$ and $\ell$, respectively. Then
$$\frac{1}{{{\tau ^2}}}\sigma \left( {\varsigma ,\ell } \right) \le \mathop {\liminf }\limits_{n \to \infty }  \sigma \left( {{\varsigma _n},{\ell _n}} \right) \le \mathop {\limsup }\limits_{n \to \infty }  \sigma \left( {{\varsigma _n},{\ell _n}} \right) \le {\tau ^2}\sigma \left( {\varsigma ,\ell } \right).$$
Especially, if $\varsigma  = \ell $, then  $\mathop {\lim }\limits_{n \to \infty } \sigma \left( {{\varsigma _n},{\ell _n}} \right) = 0.$ Also, for $z \in \mathcal{G}$, we have
$$\frac{1}{\tau }\sigma \left( {\varsigma ,z} \right) \le \mathop {\lim \inf }\limits_{n \to \infty } \sigma \left( {{\varsigma _n},z} \right) \le \mathop {\lim \sup }\limits_{n \to \infty } \sigma \left( {{\varsigma _n},z} \right) \le \tau \sigma \left( {\varsigma ,z} \right).$$
\end{lemma}



In 2010, Chistyakov put forward a new generalized metric space \cite{mms2}  titled  modular metric space. In addition, various authors have introduced many generalized metric space structures in the literature by using the metric modular concept.

Let $\varpi:\left( {0,\infty } \right) \times \mathcal{G} \times \mathcal{G} \to \left[ {0,\infty } \right]$ be a function on a non-void set $\mathcal{G}$. For simplicity, for all $\hbar > 0$ and $\varsigma ,\ell \in \mathcal{G}$, instead of $\varpi \left( {\hbar ,\varsigma ,\ell } \right)$  the ${\varpi _\hbar }\left( {\varsigma ,\ell } \right) $ expression will be used.

\begin{definition}\cite{mms2}
Let $\mathcal{G}$ be a non-empty set and $\varpi :\left( {0,\infty } \right) \times \mathcal{G} \times \mathcal{G} \to \left[ {0,\infty } \right]$ be a function. If the following axioms satisfy, then $\varpi$ is termed as a metric modular on $\mathcal{G}$: for all $\varsigma ,\ell ,\nu \in \mathcal{G}$,
\begin{itemize}
\item[$\left( {{\varpi _1}} \right)$] ${\varpi _\hbar }\left( {\varsigma ,\ell } \right)= 0$ for all $\hbar  > 0$ if and only if $\varsigma  = \ell$,
\item[$\left( {{\varpi _2}} \right)$] ${\varpi _\hbar }\left( {\varsigma ,\ell } \right) = {\varpi _\hbar }\left( {\ell,\varsigma} \right)$ for all $\hbar  > 0$,
\item[$\left( {{\varpi _3}} \right)$] ${\varpi _{\hbar  + \mu }}\left( {\varsigma ,\ell } \right) \le {\varpi _\hbar }\left( {\varsigma,\nu } \right) + {\varpi _\mu }\left( {\nu,\ell} \right)$ for all $\hbar,\mu  > 0.$
\end{itemize}
If $\varpi$ has the property $\left( {{\varpi _1}'} \right)$ given below instead of $\left( {{\varpi _1}} \right)$, then $\varpi$ is entitled a (metric) pseudomodular on $\mathcal{G}$:
\begin{itemize}
\item[$\left( {{\varpi _1}'} \right)$]${\varpi _\hbar }\left( {\varsigma,\varsigma} \right) = 0$ for all $\hbar  > 0$.
\end{itemize}

\end{definition}


By combining modular metric and $b-$metric structures, Ege and Alaca \cite{MbMS} have lately brought the concept of modular b-metric to the literature.

\begin{definition}\cite{MbMS}
Let $\mathcal{G}$ be a non-empty set and $\tau  \ge 1,\left( {\tau  \in \mathbf{R}} \right)$. A map $\kappa :\left( {0,\infty } \right) \times \mathcal{G} \times \mathcal{G} \to \left[ {0,\infty } \right]$ is labeled as a modular $b-$metric provided that  for all $\varsigma ,\ell ,\nu \in \mathcal{G}$, the  below circumstances  fulfill:
\begin{itemize}
\item[$\left( {{\kappa _1}} \right)$] ${\kappa_\hbar }\left( {\varsigma ,\ell } \right) = 0$ for all $\hbar  > 0$ if and only if $\varsigma  = \ell$,
\item[$\left( {{\kappa _2}} \right)$] ${\kappa_\hbar }\left( {\varsigma ,\ell } \right) = {\kappa_\hbar }\left( {\ell,\varsigma} \right)$ for all $\hbar  > 0$,
\item[$\left( {{\kappa _3}} \right)$] ${\kappa _{\hbar  + \mu }}\left( {\varsigma ,\ell } \right) \le \tau \left[ {{\kappa_\hbar }\left( {\varsigma,\nu} \right) + {\kappa _\mu }\left( {\nu,\ell} \right)} \right]$ for all $\hbar,\mu  > 0$.
\end{itemize}
The pair $\left( {\mathcal{G},\kappa } \right)$ is a modular $b-$metric space, which abbreviated as MbMS.
\end{definition}
In the definition of modular $b-$metric, if we choose $\tau=1$, then it is a natural extension of modular metric. Some examples of modular $b-$metric functions and modular $b-$metric spaces are given below.
\begin{example} \cite{MbMS}
Taking into account the space
$${l _p} = \left\{ {\left( {{\kappa_n}} \right) \subset \mathbf{R}:\sum\limits_{n = 1}^\infty  {{{\left| {{\kappa_n}} \right|}^p} < \infty } } \right\}\quad 0 < p < 1,$$
$\hbar  \in \left( {0,\infty } \right)$ and ${\kappa _\hbar }\left( {\varsigma,\ell} \right) = \frac{{m\left( {\varsigma,\ell} \right)}}{\hbar }$ such that
$$m\left( {\varsigma,\ell} \right) = {\left( {\sum\limits_{n = 1}^\infty  {{{\left| {{\varsigma_n} - {\ell_n}} \right|}^p}} } \right)^{\frac{1}{p}}},\quad \varsigma = {\varsigma_n},\ell = {\ell_n} \in {l _p}$$
Thereby, the $\left( {\mathcal{G},\kappa } \right)$ is an MbMS.
\end{example}

\begin{example} \cite{MbMS2}
Let $\left( {\mathcal{G},\varpi } \right)$ be a modular metric space and let $s \ge 1$ be a real number. Take ${\kappa _\hbar }\left( {\varsigma ,\ell} \right) = {\left( {{\varpi _\hbar }\left( {\varsigma ,\ell} \right)} \right)^s}$. Using the convexity of the function $J\left( t \right) = {t^s}$ for all $t \ge 0$, also from Jensen inequality, we achieve
$${\left( {\alpha + \beta } \right)^s} \le {2^{s - 1}}\left( {{\alpha^s} + {\beta^s}} \right)$$
for $\alpha ,\beta  \ge 0$. Thus, $\left( {\mathcal{G},\kappa } \right)$ is an MbMS with the constant $\tau = {2^{s - 1}}$.
\end{example}


%If $\kappa$ is a modular $b-$metric on a set $\mathcal{G}$, then a modular set is identified by
%$${\mathcal{G}_\kappa } = \left\{ {\ell  \in \mathcal{G}:\ell \mathop {\sim}\limits^{\kappa} \varsigma } \right\},$$
%where the $\mathop {\sim}\limits^{\kappa} $ is a binary relation on $\mathcal{G}$ defined by for $\varsigma ,\ell  \in \mathcal{G}$.
%Also, note that the set
%$${\mathcal{G}}_\kappa ^* = \left\{ {\varsigma  \in \mathcal{G}:\exists \hbar  = \hbar \left( \varsigma  \right) > 0\;{\rm{such}}\;{\rm{that}}\;{\kappa _\hbar }\left( {\varsigma ,{\varsigma _0}} \right) < \infty } \right\}\;\left( {{\varsigma _0} \in \mathcal{G}} \right)$$
%are mentioned as modular metric space (around ${\varsigma _0}$).

Some fundamental topological properties of an MbMS such as $\kappa-$convergence, $\kappa-$Cauchy sequences, and $\kappa-$completeness are characterized as below.
\begin{definition}
Let $\left( {\mathcal{G},\kappa } \right)$ be an MbMS and ${\left( {{\varsigma_n}} \right)_{n \in \mathbf{N}}}$ be a sequence in ${{\mathcal{G}}_\kappa ^*}$.
\begin{itemize}
\item[i.] ${\left( {{\varsigma_n}} \right)_{n \in \mathbf{N}}}$  is called $\kappa-$convergent to $\varsigma \in {{\mathcal{G}}_\kappa ^*}$ if and only if ${\kappa _\hbar }\left( {{\varsigma_n},\varsigma} \right) \to 0$, as $n \to \infty $ for all $\hbar  > 0$.

\item[ii.] ${\left( {{\varsigma_n}} \right)_{n \in \mathbf{N}}}$ in ${{\mathcal{G}}_\kappa ^*}$ is named a $\kappa-$Cauchy sequence if $\mathop {\lim }\limits_{n,m \to \infty } {\kappa _\hbar }\left( {{\varsigma_n},{\varsigma_m}} \right) = 0$ for all  $\hbar  > 0$.

\item[iii.] ${{\mathcal{G}}_\kappa ^*}$ is called $\kappa-$complete if any $\kappa-$Cauchy sequence in ${{\mathcal{G}}_\kappa ^*}$ is $\kappa-$convergent to the point of ${{\mathcal{G}}_\kappa ^*}$.
\end{itemize}
\end{definition}

We use $\Psi \left( {\left[ {0,\infty } \right)} \right)$ to symbolize the set of all continuous and strictly increasing self-mappings $\vartheta :\left[ {0,\infty } \right) \to \left[ {0,\infty } \right)$ such that  $\vartheta \left( \iota \right) = 0\;  \Leftrightarrow \; \iota = 0$.
\begin{definition}
Let $\Omega :\left[ {0,\infty } \right) \times \left[ {0,\infty } \right) \to \mathbf{R}$ be a mapping. We consider the following conditions:
\begin{itemize}
\item[$\left( {{\Omega _1}} \right)$] $\Omega \left( {0,0} \right) = 0$,
\item[$\left( {{\Omega _2}} \right)$] $\Omega \left( {\iota,\nu } \right) < \nu  - \iota$ for all $\iota,\nu  > 0$,
\item[$\left( {\Omega _2}' \right)$]$\Omega \left( {\iota,\nu } \right) < \vartheta \left( \nu  \right) - \vartheta \left( \iota \right)$ for all $\iota,\nu  > 0$ and $\vartheta  \in \Psi$
\item[$\left( {\Omega _2}'' \right)$]$\Omega \left( {\iota,\nu } \right) < \vartheta \left( \nu  \right) - \vartheta \left( {{\tau ^\lambda }\iota} \right)$ for all $\iota,\nu  > 0$, $\vartheta  \in \Psi$ and a coefficient $\lambda  \ge 1.$
\item[$\left( {{\Omega _3}} \right)$] if $\left\{ {{\iota_n}} \right\}$, $\left\{ {{\nu _n}} \right\}$ are sequences in $\left( {0,\infty } \right)$ such that $\mathop {\lim }\limits_{n \to \infty } \;{\iota_n} = \mathop {\lim }\limits_{n \to \infty } \;{\nu _n} > 0$
\begin{equation}\label{xi3}
   \mathop {\lim\sup }\limits_{n \to \infty } \; \Omega \left( {{\iota_n},{\nu _n}} \right) < 0.
\end{equation}
\item[$\left( {{\Omega _3}'} \right)$] if $\left\{ {{\iota_n}} \right\}$, $\left\{ {{\nu _n}} \right\}$ are sequences in $\left( {0,\infty } \right)$ such that $\mathop {\lim }\limits_{n \to \infty } \;{\iota_n} = \mathop {\lim }\limits_{n \to \infty } \;{\nu _n} > 0$ and ${\iota_n} \leq {\nu _n}$, thereby  (\ref{xi3}) is fulfilled.
\end{itemize}

Taking into account the condition $\Omega_i$, we say that
\begin{itemize}
    \item $i = 1,2,3$ $\Rightarrow $ simulation function in the sense of Khojasteh et al., \cite{19},
    \item $i = 2,3$ $\Rightarrow $ simulation function in the sense of Argoubi et al., \cite{8},
    \item $i = 1,2,3'$ $\Rightarrow $ simulation function in the sense of Roldan Lopez de Hierro et al., \cite{9},
    \item $i = 2',3$ $\Rightarrow $ $\Psi-$simulation function in the sense of Joonaghany et al., \cite{10},
    \item $i = 2'',3'$ $\Rightarrow $ $\Psi_\tau-$simulation function in the sense of Zoto et al., \cite{11}.
\end{itemize}
\end{definition}

Next, we present some examples of simulation functions.
\begin{example}
Let ${\Omega _i}:\left[ {0,\infty } \right) \times \left[ {0,\infty } \right) \to \mathbf{R},(i = 1,2,3,4,5)$ be some functions.
\begin{itemize}
\item[i.]${\Omega _1}\left( {\iota,\nu} \right) = \vartheta \left(\nu \right) - \phi \left( \iota \right)$ for all $\iota,\nu \in \left[ {0,\infty } \right)$, where $\vartheta ,\phi :\left[ {0,\infty } \right) \to \left[ {0,\infty } \right)$ are two continuous functions such that $\vartheta \left( \iota \right) = \phi \left( \iota \right) = 0$ if and only if $\iota = 0$ and $\vartheta \left( \iota \right) < \iota \le \phi \left( \iota \right)$ for all $\iota > 0.$

\item[ii.]${\Omega _2}\left( {\iota,\nu} \right) =\nu - \frac{{\alpha\left( {\iota,\nu} \right)}}{{\beta\left( {\iota,\nu} \right)}}$ for all $\iota,\nu \in \left[ {0,\infty } \right)$, where $\alpha, \beta:\left[ {0,\infty } \right) \to \left( {0,\infty } \right)$ are two continuous functions with respect to each variable such that  $\alpha \left( {\iota,\nu} \right) > \beta \left( {\iota,\nu} \right)$ for all $\iota,\nu > 0.$

\item[iii.]${\Omega _3} \left( {\iota ,\nu } \right) = \phi \left( \nu  \right)\vartheta \left( \nu  \right) - \vartheta \left( \iota  \right)$, where
$\phi :\left[ {0,\infty } \right) \to \left[ {0,\infty } \right)$ is a  function such that $\mathop {\lim \sup }\limits_{\iota  \to \nu } \phi \left( \iota  \right) < 1$ for each $\nu  > 0$.

\item[iv.]${\Omega _4}\left( {\iota ,\nu } \right) = \vartheta \left( \nu  \right) - \phi \left( \nu  \right) - \vartheta \left( \iota  \right)$, where $\phi :\left[ {0,\infty } \right) \to \left[ {0,\infty } \right)$ is a  function such that $\mathop {\lim \inf }\limits_{\iota  \to \nu } \phi \left( \iota  \right) > 0$ for each $\nu  > 0$.


\item[v.]${\Omega _5}\left( {\iota ,\nu } \right) = \phi \left( \nu  \right) - \vartheta \left( {{\tau ^\lambda }\iota } \right)$ for all $\alpha ,\nu  \in \left( {0,\infty } \right)$, where $\phi, \vartheta :\left[ {0,\infty } \right) \to \left[ {0,\infty } \right)$  are continuous functions and $\vartheta$ is increasing function such that $\phi \left( \nu  \right) < \vartheta \left( \nu  \right)$ for each $\nu  > 0$ and a coefficient $\lambda  \ge 1.$
\end{itemize}
Then ${\Omega _i}$ for $i = 1,2$ is a simulation function given in \cite{19}. Also, for $i = 3,4$, ${\Omega_i}$ is a simulation function given in \cite{10}. Lastly, ${\Omega _5}$ is a simulation function given in \cite{11}.
\end{example}

In the sequel, the family $\Xi$ represents the set of all simulation functions in the sense of Khojasteh et al.

\begin{definition}\cite{19}
Let $J:\mathcal{G} \to \mathcal{G}$ be a mapping on a metric space $\left( {\mathcal{G},m} \right)$ and $\Omega  \in \Xi$. Then,  $J$ is said to be $\Xi$-contraction with respect to $\Omega$ if
$$\Omega \left( {m\left( {J\varsigma,J\ell} \right),m\left( {\varsigma,\ell} \right)} \right) \ge 0$$
is fulfilled for all $\varsigma,\ell \in \mathcal{G}.$
\end{definition}
Also, if we choose $\Omega  \in \Xi$ as $\Omega\left( {\iota,\nu} \right) =q \nu - \iota$ for all $\iota,\nu \in \left[ {0,\infty } \right)$, thereby, the above definition turn into Banach contraction.

\begin{remark}
On a metric space, by considering the definition of simulation function, it can be said that $\Omega \left( {\iota,\nu} \right) < 0$ for all $\iota \ge \nu > 0$.
Also, if $J$ is a $\Xi$-contraction, it is clear that $\Omega \left( {t,v} \right) < 0$ for all $t \ge v > 0$. Therefore, if $J$ is a $\Xi$-contraction about $\Omega  \in \Xi$, then
$$m\left( {J\varsigma,J\ell} \right) < m\left( {\varsigma,\ell} \right).$$
We conclude that every $\Xi$-contraction mapping is contractive, and in turn, continuous.
\end{remark}

In 2018, Cho \cite{ze} modified the definition of simulation functions which is  called  $\mathcal{Z}$ simulation function, as indicated below.

\begin{definition}\cite{ze}
Let $\zeta  :\left[ {1,\infty } \right) \times \left[ {1,\infty } \right) \to \mathbf{R}$ be a mapping. If the circumstances below hold, then $\zeta   \in \mathcal{Z}$ is a $\mathcal{Z}$-simulation function, where $\mathcal{Z}$ denotes the family of all the mappings.
\begin{itemize}
\item[$\left( {{\zeta  _1}} \right)$] $\zeta  \left( {1,1} \right) = 1$;
\item[$\left( {{\zeta  _2}} \right)$] $\zeta  \left( {\iota,\nu} \right) < {\nu \mathord{\left/
 {\vphantom {v t}} \right.
 \kern-\nulldelimiterspace} \iota}\;,\quad \forall \iota,\nu > 1;$
\item[$\left( {{\zeta  _3}} \right)$] for any sequence $\left\{ {{\iota_n}} \right\},\left\{ {{\nu_n}} \right\} \subset \left( {1,\infty } \right)$ with ${\iota_n} \le {\nu_n}\;,\quad \forall n = 1,2,3,...$
$$\mathop {\lim }\limits_{n \to \infty } {\iota_n} = \mathop {\lim }\limits_{n \to \infty } {\nu_n} > 1\quad  \Rightarrow \quad \mathop {\lim\sup }\limits_{n \to \infty }  \zeta  \left( {{\iota_n},{\nu_n}} \right) < 1.$$
\end{itemize}
Also, note that $\zeta  \left( {\iota,\iota} \right) < 1$, for all $\iota > 1.$
\end{definition}

\begin{example}\cite{ze}
The functions ${\zeta  _1},{\zeta  _2 },{\zeta  _3 }  :\left[ {1,\infty } \right) \times \left[ {1,\infty } \right) \to \mathbf{R}$ that identified as below, are belong to $\mathcal{Z}$.
\begin{itemize}
    \item[1.] ${\zeta  _1}\left( {\iota,\nu} \right) < {{{\nu^k}} \mathord{\left/
 {\vphantom {{{\nu^k}} \iota}} \right.
 \kern-\nulldelimiterspace} \iota}$, for all $\iota,\nu \ge 1,$ where $k \in \left( {0,1} \right);$
    \item[2.] ${\zeta  _2 }\left( {\iota,\nu} \right) < {\nu \mathord{\left/
 {\vphantom {\nu \iota}} \right.
 \kern-\nulldelimiterspace} \iota}\phi \left( \nu \right)$, for all $ \iota,\nu \ge 1,$ where $\phi :\left[ {1,\infty } \right) \to \left[ {1,\infty } \right)$ is non-decreasing and lower semi-continuous such that ${\phi ^{ - 1}}\left( {\left\{ 1 \right\}} \right) = 1;$
     \item[3.]$${\zeta  _3 } \left( {\iota,\nu} \right) = \left\{ \begin{array}{l}
1,\quad \quad {\rm{if}}\;\left( {\iota,\nu} \right) = \left( {1,1} \right),\\
\frac{\nu}{{2\iota}},\quad\;\; {\rm{if}}\;\nu < \iota,\\
\frac{{{\nu^\lambda }}}{\iota},\quad\;\; {\rm{otherwise}}{\rm{,}}
\end{array} \right.$$
for all $ \iota,\nu \ge 1,$ where $\lambda \in \left( {0,1} \right).$
\end{itemize}
\end{example}

In \cite{teta}, Jleli and Samet introduced the class  $\rm{T}=\{\gamma:(0,\infty)\rightarrow (1,\infty)\}$, which the functions in this class satisfy the following features:
\begin{itemize}
    \item[$\left( {{\gamma _1}} \right)$] $\gamma $ is non-decreasing;
    \item[$\left( {{\gamma _2}} \right)$] for each sequence $\left\{ {{\iota_n}} \right\} \subset \left( {0,\infty } \right)$, $\mathop {\lim }\limits_{n \to \infty } \gamma \left( {{\iota_n}} \right) = 1$ if and only if $\mathop {\lim }\limits_{n \to \infty } {\iota_n} = {0^ + };$
    \item[$\left( {{\gamma _3}} \right)$]there exist $r \in \left( {0,1} \right)$ and $\ell  \in \left( {0,\infty } \right]$ such that $\mathop {\lim }\limits_{\iota \to {0^ + }} \frac{{\gamma \left( \iota \right)}}{{{\iota^r}}} = \ell.$
  \end{itemize}
Additionally, they proved the following theorem in the generalized metric space (Some researchers call this space Branciari metric space, others call it rectangular metric space.).
\begin{theorem}
Let $J:\mathcal{G} \to \mathcal{G}$ be a given map on a  complete generalized metric space $\left( {\mathcal{G},m} \right)$. Presume that there exist $\gamma  \in \rm{T} $ and $k \in \left( {0,1} \right)$
$$m\left( {J\varsigma,J\ell} \right) \ne 0\quad  \Rightarrow \quad \gamma \left( {m\left( {J\varsigma,J\ell} \right)} \right) \le {\left[ {\gamma \left( {m\left( {\varsigma ,\ell } \right)} \right)} \right]^k}$$
for all $\varsigma ,\ell  \in \mathcal{G}$. Thereupon, the set of $Fix\left( J \right)$ has a unique element.
\end{theorem}

By applying the below statement instead of $\left( {{\gamma _3}} \right)$,
we obtain a new modified class
\begin{itemize}
 \item[$\left( {{{\gamma _3}'}} \right)$]$\gamma $ is continuous.
\end{itemize}
We denote this class by $\Theta $ that satisfies the properties $\left( {{\gamma _1}} \right)$, $\left( {{\gamma _2}} \right)$ and $\left( {{{\gamma _3}'}} \right)$.

Some examples of the $\gamma$ functions could be given as follows.
\begin{example}
For $\chi>0$; the following functions
\begin{itemize}
    \item ${\gamma _1}\left( \chi \right) = {e^\chi}$,
    \item ${\gamma _2}\left( \chi \right) = {e^{\sqrt \chi }}$,
    \item ${\gamma _3}\left( \chi \right) = {e^{\sqrt {\chi{e^\chi}} }}$,
    \item ${\gamma _4}\left( \chi \right) = \cosh \chi$,
    \item ${\gamma _5}\left( \chi \right) = 1 + \ln \left( {1 + \chi} \right)$,
    \item ${\gamma _6}\left( \chi \right) = {e^{\chi{e^\chi}}}$,
\end{itemize}
\end{example}
belong to the class $\Theta $.

\section{Main Results}

At first, the concept of metric modular does not have to be finite. Because of this, it is necessary to discuss the following additional conditions to guarantee the existence and uniqueness of fixed points of contraction mappings on modular metric spaces and modular $b-$metric spaces.
\begin{itemize}
    \item[$(S_1)$] ${\kappa _\hbar }\left( {\varsigma ,J\varsigma } \right) < \infty $ for all $\hbar >0$ and $\varsigma  \in {\mathcal{G}}_\kappa ^*,$
    \item[$(S_2)$] ${\kappa _\hbar }\left( {\varsigma ,\ell } \right) < \infty $ for all $\hbar >0$ and $\varsigma,\ell  \in {\mathcal{G}}_\kappa ^*.$
\end{itemize}

Now, we present some sets of auxiliary functions to be used in the following discussion.

Let $\Psi^*$ be denoted the set of all continuous and strictly increasing self-mappings $\varphi$ on $\left[ {1, + \infty } \right)$ so that $\varphi \left( \iota \right) = 1\;  \Leftrightarrow \; \iota = 1$.

The symbol ${\Delta _G}$ is denoted to indicate the set of all continuous functions $G:{\left[ {0,\infty } \right)^4} \to \left[ {0,\infty } \right)$ such that
\begin{itemize}
 \item[$(G_1)$] $G$ is non-decreasing with respect to each variable;
\item[$(G_2)$] $G\left( {\iota,\iota,\iota,\iota} \right) \le \iota$ for $\iota \in \left[ {0,\infty } \right)$;
\end{itemize}

\begin{example}The followings are some examples of the function $G$, which belongs to the set of ${\Delta _G}$.
\begin{itemize}
    \item[$(G_1)$] $G\left( {{\iota_1},{\iota_2},{\iota_3},{\iota_4}} \right) = \max \left\{ {{\iota_1},{\iota_2},{\iota_3},{\iota_4}} \right\};$
    \item[$(G_2)$] $G\left( {{\iota_1},{\iota_2},{\iota_3},{\iota_4}} \right) = \max \left\{ {{\iota_1} + {\iota_2},{\iota_2} + {\iota_3},{\iota_1} + {\iota_3},{\iota_3} + {\iota_4}} \right\};$
    \item[$(G_3)$] $G\left( {{\iota_1},{\iota_2},{\iota_3},{\iota_4}} \right) = {\left[ {\max \left\{ {{\iota_1}{\iota_2},{\iota_2}{\iota_3},{\iota_1}{\iota_3},{\iota_3}{\iota_4}} \right\}} \right]^{\frac{1}{2}}};$
    \item[$(G_4)$] $G\left( {{\iota_1},{\iota_2},{\iota_3},{\iota_4}} \right) = {\left[ {\max \left\{ {{\iota_1}^q,{\iota_2}^q,{\iota_3}^q,{\iota_4}^q} \right\}} \right]^{\frac{1}{q}}},\;q > 0;$
    \item[$(G_5)$] $G\left( {{\iota_1},{\iota_2},{\iota_3},{\iota_4}} \right) = {\iota_1};$
    \item[$(G_6)$] $G\left( {{\iota_1},{\iota_2},{\iota_3},{\iota_4}} \right) = \frac{{{\iota_2} + {\iota_3}}}{2};$
    \item[$(G_7)$] $G\left( {{\iota_1},{\iota_2},{\iota_3},{\iota_4}} \right) = {\iota_1} + {\iota_2} + {\iota_3} + {\iota_4};$
    \item[$(G_8)$] $G\left( {{\iota_1},{\iota_2},{\iota_3},{\iota_4}} \right) = {p_1}{\iota_1} + {p_2}{\iota_2} + {p_3}{\iota_3} + {p_4}{\iota_4},\;{\rm{with}}\;0 < {p_1} + {p_2} + {p_3} + {p_4} < 1.$
\end{itemize}
\end{example}
Now we present a new class of functions, which serve the family of simulation functions.
\begin{definition}
Let ${\hat Z}$ be the family of all mappings $\eta  :\left[ {1,\infty } \right) \times \left[ {1,\infty } \right) \to \mathbf{R}$ and if there exists a $ \varphi \in {\Psi^*}$ and a coefficient $\lambda  \ge 1$ such that

\begin{itemize}
\item[$\left( {{\eta _1}} \right)$] $\eta \left( {1,1} \right) = 1$;
\item[$\left( {{\eta _2}} \right)$]$\eta \left( {\iota,\nu} \right) < \frac{{\varphi \left( \nu \right)}}{{\varphi \left( \iota \right)}}\;,\quad \forall \iota,\nu > 1;$
\item[$\left( {{\eta _2}'} \right)$]$\eta \left( {\iota,\nu} \right) < \frac{{\varphi \left( \nu \right)}}{{\varphi \left( {{\tau ^\lambda }\iota} \right)}}\;,\quad \forall \iota,\nu > 1;$
\item[$\left( {{\eta _3}} \right)$] for any sequence $\left\{ {{\iota_n}} \right\},\left\{ {{\nu_n}} \right\} \subset \left( {1,\infty } \right)$ with ${\iota_n} \le {\nu_n}\;,\quad \forall n = 1,2,3,...$
$$\mathop {\lim }\limits_{n \to \infty } {\iota_n} = \mathop {\lim }\limits_{n \to \infty } {\nu_n} > 1\quad  \Rightarrow \quad \mathop {\lim\sup }\limits_{n \to \infty }  \eta \left( {{\iota_n},{\nu_n}} \right) < 1.$$
\end{itemize}
Then, if the function $\eta$ satisfies $\left( {{\eta _2}} \right)$-$\left( {{\eta _3}} \right)$, we say that $\eta$ is a generalized ${\Psi^*}-$simulation function.\\
Also, if $\eta$ provides only the conditions $\left( {{\eta _2'}} \right)$-$\left( {{\eta _3}} \right)$, then we say that $\eta$ is a  generalized ${\Psi ^*} - \tau $ simulation function.\\
Besides, if we take $\varphi \left( \iota \right) = \iota$ for $t \ge 1$ and the features $\left( {{\eta _1}} \right)$-$\left( {{\eta _2}} \right)$-$\left( {{\eta _3}} \right)$ are satisfied, then $\eta  \in \mathcal{Z}$ is a generalized simulation function in the sense of S. H. Cho \cite{ze}.
\end{definition}

\begin{example}
Let ${\eta _a},{\eta _b },{\eta _c },{\eta _d }:\left[ {1,\infty } \right) \times \left[ {1,\infty } \right) \to \mathbf{R}$ be functions defined as follows,
\begin{itemize}
    \item[$(e_1)$]${\eta _a}\left( {\iota,\nu} \right) = \frac{{\alpha \varphi \left( \nu \right)}}{{\varphi \left( \tau \iota \right)}},\quad \forall \iota,\nu \ge 1;\iota  \in \left( {0,1} \right);$
    \item[$(e_2)$]${\eta _b}\left( {\iota,\nu} \right) = \frac{{\varphi \left( \nu \right)}}{{\varphi \left( {\tau^\lambda}\iota \right)\phi \left( \nu \right)}}\;,\quad \forall \iota,\nu \ge 1$ and a coefficient $\lambda  \ge 1$, where $\phi :\left[ {1,\infty } \right) \to \left[ {1,\infty } \right)$ is non-decreasing and lower semi-continuous function such that ${\phi ^{ - 1}}\left( {\left\{ 1 \right\}} \right) = 1;$
     \item[$(e_3)$]${\eta _c}\left( {\iota,\nu} \right) = \frac{{\phi \left( \nu \right)}}{{\varphi \left( {\tau^\lambda}\iota \right)}},\quad \forall \iota,\nu \ge 1$ and a coefficient $\lambda  \ge 1$, where $\phi :\left[ {1,\infty } \right) \to \left[ {1,\infty } \right)$ continuous function such that $\phi \left( \nu \right) < \varphi \left( \nu \right)$ for all $\nu>0.$
     \item[$(e_4)$]$${\eta_d} \left( {\iota,\nu} \right) = \left\{ \begin{array}{l}
1,\quad \quad \quad  \;{\rm{if}}\;\left( {\iota,\nu} \right) = \left( {1,1} \right),\\
\frac{{\varphi \left( \nu \right)}}{{k\varphi \left( \iota \right)}},\quad \;\; {\rm{if}}\;\nu < \iota,\;\\
\frac{{{{\left[ {\varphi \left( \nu \right)} \right]}^p }}}{{\varphi \left( \iota \right)}},\quad {\rm{otherwise}}{\rm{,}}
\end{array} \right.$$
$\forall \iota,\nu \ge 1,$ where $k \ge 1$ and $p \in \left( {0,1} \right).$
\end{itemize}
Then, ${\eta _a},{\eta _b },{\eta _c }, {\eta _d }$ are generalized ${\Psi ^*} - \tau $ simulation function. For $\tau=1$, the ones above are examples of ${\Psi ^*} -$simulation functions.
\end{example}

Now, we are ready to present our main theorem in this section.
\begin{definition}
Let $\kappa$ be a modular $b-$metric on a set  $ \mathcal{G}$ and $J,K:{\mathcal{G}}_\kappa ^* \to {\mathcal{G}}_\kappa ^*$  be two mappings. $J$ and $K$ are called generalized almost simulative ${{\hat Z} }_{_{\Psi ^*} }^\Theta  - $contraction mappings if there exists a generalized ${\Psi ^*} - \tau $ simulation function with respect to $\eta$ and there is a constant  $\rho  \ge 0$ as well as  $\gamma  \in \Theta $ and $G \in {\Delta _G}$ such that

$$\frac{1}{{2\tau }}\min \left\{ {{\kappa _\hbar }\left( {\varsigma ,J\varsigma } \right),{\kappa _\hbar }\left( {\ell ,K\ell } \right)} \right\} \le {\kappa _\hbar }\left( {\varsigma ,\ell } \right)$$
implies
\begin{equation}\label{2.1}
\eta \left( {\gamma \left({\tau^4} {{\kappa_\hbar }\left( {J\varsigma,K\ell} \right)} \right),{{\left[ {\gamma \left( {G\left( {\varsigma ,\ell } \right) + \rho N\left( {\varsigma ,\ell } \right)} \right)} \right]}^k}} \right) \ge 1
\end{equation}
where,
$$G\left( {\varsigma ,\ell } \right) = \left( {{\kappa_\hbar }\left( {\varsigma ,\ell } \right),{\kappa_\hbar }\left( {\varsigma,J\varsigma} \right),{\kappa_\hbar }\left( {\ell,K\ell} \right),\frac{{{\kappa _{2\hbar }}\left( {\varsigma,K\ell} \right) + {\kappa _{2\hbar }}\left( {\ell,J\varsigma} \right)}}{{2\tau}}} \right)$$
and
$$N\left( {\varsigma ,\ell } \right) = \min \left\{ {{\kappa_\hbar }\left( {\varsigma,J\varsigma} \right),{\kappa_\hbar }\left( {\ell,K\ell} \right),{\kappa_\hbar }\left( {\varsigma,K\ell} \right),{\kappa_\hbar }\left( {\ell,J\varsigma} \right)} \right\},$$
for all distinct $\varsigma ,\ell  \in {\mathcal{G}}_\kappa ^*$, $k \in \left( {0,1} \right)$ and for all $\hbar  > 0$.
\end{definition}

 \begin{theorem}\label{teo2.1}
 Let ${\mathcal{G}}_\kappa ^*$ be a $\kappa-$complete MbMS with constant $\tau \ge 1$ and let $J$ and $K$ be generalized almost simulative ${{\hat Z} }_{_{\Psi ^*} }^\Theta  - $contraction mappings. If the condition $(S_1)$ is satisfied, then there exists $u \in {\cal G}_\kappa ^*$ such that $u \in {C_{Fix}}\left( {J,K} \right)$. If, in addition, the condition $(S_2)$ is satisfied, then ${C_{Fix}}\left( {J,K} \right)=\{u\}$.
 \end{theorem}

\begin{proof}
Let ${\varsigma _0} \in {\mathcal{G}}_\kappa ^*$ be an arbitrary point and we shall construct a sequence $\left\{ {{\varsigma _n}} \right\}$ by:
$${\varsigma_{2n + 1}} = J{\varsigma_{2n}}\;{\rm{and}}\;{\varsigma_{2n + 2}} = K{\varsigma_{2n + 1}},\quad {\rm{for}}\;{\rm{all}}\;n \in \mathbf{N}.$$
If there is some $n_0 \in \mathbf{N}$ such that ${\varsigma_{{n_0}}} = {\varsigma_{{n_0} + 1}}$, then ${n_0}$ becomes a common fixed point of $J$ and $K$. Consequently, we assume that ${\varsigma_{k}} \ne {\varsigma_{k + 1}}$ for all $k \in \mathbf{N}$. Therefore, we have ${\kappa _\hbar }\left( {{\varsigma_k},{\varsigma_{k + 1}}} \right) > 0$ for all $\hbar  > 0$.

Now, we will divide the proof into four-step to make sense more clear.
\item[Step (1):] We claim that $\mathop {\lim }\limits_{k \to \infty } {\kappa _\hbar }\left( {{\varsigma_k},{\varsigma_{k + 1}}} \right) = 0$ for all $\hbar  > 0.$

Thus, at first, we must show that
\begin{equation}\label{2.2}
    {\kappa_\hbar }\left( {{\varsigma_{k + 1}},{\varsigma_{k + 2}}} \right) < {\kappa_\hbar }\left( {{\varsigma_k},{\varsigma_{k + 1}}} \right),\quad {\rm{for}}\;{\rm{all}}\;n \in \mathbf{N}.
\end{equation}
We presume that $k = 2n$ for some $n \in \mathbf{N}.$ So, we have
$$\begin{array}{l}
\frac{1}{{2\tau}}\min \left\{ {{\kappa_\hbar }\left( {{\varsigma_{2n}},J{\varsigma_{2n}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2n + 1}},K{\varsigma_{2n + 1}}} \right)} \right\} = \frac{1}{{2\tau}}\min \left\{ {{\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2n + 1}},{\varsigma_{2n + 2}}} \right)} \right\}\\
\\
\quad \quad \quad \quad  \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  \le {\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right).
\end{array}$$
By using (\ref{2.1}) and $({\eta_2}')$, we get
$$\begin{array}{l}
1 \le \eta \left( {\gamma \left({\tau ^4} {{\kappa _\hbar }\left( {J{\varsigma _{2n}},K{\varsigma _{2n + 1}}} \right)} \right),{{\left( {\gamma \left[ {G\left( {{\varsigma _{2n}},{\varsigma _{2n + 1}}} \right) + \rho N\left( {{\varsigma _{2n}},{\varsigma _{2n + 1}}} \right)} \right]} \right)}^k}} \right)\\
\\
\quad  < \frac{{\varphi \left( {{{\left( {\gamma \left[ {G\left( {{\varsigma _{2n}},{\varsigma _{2n + 1}}} \right) + \rho N\left( {{\varsigma _{2n}},{\varsigma _{2n + 1}}} \right)} \right]} \right)}^k}} \right)}}{{\varphi \left({\tau ^\lambda} {\gamma \left({\tau ^4} {{\kappa _\hbar }\left( {{\varsigma _{2n + 1}},{\varsigma _{2n + 2}}} \right)} \right)} \right)}},
\end{array}$$
that is,
$$\varphi \left( {\tau ^\lambda}{\gamma \left( {\tau ^4}{{\kappa _\hbar }\left( {{\varsigma _{2n + 1}},{\varsigma _{2n + 2}}} \right)} \right)} \right) < \varphi \left( {{{\left( {\gamma \left[ {G\left( {{\varsigma _{2n}},{\varsigma _{2n + 1}}} \right) + \rho N\left( {{\varsigma _{2n}},{\varsigma _{2n + 1}}} \right)} \right]} \right)}^k}} \right).$$

Due to $\gamma  \in \Theta $ and $k \in \left( {0,1} \right)$ and also given features of the function $\varphi$, the above inequality gives
\begin{equation}\label{2.3}
{\tau ^{4 + \lambda }}{\kappa _\hbar }\left( {{\varsigma_{2n + 1}},{\varsigma_{2n + 2}}} \right) < G\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right) + \rho N\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right),
\end{equation}
where
$$\begin{array}{l}
G\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right) = \left( {{\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2n}},J{\varsigma_{2n}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2n + 1}},K{\varsigma_{2n + 1}}} \right),\frac{{{\kappa_{2\hbar }}\left( {{\varsigma_{2n}},K{\varsigma_{2n + 1}}} \right) + {\kappa_{2\hbar }}\left( {{\varsigma_{2n + 1}},J{\varsigma_{2n}}} \right)}}{{2\tau }}} \right)\\
\\
\quad \quad \quad \quad \quad \quad  = \left( {{\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2n + 1}},{\varsigma_{2n + 2}}} \right),\frac{{{\kappa_{2\lambda }}\left( {{\varsigma_{2n}},{\varsigma_{2n + 2}}} \right)}}{{2\tau}} } \right).
\end{array}$$
From $(\kappa_3)$, note that ${\kappa_{2\hbar }}\left( {{\varsigma_{2n}},{\varsigma_{2n + 2}}} \right) \le \tau \left[ {{\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right) + {\kappa_\hbar }\left( {{\varsigma_{2n + 1}},{\varsigma_{2n + 2}}} \right)} \right]$. So, we gain
\begin{equation}\label{G}
  G\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right) = \left( \begin{array}{l}
{\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2n + 1}},{\varsigma_{2n + 2}}} \right),\\
\\
\quad \quad \quad \frac{{{\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right) + {\kappa_\hbar }\left( {{\varsigma_{2n + 1}},{\varsigma_{2n + 2}}} \right)}}{2}
\end{array} \right).
\end{equation}

Now, if we assume ${\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right) \le {\kappa_\hbar }\left( {{\varsigma_{2n + 1}},{\varsigma_{2n + 2}}} \right)$,  we deduce that
\begin{equation}\label{2.4}
   \begin{array}{l}
G\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right)\\
\\
\quad  = \left( {{\kappa_\hbar }\left( {{\varsigma_{2n + 1}},{\varsigma_{2n + 2}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2n + 1}},{\varsigma_{2n + 2}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2n + 1}},{\varsigma_{2n + 2}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2n + 1}},{\varsigma_{2n + 2}}} \right)} \right)\\
\\
\quad  \le {\kappa_\hbar }\left( {{\varsigma_{2n + 1}},{\varsigma_{2n + 2}}} \right), \quad \left( {{\rm{by}}\;G \in {\Delta _G}} \right).
\end{array}
\end{equation}
Nevertheless, we have
\begin{equation}\label{2.5}
    \begin{array}{l}
N\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right)\\
\\
\quad  = \min \left\{ {{\kappa_\hbar }\left( {{\varsigma_{2n}},J{\varsigma_{2n}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2n + 1}},K{\varsigma_{2n + 1}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2n}},K{\varsigma_{2n + 1}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2n + 1}},J{\varsigma_{2n}}} \right)} \right\}\\
\\
\quad  = \min \left\{ {{\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2n + 1}},{\varsigma_{2n + 2}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n + 2}}} \right),0} \right\} = 0.
\end{array}
\end{equation}
Consequently, by using (\ref{2.4}) and (\ref{2.5}), the inequality (\ref{2.3}) becomes
$${\tau ^{4 + \lambda }}{\kappa_\hbar }\left( {{\varsigma_{2n + 1}},{\varsigma_{2n + 2}}} \right) < {\kappa_\hbar }\left( {{\varsigma_{2n + 1}},{\varsigma_{2n + 2}}} \right).$$
This is a contradiction since $\tau,\lambda \geq 1$. Then, our assumption is false, i.e., the expression (\ref{2.2}) is proved when $k$ is an even number. Similarly, it can be shown that (\ref{2.2}) is held when $k$ is an odd number. So, we say that the sequence ${\left\{ {{\kappa_\hbar }\left( {{\varsigma_n},{\varsigma_{n + 1}}} \right)} \right\}_{n \ge 1}}$ is non-decreasing and bounded from zero. Hence there is a real number $\iota  \ge 0$ such that $\mathop {\lim }\limits_{n \to \infty } {\kappa _\hbar }\left( {{\varsigma _n},{\varsigma _{n + 1}}} \right) = \iota $ for all $\hbar >0.$

Now, we will show $\iota=0$. On the contrary, we claim that $\iota>0.$ To see this, it is enough to mention the below two cases.
\item[Case (1):] Assume that $\tau >1$.

As the expression (\ref{2.2}) holds, we conclude that
$$\begin{array}{l}
G\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right)\\
\\
\quad  = \left( {{\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right)} \right)\\
\\
\quad  \le {\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n+1}}} \right),\quad \left( {{\rm{by}}\;G \in {\Delta _G}} \right).
\end{array}$$
So, by keeping in mind (\ref{2.3}), we obtain
$${\tau ^{4 + \lambda }}{\kappa_\hbar }\left( {{\varsigma_{2n + 1}},{\varsigma_{2n + 2}}} \right) < {\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n + 1}}} \right) .$$
If we take the limit as $n\rightarrow \infty $ in the above inequality,
we face with a contradiction as ${\tau ^{4 + \lambda }}\iota  < \iota .$

\item[Case (2):] Deem that $\tau =1$. Via using the expression (\ref{2.1}), we achieve
 $$1 \le \eta \left( {\gamma \left( {{\kappa _\hbar }\left( {J{\varsigma _{2n}},K{\varsigma _{2n + 1}}} \right)} \right),{{\left( {\gamma \left[ {G\left( {{\varsigma _{2n}},{\varsigma _{2n + 1}}} \right) + \rho N\left( {{\varsigma _{2n}},{\varsigma _{2n + 1}}} \right)} \right]} \right)}^k}} \right).$$

Similar to (\ref{2.4}) and (\ref{2.5}), we get that
\begin{equation}\label{to2}
1 \le \eta \left( {\gamma \left( {{\kappa _\hbar }\left( {{\varsigma _{2n + 1}},{\varsigma _{2n + 2}}} \right)} \right),{{\left( {\gamma \left[ {{\kappa _\hbar }\left( {{\varsigma _{2n}},{\varsigma _{2n + 1}}} \right)} \right]} \right)}^k}} \right)
\end{equation}
and so, from both $({\eta_2}')$ and $\varphi$ is strictly increasing, we have
\begin{equation}\label{to1}
   \gamma \left( {{\kappa _\hbar }\left( {{\varsigma _{n + 1}},{\varsigma _{n + 2}}} \right)} \right) < {\left[ {\gamma \left( {{\kappa _\hbar }\left( {{\varsigma _n},{\varsigma _{n + 1}}} \right)} \right)} \right]^k}.
\end{equation}
Let ${\iota_n} = \gamma \left( {{\kappa_\hbar }\left( {{\varsigma_{n + 1}},{\varsigma_{n + 2}}} \right)} \right)$ and ${\nu_n} = {\left[ {\gamma \left( {{\kappa _\hbar }\left( {{\varsigma _n},{\varsigma _{n + 1}}} \right)} \right)} \right]^k}$ for all $n \in \mathbf{N}.$

Now, since $\mathop {\lim }\limits_{n \to \infty } {\kappa_\hbar }\left( {{\varsigma_n},{\varsigma_{n + 1}}} \right) = \iota  > 0$, then it follows from $(\gamma_2)$ that $\mathop {\lim }\limits_{n \to \infty } \gamma \left( {{\kappa_\hbar }\left( {{\varsigma_n},{\varsigma_{n + 1}}} \right)} \right) \ne 1$ and so $\mathop {\lim }\limits_{n \to \infty } \gamma \left( {{\kappa_\hbar }\left( {{\varsigma_n},{\varsigma_{n + 1}}} \right)} \right) > 1.$

Accordingly, from (\ref{to1}), we obtain
$$\begin{array}{l}
{\iota_n} = \gamma \left( {{\kappa _\hbar }\left( {{\varsigma _{n + 1}},{\varsigma _{n + 2}}} \right)} \right) < {\left[ {\gamma \left( {{\kappa _\hbar }\left( {{\varsigma _n},{\varsigma _{n + 1}}} \right)} \right)} \right]^k} = {\nu_n}\\
\\
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  < \left[ {\gamma \left( {{\kappa_\hbar }\left( {{\varsigma_n},{\varsigma_{n + 1}}} \right)} \right)} \right].
\end{array}$$
Hence, considering as ${n \to \infty } $ in the above, we get $\gamma \left( r \right) \le \mathop {\lim }\limits_{n \to \infty } {\nu_n} \le \gamma \left( r \right)$. This implies that $\mathop {\lim }\limits_{n \to \infty } {\iota_n} = \mathop {\lim }\limits_{n \to \infty } {\nu_n} = \gamma \left( r \right) > 1$. Hence, from $(\eta_3)$, we deduce that $\mathop {\lim\sup }\limits_{n \to \infty }  \eta ({\iota_n},{\nu_n}) < 1$. Nevertheless, it is a contradiction due to (\ref{to2}).

Consequently, in both cases, we have a contradiction.  For this reason we procure the following $\iota=0$, that is, for all $\hbar >0$,
\begin{equation}\label{2.8}
    \mathop {\lim }\limits_{n \to \infty } {\kappa_\hbar }\left( {{\varsigma_n},{\varsigma_{n + 1}}} \right) = 0.
\end{equation}

\item[Step (2):] We assert that $\left\{ {{\varsigma_n}} \right\}$ is a $\kappa-$Cauchy sequence. It is enough to prove that $\left\{ {{\varsigma_{2n}}} \right\}$ is a $\kappa -$Cauchy sequence. Unlike our claim, suppose that $\left\{ {{\varsigma_{2n}}} \right\}$ is not $\kappa -$Cauchy sequence, there exists $\varepsilon  > 0$ such that we can form two sequences $\left\{ {{\varsigma_{2{m_q}}}} \right\}$ and $\left\{ {{\varsigma_{2{n_q}}}} \right\}$ of positive integers satisfying ${n_q} > {m_q} > q$ such that ${n_q}$ smallest index for which `
\begin{equation}\label{2.9}
{\kappa_\hbar }\left( {{\varsigma_{2{m_q}}},{\varsigma_{2{n_q}}}} \right) \ge \varepsilon \quad {\rm{and}}\quad {\kappa_\hbar }\left( {{\varsigma_{2{m_q}}},{\varsigma_{2{n_q}-2}}} \right) < \varepsilon ,\quad {\rm{for}}\;{\rm{all}}\;{\hbar>0.}
\end{equation}
Now, without loss of the generality from (\ref{2.9}) and  the modular inequality, we get
$$\varepsilon  \le {\kappa _{2\hbar }}\left( {{\varsigma _{2{m_q}}},{\varsigma _{2{n_q}}}} \right) \le \tau {\kappa _\hbar }\left( {{\varsigma _{2{m_q}}},{\varsigma _{2{n_q} + 1}}} \right) + \tau {\kappa _{2\hbar }}\left( {{\varsigma _{2{n_q} + 1}},{\varsigma _{2{n_q}}}} \right).$$
Letting $q \to \infty $ and using (\ref{2.8}) in the above, we obtain
\begin{equation}\label{2.10}
    \mathop {\lim\sup }\limits_{q \to \infty } {\kappa_\hbar }\left( {{\varsigma_{2{n_q} + 1}},{\varsigma_{2{m_q}}}} \right) \ge \frac{\varepsilon }{{{\tau}}}.
\end{equation}
Similarly, we have
$${\kappa _\hbar }\left( {{\varsigma _{2{m_q} - 1}},{\varsigma _{2{n_q}}}} \right) \le \tau {\kappa _{\frac{\hbar }{2}}}\left( {{\varsigma _{2{m_q} - 1}},{\varsigma _{2{m_q}}}} \right) + {\tau ^2}{\kappa _{\frac{\hbar }{4}}}\left( {{\varsigma _{2{m_q}}},{\varsigma _{2{n_q} - 2}}} \right) +
{\tau ^3}{\kappa _{\frac{\hbar }{8}}}\left( {{\varsigma _{2{n_q} - 2}},{\varsigma _{2{n_q} - 1}}} \right) + {\tau ^3}{\kappa _{\frac{\hbar }{8}}}\left( {{\varsigma _{2{n_q} - 1}},{\varsigma _{2{n_q}}}} \right).$$
If we take the limit superior as $q \to \infty $ in the above inequality, we get
\begin{equation}\label{2.11}
    \mathop {\lim \sup }\limits_{q \to \infty } {\kappa _\hbar }\left( {{\varsigma _{2{m_q} - 1}},{\varsigma _{2{n_q}}}} \right) \le {\tau ^2}\varepsilon.
\end{equation}
Moreover, like in the above, we achieve
$${\kappa _\hbar }\left( {{\varsigma_{2{m_q}}},{\varsigma_{2{n_q}}}} \right) \le \tau {\kappa _{\frac{\hbar }{2}}}\left( {{\varsigma_{2{m_q}}},{\varsigma_{2{n_q} - 2}}} \right) + {\tau^2} {\kappa _{\frac{\hbar }{4}}}\left( {{\varsigma_{2{n_q} - 2}},{\varsigma_{2{n_q} - 1}}} \right) + {\tau^2} {\kappa _{\frac{\hbar }{4}}}\left( {{\varsigma_{2{n_q} - 1}},{\varsigma_{2{n_q}}}} \right).$$
So, by using (\ref{2.8}), letting $q \to \infty $, we procure
\begin{equation}\label{2.12}
    \mathop {\lim \sup }\limits_{q \to \infty } {\kappa _\hbar }\left( {{\varsigma_{2{m_q}-1}},{\varsigma_{2{n_q}+1}}} \right) \le {\tau^2} \varepsilon  .
\end{equation}
In conclusion, similar to the above calculations, it can be shown that
\begin{equation}\label{2.13}
   \mathop {\lim \sup }\limits_{q \to \infty } {\kappa_\hbar }\left( {{\varsigma_{2{m_q} + 1}},{\varsigma_{2{n_q} + 1}}} \right) \le {\tau ^2}\varepsilon .
\end{equation}
Besides, we suggest that for sufficiently large $q \in \mathbf{N}$, if ${n_q} > {m_q} > q$, then
\begin{equation}\label{13}
   \frac{1}{{2\tau }}\min \left\{ {{\kappa_\hbar }\left( {{\varsigma_{2{n_q}}},J{\varsigma_{2{n_q}}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2{m_q} -1}},K{\varsigma_{2{m_q} -1}}} \right)} \right\} \le {\kappa_\hbar }\left( {{\varsigma_{2{n_q}}},{\varsigma_{2{m_q} - 1}}} \right).
\end{equation}

In fact, owing to ${n_q} > {m_q}$ and ${\left\{ {{\kappa_\hbar }\left( {{\varsigma_n},{\varsigma_{n + 1}}} \right)} \right\}_{n \ge 1}}$ is non-decreasing, we acquire
$$\begin{array}{l}
{\kappa _\hbar }\left( {{\varsigma _{2{n_q}}},J{\varsigma _{2{n_q}}}} \right) = {\kappa _\hbar }\left( {{\varsigma _{2{n_q}}},{\varsigma _{2{n_q} + 1}}} \right) \le {\kappa _\hbar }\left( {{\varsigma _{2{m_q} + 1}},{\varsigma _{2{m_q}}}} \right) \le {\kappa _\hbar }\left( {{\varsigma _{2{m_q}}},{\varsigma _{2{m_q} - 1}}} \right)\\
\\
\quad \quad \quad \quad \quad \quad \quad  = {\kappa _\hbar }\left( {{\varsigma _{2{m_q} - 1}},K{\varsigma _{2{m_q} - 1}}} \right).
\end{array}$$
Hence, the left-hand side of inequality (\ref{13}) is equal to
$$\frac{1}{{2\tau }}{\kappa _\hbar }\left( {{\varsigma _{2{n_q}}},J{\varsigma _{2{n_q}}}} \right) = \frac{1}{{2\tau }}{\kappa _\hbar }\left( {{\varsigma _{2{n_q}}},{\varsigma _{2{n_q} + 1}}} \right).$$
Now, we must show that, for sufficiently large $q \in \mathbf{N}$, if ${n_q} > {m_q} > q$, then
$${\kappa _\hbar }\left( {{\varsigma _{2{n_q}}},{\varsigma _{2{n_q} + 1}}} \right) \le {\kappa _\hbar }\left( {{\varsigma _{2{n_q}}},{\varsigma _{2{m_q} - 1}}} \right).$$
According to (\ref{2.8}), there exists ${q_1} \in \mathbf{N}$ such that for any $q>{q_1}$,
$${\kappa _\hbar }\left( {{\varsigma _{2{n_q}}},{\varsigma _{2{n_q} + 1}}} \right) < \frac{\varepsilon }{2\tau }.$$
Also, there exists ${q_2} \in \mathbf{N}$ such that for any $q>{q_2}$,
$${\kappa _\hbar }\left( {{\varsigma _{2{m_q} - 1}},{\varsigma _{2{m_q}}}} \right) < \frac{\varepsilon }{2\tau }.$$
Therefore, for any $q > \max \left\{ {{q_1},{q_2}} \right\}$  and ${n_q} > {m_q} > q$, we have
$$\begin{array}{l}
\varepsilon  \le {\kappa _\hbar }\left( {{\varsigma _{2{n_q}}},{\varsigma _{2{m_q}}}} \right) \le \tau {\kappa _\hbar }\left( {{\varsigma _{2{n_q}}},{\varsigma _{2{m_q} - 1}}} \right) + \tau {\kappa _\hbar }\left( {{\varsigma _{2{m_q} - 1}},{\varsigma _{2{m_q}}}} \right)\\
\\
\quad \quad \quad \; \quad \quad \quad \quad \quad  \le \tau {\kappa _\hbar }\left( {{\varsigma _{2{n_q}}},{\varsigma _{2{m_q} - 1}}} \right) + \tau \frac{\varepsilon }{2\tau }.
\end{array}$$
So, one concludes that
$$\frac{\varepsilon }{{2\tau }} \le {\kappa _\hbar }\left( {{\varsigma _{2{n_q}}},{\varsigma _{2{m_q} - 1}}} \right).$$
Hence, we deduce that for any $q > \max \left\{ {{q_1},{q_2}} \right\}$ and ${n_q} > {m_q} > q$,
$${\kappa _\hbar }\left( {{\varsigma _{2{n_q}}},{\varsigma _{2{n_q} + 1}}} \right) \le \frac{\varepsilon }{{2\tau }} \le {\kappa _\hbar }\left( {{\varsigma _{2{n_q}}},{\varsigma _{2{m_q} - 1}}} \right)$$
that is, the expression (\ref{13}) is proved. So, from (\ref{2.1}), it implies that
$$\eta \left( {\gamma \left( {\tau ^4}{{\kappa _\hbar }\left( {J{\varsigma _{2{n_q}}},K{\varsigma _{2{m_q} - 1}}} \right)} \right),{{\left( {\gamma \left[ {G\left( {{\varsigma _{2{n_q}}},{\varsigma _{2{m_q} - 1}}} \right) + \rho N\left( {{\varsigma _{2{n_q}}},{\varsigma _{2{m_q} - 1}}} \right)} \right]} \right)}^k}} \right) \ge 1.$$
By using $({\eta_2}')$ and taking into account the properties of $\varphi$ and $\gamma$ with $k \in \left( {0,1} \right)$, we obtain
\begin{equation}\label{c}
  {\tau ^{4 + \lambda }}{\kappa_\hbar }\left( {J{\varsigma_{2{n_q}}},K{\varsigma_{2{m_q} - 1}}} \right) < G\left( {{\varsigma_{2{n_q}}},{\varsigma_{2{m_q} - 1}}} \right) + \rho N\left( {{\varsigma_{2{n_q}}},{\varsigma_{2{m_q} - 1}}} \right),
\end{equation}
where
$$\begin{array}{l}
G\left( {{\varsigma_{2{n_q}}},{\varsigma_{2{m_q} - 1}}} \right) = \left( \begin{array}{l}
{\kappa_\hbar }\left( {{\varsigma_{2{n_q}}},{\varsigma_{2{m_q} - 1}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2{n_q}}},J{\varsigma_{2{n_q}}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2{m_q} - 1}},K{\varsigma_{2{m_q} - 1}}} \right),\\
\\
\frac{{{\kappa_{2\hbar }}\left( {{\varsigma_{2{n_q}}},K{\varsigma_{2{m_q} - 1}}} \right) + {\kappa_{2\hbar }}\left( {{\varsigma_{2{m_q} - 1}},J{\varsigma_{2{n_q}}}} \right)}}{{2\tau }}
\end{array} \right)\\
\\
\quad \quad \quad \quad \quad \quad \quad \; \, = \left( \begin{array}{l}
{\kappa_\hbar }\left( {{\varsigma_{2{n_q}}},{\varsigma_{2{m_q} - 1}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2{n_q}}},{\varsigma_{2{n_q} + 1}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2{m_q} - 1}},{\varsigma_{2{m_q}}}} \right),\\
\\
\frac{{{\kappa_{2\hbar }}\left( {{\varsigma_{2{n_q}}},{\varsigma_{2{m_q}}}} \right) + {\kappa_{2\hbar }}\left( {{\varsigma_{2{m_q} - 1}},{\varsigma_{2{n_q} + 1}}} \right)}}{{2\tau }}
\end{array} \right).
\end{array}$$
By taking the limit superior as $q \to \infty $ in above and using (\ref{2.8}), (\ref{2.11}),(\ref{2.12}) and (\ref{2.13}), we derive the following:
\begin{equation}\label{Gi}
    \mathop {\limsup }\limits_{q \to \infty } G\left( {{\varsigma_{2{n_q}}},{\varsigma_{2{m_q} - 1}}} \right) \le \left( {{\tau^2}\varepsilon ,0,0,\frac{{{\tau}\varepsilon  + {\tau^2}\varepsilon }}{{2\tau}}} \right) \le {\tau^2}\varepsilon .
\end{equation}
Also,
$$\begin{array}{l}
N\left( {{\varsigma_{2{n_q}}},{\varsigma_{2{m_q} - 1}}} \right) = \min \left\{ \begin{array}{l}
{\kappa_\hbar }\left( {{\varsigma_{2{n_q}}},J{\varsigma_{2{n_q}}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2{m_q} - 1}},K{\varsigma_{2{m_q} - 1}}} \right),\\
\\
{\kappa_\hbar }\left( {{\varsigma_{2{n_q}}},K{\varsigma_{2{m_q} - 1}}} \right){\kappa_\hbar }\left( {{\varsigma_{2{m_q} - 1}},J{\varsigma_{2{n_q}}}} \right)
\end{array} \right\}\\
\\
\quad \quad \quad \quad \quad \quad  \quad  \;= \min \left\{ \begin{array}{l}
{\kappa_\hbar }\left( {{\varsigma_{2{n_q}}},{\varsigma_{2{n_q} + 1}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2{m_q} - 1}},{\varsigma_{2{m_q}}}} \right),\\
\\
{\kappa_\hbar }\left( {{\varsigma_{2{n_q}}},{\varsigma_{2{m_q}}}} \right),{\kappa_\hbar }\left( {{\varsigma_{2{m_q} - 1}},{\varsigma_{2{n_q} + 1}}} \right)
\end{array} \right\}
\end{array}$$
and so, we have
\begin{equation}\label{N}
    \mathop {\limsup  }\limits_{q \to \infty }N\left( {{\varsigma_{2{n_q}}},{\varsigma_{2{m_q} - 1}}} \right) = 0.
\end{equation}
Consequently, if we take the limit as $q \to \infty $, considering (\ref{2.10}), (\ref{Gi}) and (\ref{N}), the inequality (\ref{c}) becomes ${\tau^{4 + \lambda }}\frac{\varepsilon }{{{\tau}}} < {\tau^2}\varepsilon  + \rho 0.$ This is a contradiction, that is, $\left\{ {{\varsigma_{2n}}} \right\}$ is a $\kappa-$Cauchy sequence. Thereby, $\left\{ {{\varsigma_n}} \right\}$ is a $\kappa-$Cauchy sequence in ${{\mathcal{G}}_\kappa ^*}$. Since ${{\mathcal{G}}_\kappa ^*}$ is $\kappa-$complete MbMS, there exists $u \in {{\mathcal{G}}_\kappa ^*}$ such that
\begin{equation}\label{2.14}
 \mathop {\lim }\limits_{n \to \infty } {\varsigma_n} = u.
\end{equation}

\item[Step (3):] In this step, we will prove that $u \in {C_{Fix}}\left( {J,K} \right)$.

First of all, we shall prove that $u \in Fix\left( K \right)$. Conversely, this statement is not true.
We claim that for all $n \ge 0$, at least one of the following inequalities is true:
\begin{equation}\label{42}
\frac{1}{{2\tau }}{\kappa _\hbar }\left( {{\varsigma _{2n}},{\varsigma _{2n + 1}}} \right) \le {\kappa _\hbar }\left( {{\varsigma _{2n}},u} \right),
\end{equation}
or
\begin{equation}\label{43}
\frac{1}{{2\tau }}{\kappa _\hbar }\left( {{\varsigma _{2n+1}},{\varsigma _{2n + 2}}} \right) \le {\kappa _\hbar }\left( {{\varsigma _{2n}},u} \right).
\end{equation}
Unlike, if for some ${n_0} \ge 0$, both of them are not provided. Hence, we say that
$$\begin{array}{l}
{\kappa _\hbar }\left( {{\varsigma _{2{n_0}}},{\varsigma _{2{n_0} + 1}}} \right) \le \tau {\kappa _\hbar }\left( {{\varsigma _{2{n_0}}},u} \right) + \tau {\kappa _\hbar }\left( {u,{\varsigma _{2{n_0} + 1}}} \right)\\
\\
\quad \quad \quad \quad \quad \quad \quad  < \frac{1}{2}{\kappa _\hbar }\left( {{\varsigma _{2{n_0}}},{\varsigma _{2{n_0} + 1}}} \right) + \frac{1}{2}{\kappa _\hbar }\left( {{\varsigma _{2{n_0} + 1}},{\varsigma _{2{n_0} + 2}}} \right)\\
\\
\quad \quad \quad \quad \quad \quad \quad  < \frac{1}{2}{\kappa _\hbar }\left( {{\varsigma _{2{n_0}}},{\varsigma _{2{n_0} + 1}}} \right) + \frac{1}{2}{\kappa _\hbar }\left( {{\varsigma _{2{n_0}}},{\varsigma _{2{n_0} + 1}}} \right) = {\kappa _\hbar }\left( {{\varsigma _{2{n_0}}},{\varsigma _{2{n_0} + 1}}} \right),
\end{array}$$
such that it is a contradiction. That is why the assertion is true. From this point, one can discuss the following two subcases.
\item[Subcase (3.1):] The inequality (\ref{42}) holds for infinitely many $n \ge 0$. In this case, for infinitely many $n \ge 0$, we have

$$\begin{array}{*{20}{l}}
{\frac{1}{{2\tau }}\min \left\{ {{\kappa _\hbar }\left( {{\varsigma _{2n}},J{\varsigma _{2n}}} \right),{\kappa _\hbar }\left( {u,Ku} \right)} \right\} = \frac{1}{{2\tau }}\min \left\{ {{\kappa _\hbar }\left( {{\varsigma _{2n}},{\varsigma _{2n + 1}}} \right),{\kappa _\hbar }\left( {u,Ku} \right)} \right\}}\\
{}\\
{\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  \le {\kappa _\hbar }\left( {{\varsigma _{2n}},u} \right)}
\end{array}$$
which yields that
$$\eta \left( {\gamma \left({\tau ^4} {{\kappa _\hbar }\left( {J{\varsigma_{2n}},Ku} \right)} \right),{{\left( {\gamma \left[ {G\left( {{\varsigma_{2n}},u} \right) + \rho N\left( {{\varsigma_{2n}},u} \right)} \right]} \right)}^k}} \right) \ge 1.$$
By using $\left( {{{\eta _2}'}} \right)$ and keep in mind that $\varphi  \in {\Psi ^*}$ and $\gamma  \in \Theta $ with $k \in \left( {0,1} \right)$, then we conclude that
\begin{equation}\label{v}
    {\tau ^{4+\lambda}}{\kappa _\hbar }\left( {{\varsigma_{2n +1}},Ku,} \right) < G\left( {{\varsigma_{2n}},u} \right) + \rho N\left( {{\varsigma_{2n}},u} \right),
\end{equation}
where
$$\begin{array}{l}
G\left( {{\varsigma_{2n}},u} \right) = G\left( \begin{array}{l}
{\kappa_\hbar }\left( {{\varsigma_{2n}},u} \right),{\kappa_\hbar }\left( {{\varsigma_{2n}},J{\varsigma_{2n}}} \right),{\kappa_\hbar }\left( {u,Ku} \right),\\
\\
\frac{{{\kappa_{2\hbar }}\left( {{\varsigma_{2n}},Ku} \right) + {\kappa_{2\hbar }}\left( {u,J{\varsigma_{2n}}} \right)}}{{2\tau }}
\end{array} \right)\\
\\
\quad \quad \quad \quad \quad  \le G\left( \begin{array}{l}
{\kappa_\hbar }\left( {{\varsigma_{2n}},u} \right),{\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n+1}}} \right),{\kappa_\hbar }\left( {u,Ku} \right),\\
\\
\frac{{\tau \left[ {{\kappa_\hbar }\left( {{\varsigma_{2n}},{\varsigma_{2n + }}} \right) + {\kappa_\hbar }\left( {{\varsigma_{2n + 1 }},Ku} \right)} \right]+{\kappa_{2\hbar }}\left( {u,{\varsigma_{2n +1}}} \right)}}{{2\tau }}
\end{array} \right),
\end{array}$$
and
\begin{equation}\label{v1}
    \mathop {\limsup }\limits_{n \to \infty } G\left( {u,{\varsigma_{2n + 1}}} \right) = G\left( {0,0,{\kappa _\hbar }\left( {u,Ju} \right),\frac{{{\kappa_\hbar }\left( {u,Ku} \right)}}{2}} \right) \le {\kappa _\hbar }\left( {u,Ku} \right).
\end{equation}
Also,
$$\begin{array}{l}
N\left( {{\varsigma _{2n}},u} \right)\\
\\
\quad = \min \left\{ {{\kappa _\hbar }\left( {{\varsigma _{2n}},J{\varsigma _{2n}}} \right),{\kappa _\hbar }\left( {u,Ku} \right),{\kappa _\hbar }\left( {{\varsigma _{2n}},Ku} \right),{\kappa _\hbar }\left( {u,J{\varsigma _{2n}}} \right)} \right\}\\
\\
\quad = \min \left\{ {{\kappa _\hbar }\left( {{x_{2n}},{x_{2n + 1}}} \right),{\kappa _\hbar }\left( {u,Ku} \right),{\kappa _\hbar }\left( {{\varsigma _{2n}},Ku} \right),{\kappa _\hbar }\left( {u,{\varsigma _{2n + 1}}} \right)} \right\}
\end{array}$$
and so
\begin{equation}\label{v2}
    \mathop {\limsup }\limits_{n \to \infty } N\left( {{\varsigma _{2n}},u} \right) = \min \left\{ {0,{\kappa _\hbar }\left( {u,Ju} \right),{\kappa _\hbar }\left( {u,Ju} \right),0,0} \right\} = 0.
\end{equation}
Next, by (\ref{v1}) and (\ref{v2}), if we take the limit superior as $n \to \infty$ in (\ref{v}), then it gives a contradiction since ${\tau ^{4 + \lambda }}{\kappa _\hbar }\left( {u,Ju} \right) < {\kappa _\hbar }\left( {u,Ju} \right).$ Eventually, we have $u \in Fix\left( K \right)$. It is also clear to get $u \in Fix\left( J \right)$ by following the above similar way.
\item[Subcase (3.2):]  The inequality (\ref{42}) merely satisfies for finitely many $n \ge 0$.

Thereupon, we can find ${n_0} \ge 0$ such that (\ref{43}) holds for any $n \ge {n_0}.$ In the same way, as in Subcase (3.1), it follows that (\ref{43}) also causes a contradiction unless $u \in Fix\left( J \right)$ or $u \in Fix\left( K \right)$.

So, in both subcases, we achieve that  $u \in {C_{Fix}}\left( {J,K} \right)$.

\item[Step (4):] We claim that the  set of ${C_{Fix}}\left( {J,K} \right)$ has a unique element. Suppose on the contrary. Then, there is a point $r$ belongs to ${{\mathcal{G}}_\kappa ^*}$ by $r \ne u$ such that $r \in {C_{Fix}}\left( {J,K} \right)$. Since
$$0 = \frac{1}{{2\tau }}\min \left\{ {{\kappa_\hbar }\left( {u,Ju} \right),{\kappa_\hbar }\left( {r,Kr} \right)} \right\} \le {\kappa_\hbar }\left( {u,r} \right),$$ it implies that
\begin{equation}\label{U}
 \eta \left( {\gamma \left({\tau ^4} {{\kappa _\hbar }\left( {Ju,Kr} \right)} \right),{{\left( {\gamma \left[ {G\left( {u,r} \right) + \rho N\left( {u,r} \right)} \right]} \right)}^k}} \right) \ge 1,
\end{equation}
where
\begin{equation}\label{U1}
 G\left( {u,r} \right) = G\left( {{\kappa _\hbar }\left( {u,r} \right),{\kappa _\hbar }\left( {u,Ju} \right),{\kappa _\hbar }\left( {r,Kr} \right),\frac{{{\kappa _{2\hbar }}\left( {u,Kr} \right) + {\kappa _{2\hbar }}\left( {r,Ju} \right)}}{{2\tau }}} \right) \le {\kappa _\hbar }\left( {u,r} \right)
\end{equation}
and also,
\begin{equation}\label{U2}
\begin{array}{l}
N\left( {u,r} \right) = \min \left\{ {{\kappa _\hbar }\left( {u,Ju} \right),{\kappa _\hbar }\left( {r,Kr} \right),{\kappa _\hbar }\left( {u,Kr} \right),{\kappa _\hbar }\left( {r,Ju} \right)} \right\}\\
\\
\quad \quad \quad \quad  = \min \left\{ {0,0,{\kappa _\hbar }\left( {u,r} \right),{\kappa _\hbar }\left( {u,r} \right)} \right\} = 0.
\end{array}
\end{equation}
Now, by using $({\eta_2}')$ and the properties of $\varphi$ and $\gamma$ with $k \in \left( {0,1} \right)$, and the inequality (\ref{U}) turns into
$${\tau ^{4 + \lambda }}{\kappa _\hbar }\left( {u,r} \right) < G\left( {u,r} \right) + \rho N\left( {u,r} \right).$$
Finally, from (\ref{U1}) and (\ref{U2}), we conclude that ${\tau ^{4 + \lambda }}{\kappa _\hbar }\left( {u,r} \right) < {\kappa _\hbar }\left( {u,r} \right)$ is a contradiction, that is, our claim is false. Hence,  $u = r$ and  ${C_{Fix}}\left( {J,K} \right)=\{u\}$.
\end{proof}


\section{Consequences}
This section illustrates the applicability and validity of our main theorem and supports it with several conclusions, which permits us to cover some obtained conclusions in the literature.

Initially, if we remove the restriction
$$\frac{1}{{2\tau }}\min \left\{ {{\kappa _\hbar }\left( {\varsigma ,J\varsigma } \right),{\kappa _\hbar }\left( {\ell ,K\ell } \right)} \right\} \le {\kappa _\hbar }\left( {\varsigma ,\ell } \right),$$
 then the following consequence can be obtained directly from Theorem \ref{teo2.1}.

\begin{corollary}\label{syok}
 Let ${\mathcal{G}}_\kappa ^*$ be a $\kappa-$complete MbMS with constant $\tau \ge 1$ and $J,K:{\mathcal{G}}_\kappa ^* \to {\mathcal{G}}_\kappa ^*$  be two mappings. If there exists a generalized ${\Psi ^*} - \tau $ simulation function with respect to $\eta$, and there is a constant  $\rho  \ge 0$ as well as  $\gamma  \in \Theta $ and $G \in {\Delta _G}$ such that, for all distinct $\varsigma ,\ell  \in {\mathcal{G}}_\kappa ^*$, $k \in \left( {0,1} \right)$ and for all $\hbar  > 0$,
\begin{equation}
\eta \left( {\gamma \left({\tau^4} {{\kappa_\hbar }\left( {J\varsigma,K\ell} \right)} \right),{{\left[ {\gamma \left( {G\left( {\varsigma ,\ell } \right) + \rho N\left( {\varsigma ,\ell } \right)} \right)} \right]}^k}} \right) \ge 1,
\end{equation}
where
$G\left( {\varsigma ,\ell } \right)$ and $N\left( {\varsigma ,\ell } \right)$ are defined as in Theorem \ref{teo2.1}. So, if the condition $(S_1)$ is satisfied,  then
the set of  ${C_{Fix}}\left( {J,K} \right)$ has at least one element. Also, together with the condition $(S_2)$, the set of ${C_{Fix}}\left( {J,K} \right)$ has a unique element.
\end{corollary}

If $J = K$ in Theorem \ref{teo2.1}, the following consequence is evident, and the proof is obvious.

\begin{corollary}\label{ctek}
 Let ${\mathcal{G}}_\kappa ^*$ be a $\kappa-$complete MbMS with constant $\tau \ge 1$ and $J:{\mathcal{G}}_\kappa ^* \to {\mathcal{G}}_\kappa ^*$  be a mapping. If there exists a generalized ${\Psi ^*} - \tau $ simulation function with respect to $\eta$, and there is a constant  $\rho  \ge 0$ as well as  $\gamma  \in \Theta $ and $G \in {\Delta _G}$ such that, for all distinct $\varsigma ,\ell  \in {\mathcal{G}}_\kappa ^*$, $k \in \left( {0,1} \right)$ and for all $\hbar  > 0$,
$$\frac{1}{{2\tau }}{\kappa _\hbar }\left( {\varsigma ,J\varsigma } \right) \le {\kappa _\hbar }\left( {\varsigma ,\ell } \right)$$
implies
\begin{equation}
\eta \left( {\gamma \left({\tau^4} {{\kappa_\hbar }\left( {J\varsigma,J\ell} \right)} \right),{{\left[ {\gamma \left( {G\left( {\varsigma ,\ell } \right) + \rho N\left( {\varsigma ,\ell } \right)} \right)} \right]}^k}} \right) \ge 1,
\end{equation}
where
$$G\left( {\varsigma ,\ell } \right) = \left( \begin{array}{l}
{\kappa _\hbar }\left( {\varsigma ,\ell } \right),{\kappa _\hbar }\left( {\varsigma ,J\varsigma } \right),{\kappa _\hbar }\left( {\ell ,J\ell } \right),\\
\\
\frac{{{\kappa _{2\hbar }}\left( {\varsigma ,J\ell } \right) + {\kappa _{2\hbar }}\left( {\ell ,J\varsigma } \right)}}{{2\tau }}
\end{array} \right)$$ and
$$N\left( {\varsigma ,\ell } \right) = \min \left\{ {{\kappa_\hbar }\left( {\varsigma,J\varsigma} \right),{\kappa_\hbar }\left( {\ell,J\ell} \right),{\kappa_\hbar }\left( {\varsigma,J\ell} \right),{\kappa_\hbar }\left( {\ell,J\varsigma} \right)} \right\}.$$
Thereby, under the condition $(S_1)$, the set of $Fix\left( J \right)$ has at least one element. As well as $(S_1)$, the condition $(S_2)$ holds, the set of $Fix\left( J \right)$ consist of a unique element.
\end{corollary}

\begin{corollary}\label{max}
 Let ${\mathcal{G}}_\kappa ^*$ be a $\kappa-$complete MbMS with constant $\tau \ge 1$ and  $J,K:{\mathcal{G}}_\kappa ^* \to {\mathcal{G}}_\kappa ^*$  be two mappings. If we consider a generalized ${\Psi ^*} - \tau $ simulation function with respect to $\eta$, and  there is a constant  $\rho  \ge 0$ as well as  $\gamma  \in \Theta $ and $G \in {\Delta _G}$ such that, for all distinct $\varsigma ,\ell  \in {\mathcal{G}}_\kappa ^*$, $k \in \left( {0,1} \right)$ and for all $\hbar  > 0$,
$$\frac{1}{{2\tau }}\min \left\{ {{\kappa _\hbar }\left( {\varsigma ,J\varsigma } \right),{\kappa _\hbar }\left( {\ell ,K\ell } \right)} \right\} \le {\kappa _\hbar }\left( {\varsigma ,\ell } \right)$$
implies
\begin{equation}
\eta \left( {\gamma \left({\tau^4} {{\kappa_\hbar }\left( {J\varsigma,K\ell} \right)} \right),{{\left[ {\gamma \left( {G\left( {\varsigma ,\ell } \right) + \rho N\left( {\varsigma ,\ell } \right)} \right)} \right]}^k}} \right) \ge 1,
\end{equation}
where
$$G\left( {\varsigma ,\ell } \right) = \max \left\{ \begin{array}{l}
{\kappa _\hbar }\left( {\varsigma ,\ell } \right),{\kappa _\hbar }\left( {\varsigma ,J\varsigma } \right),{\kappa _\hbar }\left( {\ell ,K\ell } \right),\\
\\
\frac{{{\kappa _{2\hbar }}\left( {\varsigma ,K\ell } \right) + {\kappa _{2\hbar }}\left( {\ell ,K\varsigma } \right)}}{{2\tau }}
\end{array} \right\}$$
and $N\left( {\varsigma ,\ell } \right)$ is defined as in Theorem \ref{teo2.1}. So, under the condition $(S_1)$, then the set of ${C_{Fix}}\left( {J,K} \right)$ admits at least one element and together with $(S_2)$, the element of ${C_{Fix}}\left( {J,K} \right)$ is unique.
\end{corollary}
\begin{proof}
If we prefer  $G \in {\Delta _G}$ as $G\left( {{\iota_1},{\iota_2},{\iota_3},{\iota_4}} \right) = \max \left\{ {{\iota_1},{\iota_2},{\iota_3},{\iota_4}} \right\}$, then it follows from Theorem \ref{teo2.1}.
\end{proof}

As in Corollary \ref{max}, if we select the $G \in {\Delta _G}$ as $G\left( {{\iota_1},{\iota_2},{\iota_3},{\iota_4}} \right) = {\iota_1}$, then the result given below is a direct outcome of Theorem \ref{teo2.1}.

\begin{corollary}\label{t1}
 Let ${\mathcal{G}}_\kappa ^*$ be a $\kappa-$complete MbMS with constant $\tau \ge 1$ and $J,K:{\mathcal{G}}_\kappa ^* \to {\mathcal{G}}_\kappa ^*$  be two mappings. If there exists a generalized ${\Psi ^*} - \tau $ simulation function with respect to $\eta$, and  there is a constant  $\rho  \ge 0$ and  $\gamma  \in \Theta $ such that, for all distinct $\varsigma ,\ell  \in {\mathcal{G}}_\kappa ^*$, $k \in \left( {0,1} \right)$ and for all $\hbar  > 0$,
$$\frac{1}{{2\tau }}\min \left\{ {{\kappa _\hbar }\left( {\varsigma ,J\varsigma } \right),{\kappa _\hbar }\left( {\ell ,K\ell } \right)} \right\} \le {\kappa _\hbar }\left( {\varsigma ,\ell } \right)$$
implies
\begin{equation}\label{33}
\eta \left( {\gamma \left( {\tau^4} {{\kappa _\hbar }\left( {J\varsigma ,K\ell } \right)} \right),{{\left[ {\gamma \left( {{\kappa _\hbar }\left( {\varsigma ,\ell } \right) + \rho N\left( {\varsigma ,\ell } \right)} \right)} \right]}^k}} \right) \ge 1
\end{equation}
where $N\left( {\varsigma ,\ell } \right)$ is defined as in Theorem \ref{teo2.1}. With the conditions $(S_1)$ and $(S_2)$, the set of ${C_{Fix}}\left( {J,K} \right)$ admits a unique element.
\end{corollary}


Next, we give some new corollaries dependent on the choice of generalized ${\Psi ^*} - \tau $ simulation function concerning $\eta$.

Firstly, we introduce a new concept called Suzuki type $\left( {\varphi ,\gamma } \right) - $contraction in the setting of MbMS, as indicated below.

\begin{definition}
Let $\kappa$ be a modular $b-$metric on a set  $ \left( {\mathcal{G},\kappa} \right)$ and $J,K:{\mathcal{G}}_\kappa ^* \to {\mathcal{G}}_\kappa ^*$  be two mappings. The mappings $J$ and $K$ are called  Suzuki type $\left( {\varphi ,\gamma } \right) - $contraction mappings if there exists $ \varphi \in {\Psi^*}$, $\gamma  \in \Theta $ with $k \in \left( {0,1} \right)$  such that for all  $\lambda \in \left( {0,1} \right)$
$$\frac{1}{{2\tau }}\min \left\{ {{\kappa _\hbar }\left( {\varsigma ,J\varsigma } \right),{\kappa _\hbar }\left( {\ell ,K\ell } \right)} \right\} \le {\kappa _\hbar }\left( {\varsigma ,\ell } \right)$$
implies
\begin{equation}
\varphi \left( {{\tau}\gamma \left({\tau ^4} {{\kappa _\hbar }\left( {J\varsigma ,K\ell } \right)} \right)} \right) \le \lambda \varphi \left( {{{\left[ {\gamma \left( {{\kappa _\hbar }\left( {\varsigma ,\ell } \right)} \right)} \right]}^k}} \right),
\end{equation}
for all distinct $\varsigma ,\ell  \in {\mathcal{G}}_\kappa ^*$ and for all $\hbar  > 0$.
\end{definition}

\begin{corollary}\label{eta1ör}
 Let ${\mathcal{G}}_\kappa ^*$ be a $\kappa-$complete MbMS with constant $\tau \ge 1$ and let $J,K$ be a Suzuki type $\left( {\varphi ,\gamma } \right) - $contraction mappings.

 If the condition $(S_1)$ is satisfied, there is at least one element in the set of ${C_{Fix}}\left( {J,K} \right)$. Besides, together with  $(S_2)$, ${C_{Fix}}\left( {J,K} \right)$ consists of only one element.
 \end{corollary}

\begin{proof}
If we take  $\eta \left( {t,v} \right) = \frac{{\lambda \varphi \left( v \right)}}{{\varphi \left( \tau t \right)}},\quad \forall t,v > 1;\lambda  \in \left( {0,1} \right)$ as well as $G \in {\Delta _G}$ with $G\left( {{\iota_1},{\iota_2},{\iota_3},{\iota_4}} \right) = {\iota_1}$ and $\rho =0$, then it follows from Theorem \ref{teo2.1}.
\end{proof}

The following result is a generalization of the Suzuki type $\left( {\varphi ,\gamma } \right) - $contraction mapping, which is also a consequence of Theorem \ref{teo2.1}.

\begin{corollary}\label{max2}
 Let ${\mathcal{G}}_\kappa ^*$ be a $\kappa-$complete MbMS with constant $\tau \ge 1$ and  $J,K:{\mathcal{G}}_\kappa ^* \to {\mathcal{G}}_\kappa ^*$ is labeled as a generalized Suzuki type $\left( {\varphi ,\gamma } \right) - $contraction mapping, if there exists a $ \varphi \in {\Psi^*}$, $G \in {\Delta _G}$ and $\gamma  \in \Theta $ with $k \in \left( {0,1} \right)$ such that,
$$\frac{1}{{2\tau }}\min \left\{ {{\kappa _\hbar }\left( {\varsigma ,J\varsigma } \right),{\kappa _\hbar }\left( {\ell ,K\ell } \right)} \right\} \le {\kappa _\hbar }\left( {\varsigma ,\ell } \right)$$
implies
\begin{equation}
\varphi\left( {{\tau}\gamma \left({\tau ^4} {{\kappa _\hbar }\left( {J\varsigma ,K\ell } \right)} \right)} \right) \le \lambda \varphi \left( {{{\left[ {\gamma \left( {G\left( {\varsigma ,\ell } \right)} \right)} \right]}^k}} \right),
\end{equation}
where
$$G\left( {\varsigma ,\ell } \right) = \max \left\{ \begin{array}{l}
{\kappa _\hbar }\left( {\varsigma ,\ell } \right),{\kappa _\hbar }\left( {\varsigma ,J\varsigma } \right),{\kappa _\hbar }\left( {\ell ,K\ell } \right),\\
\\
\frac{{{\kappa _{2\hbar }}\left( {\varsigma ,K\ell } \right) + {\kappa _{2\hbar }}\left( {\ell ,K\varsigma } \right)}}{{2\tau }}
\end{array} \right\}$$
 and $\lambda \in \left( {0,1} \right)$, for all distinct $\varsigma ,\ell  \in {\mathcal{G}}_\kappa ^*$ and for all $\hbar  > 0$.

Provided that the condition $(S_1)$ and $(S_2)$ is satisfied, the set of ${C_{Fix}}\left( {J,K} \right)$ exactly has a unique element.
\end{corollary}

\begin{proof}
As with the proof of Corollary \ref{eta1ör}, the proof is clear if  $G\left( {{\iota_1},{\iota_2},{\iota_3},{\iota_4}} \right) = \max \left\{ {{\iota_1},{\iota_2},{\iota_3},{\iota_4}} \right\}$ is taken specifically.
\end{proof}

We also define the Suzuki type $\left( {\varphi ,\phi ,\gamma } \right) -$contraction mapping, which is an output of Theorem \ref{teo2.1}, as noted below.

\begin{definition}
Let $\kappa$ be a modular $b-$metric on a set  $ \left( {\mathcal{G},\kappa} \right)$ and $J,K:{\mathcal{G}}_\kappa ^* \to {\mathcal{G}}_\kappa ^*$  be two mappings. Then, the mappings $J$ and $K$ are called Suzuki type $\left( {\varphi ,\phi ,\gamma } \right) -$contraction mappings, if there exist $ \varphi \in {\Psi^*}$, $\gamma  \in \Theta $ with $k \in \left( {0,1} \right)$ such that, for all distinct $\varsigma ,\ell  \in {\mathcal{G}}_\kappa ^*$ and for all $\hbar  > 0$,
$$\frac{1}{{2\tau }}\min \left\{ {{\kappa _\hbar }\left( {\varsigma ,J\varsigma } \right),{\kappa _\hbar }\left( {\ell ,K\ell } \right)} \right\} \le {\kappa _\hbar }\left( {\varsigma ,\ell } \right)$$
implies
\begin{equation}
\varphi \left( {{\tau ^ \lambda}\gamma \left({\tau ^4} {{\kappa _\hbar }\left( {J\varsigma ,K\ell } \right)} \right)} \right) \le \phi \left( {{{\left[ {\gamma \left( {{\kappa _\hbar }\left( {\varsigma ,\ell } \right)} \right)} \right]}^k}} \right),
\end{equation}
where $\phi :\left[ {1,\infty } \right) \to \left[ {1,\infty } \right)$ is a continuous mapping satisfying that $\phi \left( \nu \right) < \varphi \left( \nu \right)$ for all $\nu>0$.
\end{definition}

\begin{corollary}\label{phi}
 Let ${\mathcal{G}}_\kappa ^*$ be a $\kappa-$complete MbMS with constant $\tau \ge 1$ and let $J,K$ be a Suzuki type $\left( {\varphi ,\phi ,\gamma } \right) -$contraction mappings. Then, likewise, under the  condition $(S_1)$ and condition $(S_2)$, the set of ${C_{Fix}}\left( {J,K} \right)$ exactly admits  a unique element.
\end{corollary}

\begin{proof}
If we take  $\eta \left( {t,v} \right) = \frac{{ \phi \left( v \right)}}{{\varphi \left( {\tau^\lambda} t \right)}},\quad \forall t,v > 1$ as well as $G \in {\Delta _G}$ with $$G\left( {{\iota_1},{\iota_2},{\iota_3},{\iota_4}} \right) = {\iota_1}$$
and $\rho =0$, then it follows from Theorem \ref{teo2.1}.
\end{proof}

\begin{corollary}
 Let ${\mathcal{G}}_\kappa ^*$ be a $\kappa-$complete MbMS with coefficient $\tau \ge 1$ and  $J,K:{\mathcal{G}}_\kappa ^* \to {\mathcal{G}}_\kappa ^*$  are named as generalized Suzuki type $\left( {\varphi ,\phi ,\gamma } \right) -$contraction if there exists a $ \varphi \in {\Psi^*}$, $G \in {\Delta _G}$ and $\gamma  \in \Theta $ with $k \in \left( {0,1} \right)$ such that for all distinct $\varsigma ,\ell  \in {\mathcal{G}}_\kappa ^*$ and for all $\hbar  > 0$,
$$\frac{1}{{2\tau }}\min \left\{ {{\kappa _\hbar }\left( {\varsigma ,J\varsigma } \right),{\kappa _\hbar }\left( {\ell ,K\ell } \right)} \right\} \le {\kappa _\hbar }\left( {\varsigma ,\ell } \right)$$
implies
\begin{equation}
\varphi \left( {{\tau ^ \lambda}\gamma \left({\tau ^4} {{\kappa _\hbar }\left( {J\varsigma ,K\ell } \right)} \right)} \right) \le \phi \left( {{{\left[ {\gamma \left( {G\left( {\varsigma ,\ell } \right)} \right)} \right]}^k}} \right),
\end{equation}
where
$$G\left( {\varsigma ,\ell } \right) = \max \left\{ \begin{array}{l}
{\kappa _\hbar }\left( {\varsigma ,\ell } \right),{\kappa _\hbar }\left( {\varsigma ,J\varsigma } \right),{\kappa _\hbar }\left( {\ell ,K\ell } \right),\\
\\
\frac{{{\kappa _{2\hbar }}\left( {\varsigma ,K\ell } \right) + {\kappa _{2\hbar }}\left( {\ell ,K\varsigma } \right)}}{{2\tau }}
\end{array} \right\}$$
and $\phi :\left[ {1,\infty } \right) \to \left[ {1,\infty } \right)$ is continuous mapping,   which has the property $\phi \left( \nu \right) < \varphi \left( \nu \right)$ for all $\nu>0$.

Thereupon, by considering the conditions $(S_1)$ and $(S_2)$, we say that the set of ${C_{Fix}}\left( {J,K} \right)$ holds exactly one unique element.
\end{corollary}
\begin{proof}
Similar to the proof of Corollary \ref{phi}, it is enough to take  $G \in {\Delta _G}$ as $G\left( {{\iota_1},{\iota_2},{\iota_3},{\iota_4}} \right) = \max \left\{ {{\iota_1},{\iota_2},{\iota_3},{\iota_4}} \right\}$. Thus the proof is clear.
\end{proof}

In the following theorem, we show that Corollary \ref{t1} can be satisfied without the constant $k$ and thereby, this can be considered that as a  output of Theorem \ref{teo2.1}.

\begin{theorem}\label{orn}
 Let ${\mathcal{G}}_\kappa ^*$ be a $\kappa-$complete MbMS with coefficient $\tau \ge 1$ and $J,K:{\mathcal{G}}_\kappa ^* \to {\mathcal{G}}_\kappa ^*$  be two mappings. Besides, there is a constant  $\rho  \ge 0$ and there exist $ \eta \in {\Psi^*}$, $\gamma  \in \Theta $ such that for all distinct $\varsigma ,\ell  \in {\mathcal{G}}_\kappa ^*$ and for all $\hbar  > 0$
$$\frac{1}{{2\tau }}\min \left\{ {{\kappa _\hbar }\left( {\varsigma ,J\varsigma } \right),{\kappa _\hbar }\left( {\ell ,K\ell } \right)} \right\} \le {\kappa _\hbar }\left( {\varsigma ,\ell } \right)$$
implies
\begin{equation}\label{38}
\eta \left( {\gamma \left( {{\tau ^4}{\kappa _\hbar }\left( {J\varsigma ,K\ell } \right)} \right),\gamma \left[ {{\kappa _\hbar }\left( {\varsigma ,\ell } \right) + \rho N\left( {\varsigma ,\ell } \right)} \right]} \right) \ge 1,
\end{equation}
where $N\left( {\varsigma ,\ell } \right)$ is defined as in Theorem \ref{teo2.1}.

Thereby, it follows that the set of ${C_{Fix}}\left( {J,K} \right)$ precisely admits  only one element together with the conditions $(S_1)$ and $(S_2)$.
\end{theorem}

\begin{proof}
We shall show that Theorem \ref{orn} can be achieved from Corollary \ref{t1}.  From the expression (\ref{33}) and taking into account $ \varphi \in {\Psi^*}$, we get that
$$\varphi \left( {\gamma \left( {{\tau ^4}{\kappa _\hbar }\left( {J\varsigma ,K\ell } \right)} \right)} \right) \le \varphi \left( {{{\left[ {\gamma \left( {{\kappa _\hbar }\left( {\varsigma ,\ell } \right) + \rho N\left( {\varsigma ,\ell } \right)} \right)} \right]}^k}} \right),$$
which yields
$$\varphi \left( {\gamma \left( {{\tau ^4}{\kappa _\hbar }\left( {J\varsigma ,K\ell } \right)} \right)} \right) \le \varphi \left( {\gamma \left( {{\kappa _\hbar }\left( {\varsigma ,\ell } \right) + \rho N\left( {\varsigma ,\ell } \right)} \right)} \right)$$
because $k \in \left( {0,1} \right)$.

So, under the same conditions, the above inequality can be obtained from the expression (\ref{38}), too. Hence, the proof of Theorem \ref{orn} can be achieved in a similar way  with Corollary \ref{t1}.
\end{proof}

Now, we present an example that holds the statement of Theorem \ref{orn} under the certain conditions.

\begin{example}
Let ${\cal G}_\kappa ^* = \left[ {0,1} \right]$ and take into account the modular $b-$metric by $${\kappa _\hbar }(\varsigma ,\ell ) = \frac{{{{\left| {\varsigma  - \ell } \right|}^2}}}{\hbar },$$
for all distinct $\varsigma ,\ell  \in {\mathcal{G}}_\kappa ^*$ and for all $\hbar  > 0$. Note that $\left( {{\cal G}_\kappa ^*,\kappa } \right)$ is a $\kappa-$complete MbMS with the constant $\tau=2$. Also, let the mappings $J,K:{\cal G}_\kappa ^* \to {\cal G}_\kappa ^*$ be defined by
$$J\varsigma  = \frac{\varsigma }{4}\quad {\rm{and}}\quad K\ell  = 2\ell .$$
Now, our aim is to prove the contractivity conditions
\begin{equation}\label{zi}
\frac{1}{{2\tau }}\min \left\{ {{\kappa _\hbar }\left( {\varsigma ,J\varsigma } \right),{\kappa _\hbar }\left( {\ell ,K\ell } \right)} \right\} \le {\kappa _\hbar }\left( {\varsigma ,\ell } \right)
\end{equation}
implies
\begin{equation}\label{etaa}
\eta \left( {\gamma \left( {{\tau ^4}{\kappa _\hbar }\left( {J\varsigma ,K\ell } \right)} \right),\gamma \left[ {{\kappa _\hbar }\left( {\varsigma ,\ell } \right) + \rho N\left( {\varsigma ,\ell } \right)} \right]} \right) \ge 1,
\end{equation}
where $$N\left( {\varsigma ,\ell } \right) = \min \left\{ {{\kappa_\hbar }\left( {\varsigma,J\varsigma} \right),{\kappa_\hbar }\left( {\ell,J\ell} \right),{\kappa_\hbar }\left( {\varsigma,J\ell} \right),{\kappa_\hbar }\left( {\ell,J\varsigma} \right)} \right\},$$
via $\eta \left( {\iota,\nu} \right) = \frac{{\psi {{\left( \nu \right)}^\lambda }}}{{\psi \left( \iota \right)}},\quad \forall \iota,\nu > 1,\lambda  \in \left( {0,1} \right)$ and $ \psi \in {\Psi^*}$ with $\psi \left( \iota \right) = \frac{\iota}{2}$, holds for all distinct $\varsigma ,\ell  \in {\mathcal{G}}_\kappa ^*$ and for all $\hbar  > 0$.

Notice that it is clear that $\eta$ belongs to the class of generalized ${\Psi^*}-$simulation functions (also  ${\Psi^*}-\tau$ simulations).

Also, we define the function  $\gamma :\left( {0,\infty } \right) \to \left( {1,\infty } \right)$ by $\gamma \left( \iota \right) = {e^\iota}$.

Hence, at first, we will show that the expression (\ref{zi}) is satisfied. Then, we have
$${\kappa _\hbar }\left( {\varsigma ,J\varsigma } \right) = \frac{{{{\left| {\varsigma  - J\varsigma } \right|}^2}}}{\hbar } = \frac{{{{\left| {\varsigma  - \frac{\varsigma }{4}} \right|}^2}}}{\hbar } = \frac{{9{\varsigma ^2}}}{{16\hbar }}$$
and $${\kappa _\hbar }\left( {\ell ,K\ell } \right) = \frac{{{{\left| {\ell  - K\ell } \right|}^2}}}{\hbar } = \frac{{{{\left| {\ell  - 2\ell } \right|}^2}}}{\hbar } = \frac{{{\ell ^2}}}{\hbar }.$$
Afterward, the expression (\ref{zi}) gives
\begin{equation}\label{min}
\frac{1}{4}\min \left\{ {\frac{{9{\varsigma ^2}}}{{16\hbar }},\frac{{{\ell ^2}}}{\hbar }} \right\} = \min \left\{ {\frac{{9{\varsigma ^2}}}{{64\hbar }},\frac{{{\ell ^2}}}{{4\hbar }}} \right\} \le \frac{{{{\left| {\varsigma  - \ell } \right|}^2}}}{\hbar }.
\end{equation}

Without loss of  generality, we may assume that $\varsigma  > \ell  \ge 0$.

\item[Case (1):]$\min \left\{ {\frac{{9{\varsigma ^2}}}{{64\hbar }},\frac{{{\ell ^2}}}{{4\hbar }}} \right\} = \frac{{{\ell ^2}}}{{4\hbar }}$. So, from (\ref{min}), we obtain that
$$\frac{{{\ell ^2}}}{{4\hbar }} \le \frac{{{{\left( {\varsigma  - \ell } \right)}^2}}}{\hbar }\quad  \Leftrightarrow \quad \frac{\ell }{2} \le \left| {\varsigma  - \ell } \right| = \varsigma  - \ell \quad  \Leftrightarrow \quad \ell  \le \frac{2}{3}\varsigma  < \varsigma ,$$
which is true as $\ell  < \varsigma $.
\item[Case (2):]$\min \left\{ {\frac{{9{\varsigma ^2}}}{{64\hbar }},\frac{{{\ell ^2}}}{{4\hbar }}} \right\} = \frac{{9{\varsigma ^2}}}{{64\hbar }}$. Then, by (\ref{min}), we conclude that
$$\frac{{9{\varsigma ^2}}}{{64\hbar }} \le \frac{{{{\left( {\varsigma  - \ell } \right)}^2}}}{\hbar }\quad  \Leftrightarrow \quad \frac{{3\varsigma }}{8} \le \left| {\varsigma  - \ell } \right| = \varsigma  - \ell \quad  \Leftrightarrow \quad \ell  \le \frac{5}{8}\varsigma  < \varsigma, $$
which holds because $\ell  < \varsigma .$

Consequently, in any case, the expression (\ref{zi}) is valid for all distinct $\varsigma ,\ell  \in {\mathcal{G}}_\kappa ^*$ and for all $\hbar  > 0$.

Next, we will prove that the expression (\ref{etaa}) is satisfied. Using the definitions of modular $b-$metric and the mappings $J$ and $K$, we attain
$${\kappa _\hbar }\left( {J\varsigma ,K\ell } \right) = \frac{{{{\left| {J\varsigma  - K\ell } \right|}^2}}}{\hbar } = \frac{{{{\left| {\frac{\varsigma }{4} - 2\ell } \right|}^2}}}{\hbar } = \frac{{{{\left( {\varsigma  - 8\ell } \right)}^2}}}{{16\hbar }},$$
and
\begin{equation}\label{NN}
    \begin{array}{l}
N\left( {\varsigma ,\ell } \right) = \min \left\{ {{\kappa _\hbar }\left( {\varsigma ,J\varsigma } \right),{\kappa _\hbar }\left( {\ell ,J\ell } \right),{\kappa _\hbar }\left( {\varsigma ,J\ell } \right),{\kappa _\hbar }\left( {\ell ,J\varsigma } \right)} \right\}\\
\\
\quad \quad \quad \: \,= \min \left\{ {\frac{{{{\left| {\varsigma  - \frac{\varsigma }{4}} \right|}^2}}}{\hbar },\frac{{{{\left| {\ell  - 2\ell } \right|}^2}}}{\hbar },\frac{{{{\left| {\varsigma  - 2\ell } \right|}^2}}}{\hbar },\frac{{{{\left| {\ell  - \frac{\varsigma }{4}} \right|}^2}}}{\hbar }} \right\}\\
\\
\quad \quad \quad \: \,  = \min \left\{ {\frac{{9{\varsigma ^2}}}{{16\hbar }},\frac{{{\ell ^2}}}{\hbar },\frac{{{{\left| {\varsigma  - 2\ell } \right|}^2}}}{\hbar },\frac{{{{\left| {4\ell  - \varsigma } \right|}^2}}}{{16\hbar }}} \right\}\\
\\
\quad \quad \quad \: \, = \min \left\{ {\frac{{{\ell ^2}}}{\hbar },\frac{{{{\left| {4\ell  - \varsigma } \right|}^2}}}{{16\hbar }}} \right\}.
\end{array}
\end{equation}
\item[Case (3):] We assume that $\varsigma  < 4\ell $ and so, we get $\left| {4\ell  - \varsigma } \right| = 4\ell  - \varsigma$. Using the expression (\ref{NN}), we conclude that
$$\min \left\{ {\frac{{{\ell ^2}}}{\hbar },\frac{{{{\left| {4\ell  - \varsigma } \right|}^2}}}{{16\hbar }}} \right\} = \frac{{{{\left( {4\ell  - \varsigma } \right)}^2}}}{{16\hbar }}.$$


Thereby, the expression (\ref{etaa}) provides
$$\begin{array}{l}
\eta \left( {\gamma \left( {{\tau ^3}{\kappa _\hbar }\left( {J\varsigma ,K\ell } \right)} \right),\gamma \left( {{\kappa _\hbar }\left( {\varsigma ,\ell } \right) + \rho N\left( {\varsigma ,\ell } \right)} \right)} \right)\\
\\
 = \frac{{\psi {{\left( {\gamma \left( {{\kappa _\hbar }\left( {\varsigma ,\ell } \right) + \rho N\left( {\varsigma ,\ell } \right)} \right)} \right)}^\lambda }}}{{\psi \left( {\gamma \left( {{\tau ^3}{\kappa _\hbar }\left( {J\varsigma ,K\ell } \right)} \right)} \right)}} = {2^{1 - \lambda }}\frac{{{e^{\left[ {{\kappa _\hbar }\left( {\varsigma ,\ell } \right) + \rho N\left( {\varsigma ,\ell } \right)} \right]\lambda }}}}{{{e^{{\tau ^3}{\kappa _\hbar }\left( {J\varsigma ,K\ell } \right)}}}} = {2^{1 - \lambda }}\frac{{{e^{\left[ {\frac{{{{\left( {\varsigma  - \ell } \right)}^2}}}{\hbar } + \rho \frac{{{{\left( {4\ell  - \varsigma } \right)}^2}}}{{16\hbar }}} \right]\lambda }}}}{{{e^{{2^3}\frac{{{{\left( {\varsigma  - 8\ell } \right)}^2}}}{{16\hbar }}}}}}\\
\\
\quad \quad \quad \quad \quad \quad \quad \, \quad \quad \quad  = {2^{1 - \lambda }}{e^{\left[ {\frac{{{{\left( {\varsigma  - \ell } \right)}^2}}}{\hbar } + \rho \frac{{{{\left( {4\ell  - \varsigma } \right)}^2}}}{{16\hbar }}} \right]\lambda  - \frac{{{{\left( {\varsigma  - 8\ell } \right)}^2}}}{{2\hbar }}}} \ge 1,
\end{array}$$
where $\rho  = {2^4}$ and $\lambda  = \frac{1}{2} \in \left( {0,1} \right).$ Thus, the desired state is achieved.

\item[Case (4):] We presume that $4\ell  < \varsigma $, i.e., $\left| {4\ell  - \varsigma } \right| = \varsigma- 4\ell .$ In this case, if we choose $\varsigma = 8\ell$, then we get
$$\frac{{{\ell ^2}}}{\hbar } = \frac{{{{\left| {4\ell  - \varsigma } \right|}^2}}}{{16\hbar }}.$$
Therefore, we discuss the following subcases.
\item[Subcase (4.1):] If $\varsigma > 8\ell$. In this case, by (\ref{NN}), we have
$$\min \left\{ {\frac{{{\ell ^2}}}{\hbar },\frac{{{{\left| {4\ell  - \varsigma } \right|}^2}}}{{16\hbar }}} \right\} = \frac{{{\ell ^2}}}{\hbar }.$$
Then, similar to the above, we get
$$\begin{array}{l}
\eta \left( {\gamma \left( {{\tau ^3}{\kappa _\hbar }\left( {J\varsigma ,K\ell } \right)} \right),\gamma \left( {{\kappa _\hbar }\left( {\varsigma ,\ell } \right) + \rho N\left( {\varsigma ,\ell } \right)} \right)} \right)\\
\\
 = \frac{{\psi {{\left( {\gamma \left( {{\kappa _\hbar }\left( {\varsigma ,\ell } \right) + \rho N\left( {\varsigma ,\ell } \right)} \right)} \right)}^\lambda }}}{{\psi \left( {\gamma \left( {{\tau ^3}{\kappa _\hbar }\left( {J\varsigma ,K\ell } \right)} \right)} \right)}} = {2^{1 - \lambda }}\frac{{{e^{\left[ {{\kappa _\hbar }\left( {\varsigma ,\ell } \right) + \rho N\left( {\varsigma ,\ell } \right)} \right]\lambda }}}}{{{e^{{\tau ^3}{\kappa _\hbar }\left( {J\varsigma ,K\ell } \right)}}}} = {2^{1 - \lambda }}\frac{{{e^{\left[ {\frac{{{{\left( {\varsigma  - \ell } \right)}^2}}}{\hbar } + \rho \frac{{{\ell ^2}}}{\hbar }} \right]\lambda }}}}{{{e^{{2^3}\frac{{{{\left( {\varsigma  - 8\ell } \right)}^2}}}{{16\hbar }}}}}}\\
\\
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  = {2^{1 - \lambda }}{e^{\left[ {\frac{{{{\left( {\varsigma  - \ell } \right)}^2}}}{\hbar } + \rho \frac{{{\ell ^2}}}{\hbar }} \right]\lambda  - \frac{{{{\left( {\varsigma  - 8\ell } \right)}^2}}}{{2\hbar }}}} \ge 1.
\end{array}$$

\item[Subcase (4.2):] If $\varsigma < 8\ell$. Then, by (\ref{NN}), we have $$\min \left\{ {\frac{{{\ell ^2}}}{\hbar },\frac{{{{\left| {4\ell  - \varsigma } \right|}^2}}}{{16\hbar }}} \right\} = \frac{{{{\left( {4\ell  - \varsigma } \right)}^2}}}{{16\hbar }}.$$ This case has been proved to be satisfied in Case (3).

Consequently, the mappings $J$ and $K$ satisfy the hypotheses of Theorem \ref{orn} associated with the function $\eta \left( {\iota,\nu} \right) = \frac{{\psi {{\left( \nu \right)}^\lambda }}}{{\psi \left( \iota \right)}},\quad \forall \iota,\nu > 1,\lambda  \in \left( {0,1} \right)$ and $ \psi \in {\Psi^*}$ with $\psi \left( \iota \right) = \frac{\iota}{2}$.
\end{example}

\section{ An Application to Integral Equations}
At first, we present a new result that can be applied to the nonlinear integral equation.

\begin{corollary}\label{nid}
 Let ${\mathcal{G}}_\kappa ^*$ be a $\kappa-$complete MbMS with constant $\tau \ge 1$ and $J:{\mathcal{G}}_\kappa ^* \to {\mathcal{G}}_\kappa ^*$  be a self-mapping. Presume that the following circumstances hold:
 \begin{itemize}
     \item [i.]There exists a generalized ${\Psi ^*}$ simulation function with respect to $\eta$ and also, there is a constant  $\rho  \ge 0$ and  $\gamma  \in \Theta $ , for all distinct $\varsigma ,\ell  \in {\mathcal{G}}_\kappa ^*$, $k \in \left( {0,1} \right)$ and for all $\hbar  > 0$,

\begin{equation}
\eta \left( {\gamma \left( {{\tau ^4}{\kappa _\hbar }\left( {J\varsigma ,J\ell } \right)} \right),{{\left[ {\gamma \left( {{\kappa _\hbar }\left( {\varsigma ,\ell } \right)} \right)} \right]}^k}} \right) \ge 1
\end{equation}
\item[ii.] $(S_1)$ and $(S_2)$ are provided.
 \end{itemize}
 Then, we say that the set of $Fix\left( J \right)$ has a unique element in ${\mathcal{G}}_\kappa ^*$.
\end{corollary}

\begin{proof}Without Suzuki restriction, if we select the $G \in {\Delta _G}$ as $G\left( {{\iota_1},{\iota_2},{\iota_3},{\iota_4}} \right) = {\iota_1}$ and also, $\rho =0$ and $J=K$ in Theorem \ref{teo2.1}, we obtain the desired result.
\end{proof}
We shall establish an existence theorem for the solution of the following nonlinear integral equation via Corollary \ref{nid}.
\begin{equation}\label{int}
    \varsigma \left( \iota \right) = \int\limits_{\hat a}^{\hat b} {\Sigma \left( {\iota,\nu,\varsigma \left(\nu \right)} \right)d\nu},
\end{equation}
where ${\hat a},{\hat b} \in \mathbf{R}$ by ${\hat a} < {\hat b}$, $\varsigma  \in C\left[ {{\hat a},{\hat b}} \right]$(the set of all continuous functions from $\left[ {{\hat a},{\hat b}} \right]$ into $\mathbf{R}$) and $\Sigma :\left[ {{\hat a},{\hat b}} \right] \times \left[ {{\hat a},{\hat b}} \right] \times \mathbf{R} \to \mathbf{R}$ are given mappings. We endow  ${\cal G}_\kappa ^* = C\left[ {{\hat a},{\hat b}} \right]$ with the modular $b-$metric space
$${\kappa _\hbar }\left( {x,y} \right) = \frac{{{{\left| {x\left( \iota \right) - y\left( \iota \right)} \right|}^p}}}{\lambda },\quad (p > 1),$$
for all $x,y  \in {\mathcal{G}}_\kappa ^*$ and for all $\hbar  > 0$. Evidently, $\left( {{\cal G}_\kappa ^*,\kappa } \right)$ is a $\kappa-$complete modular $b-$metric space with the constant $\tau  = {2^{p - 1}}.$

Furthermore, let $f:{\cal G}_\kappa ^* \to {\cal G}_\kappa ^*$ be defined by
$$f\left( {\varsigma \left( \iota \right)} \right) = \int\limits_{\hat a}^{\hat b} {\Sigma \left( {\iota,\nu,\varsigma \left( \nu \right)} \right)d\nu} $$
for all $\varsigma  \in {\cal G}_\kappa ^*$ and $\iota \in \left[ {{\hat a},{\hat b}} \right].$ Accordingly, the existence of a solution to (\ref{int}) is equivalent to the existence of a fixed point of $f$.

\begin{theorem}
Consider the nonlinear integral equation (\ref{int}). Presume that the following statement is satisfied:
\begin{itemize}
    \item [i.] $\Sigma :\left[ {{\hat a},{\hat b}} \right] \times \left[ {{\hat a},{\hat b}} \right] \times \mathbf{R} \to \mathbf{R}$ is continuous and non-decreasing in the third order,
    \item[ii.]there exists $p > 1$ satisfying the following condition: for each $\iota,\nu \in \left[ {{\hat a},{\hat b}} \right]$ and $\varsigma ,\ell  \in {\cal G}_\kappa ^*$ with $\varsigma \left( w \right) \le \ell \left( w \right)$ for all $w \in \left[ {{\hat a},{\hat b}} \right]$, we have
    \begin{equation}\label{int1}
       \left| {\Sigma \left( {\iota,\nu,\varsigma \left( \nu \right)} \right) - \Sigma \left( {\iota,\nu,\ell \left( \nu \right)} \right)} \right| \le \sigma \left( {\iota,\nu} \right)\left| {\varsigma \left( \nu \right) - \ell \left( \nu \right)} \right|,
    \end{equation}
    where $\sigma :\left[ {{\hat a},{\hat b}} \right] \times \left[ {{\hat a},{\hat b}} \right] \to \left[ {0,\infty } \right)$ is a continuous function defined by
    \begin{equation}\label{int2}
        \mathop {\sup }\limits_{\iota \in [{\hat a},{\hat b}]} \left( {\int\limits_{\hat a}^{\hat b} {\sigma {{\left( {\iota,\nu} \right)}^p}d\nu} } \right) \le \frac{k}{{{2^{4p - 4}}}},\quad \left( {k \in \left( {0,1} \right)} \right).
    \end{equation}
\end{itemize}
Hence, the nonlinear integral equation (\ref{int}) has a solution.
\end{theorem}
\begin{proof}
From (\ref{int1}) and (\ref{int2}), for all $\iota \in \left[ {{\hat a},{\hat b}} \right]$, we have
$$\begin{array}{l}
{e^{{2^{4p - 4}}{\kappa _\hbar }\left( {f\left( {\varsigma \left(  \iota \right)} \right),f\left( {\ell \left(  \iota \right)} \right)} \right)}}^p = {e^{{2^{4p - 4}}p{{\frac{{\left| {f\left( {\varsigma \left(  \iota \right)} \right) - f\left( {\ell \left(  \iota \right)} \right)} \right|}}{\lambda }}^p}}}\\
\\
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  = {e^{{2^{4p - 4}}.p.\frac{1}{\lambda }{{\left| {\int\limits_{\hat a}^{\hat b} {\Sigma \left( { \iota,\nu,\varsigma \left( \nu \right)} \right)d\nu}  - \int\limits_a^b {\Sigma \left( { \iota,\nu,\ell \left( \nu \right)} \right)d\nu} } \right|}^p}}}\\
\\
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  \le {e^{{2^{4p - 4}}.p.\frac{1}{\lambda }{{\left| {\int\limits_a^b {\left[ {\Sigma \left( { \iota,\nu,\varsigma \left( \nu \right)} \right) - \Sigma \left( { \iota,\nu,\ell \left( \nu \right)} \right)} \right]d\nu} } \right|}^p}}}\\
\\
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  \le {e^{{2^{4p - 4}}.p.\frac{1}{\lambda }{{\left( {\int\limits_a^b {\left| {\Sigma \left( { \iota,\nu,\varsigma \left( \nu \right)} \right) - \Sigma \left( { \iota,\nu,\ell \left( \nu \right)} \right)} \right|d\nu} } \right)}^p}}}\\
\\
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  \le {e^{{2^{4p - 4}}.p.\frac{1}{\lambda }{{\left( {\int\limits_a^b {\sigma \left( {t,r} \right)\left| {\varsigma \left( \nu \right) - \ell \left( \nu \right)} \right|d\nu} } \right)}^p}}}\\
\\
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  \le {e^{{2^{4p - 4}}.p.\left( {\int\limits_a^b {\sigma {{\left( {t,r} \right)}^p}d\nu} } \right)\frac{{{{\left| {\varsigma \left( \nu \right) - \ell \left( \nu \right)} \right|}^p}}}{\lambda }}}\\
\\
\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  \le {e^{{2^{4p - 4}}.p\frac{k}{{{2^{4p - 4}}}}{\kappa _\hbar }\left( {\varsigma \left(  \iota \right),\ell \left(  \iota \right)} \right)}} = {\left[ {{{\left( {{e^{{\kappa _\hbar }\left( {\varsigma \left(  \iota \right),\ell \left(  \iota \right)} \right)}}} \right)}^k}} \right]^p}.\\
\end{array}$$
Now, let $\eta$ be a ${\Psi ^*}-$simulation function with $\eta \left( {x,y} \right) < \frac{{\varphi \left( y \right)}}{{\varphi \left( x \right)}},\quad \left( {\forall x,y > 1} \right),$ where $\varphi \in {\Psi ^*}$ such that $\varphi \left( q \right) = {q^p},\;\left( {q > 1} \right).$ We define  $\gamma  \in \Theta $ by $ \gamma \left( \alpha  \right) = {e^\alpha },\;\left( {\alpha  > 0} \right)$.  This suggests that
$$\begin{array}{l}
\varphi \left( {\gamma \left( {{2^{4p - 4}}{\kappa _\hbar }\left( {f\left( {\varsigma \left(  \iota \right)} \right),f\left( {\ell \left(  \iota \right)} \right)} \right)} \right)} \right) = {\left( {\gamma \left( {{2^{4p - 4}}{\kappa _\hbar }\left( {f\left( {\varsigma \left(  \iota \right)} \right),f\left( {\ell \left(  \iota \right)} \right)} \right)} \right)} \right)^p}\\
\\
\quad \quad  \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  = {e^{{2^{4p - 4}}{\kappa _\hbar }\left( {f\left( {\varsigma \left(  \iota \right)} \right) - f\left( {\ell \left(  \iota \right)} \right)} \right)}}^p \le {\left[ {{{\left( {{e^{{\kappa _\hbar }\left( {\varsigma \left(  \iota \right),\ell \left(  \iota \right)} \right)}}} \right)}^k}} \right]^p}\\
\\
\quad \quad  \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  = {\left( {{{\left[ {\gamma \left( {{\kappa _\hbar }\left( {\varsigma ,\ell } \right)} \right)} \right]}^k}} \right)^p} = \varphi \left( {{{\left[ {\gamma \left( {{\kappa _\hbar }\left( {\varsigma ,\ell } \right)} \right)} \right]}^k}} \right).
\end{array}$$
With this last equation, we conclude that all conditions of Corollary \ref{nid} are held. Consequently, there exists a unique $\varsigma  \in {\cal G}_\kappa ^*$ such that $\varsigma  \in Fix\left( f \right)$, which means that $\varsigma$ is the unique solution for the integral equation (\ref{int}).
\end{proof}



\section{ Data Availability Statement:}
No data were used to support this study.

\section{Competing Interest}
   The authors declare that there is not any competing interest regarding the publication of this manuscript.

\section{Author contributions} All authors contributed equally to the writing of this paper. All authors read and
   approved the final manuscript.

\section{Funding}
   This research received no external funding.

\section{Acknowledgements}
Not applicable.










%\begin{acknowledgements}
%if you'd like to thank anyone, place your comments here
%and remove the percent signs.
%\end{acknowledgements}

%\section*{Conflict of interest}
%The authors declare that they have no conflict of interest.




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