\section{Introduction}

\qquad  Geraghty \cite{13} in 1973 came up with an
interesting generalization of Banach contraction
mapping principle using the concept of class
$\mathcal{S}$  of functions, that is $\kappa:
\mathbb{R}_{+}\rightarrow [0,1)$ with the
condition that $\kappa(t_{n})\rightarrow
1$ implies that $t_{n}\rightarrow 0$ where $\mathbb{R}_{+}$
is the set of all non-negative real number
and $t\in\mathbb{R}_{+}$ for all $n\in\mathbb{N}$ and proved
that fixed point in this class exists

\qquad In 2012, Gordji {\it et al.} \cite{27} proved some
fixed point theorems for generalized  Geraghty's
contraction in partially ordered complete metric spaces and
Bhaskar and Lakshmikantham \cite{22}  proved a fixed point
 theorem for a mixed
 monotone mapping in a metric space endowed with partial
 order, using a weak contractivity type  of assumption.

 Yolacan \cite{e} established some new fixed point
 theorems in 0-complete ordered partial metric spaces,
 and also remarked on coupled generalized Banach
 contraction mapping. Faraji {\it et al.} \cite{h}
 extended some fixed point theorems for Geraghty
 contractive mappings in b-complete b-metric spaces.
 However, Gupta {\it et al.} \cite{v}, established
some fixed point theorems in an ordered complete metric
spaces using distance function. Chaipunya {\it et al.} \cite{1}
proved some fixed point theorems of Geraghty-type
contractions concerning the existence and uniqueness
of fixed points under the setting of modular metric
spaces which also generalized the results in
Gordji {\it et al.}\cite{27}
under the influence of a modular metric spaces.

 Geraghty-type contractive mappings in
 metric spaces was generalized to the concept of
 preordered  G-metric spaces in \cite{G} and
 obtained unique fixed point results.


 \qquad In 2010, a remarkable study by Chistyakov
 \cite{12} introduced  an aspect of metric spaces called 
 modular metric spaces or parameterized metric spaces
 with the time parameter $\lambda$ (say) and his
 purpose was  to define the notion of a modular on an
 arbitrary set, and developed the theory of metric spaces
 generated by modulars, called modular metric spaces
 and, on the basis of it, defined new metric spaces of
 (multi-valued) functions of bounded generalized variation
  of a real variable with values in metric semi-groups and
  abstract convex cones.

 \qquad In the same year, Chistyakov \cite{23}, as an
 application presented an exhausting description
 of Lipschitz continuous and some other classes
 of superposition (Nemytskii) operators, acting
 in these modular metric spaces. He developed
 the theory of metric spaces generated by modulars,
 and extended the results given by Nakano \cite{x6},
 Musielak and Orlicz \cite{x3},  Musielak \cite{x2}
 to modular metric spaces.
Modular spaces are extensions of Lebesgue,
Riesz, and Orlicz spaces of integrable functions.

 \qquad Modular theories on linear spaces can be found
 by Nakano in his two monographs \cite{x6,x7},
 where he developed a spectral theory in semi-ordered
 linear spaces (vector lattices) and established the
 integral representation for projections acting in
 this modular spaces.
Nakano \cite{x6} established some modulars on
 real linear spaces which are convex functional.
 Non-convex modulars and the corresponding modular
 linear spaces were constructed by Musielak and
 Orlicz \cite{x3}. Orlicz spaces and modular linear
 spaces have already become classical tools in modern
 nonlinear functional analysis.

 \qquad The development of theory of metric spaces
generated by modulars, called modular metric spaces
attracted  many research mathematicians still 
investigating fixed point results in this area including
Chistyakov himself. Chistyakov \cite{3} also established some
fixed point theorems for contractive maps in modular spaces.
It is related to contracting, rather generalized average
velocities than metric distances, and the successive
approximations of fixed points converge to the
fixed points in a weaker sense as compared to the
metric convergence in \cite{3} and other fixed point results
in modular metric spaces can be  found in \cite{18} and \cite{11}.
 Considering applicability, these fixed point results are
 applied in solving the fixed point of nonlinear
 integral equations  see \cite{18,19} and references therein, while
 \cite{1} deals with application to partial differential equation,
 in modular metric spaces. Interested reader should see \cite{ok18,ok9,28} 
and the references therein for further studies in modular function spaces


\qquad In 2013, Azadifar {\it et al}. \cite{100} introduced the
concept of modular G-metric space and obtained some fixed point
theorems of contractive mappings defined on modular G-metric spaces.



\qquad Recently, Okeke {\it et al.} \cite{new}  proved
some existence and uniqueness of fixed point
theorems for mappings satisfying rational
inequality in modular metric spaces of which
Geraghty-type theorems in \cite{1} and Reich contraction
mapping principles are special cases. In this work, we will
examine some asymptotically $T$-regular mapping theorems
in preordered modular G-metric spaces which is an
extension of \cite{new}.

\qquad In this present paper, we 
extend the fixed point result obtained in \cite{new} to
asymptotically $T$-regular mapping theorems
in preordered modular G-metric spaces. Also  our
result shows that  the system of nonlinear 
integral equations have a unique solution in
preordered modular G-metric
spaces, $X_{\omega^{G}}.$


	
\section{Preliminaries}
	


\begin{definition}\cite{ana
A perorder set $X$ is a relation $\preceq$ that is both,
\begin{enumerate
\item[(i)] transitive  i.e; $x\preceq y$ and
$y\preceq z$ implies $x\preceq z$ and,

\item[(ii)] reflexive i.e; $x\preceq x.$
\end{enumerate
A preordered set is a pair $(X, \preceq)$  consisting
 of  a set $X$ and a preorder $\preceq$ on $X.$
\end{definition


\begin{remark
  If a preorder $\preceq$ is antisymmetic i.e; 
  $x\preceq y$ and $y\preceq x$ implies
  $x= y$, then $\preceq$
  is called a partial order.
\end{remark}


\begin{definition}\cite{13
Let $\mathcal{S}$ be the family of all
Geraghty functions, that is functions
 $\alpha:[0,\infty)\rightarrow [0,1)$
 satisfying the  condition
 $\{\alpha(t_{n})\}\rightarrow 1\implies
 \{t_{n}\}\rightarrow 0$.
\end{definition}

We will in this work  take $\mathcal{S}_{Ger}$
as the class of all Geraghty functions.
Such Geraghty class was discussed in \cite{1}

\begin{definition}\cite{1
	Let $\mathcal{S}$ be the family of all
	Geraghty functions, that is functions
	$\beta_{i}:\mathbb{R}_{+}\cup \{\infty\}
	\rightarrow [0,1)$ satisfying the  condition
	 $\beta_{i}(t_{k})\rightarrow \frac{1}{n}$
	implies that $\{t_{k}\}\rightarrow 0$ for all $i$.
\end{definition



\begin{definition}\cite{1
	Let $\Psi$ be the class of functions $\psi:
	\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ such
	that the following conditions hold;
	\begin{enumerate
		
		\item[(1)] $\psi$ is decreasing,
		
		\item[(2)] $\psi$ is continuous,
		
		\item[(3)] $\psi(t)=0$ if and only if $t=0.$
		
	\end{enumerate}
\end{definition

 Extension of Definition 2.2 above is as follows

\begin{definition}\cite{1
	Let $\bar{\Psi}$ be the class of functions
	 $\psi:\mathbb{R}_{+}\cup\{\infty\}\rightarrow
	\mathbb{R}_{+}\cup\{\infty\}$ such that the following conditions hold;
	\begin{enumerate
		\item[(a)] $\psi$ is sub-additive,
		
		\item[(b)] $\psi(t)$ is finite for $0<t<\infty,$
		
		\item[(c)] $\psi\lvert_{\mathbb{R}_{+}}\in\bar{\Psi}.$
	\end{enumerate}
\end{definition






\begin{definition}\cite{100}\label{2.1
	Let $X$ be a nonempty set, and let
	 $\omega^{G}:(0,\infty)\times X\times X\times X
	 \rightarrow [0,\infty]$ be a function satisfying;
	\begin{enumerate
		\item[(1)] $\omega^{G}_{\lambda}(x,y,z)=0$ for
		all $x,y,z\in X$ and $\lambda>0$ if $x=y=z,$
		
		\item[(2)] $\omega_{\lambda}^{G}(x,x,y)>0$ for
		all $x,y\in X$ and $\lambda>0$ with $x\neq y,$
		
		\item[(3)] $\omega_{\lambda}^{G}(x,x,y)\le\omega_{
			\lambda}^{G}(x,y,z)$ for all $x,y,z\in X$ and
		$\lambda>0$ with $z\neq y,$
		
		\item[(4)]  $\omega_{\lambda}^{G}(x,y,z)=\omega_{
			\lambda}^{G}(x,z,y)
		=\omega_{\lambda}^{G}(y,z,x)=\cdots$ for all
		 $\lambda>0$ (symmetry in all three variables),
		
		\item[(5)] $\omega_{\lambda+\mu}^{G}(x,y,z)\le\omega_{
			\lambda}^{G}(x,a,a)+\omega_{\mu}^{G}(a,y,z)$,
		for all $x,y,z,a\in X$ and $\lambda, \nu>0$,
	\end{enumerate

then the function $\omega^{G}_{\lambda}$
is called a modular G-metric on $X$.
\end{definition

\begin{remarks
	
	\begin{enumerate
		
		\item[(a)] The pair $(X, \omega^{G}) $ is
		called a modular G-metric space, and without
		any confusion we will take $X_{\omega^{G}}$ as a
		modular G-metric space.
		 From condition (5), if $\omega^{G}$ is convex,
		then we have a strong form  as,
		
		\item[(b)] $\omega^{G}_{\lambda+\mu}(x,y,z)\le\omega^{G}_{
			\frac{\lambda}{\lambda+\mu}}(x,a,a)+\omega^{G}_{
			\frac{\mu}{\lambda+\mu}}(a,y,z)$,
		
		\item[(c)]	If $x=a$, then (5) above becomes
		 $\omega_{\lambda+\mu}^{G}(a,y,z)\le\omega_{\mu}^{G}(a,y,z)$,
		
		\item[(d)] Condition (5) is called rectangle inequality.
		
	\end{enumerate
	
\end{remarks}



\begin{definition}\cite{100
	Let $(X, \omega^{G})$ be a modular G-metric space.
	The sequence $\{x_{n}\}_{n\in\mathbb{N}}$ in $X$ is
	 modular G-convergent to $x$, if it converges to
	  $x$ in the topology $\tau(\omega^{G}_{\lambda})$.
	  \newline
	A function $T: X_{\omega^{G}}\rightarrow X_{\omega^{G}} $
	at $x\in X_{\omega^{G}}$ is called modular G-continuous if
	 $\omega^{G}_{\lambda}(x_{n},x,x)\rightarrow 0$,  then
	$\omega^{G}_{\lambda}(Tx_{n},Tx,Tx)\rightarrow 0$, for all
	 $\lambda>0 .$
\end{definition}

\begin{remark
	 $\{x_{n}\}_{n\in\mathbb{N}}$  modular G-converges to
	  $x$ as $n\rightarrow\infty$, if
	   $\lim\limits_{n\rightarrow\infty}\omega^{G}_{\lambda}
	  (x_{n},x_{m},x)=0$. That is for all $\epsilon>0$ there
	  exists $n_{0}\in\mathbb{N}$ such that
	   $\omega^{G}_{\lambda}(x_{n},x_{m},x)<\epsilon$ for all
	  $n,m\geq n_{0}$. Here we say that $x$ is modular G-limit of
	   $\{x_{n}\}_{n\in\mathbb{N}}$.
\end{remark

\begin{definition}\cite{100
	Let $(X, \omega^{G})$ be a modular G-metric space, then $\{
	 x_{n}\}_{n\in\mathbb{N}}\subseteq X_{\omega^{G}}$ is
	 said to be  modular G-Cauchy if for every  $\epsilon>0$,
	 there exists $n_{\epsilon}\in \mathbb{N}$ such that
	$\omega^{G}_{\lambda}(x_{n},x_{m},x_{l})<\epsilon$ for all
	 $n,m,l\geq n_{\epsilon}$ and $\lambda>0$.\\
	A modular G-metric space $X_{\omega^{G}}$ is said to be
	modular G-complete if every modular G-Cauchy sequence
	in $X_{\omega^{G}}$ is  modular G-convergent in $X_{\omega^{G}}$.
\end{definition


\begin{proposition}\cite{100}\label{2.2
	Let $(X, \omega^{G})$ be a modular G-metric space,
	for any $x,y,z,a\in X$, it follows that:
	\begin{enumerate}
		\item [(1)] If $\omega_{\lambda}^{G}(x,y,z)=0$ for all
		 $\lambda>0$, then $x=y=z.$
		
		\item [(2)] $ \omega_{\lambda}^{G}(x,y,z)\le
		 \omega_{\frac{\lambda}{2}}^{G}(x,x,y)+\omega_{
			\frac{\lambda}{2}}^{G}(x,x,z)$ for  all $\lambda>0 .$
		
		\item [(3)]  $\omega_{\lambda}^{G}(x,y,y)\le
		 2\omega_{\frac{\lambda}{2}}^{G}(x,x,y)$ for  all $\lambda>0.$
		
		\item [(4)] $\omega_{\lambda}^{G}(x,y,z)\le
		 \omega_{\frac{\lambda}{2}}^{G}(x,a,z)+\omega_{\frac{
				\lambda}{2}}^{G}(a,y,z)$ for  all $\lambda>0.$
			
		\item[(5)] $\omega_{\lambda}^{G}(x,y,z)\le
		 \frac{2}{3}(\omega_{\frac{\lambda}{2}}^{G}(x,y,a)+
		\omega_{\frac{\lambda}{2}}^{G}(x,a,z) +\omega_{
			\frac{\lambda}{2}}^{G}(a,y,z))$ for  all $\lambda>0.$
		
		\item[(6)] $\omega_{\lambda}^{G}(x,y,z)\le
		 \omega_{\frac{\lambda}{2}}^{G}(x,a,a)+\omega_{\frac{
				\lambda}{4}}^{G}(y,a,a)
		+\omega_{\frac{\lambda}{4}}^{G}(z,a,a)$ for  all $\lambda>0.$
	\end{enumerate
\end{proposition


\begin{proposition}\cite{100}\label{2.3
	Let $(X, \omega^{G})$ be a modular G-metric space and 
	$ \{x_{n}\}_{n\in\mathbb{N}}$ be a sequence in
	$X_{\omega^{G}}.$  Then the following are equivalent:
	\begin{enumerate
		\item[(1)] $\{x_{n}\}_{n\in\mathbb{N}}$ is $\omega^{G}$-convergent to $x$,
		
		\item[(2)] $\omega^{G}_{\lambda}(x_{n},x)\rightarrow 0$ as $n\rightarrow \infty,$ i.e;
		 $\{x_{n}\}_{n\in\mathbb{N}}$ converges to $x$ relative to modular metric
		  $\omega^{G}_{\lambda}(.)$,
		
		\item[(3)] $\omega^{G}_{\lambda}(x_{n},x_{n},x)\rightarrow0$ as $n\rightarrow
		 \infty$ for all $\lambda>0,$
		
		\item[(4)] $\omega^{G}_{\lambda}(x_{n},x,x)\rightarrow 0$ as $n\rightarrow
		 \infty$ for all $\lambda>0,$
		
		\item[(5)] $\omega^{G}_{\lambda}(x_{m},x_{n},x)\rightarrow 0$ as $m,n\rightarrow
		 \infty$ for all $\lambda>0.$
	\end{enumerate
\end{proposition

We give the following definitions
which will form the basis of the results
 in this section 3 below

\begin{definition
	An ordered modular G-metric space is a
triple $(X,\omega^{G},\preccurlyeq)$ where
	 $(X,\omega)$ is a modular metric space
and $\preccurlyeq$ is a partial order on
	 $X_{\omega^{G}}.$ If $\preceq$ is a
preorder on $X_{\omega^{G}}$, then
	 $(X,\omega^{G},\preceq)$ is a
preordered modular G-metric space.
\end{definition

\begin{definition
	A sequence $\{T^{n}x_{0}\}_{n\in\mathbb{N}}$ in a preordered
	 modular G-metric space  $(X,\omega^{G},\preceq)$
	 is asymptotically $T$-regular at $x_{0}\in X$ if for all
	 $\lambda>0$, $\lim\limits_{n\rightarrow\infty}\omega_{\lambda}
	 (T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})=0$.
\end{definition






	
\section{Main Results}
	
\begin{theorem}\label{3.1}
Let $(X,\omega^{G},\preceq)$ be a complete preordered
 modular G-metric space.
 Let $T:X_{\omega^{G}}\rightarrow X_{\omega^{G}}$
  be a non-decreasing, asymptotically $T$-regular map on some point
  $x_{0}\in X_{\omega^{G}}$ 
  such that for each $\lambda>0$, there
 is $\nu(\lambda)\in\lbrack 0,\lambda)$ such that the following
  conditions hold:
\begin{align}\label{a}
(1)~~\psi(\omega^{G}_{\lambda}(Tx,Ty,Ty))\nota
\le
\kappa_{1}(\psi(\omega^{G}_{\lambda}(x,y,y))
\psi(\omega^{G}_{\lambda+\nu(\lambda)}(x,y,y))\\ \nota
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}(x,y,y))
\psi(\omega^{G}_{\lambda}(x,Tx,Tx))\\ \notag
+& \kappa_{3}(\psi(\omega^{G}_{\lambda}(x,y,y))
\psi(\omega^{G}_{\lambda}(y,Ty,Ty))\\ \notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}(x,y,y))
\psi(\omega^{G}_{\lambda}(Tx,Ty,Ty))\\ \notag
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}(x,y,y))
\psi(\omega^{G}_{\lambda}(Tx,y,y))\\ \notag
+&\kappa_{6}(\psi(\omega^{G}_{\lambda}(x,y,y)))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(y,Ty,Ty
[1+\omega^{G}_{\lambda}(x,y,y)]}{1+\omega^{G}_{\lambda}(x,Tx,Tx)}}\\ \nota
+&\kappa_{7}(\psi(\omega^{G}_{\lambda}(x,y,y)))\\\nota
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(y,Ty,Ty
[1+\omega^{G}_{\lambda}(x,Tx,Tx)]}{1+\omega^{G}_{\lambda}(x,y,y)}}\\ \nota
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}(x,y,y)))\\\nota
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x,Tx,Tx
[1+\omega^{G}_{\lambda}(x,y,y)]}{1+\omega^{G}_{\lambda}(y,Tx,Tx
+\omega^{G}_{\lambda}(x,y,y)}}\\ \nota
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}(x,y,y)))\\\nota
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x,y,y
[1+\omega^{G}_{\lambda}(y,Ty,Ty)]}{1+\omega^{G}_{\lambda}(Tx,Tx,y
+\omega^{G}_{\lambda}(Tx,Ty,Ty)}}\\ \nota
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}(x,y,y)))\\\nota
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(Tx,Ty,Ty
[1+\omega^{G}_{\lambda}(x,Ty,Ty)]}{1+\omega^{G}_{\lambda}(Tx,Tx,y
+\omega^{G}_{\lambda}(x,Ty,Ty)}}\\ \nota
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}(x,y,y)))\\%
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(Tx,Ty,Ty
[1+\omega^{G}_{\lambda}(y,Ty,Ty)+\omega^{G}_{\lambda}(Tx,Tx,y)]
{1+\omega^{G}_{\lambda}(Tx,Tx,y)+\omega^{G}_{\lambda}(Tx,Ty,Ty)}}
\end{align

where $\psi\in \bar{\Psi}$ , $\{\kappa_{1},\kappa_{2},
\cdots, \kappa_{11}\}\in \mathcal{S}_{Ger}$ and
$\max\{\sup_{t\geq 0}\kappa_{1}(t),\sup_{t\geq 0}\kappa_{2}(t),
\cdots, \sup_{t\geq 0}\kappa_{11}(t)\}<1.$
Assume that if a non-decreasing sequence $\{x_{n}\}_{n\in\mathbb{N}}$
converges to $x^{\ast}$, then $x_{n}\preceq x^{\ast}$ for
each $n\in\mathbb{N}$
\newlin
$(2)$ if $\psi$ is sub-additive and for any $x,y\in X_{\omega^{G}}$
there exists $z\in X_{\omega^{G}}$ with $z\preceq Tz$ and
$\omega^{G}_{\lambda}(z,Tz,Tz)$ is finite  for all $\lambda>0$
such that $z$ is comparable to both $x~{\rm and}~y$
Then $T$ has a  fixed point  $x^{\ast}\in X_{\omega^{G}}$
and the sequence define by $\{T^{n}x_{0}\}_{n\ge 1}$ converges to $x^{\ast}$
Moreover the fixed point of $T$ is unique.
\end{theorem}

\begin{proof}
(a)
Let $x_{0}\in X_{\omega^{G}}$ be such
that $x_{0}\preceq Tx_{0}$ and let
$x_{n}=Tx_{n-1}=T^{n}x_{0}$ for all
$n\in \mathbb{N}.$ Regarding that $T$ is
non-decreasing mapping, we have that
$x_{0}\preceq Tx_{0}=x_{1}$,
which implies that $x_{1}=Tx_{0}
\preceq Tx_{1}=x_{2}$. Inductively,
we have
\begin{equation}\label{1111
x_{0}\preceq x_{1}\preceq x_{2}\preceq \cdots \preceq x_{n-1}
\preceq x_{n}\preceq x_{n+1}\preceq \cdots.
\end{equation
	
Assume that there exists $n_{0}\in\mathbb{N}$
such that $x_{n_{0}}=x_{n_{0}+1}$. Since
$x_{n_{0}}=x_{n_{0}+1}=Tx_{n_{0}}$, then
$x_{n_{0}}$ is the fixed point of $T$.
Now suppose that $x_{n} \precneqq x_{n+1}$ for
all $n\in\mathbb{N}$, thus  by inequalit
\ref{1111}, we have that
\begin{equation}\label{22
x_{0}\prec x_{1}\prec x_{2}\prec \cdots
\prec x_{n-1}\prec x_{n}\prec x_{n+1}\prec \cdots.
\end{equation
Now for each $\lambda>0,$ and $x_{0}\prec
Tx_{0}$ for all $n\in\mathbb{N}$
implies that $\omega^{G}_{\lambda}(x_{0},Tx_{0},Tx_{0})>0.$
Again, let $x_{0}\in X_{\omega^{G}}$  such that
$\omega^{G}_{\lambda}(x_{0},Tx_{0},Tx_{0})
<\infty~\forall~\lambda>0$. 
\newlin
(b)	
Now, we show  that for all $n\in \mathbb{N}$, the sequence
$\omega^{G}_{\lambda}(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})= 0$
for all $\lambda>0$, as  $n\rightarrow\infty$. This means that
the sequence $\{T^{n}x_{0}\}_{n\ge 1}$ is asymptotically $T$-regular
on some point $x_{0}\in X_{\omega^{G}}$. Assume that,
for each $n\in\mathbb{N}$, there exists $\lambda_{n}>0$ such that
$\omega^{G}_{\lambda_{n}}(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})
\neq 0$.  Otherwise we are done.
For each $n\ge 1$, if $0<\lambda<\lambda_{n}$, then we have 
$\omega^{G}_{\lambda}(T^{n}x_{0},T^{n}x_{0},T^{n+1}x_{0})\neq 0$.
Since $T^{n}x_{0}\preceq T^{n+1}x_{0}$, we have from condition (1)
of \autoref{3.1},then we have that $\psi(\omega^{G}_{\lambda_{n}}(T^{n}x_{0},
T^{n+1}x_{0},T^{n+1}x_{0}))\le \psi(\omega^{G}_{\lambda}(T^{n}x_{0},
T^{n+1}x_{0},T^{n+1}x_{0}))
=\psi(\omega^{G}_{\lambda}(TT^{n-1}x_{0},TT^{n}x_{0},TT^{n}x_{0}))$.
Take $x=T^{n-1}x_{0}$  and $ y=T^{n}x_{0}$, assume that we are taking 
$d_{n}=\psi(\omega^{G}_{\lambda}
(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0}))$,
hence we have that
\begin{align}\label{b
d_{n}\notag
\le&
\kappa_{1}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))
\psi(\omega^{G}_{\lambda+\nu(\lambda)}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0}))\\\notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))
\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},TT^{n-1}x_{0},TT^{n-1}x_{0})) \\\notag
+&\kappa_{3}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))
\psi(\omega^{G}_{\lambda}
(T^{n}x_{0},TT^{n}x_{0},TT^{n}x_{0}))\\ \notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))
\psi(\omega^{G}_{\lambda}
(TT^{n-1}x_{0},TT^{n}x_{0},TT^{n}x_{0}))\\ \notag
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))
\psi(\omega^{G}_{\lambda}
(TT^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0}))\\ \notag
+&\kappa_{6}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}
(T^{n}x_{0},TT^{n}x_{0},TT^{n}x_{0})
[1+\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})]}
{1+\omega^{G}_{\lambda}
(T^{n-1}x_{0},TT^{n-1}x_{0},TT^{n-1}x_{0})}}\\ \notag
+&\kappa_{7}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}
(T^{n}x_{0},TT^{n}x_{0},TT^{n}x_{0})
[1+\omega^{G}_{\lambda}
(T^{n-1}x_{0},TT^{n-1}x_{0},TT^{n-1}x_{0})]}
{1+\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})}}\\\notag
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}
(T^{n-1}x_{0},TT^{n-1}x_{0},TT^{n-1}x_{0})
[1+\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})]}
{1+\omega^{G}_{\lambda}
(T^{n}x_{0},TT^{n-1}x_{0},TT^{n-1}x_{0})
+\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})}}\\ \notag
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})
[1+\omega^{G}_{\lambda}
(T^{n}x_{0},TT^{n}x_{0},TT^{n}x_{0})]}
{1+\omega^{G}_{\lambda}
(TT^{n-1}x_{0},TT^{n-1}x_{0},T^{n}x_{0})
+\omega^{G}_{\lambda}
(TT^{n-1}x_{0},TT^{n}x_{0},TT^{n}x_{0})}}\\ \notag
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}
(TT^{n-1}x_{0},TT^{n}x_{0},TT^{n}x_{0})
[1+\omega^{G}_{\lambda}
(T^{n-1}x_{0},TT^{n}x_{0},TT^{n}x_{0})]}
{1+\omega^{G}_{\lambda}
(TT^{n-1}x_{0},TT^{n-1}x_{0},T^{n}x_{0})
+\omega^{G}_{\lambda}
(T^{n-1}x_{0},TT^{n}x_{0},TT^{n}x_{0})}} \\\notag	
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))
\times\notag
\end{align
\begin{equation
\psi\Set{\frac{\omega^{G}_{\lambda}
(TT^{n-1}x_{0},TT^{n}x_{0},TT^{n}x_{0})
[1+\omega^{G}_{\lambda}
(T^{n}x_{0},TT^{n}x_{0},TT^{n}x_{0})
+\omega^{G}_{\lambda}
(TT^{n-1}x_{0},TT^{n-1}x_{0},T^{n}x_{0})]}
{1+\omega^{G}_{\lambda}
(TT^{n-1}x_{0},TT^{n-1}x_{0},T^{n}x_{0})
+\omega^{G}_{\lambda}
(TT^{n-1}x_{0},TT^{n}x_{0},TT^{n}x_{0})}}.
\end{equation

 We get,

\begin{align}\label{c
d_{n}\notag
\le&
\kappa_{1}\big(\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0}\big)\big)\big)
\psi\big(\omega^{G}_{\lambda+\nu(\lambda)}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0}))\\ \notag
+&\kappa_{2}\big(\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))
\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0}))\\ \notag
+&\kappa_{3}\big(\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))
\psi\big(\omega^{G}_{\lambda}
\big(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0}))\\ \notag
+&\kappa_{4}\big(\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))
\psi\big(\omega^{G}_{\lambda}
\big(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0}))\\ \notag
+&\kappa_{5}\big(\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))
\psi\big(\omega^{G}_{\lambda}
\big(T^{n}x_{0},T^{n}x_{0},T^{n}x_{0}))\\\notag
+&\kappa_{6}\big(\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}
\big(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})
\big[1+\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})]}
{1+\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})}}\\ \notag
+&\kappa_{7}\big(\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}
\big(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})
\big[1+\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})]}
{1+\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})}}\\\notag
+&\kappa_{8}\big(\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})
\big[1+\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})]}
{1+\omega^{G}_{\lambda}
\big(T^{n}x_{0},T^{n}x_{0},T^{n}x_{0})
+\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})}}\\ \notag
+&\kappa_{9}\big(\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})
\big[1+\omega^{G}_{\lambda}
\big(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})]}
{1+\omega^{G}_{\lambda}\big(T^{n}x_{0},T^{n}x_{0},T^{n}x_{0})
+\omega^{G}_{\lambda}
\big(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})}}\\ \notag
+&\kappa_{10}\big(\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}
\big(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})
\big[1+\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})]}
{1+\omega^{G}_{\lambda}
\big(T^{n}x_{0},T^{n}x_{0},T^{n}x_{0})
+\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})}}\\ \notag
+&\kappa_{11}\big(\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\notag
\end{align
\begin{equation
\qquad\times\psi\Set{\frac{\omega^{G}_{\lambda}
\big(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})
\big[1+\omega^{G}_{\lambda}
\big(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})
+\omega^{G}_{\lambda}
\big(T^{n}x_{0},T^{n}x_{0},T^{n}x_{0})]}
{1+\omega^{G}_{\lambda}
\big(T^{n}x_{0},T^{n}x_{0},T^{n}x_{0})
+\omega^{G}_{\lambda}
\big(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})}}.
\end{equation

So applying property 1 of Definition \ref{2.1}, we have

\begin{align
\psi(\omega^{G}_{\lambda}
(T^{n}x_{0}, T^{n+1}x_{0},T^{n+1}x_{0}))\notag
\le&
\kappa_{1}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0}, T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi(\omega^{G}_{\lambda+\nu(\lambda)}
(T^{n-1}x_{0}, T^{n}x_{0},T^{n}x_{0}))\\\notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0}, T^{n}x_{0},T^{n}x_{0}))\\\notag
+&\kappa_{3}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi(\omega^{G}_{\lambda}
(T^{n}x_{0}, T^{n+1}x_{0},T^{n+1}x_{0}))\\\notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi(\omega^{G}_{\lambda}
(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0}))\\\notag
+&\kappa_{6}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi(\omega^{G}_{\lambda}
(T^{n}x_{0}, T^{n+1}x_{0},T^{n+1}x_{0}))\\\notag
+&\kappa_{7}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi(\omega^{G}_{\lambda}
(T^{n}x_{0}, T^{n+1}x_{0},T^{n+1}x_{0}))\\\notag
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0}, T^{n}x_{0},T^{n}x_{0}))\\\notag
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0}, T^{n}x_{0},T^{n}x_{0}))\\\notag
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi(\omega^{G}_{\lambda}
(T^{n}x_{0}, T^{n+1}x_{0},T^{n+1}x_{0}))\\\notag
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\
\times&\psi(\omega^{G}_{\lambda}
(T^{n}x_{0}, T^{n+1}x_{0},T^{n+1}x_{0})),
\end{align

so that

\begin{align}\label{3.7
\psi(\omega^{G}_{\lambda}
(T^{n}x_{0}, T^{n+1}x_{0},T^{n+1}x_{0}))\notag
\le&
\kappa_{1}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0}, T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0}, T^{n}x_{0},T^{n}x_{0}))\\\notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0}, T^{n}x_{0},T^{n}x_{0}))\\\notag
+&\kappa_{3}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi(\omega^{G}_{\lambda}
(T^{n}x_{0}, T^{n+1}x_{0},T^{n+1}x_{0}))\\\notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi(\omega^{G}_{\lambda}
(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0}))\\\notag
+&\kappa_{6}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi(\omega^{G}_{\lambda}
(T^{n}x_{0}, T^{n+1}x_{0},T^{n+1}x_{0}))\\\notag
+&\kappa_{7}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi(\omega^{G}_{\lambda}
(T^{n}x_{0}, T^{n+1}x_{0},T^{n+1}x_{0}))\\\notag
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0}, T^{n}x_{0},T^{n}x_{0}))\\\notag
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0}, T^{n}x_{0},T^{n}x_{0}))\\\notag
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
\times&\psi(\omega^{G}_{\lambda}
(T^{n}x_{0}, T^{n+1}x_{0},T^{n+1}x_{0}))\\\notag
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\
\times&\psi(\omega^{G}_{\lambda}
(T^{n}x_{0}, T^{n+1}x_{0},T^{n+1}x_{0})).
\end{align
		
Thus,

\begin{eqnarray}\notag
&&\bigg(1-\bigg(\kappa_{3}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))
+\kappa_{4}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
&&+\kappa_{6}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\notag
+\kappa_{7}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
&&+\kappa_{10}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))
+\kappa_{11}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\bigg)
\bigg)\\\nota
&&\times\psi(\omega^{G}_{\lambda}
(T^{n}x_{0}, T^{n+1}x_{0},T^{n+1}x_{0}))\\\notag
&&\le\notag
\bigg(\kappa_{1}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\notag
+\kappa_{2}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
&&+\kappa_{8}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))
+\kappa_{9}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))
\bigg)\\&&\times\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0}, T^{n}x_{0},T^{n}x_{0})).
\end{eqnarray

Therefore,

\begin{align}\label{3.9
\psi(\omega^{G}_{\lambda}
(T^{n}x_{0}, T^{n+1}x_{0},T^{n+1}x_{0}))\notag
\le& \rho \psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0}, T^{n}x_{0},T^{n}x_{0}))\\\notag
\le&\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0}, T^{n}x_{0},T^{n}x_{0}))\\\notag
\le&\psi(\omega^{G}_{\lambda}
(T^{n-2}x_{0}, T^{n-1}x_{0},T^{n-1}x_{0}))\\\notag
\vdots&\\\notag
\le&\psi(\omega^{G}_{\lambda}(x_{0}, Tx_{0},Tx_{0}))\\
< &\infty,
\end{align

where $\rho=\dfrac{\rho_{1}}{\rho_{2}},$ where,

\begin{align
\rho_{1}:=\notag
&1-\big(\kappa_{3}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\notag
+\kappa_{4}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
&+\kappa_{6}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\notag
+\kappa_{7}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
&+\kappa_{10}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))
+\kappa_{11}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0}))),
\end{align

and

\begin{align
\rho_{2}:=\notag
&\kappa_{1}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\notag
+\kappa_{2}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\\\notag
&+\kappa_{8}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))
+\kappa_{9}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0}))).
\end{align

Hence $\{\psi(\omega_{\lambda}
(T^{n}x_{0}, T^{n+1}x_{0},T^{n+1}x_{0}))\}_{n\ge 1}$
is non-increasing and bounded below for which the 
sequence converges to some real number
$\gamma\ge 0$ (say); that is $\gamma\in [0,\infty)$. 
We claim that $\gamma=0$.
Suppose not, then  $\gamma>0$, then  for any $\lambda>0$,
using inequalities \ref{3.9} and \ref{3.7}, 
we can see clearly that
\begin{align
\psi(\omega^{G}_{\lambda}
(T^{n}x_{0}, T^{n+1}x_{0},T^{n+1}x_{0}))\notag
&\le
\bigg(\kappa_{1}\big(\psi\big(
\omega^{G}_{\lambda}\big(T^{n-1}x_{0}, T^{n}x_{0},
T^{n}x_{0}\big)\big)\big)\notag
+ \kappa_{2}\big(\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},
T^{n}x_{0}\big)\big)\big)\\\notag
&+ \kappa_{3}\big(\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},
T^{n}x_{0}\big)\big)\big)\notag
+\kappa_{4}
\big(\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0}\big)\big)\big)\\\notag
&+\kappa_{6}\big(\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0}\big)\big)\big)\\\notag
&+\sum_{k=7}^{11}\bigg(\kappa_{k}\big(\psi\big(
\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0}\big)\big)\big)\bigg)\bigg)\\
&\times\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})).\\\notag
\end{align
So that
\begin{align}
\psi(\omega_{\lambda}(T^{n}x_{0}, T^{n+1}x_{0},T^{n+1}x_{0}))
&\le\notag
\sum_{k=1}^{4}\kappa_{k}\big(\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0}\big)\big)\big)\\\notag 
&+\sum_{k=6}^{11}\kappa_{k}\big(\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0}\big)\big)\big)\\
&\times\psi(\omega^{G}_{\lambda}(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})),
\end{align

as $n\rightarrow\infty$,

\begin{equation
1\le\liminf_{n\rightarrow\infty}\biggl(\sum_{k=1}^{4}
\kappa_{k}\big(\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0}\big)\big)\big) 
+\sum_{k=6}^{11}\kappa_{k}\big(\psi\big(\omega^{G}_{\lambda}
\big(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0}\big)\big)\big)\biggr).
\end{equation
	
So, we have that

\begin{equation
\lim\limits_{n\rightarrow\infty}
\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0}))
=0~\forall ~\lambda>0,
\end{equation}

hence,

\begin{equation
\lim\limits_{n\rightarrow\infty}\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})=0~~ 
\forall ~\lambda>0.
\end{equation
	
This is a  contradiction to our claim.
Hence,

\begin{equation
\lim\limits_{n\rightarrow\infty}
\psi(\omega^{G}_{\lambda}
(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0}))
=0~ \forall ~\lambda>0,
\end{equation

moreover, we have

\begin{equation}\label{A
\lim\limits_{n\rightarrow\infty}\omega^{G}_{\lambda}
(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})=0 ~\forall~\lambda>0.
\end{equation

This shows that the sequence
$\{T^{n}x_{0}\}_{n\in\mathbb{N}}$ is asymptotically $T$-regular
on some point $x_{0}\in X_{\omega^{G}}$.
\newlin
(c)	
We now show that $\{T^{n}x_{0}\}_{n\in\mathbb{N}}$
is a modular G-Cauchy sequence in $X_{\omega^{G}}$.
By contradiction, suppose, if possible, that there
 exists real number,
$\epsilon>0$ and two modular G-sub-sequences
$\{T^{m_{k}}x_{0}\}_{k\in\mathbb{N}}$
and $\{T^{n_{k}}x_{0}\}_{k\in\mathbb{N}}$ of the modular G-sequence
$\{T^{n}x_{0}\}_{n\in\mathbb{N}}$
such that for $\lambda_{0}>0$ and $n_{k}>m_{k}>k$, 
we have that $\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})
\ge6\epsilon$, but
$\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})
<\epsilon$.
Now, $T^{m_{k}}x_{0}\preceq T^{n_{k}}x_{0}$,
we have $6\epsilon\le \omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})$
which implies that  $6\psi(\epsilon)\le\psi(
\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))
=\psi(\omega^{G}_{\lambda_{0}}
(TT^{m_{k}-1}x_{0},TT^{n_{k}-1}x_{0},TT^{n_{k}-1}x_{0}))$.
Set $x=T^{m_{k}-1}x_{0}$ and $y=T^{n_{k}-1}x_{0}$ 
into inequality \ref{a}, then we have
\begin{align}\label{d
6\psi(\epsilon)\notag
&\le
\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&\le
\kappa_{1}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))
\psi(\omega^{G}_{\lambda_{0}+\nu(\lambda_{0})}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\kappa_{2}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))
\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},TT^{m_{k}-1}x_{0},TT^{m_{k}-1}x_{0}))\\ \notag
&+\kappa_{3}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))
\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},TT^{n_{k}-1}x_{0},TT^{n_{k}-1}x_{0}))\\\notag
&+\kappa_{4}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))
\psi(\omega^{G}_{\lambda_{0}}
(TT^{m_{k}-1}x_{0},TT^{n_{k}-1}x_{0},TT^{n_{k}-1}x_{0}))\\ \notag
&+\kappa_{5}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))
\psi(\omega^{G}_{\lambda_{0}}
(TT^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\kappa_{6}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\\\notag
&\times\psi\Set{\frac{\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},TT^{n_{k}-1}x_{0},TT^{n_{k}-1}x_{0})
[1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})]}
{1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},TT^{m_{k}-1}x_{0},TT^{m_{k}-1}x_{0})}}\\ \notag
&+\kappa_{7}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\\\notag
&\times\psi\Set{\frac{\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},TT^{n_{k}-1}x_{0},TT^{n_{k}-1}x_{0})
[1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},TT^{m_{k}-1}x_{0},TT^{m_{k}-1}x_{0})]}
{1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})}}\\ \notag
&+\kappa_{8}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\\\notag
&\times\psi\Set{\frac{\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},TT^{m_{k}-1}x_{0},TT^{m_{k}-1}x_{0})
[1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})]}
{1+\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},TT^{m_{k}-1}x_{0},TT^{m_{k}-1}x_{0})
+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})}}\\ \notag
&+\kappa_{9}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\\\notag
&\times\psi\Set{\frac{\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})
[1+\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},TT^{n_{k}-1}x_{0},TT^{n_{k}-1}x_{0})]}
{1+\omega^{G}_{\lambda_{0}}
(TT^{m_{k}-1}x_{0},TT^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0})
+\omega^{G}_{\lambda_{0}}
(TT^{m_{k}-1}x_{0},TT^{n_{k}-1}x_{0},TT^{n_{k}-1}x_{0})}}\\ \notag
&+\kappa_{10}(\psi(\omega^{G}_{\lambda_{o}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\\\notag
&\times\psi\Set{\frac{\omega^{G}_{\lambda_{0}}
(TT^{m_{k}-1}x_{0},TT^{n_{k}-1}x_{0},TT^{n_{k}-1}x_{0})
[1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},TT^{n_{k}-1}x_{0},TT^{n_{k}-1}x_{0})]}
{1+\omega^{G}_{\lambda_{0}}
(TT^{m_{k}-1}x_{0},TT^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0})
+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},TT^{n_{k}-1}x_{0},TT^{n_{k}-1}x_{0})}} \\
&+\kappa_{11}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))
\end{align}
$ \times\psi\Set{\frac{\omega^{G}_{\lambda_{0}}
(TT^{m_{k}-1}x_{0},TT^{n_{k}-1}x_{0},TT^{n_{k}-1}x_{0})
[1+\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},TT^{n_{k}-1}x_{0},TT^{n_{k}-1}x_{0})
+\omega^{G}_{\lambda_{0}}
(TT^{m_{k}-1}x_{0},TT^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0})]}
{1+\omega^{G}_{\lambda_{0}}
(TT^{m_{k}-1}x_{0},TT^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0})
+\omega^{G}_{\lambda_{0}}
(TT^{m_{k}-1}x_{0},TT^{n_{k}-1}x_{0},TT^{n_{k}-1}x_{0})}}.$
So, we have that
\begin{align}\label{
6\psi(\epsilon)\notag
&\le
\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&\l
\kappa_{1}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))
\psi(\omega^{G}_{\lambda_{0}+\nu(\lambda_{0})}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\kappa_{2}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))
\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0}))\\ \notag
&+ \kappa_{3}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))
\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\kappa_{4}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))
\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\ \notag
&+\kappa_{5}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))
\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\kappa_{6}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\\\notag
&\times\psi\Set{\frac{\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})
[1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})]}
{1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0})}}\\ \notag
&+\kappa_{7}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\\\notag
&\times\psi\Set{\frac{\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})
[1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0})]}
{1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})}}\\ \notag
&+\kappa_{8}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\\\notag
&\times\psi\Set{\frac{\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0})
[1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})]}
{1+\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0})
+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})}}\\ \notag
&+\kappa_{9}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\\\notag
&\times\psi\Set{\frac{\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})
[1+\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})]}
{1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{m_{k}}x_{0},T^{n_{k}-1}x_{0})
+\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})}}\\ \notag
&+\kappa_{10}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\\\notag
&\times\psi\Set{\frac{\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})
[1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})]}
{1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{m_{k}}x_{0},T^{n_{k}-1}x_{0})
+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})}} \\\notag
&+\kappa_{11}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\notag
\end{align
\begin{equation}\label{e
\times\psi\Set{\frac{\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})
[1+\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})
+\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{m_{k}}x_{0},T^{n_{k}-1}x_{0})]}
{1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{m_{k}}x_{0},T^{n_{k}-1}x_{0})
+\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})}}.
\end{equation

We rewrite inequality \ref{e} as

\begin{align}\label{f
6\psi(\epsilon)\notag
\le&
\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
\le&
\psi(\omega^{G}_{\lambda_{0}+\nu(\lambda_{0})}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
+&\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0}))\\ \notag
+& \psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
+&\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\ \notag
+&\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
+&\psi\Set{\frac{\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})
[1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})]}
{1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0})}}\\ \notag
+&\psi\Set{\frac{\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})
[1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0})]}
{1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})}}\\ \notag
+&\psi\Set{\frac{\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0})
[1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})]}
{1+\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0})
+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})}}\\ \notag
+&\psi\Set{\frac{\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})
[1+\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})]}
{1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{m_{k}}x_{0},T^{n_{k}-1}x_{0})
+\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})}}\\ \notag
+&\psi\Set{\frac{\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})
[1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})]}
{1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{m_{k}}x_{0},T^{n_{k}-1}x_{0})
+\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})}}
\end{align
\begin{equation
+\psi\Set{\frac{\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})
[1+\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})
+\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{m_{k}}x_{0},T^{n_{k}-1}x_{0})]}
{1+\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{m_{k}}x_{0},T^{n_{k}-1}x_{0})
+\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})}}.
\end{equation

Using the fact that $\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0})\le 
\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k-1}}x_{0},T^{n_{k-1}}x_{0}),$
\newlin
$\omega^{G}_{\lambda_{0}}(T^{m_{k}}x_{0},
T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})<\epsilon$
and using property 5 of Definition \ref{2.1}, we have
\begin{align
6\psi(\epsilon)\notag
&\le
\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&\le\psi(\omega^{G}_{\nu(\lambda_{0})}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0})
+\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0}))
+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))
+\psi(\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))
+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))
+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0})
+\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\psi(\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})
+\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\
&+\psi(\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})
+\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})),
\end{align
so that
\begin{align
6\psi(\epsilon)\notag
&\le
\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&\le\psi(\omega^{G}_{\nu(\lambda_{0})}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0}))
+\psi(\epsilon)\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0}))
+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\epsilon)+\psi(\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))
+\psi(\epsilon)\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))
+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0}))
+\psi(\epsilon)\\\notag
&+\psi(\epsilon)+\psi(\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})\\
&+\psi(\epsilon)+\psi(\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})),
\end{align

as $k\rightarrow\infty$, we obtain that
$6\psi(\epsilon)\le\lim\limits_{k\rightarrow\infty}
\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))
\le6\psi(\epsilon)$
which implies that $\lim\limits_{k\rightarrow\infty}
\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))
=6\psi(\epsilon)$. 
\newlin

Therefore, we have

\begin{equation}
\lim\limits_{k\rightarrow\infty}
\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})
=6\epsilon.
\end{equation
		
		
Again, we can see that

\begin{align
\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\notag
&\le\psi(\omega^{G}_{\lambda_{0}+\nu(\lambda_{0})}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&\le
\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&\le
\psi(\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\psi(\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\psi(\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\psi(\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\psi(\omega^{G}_{\frac{\lambda_{0}}{2}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&\le
\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{m_{k}}x_{0},T^{m_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0}))\\\notag
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))\\
&+\psi(\omega^{G}_{\lambda_{0}}
(T^{n_{k}-1}x_{0},T^{n_{k}}x_{0},T^{n_{k}}x_{0})).
\end{align

Letting $k\rightarrow\infty$, we deduce that
$\psi(6\epsilon)\le\lim\limits_{k\rightarrow\infty}
\psi(\omega^{G}_{\lambda_{0}}(T^{m_{k}-1}x_{0},
T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))
\le\psi(6\epsilon)$, for which we have that
$\lim\limits_{k\rightarrow\infty}
\psi(\omega^{G}_{\lambda_{0}}(T^{m_{k}-1}x_{0},
T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))
=\psi(6\epsilon)$. 
\newline
Hence $\lim\limits_{k\rightarrow
\infty}\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},
T^{n_{k}-1}x_{0})=6\epsilon.$
Thus, it follows that
\begin{align
1\notag
&\le\lim\limits_{k\rightarrow
\infty}\inf\biggl(\kappa_{1}
(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\notag
+\kappa_{2}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\\\notag
&+\kappa_{3}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\notag
+\kappa_{4}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\\\notag
&+\kappa_{5}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\notag
+\kappa_{6}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\\\notag
&+\kappa_{7}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\notag
+\kappa_{8}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\\\notag
&+\kappa_{9}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\notag
+\kappa_{10}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\\
&+\kappa_{11}(\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})))\bigg).
\end{align

Therefore,
$\lim\limits_{k\rightarrow\infty}\psi(\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0}))=0$,
which implies that
\begin{equation}
\lim\limits_{k\rightarrow\infty}\omega^{G}_{\lambda_{0}}
(T^{m_{k}-1}x_{0},T^{n_{k}-1}x_{0},T^{n_{k}-1}x_{0})=0,
\end{equation

which is a contradiction.
Hence $\{T^{n}x_{0}\}_{n\in\mathbb{N}}$ is a
modular G-Cauchy sequence in $X_{\omega^{G}}$.
\newlin
(d)	
Since $X_{\omega^{G}}$ is complete modular G-metric space,
there exists $x^{\ast}\in X_{\omega^{G}}$ such that
$T^{n}x_{0}\rightarrow x^{\ast}\in X_{\omega^{G}}$.
Now we show that $x^{\ast}$ is a fixed point of $T$
for $\lambda>0$, suppose that $Tx^{\ast}\neq x^{\ast}$,
applying condition 5 of Definition \ref{2.1}, we have that
\begin{align} \label{do
\psi(\omega^{G}_{\lambda}
(T^{n}x_{0},Tx^{\ast},Tx^{\ast}))\notag
&\le\psi(\omega^{G}_{\frac{\lambda}{2}}
(T^{n+1}x_{0},Tx^{\ast},Tx^{\ast}))\notag
+\psi(\omega^{G}_{\frac{\lambda}{2}}
(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0}))\\\notag
&=\psi(\omega^{G}_{\frac{\lambda}{2}}
(T^{n+1}x_{0},Tx^{\ast},Tx^{\ast}))\notag
+\psi(\omega^{G}_{\frac{\lambda}{2}}
(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0}))\\
&\le\psi(\omega^{G}_{\lambda}
(T^{n+1}x_{0},Tx^{\ast},Tx^{\ast}))
+\psi(\omega^{G}_{\lambda}
(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})).
\end{align

Putting  $x=T^{n}x_{0}, y=x^{\ast}$ in the fist 
quantity in inequality \ref{do} 
into inequality \ref{a}, take $g_{n}=\psi(\omega^{G}_{\lambda}
(T^{n}x_{0},Tx^{\ast},Tx^{\ast}))$, we have
\begin{align}\label{g
g_{n}\notag
\le&\kappa_{1}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda+\nu(\lambda)}(T^{n}x_{0},x^{\ast},x^{\ast}))\\ \notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}x_{0},TT^{n}x_{0},TT^{n}x_{0}))\\ \notag
+& \kappa_{3}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast}))\\ \notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(TT^{n}x_{0},Tx^{\ast},Tx^{\ast}))\\ \notag
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(TT^{n}x_{0},x^{\ast},x^{\ast}))\\ \notag
+&\kappa_{6}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})]}
{1+\omega^{G}_{\lambda}(T^{n}x_{0},TT^{n}x_{0},TT^{n}x_{0})}}\\ \notag
+&\kappa_{7}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(T^{n}x_{0},TT^{n}x_{0},TT^{n}x_{0})]}
{1+\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})}}\\ \notag
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(T^{n}x_{0},TT^{n}x_{0},TT^{n}x_{0})
[1+\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})]}
{1+\omega^{G}_{\lambda}(x^{\ast},TT^{n}x_{0},TT^{n}x_{0})
+\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})}}\\ \notag
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})
[1+\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})]}
{1+\omega^{G}_{\lambda}(TT^{n}x_{0},TT^{n}x_{0},x^{\ast})
+\omega^{G}_{\lambda}(TT^{n}x_{0},Tx^{\ast},Tx^{\ast})}}\\ \notag
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(TT^{n}x_{0},Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(T^{n}x_{0},Tx^{\ast},Tx^{\ast})]}
{1+\omega^{G}_{\lambda}(TT^{n}x_{0},TT^{n}x_{0},x^{\ast})
+\omega^{G}_{\lambda}(T^{n}x_{0},Tx^{\ast},Tx^{\ast})}}\\ \notag
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(TT^{n}x_{0},Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})
+\omega^{G}_{\lambda}(TT^{n}x_{0},TT^{n}x_{0},x^{\ast})]}
{1+\omega^{G}_{\lambda}(TT^{n}x_{0},TT^{n}x_{0},x^{\ast})
+\omega^{G}_{\lambda}(TT^{n}x_{0},Tx^{\ast},Tx^{\ast})}}\\\notag
+&\psi(\omega^{G}_{\lambda}
(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0}))\\\notag
=&\kappa_{1}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda+\nu(\lambda)}(T^{n}x_{0},x^{\ast},x^{\ast}))\\ \notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0}))\\ \notag
+& \kappa_{3}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast}))\\ \notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}x_{0},Tx^{\ast},Tx^{\ast}))\\ \notag
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}x_{0},x^{\ast},x^{\ast}))\\ \notag
+&\kappa_{6}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})]}
{1+\omega^{G}_{\lambda}(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})}}\\ \notag
+&\kappa_{7}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})]}
{1+\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})}}\\ \notag
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})
[1+\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})]}
{1+\omega^{G}_{\lambda}(x^{\ast},T^{n+1}x_{0},T^{n+1}x_{0})
+\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})}}\\ \notag
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})
[1+\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})]}
{1+\omega^{G}_{\lambda}(T^{n+1}x_{0},T^{n+1}x_{0},x^{\ast})
+\omega^{G}_{\lambda}(T^{n+1}x_{0},Tx^{\ast},Tx^{\ast})}}\\ \notag
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(T^{n+1}x_{0},Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(T^{n}x_{0},Tx^{\ast},Tx^{\ast})]}
{1+\omega^{G}_{\lambda}(T^{n+1}x_{0},T^{n+1}x_{0},x^{\ast})
+\omega^{G}_{\lambda}(T^{n}x_{0},Tx^{\ast},Tx^{\ast})}}\\ \notag
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}(T^{n}x_{0},x^{\ast},x^{\ast})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(T^{n+1}x_{0},Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})
+\omega^{G}_{\lambda}(T^{n+1}x_{0},T^{n+1}x_{0},x^{\ast})]}
{1+\omega^{G}_{\lambda}(T^{n+1}x_{0},T^{n+1}x_{0},x^{\ast})
+\omega^{G}_{\lambda}(T^{n+1}x_{0},Tx^{\ast},Tx^{\ast})}}\\
+&\psi(\omega^{G}_{\lambda}
(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})).\\\notag
\end{align

Letting $n\rightarrow\infty$, we have that

\begin{align}
\psi(\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast}))\notag
\le &\kappa_{1}(\psi(\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda+\nu(\lambda)}(x^{\ast},x^{\ast},x^{\ast}))\\ \notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast}))\\ \notag
+& \kappa_{3}(\psi(\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast}))\\ \notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast}))\\ \notag
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast}))\\ \notag
+&\kappa_{6}(\psi(\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})]}
{1+\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})}}\\ \notag
+&\kappa_{7}(\psi(\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})]}
{1+\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})}}\\ \notag
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})
[1+\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})]}
{1+\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})
+\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})}}\\ \notag
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})
[1+\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})]}
{1+\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})
+\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})}}\\ \notag
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})]}
{1+\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})
+\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})}}\\ \notag
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})
+\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})]}
{1+\omega^{G}_{\lambda}(x^{\ast},x^{\ast},x^{\ast})
+\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})}}\\
+&\psi(\omega^{G}_{\lambda}
(x^{\ast},x^{\ast},x^{\ast})).
\end{align

Using condition 1 of Definition \ref{2.1}, we have
\begin{equation}\label{mee}
  \psi(\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast}))
  \le(\kappa_{3}(0)+\kappa_{4}(0)+
  \kappa_{6}(0)+\kappa_{7}(0)+\kappa_{10}(0)+\kappa_{11}(0))
  \psi(\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})).
\end{equation
So that
\begin{equation}\label{m3}
  (1-(\kappa_{3}(0)+\kappa_{4}(0)+
  \kappa_{6}(0)+\kappa_{7}(0)+\kappa_{10}(0)+\kappa_{11}(0))) 
  \psi(\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast}))\le 0,
\end{equation}
where, $\kappa_{3}(0)+\kappa_{4}(0)+
  \kappa_{6}(0)+\kappa_{7}(0)+\kappa_{10}(0)+\kappa_{11}(0)<1$
  and for all $\lambda>0$.
  Therefore,
$\psi(\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast}))
\le 0$ which implies that $\omega^{G}_{\lambda}
(x^{\ast},Tx^{\ast},Tx^{\ast})\le 0$.
Hence $Tx^{\ast}=x^{\ast}$. Therefore, 
$x^{\ast}$ is a fixed point of $T$.
Lastly, for the uniqueness, we know that $T$ has
a fixed point $x^{\ast}\in X_{\omega^{G}}$. 
Suppose that there is another fixed point of
$T$ i.e; $y^{\ast}\in X_{\omega^{G}}$, thus 
condition (2) of \autoref{3.1} says that if
$z\in X_{\omega^{G}}$ with $z\preceq Tz$ and 
it is comparable to both
$x^{\ast}$ and $y^{\ast}$ and $T^{n}z$ is 
also comparable to $x^{\ast}$
and $y^{\ast}$ for each $n\in\mathbb{N}$. 
Now for $\lambda>0$, then
$\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))$ and  
$\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))$ 
are finite.
\newlin
(e)
We now show that $x^{\ast}=y^{\ast}$. Indeed, using inequality \ref{a},
we have by taking $x=T^{n}z$ and $y=x^{\ast}$. First consider 
$\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))<\infty,$
so that we have the following;
\begin{align}
\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\notag
=&\psi(\omega_{\lambda}(T^{n+1}z,Tx^{\ast},Tx^{\ast}))\\\notag
=&\psi(\omega^{G}_{\lambda}(TT^{n}z,Tx^{\ast},Tx^{\ast}))\\\notag
\le&\kappa_{1}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda+\nu(\lambda)}(T^{n}z,x^{\ast},x^{\ast})) \notag
+\kappa_{2}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\\\notag\times&\psi(\omega^{G}_{\lambda}(T^{n}z,TT^{n}z,TT^{n}z)) \notag
+\kappa_{3}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast}))\\ \notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(TT^{n}z,Tx^{\ast},Tx^{\ast}))\\ \notag
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(TT^{n}z,x^{\ast},x^{\ast}))\notag
+\kappa_{6}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})]}
{1+\omega^{G}_{\lambda}(T^{n}z,TT^{n}z,T^{n}z)}}\\ \notag
+&\kappa_{7}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(T^{n}z,TT^{n}z,TT^{n}z)]}
{1+\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})}}\\ \notag
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(T^{n}z,TT^{n}z,TT^{n}z)
[1+\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})]}
{1+\omega^{G}_{\lambda}(x^{\ast},TT^{n}z,TT^{n}z)
+\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})}}\\ \notag
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})
[1+\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})]}
{1+\omega^{G}_{\lambda}(TT^{n}z,TT^{n}z,x^{\ast})
+\omega^{G}_{\lambda}(TT^{n}z,Tx^{\ast},Tx^{\ast})}}\\ \notag
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(TT^{n}z,Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(T^{n}z,Tx^{\ast},Tx^{\ast})]}
{1+\omega^{G}_{\lambda}(TT^{n}z,TT^{n}z,x^{\ast})
+\omega^{G}_{\lambda}(T^{n}z,Tx^{\ast},Tx^{\ast})}}\\ \notag
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))\\
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(TT^{n}z,Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})
+\omega^{G}_{\lambda}(TT^{n}z,TT^{n}z,x^{\ast})]}
{1+\omega^{G}_{\lambda}(TT^{n}z,TT^{n}z,x^{\ast})
+\omega^{G}_{\lambda}(TT^{n}z,Tx^{\ast},Tx^{\ast})}}.
\end{align

So that,

\begin{align
\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\notag
\le&\kappa_{1}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda+\nu(\lambda)}(T^{n}z,x^{\ast},x^{\ast}))\\ \notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,T^{n+1}z,T^{n+1}z))\\ \notag
+&\kappa_{3}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast}))\\ \notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,Tx^{\ast},Tx^{\ast}))\\ \notag
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\\\notag
+&\kappa_{6}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})]}
{1+\omega^{G}_{\lambda}(T^{n}z,T^{n+1}z,T^{n}z)}}\\ \notag
+&\kappa_{7}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(T^{n}z,T^{n+1}z,T^{n+1}z)]}
{1+\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})}}\\ \notag
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(T^{n}z,T^{n+1}z,T^{n+1}z)
[1+\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})]}
{1+\omega^{G}_{\lambda}(x^{\ast},T^{n+1}z,T^{n+1}z)
+\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})}}\\ \notag
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})
[1+\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})]}
{1+\omega^{G}_{\lambda}(T^{n+1}z,T^{n+1}z,x^{\ast})
+\omega^{G}_{\lambda}(T^{n+1}z,Tx^{\ast},Tx^{\ast})}}\\ \notag
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(T^{n+1}z,Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(T^{n}z,Tx^{\ast},Tx^{\ast})]}
{1+\omega^{G}_{\lambda}(T^{n+1}z,T^{n+1}z,x^{\ast})
+\omega^{G}_{\lambda}(T^{n}z,Tx^{\ast},Tx^{\ast})}}\\ \notag
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))\\
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(T^{n+1}z,Tx^{\ast},Tx^{\ast})
[1+\omega^{G}_{\lambda}(x^{\ast},Tx^{\ast},Tx^{\ast})
+\omega^{G}_{\lambda}(T^{n+1}z,T^{n+1}z,x^{\ast})]}
{1+\omega^{G}_{\lambda}(T^{n+1}z,T^{n+1}z,x^{\ast})
+\omega^{G}_{\lambda}(T^{n+1}z,Tx^{\ast},Tx^{\ast})}},
\end{align}

using the fact that $Tx^{\ast}=x^{\ast}$, we get

\begin{align}\label{B
\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\notag
\le&\kappa_{1}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda+\nu(\lambda)}(T^{n}z,x^{\ast},x^{\ast})) \notag
+\kappa_{2}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\\\notag\times&\psi(\omega^{G}_{\lambda}(T^{n}z,T^{n+1}z,T^{n+1}z)) \notag
+\kappa_{4}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\\ \notag
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\notag
+\kappa_{8}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\\\notag\times&\psi(\omega^{G}_{\lambda}(T^{n}z,T^{n+1}z,T^{n+1}z))\notag
+\kappa_{9}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega_{\lambda}(T^{n}z,x^{\ast},x^{\ast}))\\\notag
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))
+\kappa_{11}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\\\times&\psi(\omega^{G}_{\lambda}(
T^{n}z,x^{\ast},x^{\ast})).
\end{align

Applying condition 6 of Proposition \ref{2.2}, we get

\begin{align
\omega^{G}_{\lambda}
(T^{n}z,T^{n+1}z,T^{n+1}z)\notag
\le&\omega^{G}_{\frac{\lambda}{2}}
(T^{n}z,x^{\ast},x^{\ast})
+\omega^{G}_{\frac{\lambda}{4}}
(T^{n+1}z,x^{\ast},x^{\ast})
+\omega^{G}_{\frac{\lambda}{4}}
(T^{n+1}z,x^{\ast},x^{\ast})\\\notag
=&\omega^{G}_{\frac{\lambda}{2}}
(T^{n}z,x^{\ast},x^{\ast})
+2\omega^{G}_{\frac{\lambda}{4}}
(T^{n+1}z,x^{\ast},x^{\ast})\\
\le& \omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})
+2\omega^{G}_{\lambda}
(T^{n+1}z,x^{\ast},x^{\ast}).
\end{align

Therefore inequality \ref{B} and 
noticing that $\psi$(.) is sub-additive, we have
\begin{align}\label{
\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\notag
\le&\kappa_{1}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast}))\\ \notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast}))
+2\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\\ \notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\\ \notag
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\\\notag
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})
+2\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\\\notag
+&\alpha_{9}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega_{\lambda}(T^{n}z,x^{\ast},x^{\ast}))\\\notag
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\\\notag
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast}))\\\notag
\le&\kappa_{1}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast}))\\ \notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
+2\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\\ \notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\\ \notag
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\\\notag
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast}))
+2\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast})))\\\notag
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega_{\lambda}(T^{n}z,x^{\ast},x^{\ast}))\\\notag
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\\\notag
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast}))\\\notag
=&\kappa_{1}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast}))\\ \notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast}))\\\notag
+&2\kappa_{2}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\\ \notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\\ \notag
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\\\notag
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast}))\\\notag
+&2\kappa_{8}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\\\notag
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega_{\lambda}(T^{n}z,x^{\ast},x^{\ast}))\\\notag
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,x^{\ast},x^{\ast}))\\
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast})),
\end{align

so that
		
\begin{eqnarray
&&\bigg(1-\bigg(2\kappa_{2}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\notag
+\kappa_{4}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\notag
+\kappa_{5}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\\
&&+2\kappa_{8}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\notag
+\kappa_{10}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\bigg)\bigg)\notag
\psi(\omega^{G}_{\lambda}
(T^{n+1}z, x^{\ast},x^{\ast}))\\\notag
&&\le\notag
\bigg(\kappa_{1}(\psi(\omega^{G}_{\lambda}
(T^{n},x^{\ast},x^{\ast})))\notag
+\kappa_{2}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\notag
+\kappa_{8}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\\
&&+\kappa_{9}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))
+\kappa_{11}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\bigg)
\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})).
\end{eqnarray
		
Therefore,

\begin{align
\psi(\omega^{G}_{\lambda}
(T^{n+1}z,x^{\ast},x^{\ast}))\notag
\le& \delta\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast}))\\\notag
\le&\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast}))\\\notag
\le&\psi(\omega^{G}_{\lambda}
(T^{n-1}z,x^{\ast},x^{\ast}))\\\notag
\vdots&\\\notag
\le&\psi(\omega^{G}_{\lambda}
(z,x^{\ast},x^{\ast}))\\
< &\infty,
\end{align}

where $\delta=\dfrac{\delta_{1}}{\delta_{2}}.$

\begin{eqnarray
\delta_{1}:=&& 1-\bigg(2\kappa_{2}(
\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\notag
+\kappa_{4}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\notag
+\kappa_{5}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\\
&&+2\kappa_{8}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))
+\kappa_{10}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\bigg),
\end{eqnarray

an

\begin{eqnarray
\delta_{2}:=&&\bigg(\kappa_{1}(\psi(\omega^{G}_{\lambda}
(T^{n},x^{\ast},x^{\ast})))\notag
+\kappa_{2}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\notag
+\kappa_{8}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\\
&&+\kappa_{9}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))
+\kappa_{11}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\bigg).
\end{eqnarray
		
Therefore, $\{\psi(\omega^{G}_{\lambda}
(T^{n+1}z,x^{\ast},x^{\ast}))\}_{n\ge 1}$
is non-increasing sequence and bounded below.
 So, it converges to some real number $\beta\ge 0$ (say).
For $\beta>0$. Recall that $\lim\limits_{n\rightarrow\infty}
\omega^{G}_{\lambda}(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})=0 $
for all $\lambda>0$, that is $\{T^{n}x_{0}\}_{n\in\mathbb{N}}$
is asymptotically $T$-regular on some point $x_{0}\in
X_{\omega^{G}}$, Thus, letting $n\rightarrow \infty$ 
in the inequality \ref{B},
and using the fact that $\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast}))\leq \psi(\omega^{G}_{\lambda}
(T^{n+1}z,x^{\ast},x^{\ast}))$, we have  that
\begin{align
1\notag &\le\lim\limits_{n\rightarrow\infty}
\inf\bigg(\kappa_{1}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\notag
+\kappa_{4}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\\\notag
&+\kappa_{5}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\notag
+\kappa_{9}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\\\notag
&+\kappa_{10}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))
+\kappa_{11}(\psi(\omega^{G}_{\lambda}
(T^{n}z,x^{\ast},x^{\ast})))\bigg).
\end{align

Therefore, $\lim\limits_{n\rightarrow\infty} 
\psi(\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast}))=0$, 
for which by condition 4 of Proposition
\ref{2.3}, we have  that

\begin{equation
\lim\limits_{n\rightarrow\infty}
\omega^{G}_{\lambda}(T^{n}z,x^{\ast},x^{\ast}))=0 
~\forall~\lambda>0.
\end{equation

 Hence $T^{n}z\rightarrow x^{\ast}$ as
  $n\rightarrow\infty.$
		
Again, $T^{n}z$ is also comparable with
$y^{\ast}$ for each $n\in\mathbb{N}$.
Now for $\lambda>0$,
using inequality \ref{a}, we have by 
taking $x=T^{n}z$ and $y=y^{\ast}.$
Again, consider $\psi(\omega^{G}_{
\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))<\infty,$
so that we have the following

\begin{align
\psi(\omega^{G}_{\lambda}
(T^{n+1}z,y^{\ast},y^{\ast}))\notag
=&\psi(\omega_{\lambda}
(T^{n+1}z,Ty^{\ast},Ty^{\ast}))\\\notag
=&\psi(\omega^{G}_{\lambda}
(TT^{n}z,Ty^{\ast},Ty^{\ast}))\\\notag
\le&\kappa_{1}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda+\nu(\lambda)}
(T^{n}z,y^{\ast},y^{\ast}))\\ \notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}
(T^{n}z,TT^{n}z,TT^{n}z))\\ \notag
+&\kappa_{3}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}
(y^{\ast},Ty^{\ast},Ty^{\ast}))\\ \notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}
(TT^{n}z,Ty^{\ast},Ty^{\ast}))\\ \notag
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}
(TT^{n}z,y^{\ast},y^{\ast}))\\\notag
+&\kappa_{6}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}
(y^{\ast},Ty^{\ast},Ty^{\ast})
[1+\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})]}
{1+\omega^{G}_{\lambda}
(T^{n}z,TT^{n}z,T^{n}z)}}\\ \notag
+&\kappa_{7}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}
(y^{\ast},Ty^{\ast},Ty^{\ast})
[1+\omega^{G}_{\lambda}(T^{n}z,TT^{n}z,TT^{n}z)]}
{1+\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})}}\\ \notag
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}
(T^{n}z,TT^{n}z,TT^{n}z)
[1+\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})]}
{1+\omega^{G}_{\lambda}(y^{\ast},TT^{n}z,TT^{n}z)
+\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})}}\\ \notag
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast}) 
[1+\omega^{G}_{\lambda}(y^{\ast},Ty^{\ast},Ty^{\ast})]}
{1+\omega^{G}_{\lambda}(TT^{n}z,TT^{n}z,y^{\ast})
+\omega^{G}_{\lambda}(TT^{n}z,Ty^{\ast},Ty^{\ast})}}\\ \notag
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}
(TT^{n}z,Ty^{\ast},Ty^{\ast})
[1+\omega^{G}_{\lambda}(T^{n}z,Ty^{\ast},Ty^{\ast})]}
{1+\omega^{G}_{\lambda}(TT^{n}z,TT^{n}z,y^{\ast})
+\omega^{G}_{\lambda}(T^{n}z,Ty^{\ast},Ty^{\ast})}}\\ \notag
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))\\%
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(
TT^{n}z,Ty^{\ast},Ty^{\ast})
[1+\omega^{G}_{\lambda}(y^{\ast},Ty^{\ast},Ty^{\ast})
+\omega^{G}_{\lambda}(TT^{n}z,TT^{n}z,y^{\ast})]}
{1+\omega^{G}_{\lambda}(TT^{n}z,TT^{n}z,y^{\ast})
+\omega^{G}_{\lambda}(TT^{n}z,Ty^{\ast},Ty^{\ast})}},
\end{align

so that

\begin{align
\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\notag
\le&\kappa_{1}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda+\nu(\lambda)}(T^{n}z,y^{\ast},y^{\ast}))\\ \notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,T^{n+1}z,T^{n+1}z))\\ \notag
+&\kappa_{3}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(y^{\ast},Ty^{\ast},Ty^{\ast}))\\ \notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,Ty^{\ast},Ty^{\ast}))\\ \notag
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\\\notag
+&\kappa_{6}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(y^{\ast},Ty^{\ast},Ty^{\ast})
[1+\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})]}{1
+\omega^{G}_{\lambda}(T^{n}z,T^{n+1}z,T^{n}z)}}\\ \notag
+&\kappa_{7}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(y^{\ast},Ty^{\ast},Ty^{\ast})
[1+\omega^{G}_{\lambda}(T^{n}z,T^{n+1}z,T^{n+1}z)]}{1
+\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})}}\\ \notag
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(T^{n}z,T^{n+1}z,T^{n+1}z)
[1+\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})]}
{1+\omega^{G}_{\lambda}(y^{\ast},T^{n+1}z,T^{n+1}z)
+\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})}}\\ \notag
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})
[1+\omega^{G}_{\lambda}(y^{\ast},Ty^{\ast},Ty^{\ast})]}
{1+\omega^{G}_{\lambda}(T^{n+1}z,T^{n+1}z,y^{\ast})
+\omega^{G}_{\lambda}(T^{n+1}z,Ty^{\ast},Ty^{\ast})}}\\\notag
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))\\ \notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(T^{n+1}z,Ty^{\ast},Ty^{\ast})
[1+\omega^{G}_{\lambda}(T^{n}z,Ty^{\ast},Ty^{\ast})]}
{1+\omega^{G}_{\lambda}(T^{n+1}z,T^{n+1}z,y^{\ast})
+\omega^{G}_{\lambda}(T^{n}z,Ty^{\ast},Ty^{\ast})}}\\ \notag
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))\\
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(T^{n+1}z,Ty^{\ast},Ty^{\ast})
[1+\omega^{G}_{\lambda}(y^{\ast},Ty^{\ast},Ty^{\ast})
+\omega^{G}_{\lambda}(T^{n+1}z,T^{n+1}z,y^{\ast})]}
{1+\omega^{G}_{\lambda}(T^{n+1}z,T^{n+1}z,y^{\ast})
+\omega^{G}_{\lambda}(T^{n+1}z,Ty^{\ast},Ty^{\ast})}},
\end{align
	
using the fact that $Ty^{\ast}=y^{\ast}$, we get

\begin{align}\label{C
\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\notag
\le&\kappa_{1}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda+\nu(\lambda)}(T^{n}z,y^{\ast},y^{\ast}))\\ \notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,T^{n+1}z,T^{n+1}z))\\\notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\\ \notag
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\\\notag
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,T^{n+1}z,T^{n+1}z))\\\notag
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega_{\lambda}(T^{n}z,y^{\ast},y^{\ast}))\\\notag
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\\
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})).
\end{align

Now, applying condition 6 of Proposition \ref{2.2}, we get

\begin{align
\omega^{G}_{\lambda}(T^{n}z,T^{n+1}z,T^{n+1}z)\notag 
\le&\omega^{G}_{\frac{\lambda}{2}}(T^{n}z,y^{\ast},y^{\ast})
+\omega^{G}_{\frac{\lambda}{4}}(T^{n+1}z,y^{\ast},y^{\ast})
+\omega^{G}_{\frac{\lambda}{4}}(T^{n+1}z,y^{\ast},y^{\ast})\\\notag
=&\omega^{G}_{\frac{\lambda}{2}}(T^{n}z,y^{\ast},y^{\ast})
+2\omega^{G}_{\frac{\lambda}{4}}(T^{n+1}z,y^{\ast},y^{\ast})\\
\le& \omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})
+2\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}).
\end{align

Therefore, from inequality \ref{C} and 
that $\psi$(.) is sub-additive, we have
	
\begin{align}\label{}
\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\notag
\le&\kappa_{1}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast}))\\ \notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast}))
+2\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\\ \notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\\ \notag
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\\\notag
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})
+2\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\\\notag
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega_{\lambda}(T^{n}z,y^{\ast},y^{\ast}))
\\\notag
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\\\notag
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast}))\\\notag
\le&\kappa_{1}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast}))\\ \notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
+2\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\\ \notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\\ \notag
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\\\notag
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast}))
+2\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast})))\\\notag
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega_{\lambda}(T^{n}z,y^{\ast},y^{\ast}))
\\\nota
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\\\notag
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast}))\\\notag
=&\kappa_{1}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast}))\\\notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast}))\\\notag
+&2\kappa_{2}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast})\\ \notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\\ \notag
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\\\notag
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast}))\\\notag
+&2\kappa_{8}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\\\notag
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega_{\lambda}(T^{n}z,y^{\ast},y^{\ast}))
\\\nota
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))\\
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})))
\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast})),
\end{align

so that
	
\begin{eqnarray
&&\bigg(1-\bigg(2\kappa_{2}(\psi(\omega^{G}_{
\lambda}(T^{n}z,y^{\ast},y^{\ast})))\notag
+\kappa_{4}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))\notag
+\kappa_{5}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))\\
&&+2\kappa_{8}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))\notag
+\kappa_{10}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))\bigg)\bigg)\notag
\psi(\omega^{G}_{\lambda}(T^{n+1}z,
 y^{\ast},y^{\ast}))\\\notag
\end{eqnarray

\begin{eqnarray
\le\notag
&&\bigg(\kappa_{1}(\psi(
\omega^{G}_{\lambda}(T^{n},
y^{\ast},y^{\ast})))\notag
+\kappa_{2}(\psi(\omega^{G}_{
\lambda}(T^{n}z,y^{\ast},y^{\ast})))\notag
+\kappa_{8}(\psi(\omega^{G}_{
\lambda}(T^{n}z,y^{\ast},y^{\ast})))\\
&&+\kappa_{9}(\psi(\omega^{G}_{
\lambda}(T^{n}z,y^{\ast},y^{\ast})))
+\kappa_{11}(\psi(\omega^{G}_{
\lambda}(T^{n}z,y^{\ast},y^{\ast})))\bigg)
\psi(\omega^{G}_{\lambda}(T^{n}z,
y^{\ast},y^{\ast})).
\end{eqnarray
	
Therefore,

\begin{align}
\psi(\omega^{G}_{\lambda}(
T^{n+1}z,y^{\ast},y^{\ast}))\notag
\le& \tau\psi(\omega^{G}_{
\lambda}(T^{n}z,y^{\ast},y^{\ast}))\\\notag
\le&\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast}))\\\notag
\le&\psi(\omega^{G}_{\lambda}
(T^{n-1}z,y^{\ast},y^{\ast}))\\\notag
\vdots&\\\notag
\le&\psi(\omega^{G}_{\lambda}(z,y^{\ast},y^{\ast}))\\
< &\infty,
\end{align

where $\tau=\dfrac{\tau_{1}}{\tau_{2}}.$

\begin{eqnarray
\tau_{1}:=&& 1-\bigg(2\kappa_{2}(
\psi(\omega^{G}_{\lambda}(T^{n}z,
y^{\ast},y^{\ast})))\notag
+\kappa_{4}(\psi(\omega^{G}_{
\lambda}(T^{n}z,y^{\ast},y^{\ast})))\notag
+\kappa_{5}(\psi(\omega^{G}_{
\lambda}(T^{n}z,y^{\ast},y^{\ast})))\\
&&+2\kappa_{8}(\psi(\omega^{G}_{
\lambda}(T^{n}z,y^{\ast},y^{\ast})))\notag
+\kappa_{10}(\psi(\omega^{G}_{
\lambda}(T^{n}z,y^{\ast},y^{\ast})))\bigg),
\end{eqnarray

and

\begin{eqnarray}
\tau_{2}:=&&\bigg(\kappa_{1}(
\psi(\omega^{G}_{\lambda}(T^{n},
y^{\ast},y^{\ast})))\notag
+\kappa_{2}(\psi(\omega^{G}_{
\lambda}(T^{n}z,y^{\ast},y^{\ast})))\notag
+\kappa_{8}(\psi(\omega^{G}_{
\lambda}(T^{n}z,y^{\ast},y^{\ast})))\\
&&+\kappa_{9}(\psi(\omega^{G}_{
\lambda}(T^{n}z,y^{\ast},y^{\ast})))
+\kappa_{11}(\psi(\omega^{G}_{
\lambda}(T^{n}z,y^{\ast},y^{\ast})))\bigg).
\end{eqnarray
	
Therefore, $\{\psi(\omega^{G}_{\lambda}
(T^{n+1}z,y^{\ast},y^{\ast}))\}_{n\ge 1}$
is non-increasing sequence and bounded below.
So, it  converges to some real number 
$\sigma\ge 0$ (say). For $\sigma>0$. Recall that 
$\lim\limits_{n\rightarrow\infty}\omega^{G}_{
\lambda}(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})=0 $
for all $\lambda>0$. Because the sequence $\{T^{n}x_{0}\}_{n\in\mathbb{N}}$
is asymptotically $T$-regular on some point $x_{0}\in X_{\omega^{G}}.$
Thus, letting $n\rightarrow
\infty$ in the inequality \ref{C},
and using the fact that $\psi(\omega^{G}_{
\lambda}(T^{n}z,y^{\ast},y^{\ast}))\leq 
\psi(\omega^{G}_{\lambda}(T^{n+1}z,y^{\ast},y^{\ast}))$,
we have  that
\begin{align
1\notag
&\le\lim\limits_{n\rightarrow\infty}\inf\biggl\{
\kappa_{1}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))\notag
+\kappa_{4}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))\notag
+\kappa_{5}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))\\\notag
&+\kappa_{9}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))\notag
+\kappa_{10}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))
+\kappa_{11}(\psi(\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast})))\biggr\}.
\end{align

Therefore, $\lim\limits_{n\rightarrow\infty}
\psi(\omega^{G}_{\lambda}(T^{n}z,y^{\ast},y^{\ast}))=0$,
for which by condition 4 of Proposition
\ref{2.3}, we have that

\begin{equation}
\lim\limits_{n\rightarrow\infty}\omega^{G}_{\lambda}
(T^{n}z,y^{\ast},y^{\ast}))=0 ~\forall~\lambda>0.
\end{equation

Hence $T^{n}z\rightarrow y^{\ast}$ as
$n\rightarrow\infty$.
\newlin
(f)
Now, suppose, if possible, that
$\lim\limits_{n\rightarrow\infty}T^{n}z$
exists and not unique.
Let $\lim\limits_{n\rightarrow\infty}T^{n}z=x^{\ast}$
and $\lim\limits_{n\rightarrow\infty}T^{n}z
=y^{\ast}$ as we have seen above,
where $x^{\ast}\neq y^{\ast}$.
For each $\lambda>0$, $x^{\ast}\neq y^{\ast}
\Rightarrow\omega^{G}_{\lambda}(x^{\ast},
y^{\ast},y^{\ast})>0$. If we take $\psi(
\epsilon_{1})=\frac{1}{3}\psi(\omega_{\lambda}(x^{\ast},
y^{\ast},y^{\ast}))>0,$
then for $\lambda>0$,
$\lim\limits_{n\rightarrow\infty}T^{n}z
=x^{\ast}\Rightarrow$ given $\epsilon_{1}>0,
~\exists~ m_{1}\in\mathbb{N}$ such that 
$\psi(\omega^{G}_{\frac{\lambda}{2}}(x^{\ast},T^{n}z,T^{n}z))
<\psi(\epsilon_{1})$ for $n>m_{1}$.
Again,
$\lim\limits_{n\rightarrow\infty}T^{n}z=y^{\ast}
\Rightarrow$ given $\epsilon_{1}>0,\exists~ m_{2}
\in\mathbb{N}$ such that $\psi(\omega^{G}_{\frac{
\lambda}{4}}(y^{\ast},T^{n}z,T^{n}z))<\psi(\epsilon_{1})$
for $n>m_{2}$.
Set $m=\max\{m_{1},m_{2}\}$, then for $n\geq m$,
by condition 6 of Proposition \ref{2.2}, we have

\begin{eqnarray
\psi(\omega^{G}_{\lambda}(x^{\ast},
y^{\ast},y^{\ast}))\notag
&\le&\psi(\omega^{G}_{\frac{\lambda}{2}}
(x^{\ast},T^{n}z,T^{n}z)
+2\omega^{G}_{\frac{\lambda}{4}}
(y^{\ast},T^{n}z,T^{n}z))\\\notag
&\le&\psi(\omega^{G}_{\frac{\lambda}{2}}
(x^{\ast},T^{n}z,T^{n}z))
+2\psi(\omega^{G}_{\frac{\lambda}{4}}
(y^{\ast},T^{n}z,T^{n}z))\\\notag
&<&\psi(\epsilon_{1})+2\psi(\epsilon_{1})\\\notag
&=&3\psi(\epsilon_{1}),
\end{eqnarray

which we can see clearly that $\psi(\omega^{G}_{\lambda}
(x^{\ast}, y^{\ast},y^{\ast}))<
\psi(\omega^{G}_{\lambda}(x^{\ast}, y^{\ast},y^{\ast}))$
for all $\lambda>0$. This is a contradiction.
Hence, $x^{\ast}= y^{\ast}$.
Therefore the fixed point of $T$ is unique and
 hence, the proof is completed.
		
\end{proof

\begin{corollary}\label{aaa
Let $(X,\omega^{G},\preceq)$ be 
a complete preordered
modular G-metric space.
Let $T:X_{\omega^{G}}\rightarrow X_{\omega^{G}}$
be a non-decreasing, asymptotically $T$-regular map on
some point $x_{0}\in X_{\omega^{G}}$
such that for each $\lambda>0$,
there is $\nu(\lambda)\in
\lbrack 0,\lambda)$ such that
the following conditions hold:
\begin{enumerate
\item[(1)]
\begin{align}\label{bb}
\psi(\omega^{G}_{\lambda}(Tx,Ty,Ty))\notag
\le&
\kappa_{1}(\psi(\omega^{G}_{\lambda}(x,y,y)))
\psi(\omega^{G}_{\lambda+\nu(\lambda)}(x,y,y))\\ \notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}(x,y,y)))
\psi(\omega^{G}_{\lambda}(x,Tx,Tx))\\
+& \kappa_{3}(\psi(\omega^{G}_{\lambda}(x,y,y)))
\psi(\omega^{G}_{\lambda}(y,Ty,Ty)),
\end{align
where $\psi\in \bar{\Psi}$ and $\{\kappa_{1},\kappa_{2},
\kappa_{3}\}\in \mathcal{S}_{Ger}$ with $\kappa_{1}(t)
+2\max\{\sup_{t\geq 0}\kappa_{2}(t), \sup_{t\geq 0}
\kappa_{3}(t)\}<1,$
and distinct $x,y\in X_{\omega^{G}}.$
Assume that if a non-decreasing sequence $\{x_{n}\}_{n\in
\mathbb{N}}$ converges to $x^{\ast}$,
then $x_{n}\preceq x^{\ast}$ for each $n\in\mathbb{N}$;

\item[(2)
if $\psi$ is sub-additive and for any $x,y\in X_{\omega^{G}}$,
there exists $z\in X_{\omega^{G}}$ with $z\preceq Tz$ and
$\omega^{G}_{\lambda}(z,Tz,Tz)$ is finite  for all $\lambda>0$
such that $z$ is comparable to both $x~{\rm and}~y$.
Then $T$ has a fixed point in $x^{\ast}\in X_{\omega^{G}}$ and
the sequence define by $\{T^{n}x_{0}\}_{n\ge 1}$
converges to $x^{\ast}$.
Furthermore, the fixed point of $T$ is unique.
\end{enumerate
\end{corollary

\begin{proof}
Since $\{\kappa_{1},\kappa_{2}, \kappa_{3}\}\in
\mathcal{S}_{Ger}$,
then by \autoref{3.1}, 
fixed point of $T$ is unique.
	
\end{proof

We have the following variant form on \autoref{3.1}

\begin{theorem}\label{3.2
Let $(X,\omega^{G},\preceq)$ be a
complete preordered modular G-metric space.
Let $T:X_{\omega^{G}}\rightarrow X_{\omega^{G}}$
be a non-decreasing, asymptotically $T$-regular map on
some point $x_{0}\in X_{\omega^{G}}$
such that for each $\lambda>0$ and  
for some positive integer $m\ge 1$,
there is $\nu(\lambda)\in [0,\lambda)$ 
such that the following conditions hold:
\begin{align}\label{7
(1)~~ f_{m}\notag
\le&
\kappa_{1}(\psi(\omega^{G}_{\lambda}(x,y,y)))
\psi(\omega^{G}_{\lambda+\nu(\lambda)}(x,y,y)) \notag
+\kappa_{2}(\psi(\omega^{G}_{\lambda}(x,y,y)))
\psi(\omega^{G}_{\lambda}(x,T^{m}x,T^{m}x))\\ \notag
+& \kappa_{3}(\psi(\omega^{G}_{\lambda}(x,y,y)))
\psi(\omega^{G}_{\lambda}(y,T^{m}y,T^{m}y)) \notag
+\kappa_{4}(\psi(\omega^{G}_{\lambda}(x,y,y)))
\psi(\omega^{G}_{\lambda}(T^{m}x,T^{m}y,T^{m}y))\\ \notag
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}(x,y,y)))
\psi(\omega^{G}_{\lambda}(T^{m}x,y,y)) \notag
+\kappa_{6}(\psi(\omega^{G}_{\lambda}(x,y,y)))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(y,T^{m}y,T^{m}y)
[1+\omega^{G}_{\lambda}(x,y,y)]}{1
+\omega^{G}_{\lambda}(x,T^{m}x,T^{m}x)}}\\ \notag
+&\kappa_{7}(\psi(\omega^{G}_{\lambda}(x,y,y)))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(y,T^{m}y,T^{m}y)
[1+\omega^{G}_{\lambda}(x,T^{m}x,T^{m}x)]}{1
+\omega^{G}_{\lambda}(x,y,y)}}\\ \notag
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}(x,y,y)))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x,T^{m}x,T^{m}x)
[1+\omega^{G}_{\lambda}(x,y,y)]}{1+
\omega^{G}_{\lambda}(y,T^{m}x,T^{m}x)
+\omega^{G}_{\lambda}(x,y,y)}}\\ \notag
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}(x,y,y)))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x,y,y)
[1+\omega^{G}_{\lambda}(y,T^{m}y,T^{m}y)]}
{1+\omega^{G}_{\lambda}(T^{m}x,T^{m}x,y)
+\omega^{G}_{\lambda}(T^{m}x,T^{m}y,T^{m}y)}}\\ \notag
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}(x,y,y)))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(T^{m}x,T^{m}y,T^{m}y)
[1+\omega^{G}_{\lambda}(x,T^{m}y,T^{m}y)]}
{1+\omega^{G}_{\lambda}(T^{m}x,T^{m}x,y)
+\omega^{G}_{\lambda}(x,T^{m}y,T^{m}y)}}\\ \notag
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}(x,y,y)))\\
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(T^{m}x,T^{m}y,T^{m}y)
[1+\omega^{G}_{\lambda}(y,T^{m}y,T^{m}y)
+\omega^{G}_{\lambda}(T^{m}x,T^{m}x,y)]}
{1+\omega^{G}_{\lambda}(T^{m}x,T^{m}x,y)
+\omega^{G}_{\lambda}(T^{m}x,T^{m}y,T^{m}y)}},
\end{align

where $f_{m}=\psi(\omega^{G}_{\lambda}(T^{m}x,T^{m}y,T^{m}y))$
and $\psi\in \bar{\Psi}$ , $\{\kappa_{1},\kappa_{2}, \cdots,
\kappa_{11}\}\in \mathcal{S}_{Ger}$ and
\newlin
$\max\{\sup_{t\geq 0}\kappa_{1}(t),\sup_{t\geq 0}\kappa_{2}(t), \cdots,
\sup_{t\geq 0}\kappa_{11}(t)\}<1.$
Assume that if a non-decreasing sequence $\{x_{n}\}_{n\in\mathbb{N}}$
converges to $x^{\ast}$, then $x_{n}\preceq x^{\ast}$ for
each $n\in\mathbb{N}$,
 \newlin
$(2)$ if $\psi$ is sub-additive and for any $x,y\in X_{\omega^{G}}$,
there exists $z\in X_{\omega^{G}}$ with $z\preceq Tz$ and
$\omega^{G}_{\lambda}(z,Tz,Tz)$ is finite  for all $\lambda>0$ such
that $z$ is comparable to both $x~{\rm and}~y$.
Then $T$ has a  fixed point $x^{\ast}\in X_{\omega^{G}}$ for
some positive integer $m\ge 1$ in $X_{\omega^{G}}$
and the sequence define by $\{T^{n}x_{0}\}_{n\ge 1}$ 
converges to $x_{\ast}$.
Moreover the fixed point of $T$ is unique.
\end{theorem

\begin{proof}
By \autoref{3.1}, $T^{m}$ has a  fixed point say
$u_{\ast}\in X_{\omega^{G}}$ for some positive
integer $m\ge 1$ by using inequality \ref{7}.
Now $T^{m}(Tu_{\ast})=T^{m+1}u_{\ast}=T(T^{m}
u_{\ast})=Tu_{\ast}$,
so $Tu_{\ast}$ is a fixed point of $T^{m}$.
By the uniqueness of fixed point of $T^{m}$,
we have $Tu_{\ast}=u_{\ast}$. Therefore, $u_{\ast}$
is a fixed point of $T$. Since fixed point of $T$ is
also fixed point of $T^{m}$, hence $T$ has a unique
fixed point in $X_{\omega^{G}}$.

\end{proof





\begin{theorem}\label{qq
Let $(X,\omega^{G},\preceq)$ be a complete
preordered modular G-metric space.
Let $T:X_{\omega^{G}}\rightarrow X_{\omega^{G}}$
be a non-decreasing, asymptotically $T$-regular map on
some point $x_{0}\in X_{\omega^{G}}$
such that for each $\lambda>0$ and  for 
some positive integer $m\ge 1$,
there is $\nu(\lambda)\in [0,\lambda)$ 
such that the following conditions hold:
\begin{align}\label{aa
(1)~\psi(\omega^{G}_{\lambda}(Tx,Ty,Tz))\notag
\le&
\kappa_{1}(\psi(\omega^{G}_{\lambda}(x,y,z)))
\psi(\omega^{G}_{\lambda+\nu(\lambda)}(x,y,z)) \nota
+\kappa_{2}(\psi(\omega^{G}_{\lambda}(x,y,z)))
\psi(\omega^{G}_{\lambda}(x,Tx,Tx))\\ \notag
+& \kappa_{3}(\psi(\omega^{G}_{\lambda}(x,y,z)))
\psi(\omega^{G}_{\lambda}(y,Ty,Ty)) \notag
+\kappa_{4}(\psi(\omega^{G}_{\lambda}(x,y,z)))
\psi(\omega^{G}_{\lambda}(z,Tz,Tz))\\ \notag
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}(x,y,z))
\psi(\omega^{G}_{\lambda}(Tx,y,z)) \notag
+\kappa_{6}(\psi(\omega^{G}_{\lambda}(x,y,z)))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(y,Ty,Ty)
[1+\omega^{G}_{\lambda}(x,y,z)]}{1+\omega^{G}_{
\lambda}(x,Tx,Tx)}}\\ \notag
+&\kappa_{7}(\psi(\omega^{G}_{\lambda}(x,y,z)))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(y,Ty,Ty)
[1+\omega^{G}_{\lambda}(x,Tx,Tx)]}{1+\omega^{G}_{
\lambda}(x,y,z)}}\\ \notag
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}(x,y,z)))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x,Tx,Tx)
[1+\omega^{G}_{\lambda}(x,y,z)]}{1+\omega^{G}_{
\lambda}(y,Tx,Tx)
+\omega^{G}_{\lambda}(x,y,z)}}\\ \notag
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}(x,y,z)))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x,y,z)
[1+\omega^{G}_{\lambda}(y,Ty,Ty)]}{1+\omega^{G}_{
\lambda}(Tx,Tx,y)
+\omega^{G}_{\lambda}(Tx,Ty,Ty)}}\\ \notag
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}(x,y,z)))\\\notag
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(Tx,Ty,Ty)
[1+\omega^{G}_{\lambda}(x,Ty,Ty)]}{1+\omega^{G}_{
\lambda}(Tx,Tx,y)
+\omega^{G}_{\lambda}(x,Ty,Ty)}}\\ \notag
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}(x,y,z)))\\
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(Tx,Ty,Ty)
[1+\omega^{G}_{\lambda}(y,Ty,Ty)+\omega^{G}_{
\lambda}(Tx,Tx,y)]}
{1+\omega^{G}_{\lambda}(Tx,Tx,y)+\omega^{G}_{
\lambda}(Tx,Ty,Ty)}},
\end{align

where $\psi\in \bar{\Psi}$ , $\{\kappa_{1},
\kappa_{2}, \cdots, \kappa_{11}\}
\in \mathcal{S}_{Ger}$ and
$\max\{\sup_{t\geq 0}\kappa_{1}(t),
\sup_{t\geq 0}\kappa_{2}(t), \cdots,
\sup_{t\geq 0}\kappa_{11}(t)\}<1.$
Assume that if a non-decreasing sequence
$\{x_{n}\}_{n\in\mathbb{N}}$
converges to $x^{\ast}$, then $x_{n}\preceq x^{\ast}$
 for each $n\in\mathbb{N}$,
 \newline
$(2)$ if $\psi$ is sub-additive and for any
$x,y,z\in X_{\omega^{G}}$,
there exists $u\in X_{\omega^{G}}$ with
$u\preceq Tu$ and
$\omega^{G}_{\lambda}(u,Tu,Tu)$ is finite 
for all $\lambda>0$
such that $u$ is comparable to  $x,y~{\rm and}~z$.
Then $T$ has a fixed point in $ x_{\ast}
\in X_{\omega^{G}}$
and the sequence define by $\{T^{n}x_{0}\}_{n\ge 1}$
 converges to $x_{\ast}$.
Moreover the fixed point of $T$ is unique.
\end{theorem


\begin{proof}
(a)
Let $x_{0}\in X_{\omega^{G}}$ be such that
 $x_{0}\preceq Tx_{0}$ and let
$x_{n}=Tx_{n-1}=T^{n}x_{0}$ for all $n\in 
\mathbb{N}.$ Regarding that $T$ is
non-decreasing mapping, we have that $x_{0}
\preceq Tx_{0}=x_{1}$,
which implies that $x_{1}=Tx_{0}\preceq
Tx_{1}=x_{2}$. Inductively,
we have
\begin{equation}\label{pp
x_{0}\preceq x_{1}\preceq x_{2}\preceq \cdots \preceq x_{n-1}
\preceq x_{n}\preceq x_{n+1}\preceq \cdots.
\end{equation
	
Assume that there exists $n_{0}\in\mathbb{N}$
such that $x_{n_{0}}=x_{n_{0}+1}$. Since
$x_{n_{0}}=x_{n_{0}+1}=Tx_{n_{0}}$, then
$x_{n_{0}}$ is the fixed point of $T$.
Now suppose that $x_{n} \precneqq x_{n+1}$ for
all $n\in\mathbb{N}$, thus  by inequalit
\ref{pp}, we have that
\begin{equation}\label{ppp
x_{0}\prec x_{1}\prec x_{2}\prec \cdots
\prec x_{n-1}\prec x_{n}\prec x_{n+1}\prec \cdots.
\end{equation
Now for each $\lambda>0,$ and $x_{0}\prec
Tx_{0}$ for all $n\in\mathbb{N}$
implies that $\omega^{G}_{\lambda}(x_{0},Tx_{0},Tx_{0})>0.$
Again, let $x_{0}\in X_{\omega^{G}}$  such that
$\omega^{G}_{\lambda}(x_{0},Tx_{0},Tx_{0})
<\infty~\forall~\lambda>0$. 
\newlin
(b)	
Next, we show  that for all $n\in \mathbb{N}$, 
$\omega^{G}_{\lambda}(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})= 0$
for all $\lambda>0$, as  $n\rightarrow\infty$. This means that
 the sequence $\{T^{n}x_{0}\}_{n\in\mathbb{N}}$ is asymptotically
 $T$-regular on some point $x_{0}\in X_{\omega^{G}}$. Assume that,
for each $n\in\mathbb{N}$, there exists $\lambda_{n}>0$ such that
$\omega^{G}_{\lambda_{n}}(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})
\neq 0$.
Otherwise we are done. For each $n\ge 1$, if $0<\lambda<\lambda_{n}$,
then we have $\omega^{G}_{\lambda}(T^{n}x_{0},
T^{n}x_{0},T^{n+1}x_{0})\neq 0$.
Since $T^{n}x_{0}\preceq T^{n+1}x_{0}$, we have from condition (1)
of \autoref{qq}, then we have that $\psi(\omega^{G}_{\lambda_{n}}(
T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0}))\le \psi(\omega^{G}_{\lambda}(
T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0}))
=\psi(\omega^{G}_{\lambda}(TT^{n-1}x_{0},TT^{n}x_{0},TT^{n}x_{0}))$.
Take $x=T^{n-1}x_{0}$  and $ y=T^{n}x_{0}=z$,
by inequality \ref{aa}, we see that  $\psi(\omega^{G}_{\lambda}(
T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0}))=0.$  On taking
limit, we have $\omega^{G}_{\lambda}(T^{n}x_{0},
T^{n+1}x_{0},T^{n+1}x_{0})=0$
as $n\rightarrow \infty.$ In fact by
\autoref{3.1}, $T$ has a unique
fixed in $X_{\omega^{G}}.$
	
\end{proof


\begin{corollary}\label{n
Let $(X,\omega^{G},\preceq)$ be a complete 
preordered modular G-metric space.
Let $T:X_{\omega^{G}}\rightarrow X_{\omega^{G}}$
be a non-decreasing, asymptotically $T$-regular map
on some point $x_{0}\in X_{\omega^{G}}$ 
such that for each $\lambda>0$, there is
$\nu(\lambda)\in [0,\lambda)$ such that the following
conditions hold:
	
\begin{align}\label{o}
(1) \psi(\omega^{G}_{\lambda}(Tx,Ty,Tz))\notag
\le
\kappa_{1}(\psi(\omega^{G}_{\lambda}(x,y,z)))
\psi(\omega^{G}_{\lambda+\nu(\lambda)}(x,y,z)) \notag
+\kappa_{2}(\psi(\omega^{G}_{\lambda}(x,y,z)))
\psi(\omega^{G}_{\lambda}(x,Tx,Tx))\\
+& \kappa_{3}(\psi(\omega^{G}_{\lambda}(x,y,z)))
\psi(\omega^{G}_{\lambda}(y,Ty,Ty))
+\kappa_{4}(\psi(\omega^{G}_{\lambda}(x,y,z)))
\psi(\omega^{G}_{\lambda}(z,Tz,Tz)),
\end{align

where $\psi\in \bar{\Psi}$ and
$\{\kappa_{1},\kappa_{2}, \kappa_{3},\kappa_{4}\}\in \mathcal{S}_{Ger}$ with
$\kappa_{1}(t)+2\max\{\sup_{t\geq 0}\kappa_{2}(t),
\sup_{t\geq 0}\kappa_{3}(t),\\\sup_{t\geq 0}\kappa_{4}(t)\}<1,$
and $x,y,z\in X_{\omega^{G}}.$
Assume that if a non-decreasing sequence $\{x_{n}\}_{n\in\mathbb{N}}$
converges to $x$, then $x_{n}\preceq x$ for each $n\in\mathbb{N}$;

(2)~ if $\psi$ is sub-additive and for any $x,y,z\in X_{\omega^{G}}$,
there exists $w\in X_{\omega^{G}}$ with $w\preceq Tw$ and
$\omega^{G}_{\lambda}(w,Tw,Tw)$ is finite  for all $\lambda>0$
such that $w$ is comparable to both $x~{\rm ,}~y~ and~z$.
Then $T$ has a fixed point in $x\in X_{\omega^{G}}$ and the
sequence define by $\{T^{n}x_{0}\}_{n\ge 1}$ converges to $x$.
Moreover the fixed point of $T$ is unique.
\end{corollary
	
\begin{proof}
(a)
Let $x_{0}\in X_{\omega^{G}}$ be such that
$x_{0}\preceq Tx_{0}$ and let
$x_{n}=Tx_{n-1}=T^{n}x_{0}$ for all $n\in \mathbb{N}.$
Regarding that $T$ is
non-decreasing mapping, we have that $x_{0}\preceq Tx_{0}=x_{1}$,
which implies that $x_{1}=Tx_{0}\preceq Tx_{1}=x_{2}$.
Inductively,
 we have
\begin{equation}\label{3
x_{0}\preceq x_{1}\preceq x_{2}\preceq \cdots \preceq x_{n-1}
\preceq x_{n}\preceq x_{n+1}\preceq \cdots.
\end{equation
	
Assume that there exists $n_{0}\in\mathbb{N}$
such that $x_{n_{0}}=x_{n_{0}+1}$. Since
$x_{n_{0}}=x_{n_{0}+1}=Tx_{n_{0}}$, then
$x_{n_{0}}$ is the fixed point of $T$.
Now suppose that $x_{n} \precneqq x_{n+1}$ for
all $n\in\mathbb{N}$, thus  by inequalit
\ref{3}, we have that
\begin{equation}\label{4
x_{0}\prec x_{1}\prec x_{2}\prec \cdots
\prec x_{n-1}\prec x_{n}\prec x_{n+1}\prec \cdots.
\end{equation
Now for each $\lambda>0,$ and $x_{0}\prec
Tx_{0}$ for all $n\in\mathbb{N}$
implies that $\omega^{G}_{\lambda}(x_{0},Tx_{0},Tx_{0})>0.$
Again, let $x_{0}\in X_{\omega^{G}}$  such that
$\omega^{G}_{\lambda}(x_{0},Tx_{0},Tx_{0})
<\infty~\forall~\lambda>0$. 
\newlin
(b)	
Now, we show  that for all $n\in \mathbb{N}$, 
$\omega^{G}_{\lambda}(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})= 0$
for all $\lambda>0$, as  $n\rightarrow\infty$. It implies that
the sequence $\{T^{n}x_{0}\}_{n\in\mathbb{N}}$ is asymptotically
$T$-regular on some point $x_{0}\in X_{\omega^{G}}$. Assume that,
for each $n\in\mathbb{N}$, there exists $\lambda_{n}>0$ such that
$\omega^{G}_{\lambda_{n}}(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0})
\neq 0$.
Otherwise we are done. For each $n\ge 1$, if $0<\lambda<\lambda_{n}$,
then we have $\omega^{G}_{\lambda}(T^{n}x_{0},T^{n}x_{0},T^{n+1}x_{0})\neq 0$.
Since $T^{n}x_{0}\preceq T^{n+1}x_{0}$, we have from inequality \ref{o} that
$\psi(\omega^{G}_{\lambda_{n}}(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0}))\le 
\psi(\omega^{G}_{\lambda}(T^{n}x_{0},T^{n+1}x_{0},T^{n+1}x_{0}))=
\psi(\omega^{G}_{\lambda}(TT^{n-1}x_{0},TT^{n}x_{0},TT^{n}x_{0}))$.
Take $x=T^{n-1}x_{0}$  and $ y=T^{n}x_{0}=z$, then inequality \ref{o}
becomes
\begin{align}\label{p
\psi(\omega^{G}_{\lambda_{n}}(T^{n}x_{0},
T^{n+1}x_{0},T^{n+1}x_{0}))\notag
\le&
\psi(\omega^{G}_{\lambda}(T^{n}x_{0},
T^{n+1}x_{0},T^{n+1}x_{0}))\\\notag
\le&
\kappa_{1}(\psi(\omega^{G}_{\lambda}(
T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\notag
\psi(\omega^{G}_{\lambda+\nu(\lambda)}(
T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0}))\\ \notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\notag
\psi(\omega^{G}_{\lambda}(T^{n-1}x_{0},
TT^{n-1}x_{0},TT^{n-1}x_{0}))\\\notag
+& \kappa_{3}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\notag
\psi(\omega^{G}_{\lambda}(T^{n}x_{0},
TT^{n}x_{0},TT^{n}x_{0}))\\\notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\notag
\psi(\omega^{G}_{\lambda}(T^{n}x_{0},
TT^{n}x_{0},TT^{n}x_{0}))\\\notag
=& 
\kappa_{1}(\psi(\omega^{G}_{\lambda}(
T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\notag
\psi(\omega^{G}_{\lambda+\nu(\lambda)}(
T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0}))\\ \notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}(
T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\notag
\psi(\omega^{G}_{\lambda}(T^{n-1}x_{0},
T^{n}x_{0},T^{n}x_{0}))\\\notag
+& \kappa_{3}(\psi(\omega^{G}_{\lambda}(
T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\notag
\psi(\omega^{G}_{\lambda}(T^{n}x_{0},
T^{n+1}x_{0},T^{n+1}x_{0}))\\\notag
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}(
T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\notag
\psi(\omega^{G}_{\lambda}(T^{n}x_{0},
T^{n+1}x_{0},T^{n+1}x_{0}))\\\notag
\le&
\kappa_{1}(\psi(\omega^{G}_{\lambda}(
T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\notag
\psi(\omega^{G}_{\lambda}(T^{n-1}x_{0},
T^{n}x_{0},T^{n}x_{0}))\\ \notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}(
T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\notag
\psi(\omega^{G}_{\lambda}(T^{n-1}x_{0},
T^{n}x_{0},T^{n}x_{0}))\\\notag
+& \kappa_{3}(\psi(\omega^{G}_{\lambda}(
T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))\notag
\psi(\omega^{G}_{\lambda}(T^{n}x_{0},
T^{n+1}x_{0},T^{n+1}x_{0}))\\
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}(
T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))
\psi(\omega^{G}_{\lambda}(T^{n}x_{0},
T^{n+1}x_{0},T^{n+1}x_{0})),
\end{align

which implies that

\begin{align}\label{q}
\psi(\omega^{G}_{\lambda}(T^{n}x_{0},
T^{n+1}x_{0},T^{n+1}x_{0}))\notag
\le&h\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0}))\\\notag
\le&
\psi(\omega^{G}_{\lambda}(T^{n-1}x_{0},
T^{n}x_{0},T^{n}x_{0}))\\\notag
\vdots\\\notag
\le&
\psi(\omega^{G}_{\lambda}(x_{0},Tx_{0},Tx_{0}))\\
<&\infty,
\end{align
where
\begin{equation
h:=\dfrac{\kappa_{1}(\psi(\omega^{G}_{\lambda
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))
+\kappa_{2}(\psi(\omega^{G}_{\lambda}
(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0})))}{1-(\kappa_{3}
(\psi(\omega^{G}_{\lambda}(T^{n-1}x_{0},
T^{n}x_{0},T^{n}x_{0})))+\kappa_{4}(\psi(
\omega^{G}_{\lambda}(T^{n-1}x_{0},T^{n}x_{0},T^{n}x_{0}))))}.
\end{equation}
Therefore, $\{\psi(\omega^{G}_{\lambda}(T^{n}x_{0},
T^{n+1}x_{0},T^{n+1}x_{0}))\}_{n\ge 1}$ is non-increasing 
and bounded below. So, the sequence converges to some 
real number $M\ge 0$, by \autoref{3.1}, the proof is completed.
		
	
\end{proof
	
\begin{theorem}\label{nn
Let $(X,\omega^{G},\preceq)$ be a complete preordered modular
G-metric space.
Let $T:X_{\omega^{G}}\rightarrow X_{\omega^{G}}$
be a non-decreasing, asymptotically $T$-regular map on
some point $x_{0}\in X_{\omega^{G}}
such that for each $\lambda>0$ and some positive integer $m\ge 1$,
there is $\nu(\lambda)\in [0,\lambda)$ such that the 
following conditions hold:
\begin{enumerate
\item[(1)]	
\begin{align}\label{oo
\psi(\omega^{G}_{\lambda}(T^{m}x,T^{m}y,T^{m}z))\notag
\le&
\kappa_{1}(\psi(\omega^{G}_{\lambda}(x,y,z)))
\psi(\omega^{G}_{\lambda+\nu(\lambda)}(x,y,z))\\ \notag
+&\kappa_{2}(\psi(\omega^{G}_{\lambda}(x,y,z)))
\psi(\omega^{G}_{\lambda}(x,T^{m}x,T^{m}x))\\\notag
+& \kappa_{3}(\psi(\omega^{G}_{\lambda}(x,y,z)))
\psi(\omega^{G}_{\lambda}(y,T^{m}y,T^{m}y))\\
+&\kappa_{4}(\psi(\omega^{G}_{\lambda}(x,y,z)))
\psi(\omega^{G}_{\lambda}(z,T^{m}z,T^{m}z)),
\end{align
where $\psi\in \bar{\Psi}$ and $\{\kappa_{1},
\kappa_{2}, \kappa_{3},\kappa_{4}\}\in \mathcal{S}_{Ger}$ with
$\kappa_{1}(t)+2\max\{\sup_{t\geq 0}\kappa_{2}(t),
\sup_{t\geq 0}\kappa_{3}(t),\\\sup_{t\geq 0}\kappa_{4}(t)\}<1,$
and $x,y,z\in X_{\omega^{G}}.$
Assume that if a non-decreasing sequence
$\{x_{n}\}_{n\in\mathbb{N}}$ converges to $x$, then $x_{n}
\preceq x$ for each $n\in\mathbb{N}$;
\item[(2)] if $\psi$ is sub-additive and for any
$x,y,z\in X_{\omega^{G}}$, there exists $w\in X_{\omega^{G}}$
with $w\preceq Tw$ and $\omega^{G}_{\lambda}(w,Tw,Tw)$ is
finite  for all $\lambda>0$ such that $w$ is comparable to
both $x~{\rm ,}~y~ and~z$.
Then $T$ has a  fixed point $x\in X_{\omega^{G}}$ for some positive
integer $m\ge 1$ in $X_{\omega^{G}}$ and the sequence define
by $\{T^{n}x_{0}\}_{n\ge 1}$ converges to $x$.
Furthermore, the fixed point of $T$ is unique.
\end{enumerate
\end{theorem

\begin{proof}
By Corollary \ref{n}, $T^{m}$ has
a  fixed point say $u\in X_{\omega^{G}}$
by using inequality \ref{oo} for some
positive integer $m\ge 1$. Now
$T^{m}(Tu)=T^{m+1}u=T(T^{m}u)=Tu$,
so $Tu$ is a fixed point of $T^{m}$.
By the uniqueness of fixed point of
$T^{m}$, we have $Tu=u$. Therefore,
$u$ is a fixed point of $T$. Since
fixed point of $T$ is also fixed
point of $T^{m}$, hence $T$ has a
unique fixed point in $X_{\omega^{G}}$.

\end{proof



\begin{theorem}\label{}
Let $(X,\omega^{G},\preceq)$ be a complete  preordered
modular G-metric space.
Let $T:X_{\omega^{G}}\rightarrow X_{\omega^{G}}$
be a non-decreasing, asymptotically $T$-regular map on
some point $x_{0}\in X_{\omega^{G}}$
such that for each $\lambda>0$
and  for some positive integer $m\ge 1$, there is
$\nu(\lambda)\in [0,\lambda)$ such that the following
conditions hold:
\begin{align}\label{k}
(1)~ h_{n}\nota
\le
\kappa_{1}(\psi(\omega^{G}_{\lambda}(x,y,z))
\psi(\omega^{G}_{\lambda+\nu(\lambda)}(x,y,z)) \nota
+\kappa_{2}(\psi(\omega^{G}_{\lambda}(x,y,z))
\psi(\omega^{G}_{\lambda}(x,T^{m}x,T^{m}x))\\ \nota
+& \kappa_{3}(\psi(\omega^{G}_{\lambda}(x,y,z))
\psi(\omega^{G}_{\lambda}(y,T^{m}y,T^{m}y)) \nota
+\kappa_{4}(\psi(\omega^{G}_{\lambda}(x,y,z))
\psi(\omega^{G}_{\lambda}(z,T^{m}z,T^{m}z))\\ \nota
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}(x,y,z))
\psi(\omega^{G}_{\lambda}(T^{m}x,y,z)) \nota
+\kappa_{6}(\psi(\omega^{G}_{\lambda}(x,y,z)))\\\nota
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(y,T^{m}y,T^{m}y
[1+\omega^{G}_{\lambda}(x,y,z)]
{1+\omega^{G}_{\lambda}(x,T^{m}x,T^{m}x)}}\\ \nota
+&\kappa_{7}(\psi(\omega^{G}_{\lambda}(x,y,z)))\\\nota
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(y,T^{m}y,T^{m}y
[1+\omega^{G}_{\lambda}(x,T^{m}x,T^{m}x)]
{1+\omega^{G}_{\lambda}(x,y,z)}}\\ \nota
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}(x,y,z)))\\\nota
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x,T^{m}x,T^{m}x
[1+\omega^{G}_{\lambda}(x,y,z)]
{1+\omega^{G}_{\lambda}(y,T^{m}x,T^{m}x
+\omega^{G}_{\lambda}(x,y,z)}}\\ \nota
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}(x,y,z)))\\\nota
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(x,y,z
[1+\omega^{G}_{\lambda}(y,T^{m}y,T^{m}y)]
{1+\omega^{G}_{\lambda}(T^{m}x,T^{m}x,y
+\omega^{G}_{\lambda}(T^{m}x,T^{m}y,T^{m}y)}}\\ \nota
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}(x,y,z)))\\\nota
\times&\psi\Set{\frac{\omega^{G}_{\lambda
(T^{m}x,T^{m}y,T^{m}y)[1+\omega^{G}_{\lambda}(x,T^{m}y,T^{m}y)]
{1+\omega^{G}_{\lambda}(T^{m}x,T^{m}x,y
+\omega^{G}_{\lambda}(x,T^{m}y,T^{m}y)}}\\ \nota
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}(x,y,z)))\\%
\times&\psi\Set{\frac{\omega^{G}_{\lambda
(T^{m}x,T^{m}y,T^{m}y
[1+\omega^{G}_{\lambda}(y,T^{m}y,T^{m}y
+\omega^{G}_{\lambda}(T^{m}x,T^{m}x,y)]
{1+\omega^{G}_{\lambda}(T^{m}x,T^{m}x,y
+\omega^{G}_{\lambda}(T^{m}x,T^{m}y,T^{m}y)}}
\end{align
where $h_{n}=\psi(\omega^{G}_{\lambda}(T^{m}x,T^{m}y,T^{m}z))$,
$\psi\in \bar{\Psi}$ , $\{\kappa_{1},\kappa_{2}, \cdots, \kappa_{11}\}
\in \mathcal{S}_{Ger}$ and
$\max\{\sup_{t\geq 0}\kappa_{1}(t),\\\sup_{t\geq 0}\kappa_{2}(t),
\cdots, \sup_{t\geq 0}\kappa_{11}(t)\}<1.$
Assume that if a non-decreasing sequence $\{x_{n}\}_{n\in\mathbb{N}}$
converges to $x^{\ast}$, then $x_{n}\preceq x^{\ast}$ for each
$n\in\mathbb{N}$
\newline
$(2)$ if $\psi$ is sub-additive and for any $x,y,z\in X_{\omega^{G}}$,
there exists $u\in X_{\omega^{G}}$ with $u\preceq Tu$ and
$\omega^{G}_{\lambda}(u,Tu,Tu)$ is finite  for all $\lambda>0$
such that $u$ is comparable to  $x,y~{\rm and}~z$.
Then $T$ has a  fixed point $x^{\ast}\in X_{\omega^{G}}$
for some positive integer
$m\ge 1$ in $X_{\omega^{G}}$ and the sequence define by
$\{T^{n}x_{0}\}_{n\ge 1}$ converges to $x_{\ast}$.
Furthermore, the fixed point of $T$ is unique.
\end{theorem}


\begin{proof}
By \autoref{qq}, $T^{m}$ has a  fixed point
say $w_{\ast}\in X_{\omega^{G}}$ by using inequality
\ref{k} for some positive integer $m\ge 1$. Now $T^{m}(Tw_{\ast})
=T^{m+1}w_{\ast}=T(T^{m}w_{\ast})=Tw_{\ast}$,
so $Tw_{\ast}$ is a fixed point of $T^{m}$.
By the uniqueness of fixed point of $T^{m}$, we have
$Tw_{\ast}=w_{\ast}$. Therefore, $w_{\ast}$ is a
fixed point of $T$. Since fixed point of $T$ is also
fixed point of $T^{m}$, hence $T$ has a unique fixed
point in $X_{\omega^{G}}$.

\end{proof






\section{Applications to nonlinear integral equations}
We construct a system of nonlinear integral
equations that will satisfies the conditions of
\autoref{3.1} as follows.

Consider the following nonlinear integral equations

\begin{equation}\label{L}
u(t)=\int_{0}^{\Omega}H_{1}(t,s,u(s))ds+\mu_{1}(t),
\end{equation}
and

\begin{equation}\label{LL}
u(t)=\int_{0}^{\Omega}H_{2}(t,s,u(s))ds+\mu_{2}(t),~~\forall~~t\in[0,\Omega],
\end{equation}

where $H_{1},H_{2}:[0,\Omega]\times[0,\Omega]\times\mathbb{R}\rightarrow
\mathbb{R}$ and $\mu:\mathbb{R}\to\mathbb{R}$ are continuous, $\Omega>0$.
For all $t,s\in[0,\Omega]$, take

\begin{equation}
	H_{1}(t,s,u(t))\le
H_{2}\biggl(t,s,\int_{0}^{\Omega}H_{1}(s,z,u(z))ds+\mu_{1}(s)\biggr)
\end{equation}

and
 \begin{equation
H_{2}(t,s,u(t))\le H_{1}\biggl(t,s,\int_{0}^{\Omega}H_{2}(s,z,u(z))ds
+\mu_{2}(s)\biggr).
\end{equation}

Again let $X_{\omega^{G}}=C[0,\Omega]$ be the
space of continuous functions
defined on $[0,\Omega]$. $X_{\omega^{G}}=C[0,\Omega]$ 
can also be equipped with a
preorder  $\preceq$ given by for all $x,y\in
 X_{\omega^{G}}$
given by $x\preceq y\iff x(t)\le y(t)$ for all
 $t\in [0,\Omega]$.
Suppose that there exists a continuous functions
$B:[0,\Omega]\times[0,\Omega]\to\mathbb{R}_{+}$,
 $D_{k}:[0,\Omega]\times[0,\Omega]\rightarrow
\mathbb{R}_{+}$ and $\kappa_{k}:[0,\infty)\rightarrow [0,1)$ 
for all $k\ge 1$ such that

\begin{equation}
\norm{H_{1}(t,s,x)-H_{2}(t,s,y)}\le \sum_{k=1}^{n}B(t,s)
\dfrac{\sin\norm{\frac{1}{\lambda}(x-y)}}{
\norm{\frac{1}{\lambda}(x-y)}}D_{k}(x,y),
\end{equation}

where $$\sup_{t\in [0,\Omega]}\int_{0}^{\Omega}
B^2(t,s)ds\le \dfrac{1}{\Omega},$$ $\lambda>0$. 

Now for any $\lambda>0$, we define

\begin{equation}\label{0
\omega^{G}_{\lambda}(x,y,z):=\frac{1}{2(1+\lambda)}
\sup_{t\in[0,\Omega]}\{\norm{x(t)-y(t)}+
\norm{y(t)-z(t)}+\norm{x(t)-z(t)}\},
\end{equation}

so that

\begin{equation}\label{00
\omega^{G}_{\lambda}(x,y,y):=\frac{1}{(1+\lambda)}
\sup_{t\in[0,\Omega]}\{\norm{x(t)-y(t)}\}.
\end{equation

In fact \autoref{0} and \autoref{00} satisfies all
the conditions in Definition \ref{2.1} endowed with
 $X_{\omega^{G}}=(X_{\omega^{G}},\omega^{G})
=C([0,\Omega],[0,\Omega]).$


Let
\begin{equation}\label{12
G_{u}=\int_{0}^{\Omega}H_{1}(t,s,u(s))ds-\mu_{1}(t),
\end{equation}
and \begin{equation}\label{13}
G_{v}=\int_{0}^{\Omega}H_{2}(t,s,v(s))ds-\mu_{2}(t),
\end{equation}


Now, for distinct $x,y\in X_{\omega^{G}}$, let
 $W=\frac{1}{1+\lambda}\sup_{t\in[0,\Omega]
\bigl\{\norm{G_{x}(t)+\mu_{1}(t)-G_{y}(t)-\mu_{2}(t)}\bigr\}>0$, so that by
 Cauchy-Schwartz inequality,
\begin{align}\label{4.1
W\notag
=&\frac{1}{1+\lambda}\sup_{t\in[0,\Omega]}
\biggl\{\norm{\int_{0}^{\Omega}H_{1}(t,s,x(s))ds
-\int_{0}^{\Omega}H_{2}(t,s,y(s))ds}\biggr\}\\\notag
\le&\frac{1}{1+\lambda}\sup_{t\in[0,\Omega]}\biggl\{\int_{0}^{\Omega}
\norm{H_{1}(t,s,x(s))-H_{2}(t,s,y(s))}ds\biggr\}\\\notag
\le&\frac{1}{1+\lambda}\sup_{t\in[0,\Omega]
\biggl\{\int_{0}^{\Omega}\sum_{k=1}^{n}B(t,s)
\sqrt{\dfrac{\sin^{2}\norm{\frac{1}{\lambda}(x(s)-y(s))}}
{\norm{\frac{1}{\lambda}(x(s)-y(s))}^{2}}}D_{k}(x(s),y(s))ds\biggr\}\\\notag
\le&\frac{1}{1+\lambda}\sup_{t\in[0,\Omega]}
\biggl\{\sum_{k=1}^{n}\int_{0}^{\Omega}B(t,s)
\sqrt{\dfrac{\sin^{2}\norm{\frac{1}{\lambda}(x(s)-y(s))}}
{\norm{\frac{1}{\lambda}(x(s)-y(s))}^{2}}}D_{k}(x(s),y(s))ds\biggr\}\\\notag
\le&\frac{1}{1+\lambda}\sup_{t\in[0,\Omega]}\sum_{k=1}^{n}
\biggl\{\bigg(\int_{0}^{\Omega}B(t,s)^{2}ds\bigg)^{\frac{1}{2}}\\\notag
\times&\bigg(\int_{0}^{\Omega}\dfrac{\sin^{2}\norm{
\frac{1}{\lambda}(x(s)-y(s))}}{\norm{\frac{1}{\lambda}(x(s)-y(s))}^{2}}
D_{k}(x(s),y(s))^{2}ds\bigg)^{\frac{1}{2}}\biggr\}\\\notag
\le&\frac{1}{1+\lambda}\sup_{t\in[0,\Omega]}\sum_{k=1}^{n}
\biggl\{\bigg(\dfrac{1}{\Omega}\bigg)^{\frac{1}{2}}\notag
\bigg(\int_{0}^{\Omega}\dfrac{\sin^{2}\norm{
\frac{1}{\lambda}(x(s)-y(s))}}{\norm{\frac{1}{\lambda}(x(s)-y(s))}^{2}}
D_{k}(x(s),y(s))^{2}ds\bigg)^{\frac{1}{2}}\biggr\}\\\notag
\le&\frac{1}{1+\lambda}\sup_{t\in[0,\Omega]}\sum_{k=1}^{n}
\biggl\{\bigg(\dfrac{1}{\Omega}\bigg)^{\frac{1}{2}}\notag
\bigg(\dfrac{\sin\omega^{G}_{\lambda}(x,y,y)}{\omega^{G}_{\lambda}(x,y,y)}
D_{k}(x,y)\sqrt{\Omega}\bigg)\biggr\}\\
=&\frac{1}{1+\lambda}\sup_{t\in[0,\Omega]}\sum_{k=1}^{n}
\biggl\{\bigg(\dfrac{\sin\omega^{G}_{\lambda}(x,y,y)}
{\omega^{G}_{\lambda}(x,y,y)}D_{k}(x,y)\bigg)\biggr\}.
\end{align

Now, take $\omega^{G}_{\lambda}(x,y,y):=\theta$, so that
$\frac{\sin\theta}{\theta}\rightarrow 1$ as 
$\theta\rightarrow 0$. This agrees 
with the Definition \ref{2.2}. Hence for all 
$k\in\mathbb{N}$, 
$\kappa_{k}(\theta)=\frac{\sin\theta}{\theta}$.
Therefore, from inequality \ref{4.1}, we get

\begin{equation
\frac{1}{1+\lambda}\sup_{t\in[0,\Omega]}
\bigl\{\norm{G_{x}(t)+\mu_{1}(t)-G_{y}(t)-\mu_{2}(t)}\bigr\}\le
\sup_{t\in[0,\Omega]}\sum_{k=1}^{n}
\biggl\{\kappa_{k}\big(\omega^{G}_{\lambda}(x,y,y)\big)D_{k}(x,y)(t)\biggr\}.
\end{equation

 For the application here, we have
 \begin{equation}\label{z}
 \frac{1}{1+\lambda}\sup_{t\in[0,\Omega]}
 \bigl\{\norm{G_{x}(t)+\mu_{1}(t)-G_{y}(t)-\mu_{2}(t)}\bigr\}\le
 \sup_{t\in[0,\Omega]}\sum_{k=1}^{11}
 \biggl\{\kappa_{k}\big(\omega^{G}_{\lambda}(x,y,y)\big)D_{k}(x,y)(t)\biggr\},
 \end{equation

where
\begin{equation*
D_{1}(x,y)(t)=\frac{1}{1+\lambda+\nu(\lambda)}
\norm{x(t)-y(t)}
\end{equation*

 for $\nu(\lambda)\in [0,\lambda).$

\begin{equation*
D_{2}(x,y)(t)=\dfrac{1}{1+\lambda}
\norm{G_{x}(t)+\mu_{1}(t)-x(t)}.
\end{equation*}

\begin{equation*
D_{3}(x,y)(t)=\dfrac{1}{1+\lambda}
\norm{G_{y}(t)+\mu_{2}(t)-y(t)}.
\end{equation*

\begin{equation*
D_{4}(x,y)(t)=\dfrac{1}{1+\lambda}
\norm{G_{x}(t)+\mu_{1}(t)-G_{y}(t)-\mu_{2}(t)}
\end{equation*

\begin{equation*
D_{5}(x,y)(t)=\dfrac{1}{1+\lambda}
\norm{G_{x}(t)+\mu_{1}(t)-y(t)}.
\end{equation*

\begin{equation*
D_{6}(x,y)(t)=\dfrac{\frac{1}{1+\lambda}
\norm{G_{y}(t)+\mu_{2}(t)-y(t)}\bigg(1+\frac{1}{1+\lambda}
\norm{x(t)-y(t)}\bigg)}{1+\frac{1}{1+\lambda}\norm{G_{x}(t)
+\mu_{1}(t)-x(t)}}.
\end{equation*

\begin{equation*
D_{7}(x,y)(t)=\dfrac{\frac{1}{1+\lambda}
\norm{G_{y}(t)+\mu_{2}(t)-y(t)}\bigg(1+
\frac{1}{1+\lambda}\norm{G_{x}(t)+\mu_{1}(t)-x(t)}\bigg)}
{1+\frac{1}{1+\lambda}\norm{x(t)-y(t)}}.
\end{equation*

\begin{equation*
D_{8}(x,y)(t)=\dfrac{\frac{1}{1+\lambda}\norm{G_{x}(t)
+\mu_{1}(t)-x(t)}\bigg(1+\frac{1}{1+\lambda}
\norm{x(t)-y(t)}\bigg)}{1+\frac{1}{1+\lambda}
\bigg(\norm{G_{x}(t)+\mu_{1}(t)-y(t)}+\norm{x(t)-y(t)}\bigg)}.
\end{equation*

\begin{equation*
D_{9}(x,y)(t)=\dfrac{\frac{1}{1+\lambda}
\norm{x(t)-y(t)}\bigg(1+\frac{1}{1+\lambda}
\norm{G_{y}(t)+\mu_{2}(t)-y(t)}\bigg)}{1+
\frac{1}{1+\lambda}\bigg(\norm{G_{x}(t)
+\mu_{1}(t)-y(t)}+\norm{G_{x}(t)+\mu_{1}(t)-G_{y}(t)
-\mu_{2}(t)}\bigg)}.
\end{equation*

\begin{equation*
D_{10}(x,y)(t)=\dfrac{\frac{1}{1+\lambda}
\norm{G_{x}(t)+\mu_{1}(t)-G_{y}(t)-\mu_{2}(t)}
\bigg(1+\frac{1}{1+\lambda}\norm{G_{y}(t)+
\mu_{2}(t)-x(t)}\bigg)}{1+\frac{1}{1+\lambda}
\bigg(\norm{G_{x}(t)+\mu_{1}(t)-y(t)}+\norm{G_{y}(t)
-\mu_{2}(t)-x(t)}\bigg)}.
\end{equation*}

$D_{11}(x,y)(t)=$
\begin{equation*
\dfrac{\frac{1}{1+\lambda}\norm{G_{x}(t)
+\mu_{1}(t)-G_{y}(t)-\mu_{2}(t)}\bigg(1
+\frac{1}{1+\lambda}\norm{G_{y}(t)+\mu_{2}(t)-y(t)}
+\norm{G_{x}(t)+\mu_{1}(t)-y(t)}\bigg)}{1+\frac{1}
{1+\lambda}\bigg(\norm{G_{x}(t)+\mu_{1}(t)-y(t)}
+\norm{G_{x}(t)+\mu_{1}(t)-G_{y}(t)-\mu_{2}(t)}\bigg)}
\end{equation*


\begin{theorem
Let $X_{\omega^{G}}=C([0,\Omega],[0,\Omega])$ be a
 complete preordered modular G-metric space  and 
 $\omega^{G}:(0,\infty)\times X_{\omega^{G}}\times 
 X_{\omega^{G}}\times X_{\omega^{G}}\to [0,\Omega]\cup\{\infty\}$ 
 be define by
 \begin{equation
\omega^{G}_{\lambda}(x,y,y):=\frac{1}{(1+\lambda)}
\sup_{t\in[0,\Omega]}\{\norm{x(t)-y(t)}\}, \lambda>0.
\end{equation
 Let $G_{x}, G_{y}:[0,\Omega]\times[0,\Omega]
 \times\mathbb{R}\rightarrow
 \mathbb{R}$ and $\mu:\mathbb{R}\to\mathbb{R}$ 
 are continuous, $\Omega>0$.
 For all $t,s\in[0,\Omega]$, are such that
 $G_{x},G_{y}\in X_{\omega^{G}}$ for all
 $x,y\in X_{\omega^{G}}$ and $G_{x}$, $G_{y}$
 satisfying \autoref{12} and \autoref{13}
 respectively for all  $t,s\in[0,\Omega]$.
 Suppose that there exists non-negative real numbers
  $\kappa_{1},\kappa_{2},\cdots,\kappa_{11}>0$ with
   $\sum_{k=1}^{11}\kappa_{k}(\psi(\omega^{G}_{
 \lambda}(x,y,y)))\in\mathcal{S}_{Ger}$ and
 $\max\{\sup_{t\geq 0}\kappa_{1}(t),\sup_{t\geq 0}\kappa_{2}(t),
 \cdots, \\\sup_{t\geq 0}\kappa_{11}(t)\}<1.$
 for all $\lambda>0$, $\psi\in \bar{\Psi}$ such
 that inequality \ref{z} is satisfied for every 
 $x,y\in X_{\omega^{G}}.$ Moreover if $\psi$ is
 sub-additive and for any $x,y\in X_{\omega^{G}}$, 
 there exists $u\in X_{\omega^{G}}$  such that $u$ 
 is comparable to both $x~{\rm and}~y$.
Then the system of integral Equations \ref{L} and \ref{LL}
have a unique solution in $X_{\omega^{G}}.$
\end{theorem


\begin{proof}
Let $T:X_{\omega^{G}}\rightarrow X_{\omega^{G}}$ be non-decreasing
and asymptotically $T$-regular map on some point $x_{0}\in X_{\omega^{G}}$
 define by
$Tx=G_{x}+\mu_{1}$ and $Ty=G_{y}+\mu_{2}$. Then for $\lambda>0$,
there exists $u\in X_{\omega^{G}}$
with $u\preceq Tu$ and $\omega^{G}_{\lambda}(u,Tu,Tu)$ is finite  
for all $\lambda>0$.
So from inequality 
\ref{z}, we get by noticing that $\psi$ is continuous
and sub-additive and there exists $\nu\in [0,\lambda)$ such that
			
\begin{align}\label{
\psi(\omega^{G}_{\lambda}(Tx,Ty,Ty))\nota
\le
\kappa_{1}(\psi(\omega^{G}_{\lambda}(x,y,y))
\psi(\omega^{G}_{\lambda+\nu(\lambda)}(x,y,y)) \nota
+\kappa_{2}(\psi(\omega^{G}_{\lambda}(x,y,y))
\psi(\omega^{G}_{\lambda}(x,Tx,Tx))\\ \nota
+& \kappa_{3}(\psi(\omega^{G}_{\lambda}(x,y,y))
\psi(\omega^{G}_{\lambda}(y,Ty,Ty)) \nota
+\kappa_{4}(\psi(\omega^{G}_{\lambda}(x,y,y))
\psi(\omega^{G}_{\lambda}(Tx,Ty,Ty))\\ \nota
+&\kappa_{5}(\psi(\omega^{G}_{\lambda}(x,y,y))
\psi(\omega^{G}_{\lambda}(Tx,y,y))\\ \nota
+&\kappa_{6}(\psi(\omega^{G}_{\lambda}(x,y,y)))\nota
\psi\Set{\frac{\omega^{G}_{\lambda}(y,Ty,Ty
[1+\omega^{G}_{\lambda}(x,y,y)]}{1+\omega^{G}_{\lambda}(x,Tx,Tx)}}\\ \nota
+&\kappa_{7}(\psi(\omega^{G}_{\lambda}(x,y,y)))\nota
\psi\Set{\frac{\omega^{G}_{\lambda}(y,Ty,Ty
[1+\omega^{G}_{\lambda}(x,Tx,Tx)]}{1+\omega^{G}_{\lambda}(x,y,y)}}\\ \nota
+&\kappa_{8}(\psi(\omega^{G}_{\lambda}(x,y,y)))\nota
\psi\Set{\frac{\omega^{G}_{\lambda}(x,Tx,Tx
[1+\omega^{G}_{\lambda}(x,y,y)]}{1+\omega^{G}_{\lambda}(y,Tx,Tx
+\omega^{G}_{\lambda}(x,y,y)}}\\ \nota
+&\kappa_{9}(\psi(\omega^{G}_{\lambda}(x,y,y)))\nota
\psi\Set{\frac{\omega^{G}_{\lambda}(x,y,y
[1+\omega^{G}_{\lambda}(y,Ty,Ty)]}{1+\omega^{G}_{\lambda}(Tx,Tx,y
+\omega^{G}_{\lambda}(Tx,Ty,Ty)}}\\ \nota
+&\kappa_{10}(\psi(\omega^{G}_{\lambda}(x,y,y)))\nota
\psi\Set{\frac{\omega^{G}_{\lambda}(Tx,Ty,Ty
[1+\omega^{G}_{\lambda}(x,Ty,Ty)]}{1+\omega^{G}_{\lambda}(Tx,Tx,y
+\omega^{G}_{\lambda}(x,Ty,Ty)}}\\ \nota
+&\kappa_{11}(\psi(\omega^{G}_{\lambda}(x,y,y)))\\%
\times&\psi\Set{\frac{\omega^{G}_{\lambda}(Tx,Ty,Ty
[1+\omega^{G}_{\lambda}(y,Ty,Ty)+\omega^{G}_{\lambda}(Tx,Tx,y)]
{1+\omega^{G}_{\lambda}(Tx,Tx,y)+\omega^{G}_{\lambda}(Tx,Ty,Ty)}}
\end{align}
By \autoref{3.1}, then the system of integral Equations \ref{L} and \ref{LL}
have a unique solution in $X_{\omega^{G}}.$	
\end{proof


\begin{corollary}
	Let $X_{\omega^{G}}=C([0,\Omega],[0,\Omega])$ be a
	complete preordered modular G-metric space  and
	 $\omega^{G}:(0,\infty)\times X_{\omega^{G}}\times
	  X_{\omega^{G}}\times X_{\omega^{G}}\to [0,\Omega]
	  \cup\{\infty\}$ be define by\begin{equation}
	\omega^{G}_{\lambda}(x,y,y):=\frac{1}{(1+\lambda)}
	\sup_{t\in[0,\Omega]}\{\norm{x(t)-y(t)}\}, \lambda>0.
	\end{equation}
	Let $G_{x}, G_{y}:[0,\Omega]\times[0,\Omega]
	\times\mathbb{R}\rightarrow
	\mathbb{R}$ and $\mu:\mathbb{R}\to\mathbb{R}$ are
	 continuous, $\Omega>0$.
	For all $t,s\in[0,\Omega]$, are such that 
	$G_{x},G_{y}\in X_{\omega^{G}}$ for all 
	$x,y\in X_{\omega^{G}}$ and $G_{x}$, $G_{y}$ 
	satisfying \autoref{12} and \autoref{13}
	 respectively for all  $t,s\in[0,\Omega]$.
	 Suppose that there exists non-negative real numbers 
	  $\kappa_{1},\kappa_{2},\kappa_{3}>0$ with
	 $\sum_{k=1}^{3}\kappa_{k}(\psi(\omega^{G}_{
	 	\lambda}(x,y,y)))\in\mathcal{S}_{Ger}$
	and $\kappa_{1}(t)
	+2\max\{\sup_{t\geq 0}\kappa_{2}(t), \sup_{t\geq 0}
	\kappa_{3}(t)\}<1,$
	for distinct $x,y\in X_{\omega^{G}}$ and for all
	 $\lambda>0$, $\psi\in \bar{\Psi}$ such that
	 inequality \ref{z} is satisfied for every
	 $x,y\in X_{\omega^{G}}.$ Moreover if $\psi$ is
	 sub-additive and for any $x,y\in X_{\omega^{G}}$,
	 there exists $u\in X_{\omega^{G}}$ such that $u$ is
	 comparable to both $x~{\rm and}~y$.
	Then the system of integral Equations \ref{L} and \ref{LL}
	have a unique solution in $X_{\omega^{G}}.$
\end{corollary}

\begin{proof}
Let $T:X_{\omega^{G}}\rightarrow X_{\omega^{G}}$ 
be non-decreasing
and asymptotically $T$-regular map on some point 
$x_{0}\in X_{\omega^{G}}$
define by
$Tx=G_{x}+\mu_{1}$ and $Ty=G_{y}+\mu_{2}$.
Then for $\lambda>0$,
there exists $u\in X_{\omega^{G}}$
with $u\preceq Tu$ and $\omega^{G}_{\lambda}(u,Tu,Tu)$ 
is finite  for all $\lambda>0$.
So from inequality \ref{z}, we have
$\omega^{G}_{\lambda}(Tx,Ty,Ty)=\frac{1}
{1+\lambda}\sup_{t\in[0,\Omega]}
\bigl\{\norm{G_{x}(t)+\mu_{1}(t)-G_{y}(t)-
	\mu_{2}(t)}\bigr\}$,
$\omega^{G}_{\lambda}(x,y,y)=\frac{1}{1+
\lambda+\nu(\lambda)}\norm{x(t)-y(t)}$,
for $\nu(\lambda)\in [0,\lambda).$ 
$\omega^{G}_{\lambda}(x,Tx,Tx)=\dfrac{1}{1+\lambda}
\norm{G_{x}(t)+\mu_{1}(t)-x(t)},$ and
$\omega^{G}_{\lambda}(y,Ty,Ty)=\dfrac{1}{1+\lambda}
\norm{G_{y}(t)+\mu_{2}(t)-y(t)}.$
Since $\psi$ is continuous
and sub-additive, then by inequality \ref{z} for 
$k=1,2,3$, we have
\begin{align}\label{
\psi(\omega^{G}_{\lambda}(Tx,Ty,Ty))\notag
\le
\kappa_{1}(\psi(\omega^{G}_{\lambda}(x,y,y)))
\psi(\omega^{G}_{\lambda+\nu(\lambda)}(x,y,y)) \notag
+\kappa_{2}(\psi(\omega^{G}_{\lambda}(x,y,y)))
\psi(\omega^{G}_{\lambda}(x,Tx,Tx))\\ \notag
+& \kappa_{3}(\psi(\omega^{G}_{\lambda}(x,y,y)))
\psi(\omega^{G}_{\lambda}(y,Ty,Ty)).
\end{align
By Corollary \ref{aaa}, then the system of 
integral Equations \ref{L} and \ref{LL}
have a unique solution in $X_{\omega^{G}}.$
\end{proof}
\newline	
\textbf{Data Availability}\newline
The data used to support the findings of this study are included within the
article.\newline
\newline
\textbf{Conflicts of Interest}\newline
The authors declare no conflict of interest.\newline
\newline
\textbf{Authors Contributions}\newline
All authors contributed equally to the writing of this paper.







