\section{Introduction}
Suppose that $W$ is a non-void set endowed with a metric $d$. The set $CB(W)$ represents the collection of all non-void bounded closed subsets of $W$. A map $H:CB(W)\times CB(W) \to \mathbb{R}$ is described as a Hausdorff metric on $CB(W)$, whenever for $Y$ and $Z$ lies in $CB(W)$, 
\begin{center}
	$\displaystyle{	H(Y,Z)= \max\left\{ \sup_{u\in Y}  D(u,Z),\,\,\sup_{v\in Z}D(v,Y)\right\}}$,
\end{center}
whereas the notation $D(u,Z)$ describes the $\inf_{v\in Z}d(u,v)$. Further, $CL(W)$ expresses the collection of all non-void subsets of $W$, which are closed. Given a set-valued map $F:W\to CL(W)$, an element $w^*\in W$ is stated as a fixed point for $F$, if it lie in $\{w\in W: w\in Fw\}$. The occurrence of fixed points for a set-valued contraction first derived by Nadler \cite{nadler} in the year of 1969.
Later, Mizoguchi and Takahashi \cite{mizo} improved the Nadler's \cite{nadler} result for a set-valued non-linear contraction in the year of 1989. The authors \cite{mizo} deduced that if $F:W\to CB(W)$ and for each $v,w$ in $W$ fulfils $H(Fv, Fw)\leq k(d(v,w))d(v,w)$, where $k \in U=\{h:[0,\infty)\to [0,1)| \limsup_{r\to t^+}h(r)<1,\forall \,0\leq t<\infty\}$, then occurrence of a fixed point for the map $F$ is guaranteed whenever $(W,d)$ is complete. There are several generalization of this result in the literature, which can be found in \cite{berinde,mam, mam_feng}. In the year 2006, Feng and Liu \cite{feng} extended the well known Nadler's \cite{nadler} result for a set-valued generalized contraction $F$ from $W$ into $CL(W)$, where the Hausdorff distance function is not involved.
\begin{theorem}\cite{feng}\label{feng}
	For the metric space $(W,d)$, a set-valued map $F:W\to CL(W)$ in order that for each $v\in W$, $w\in Fv$ fulfils
	\begin{equation}
		D(w,Fw)\leq \theta d(v,w)~\textrm{where}~0\leq \theta<1 .
	\end{equation}
	Then there is an element $w^*\in Fw^*$ if the map $h: W\to \mathbb{R}$ given by $h(v)=D(v,Fv)$ is lower semicontinuous and $(W,d)$ is complete.       
\end{theorem}


For some more results about the fixed points for set-valued generalized contraction can be found in \cite{azam,klim}.

In the parallel development, many mathematicians scrutinized common fixed point results for a couple of single valued mappings on a metric space, one may see \cite{abbas, Doric}, whereas in \cite{khan,kumam}, the authors studied the common fixed point theory for a couple of single valued and set-valued mappings.
In the year 2007, Kamran \cite{kamran} established a common fixed point result for a single valued and set-valued generalized contractive mappings. This result actually enables the author \cite{kamran} to generalize several fixed point and common fixed point results on metric space available in the literature. Indeed, the author \cite{kamran} established the below stated result. 
\begin{theorem}\cite{kamran}\label{thm1}
	For the metric space $(W,d)$, the maps $F:W\to CB(W)$, $f:W\to W$ in order that for each $u\in W$, $Fu\subset fW$ and 
	\begin{equation}\label{eqn1}
		H(Fv,Fw)\leq k(d(fv,fw))d(fv,fw)+M\,D(fv,Fw)~\text{for each}~ v,w \in W,
	\end{equation}
	where $k\in U$, $M\geq 0$. Then occurrence of $w^*$ having $fw^*\in Fw^*$ is guaranteed if $fW$ is complete. Moreover, $w^*=fw^*\in Fw^*$, whenever $ffw^*=fw^*$ and $\{ffw^*\}\subset Ffw^*$.
\end{theorem}
\noindent We note that, the above mentioned result deduces Mizoguchi-Takahashi's \cite{mizo} theorem by considering $f$ to be an identity map and $M=0$.


On the other side, Jachymski \cite{Jach} extended the renowned Banach contraction principle via $G$-contraction on a metric space having  graph $G$. With the aid of this result, the author \cite{Jach} enables to extend and unify some fixed point results proved in metric spaces as well as metric spaces having partial order.  
Followed by Jachymski \cite{Jach}, an extension of the Theorem \ref{feng} due to Feng and Liu \cite{feng} was derived by Sultana and Vetrivel \cite{mam_feng}  in the year 2017, on a metric space consisting with directed graph $G$. 
The authors \cite{mam_feng} demonstrated the below stated result.  
\begin{theorem}\cite{mam_feng}\label{mam}
	Let $(W,d)$ along with a directed graph $G$ and a map $F:W\to CL(W)$ in order that for every $v \in W$, $w\in Fv$ with $(v,w)\in E(G)$ fulfils
	\begin{enumerate}
		\item [(i)] $D(w,Fw)\leq k(d(v,w))d(v,w)$ for $k\in U$,
		\item [(ii)] for any $p\in Fw$ having $d(p,w)\leq d(v,w)$ indicates $(p,w)\in E(G)$. 
	\end{enumerate}
	Suppose that $w_0\in W$,  $w_1\in Fw_0$ with $(w_0,w_1)\in E(G)$. Moreover, assume that if any sequence $\{w_n\}_n\subset W$ with $(w_{n-1},w_{n})\in E(G)$ for every $n$, and $w_n\to w$ then there is $\{w_{n_r}\}_{r\in \mathbb{N}}$ in order that $D(w,Fw)\leq \liminf_{r\to \infty}D(w_{n_r},F(w_{n_r}))$. Then there is an element $w^*\in Fw^*$ if $(W,d)$ is complete.
\end{theorem}
\noindent As an application of the Theorem \ref{mam}, the authors \cite{mam_feng} improved the Kelisky and Rivlin \cite[Theorem 1]{KR8} result about the convergence of iterates for Bernstein operator on a complete normed linear space. In the literature, there are several fixed point theorems using the notion of graph theory, which can be found in \cite{frigon,khan, mam}.  


Inspired by Kamran \cite{kamran} and Sultana-Vetrivel \cite{mam_feng}, 
in this present manuscript, we derive a common fixed point result on a metric space having graphical structure for a set-valued mapping, which is supposed to be closed valued instead of bounded and closed valued. 
The main result is indeed an extension of the Theorem \ref{thm1} due to Kamran \cite{kamran} and the Theorem \ref{mam} due to Sultana-Vetrivel \cite{mam_feng}. 
As an implementation of our result,  we establish the convergence of iterates for some nonlinear operators, which enables us to derive the convergence of iterates for a nonlinear $q$-analogue Bernstein operator. As an another application, we derive a sufficient criteria for the presence of a solution for a fractional order differential equation. An invariant best approximation result is also discussed in this article.
\bigskip
\section{Preliminaries}
In this part, we utilize some essential symbols and definitions, which is needful all throughout this article.
An element $w^*$ is described as a coincidence point for a couple of mappings $F:W\to CL(W)$ and $f:W\to W$, if $\{fw^*\}\subset Fw^*$. Furthermore, an element $w^*$ is described as a common fixed point for the map $f$ coupled with $F$ if $w^*=fw^*\in Fw^*$. Additionally, the map $f$ is stated as a $F$-weakly commuting \cite{kamran} at an element $z^*$ in $W$ if $ffz^*\in Ffz^*$. Up to the end, the collection of common fixed points and coincidence points for the maps $f$ and $F$ are expressed by $Fix(f,F)$ and $Coin(f,F)$ respectively.



Now we present some fundamental terminology of graph theory (see \cite{John}), which is mandatory up to the end of this article. For the metric space $(W,d)$, let us assume a directed graph $G$ which is a pair of $(V(G),E(G))$, whereas the collection of all vertices are symbolized by $V(G)$, which is actually whole set $W$ and the collection of edges $E(G)$ includes the set $\Delta=\{(s,s):(s,s)\in W\times W\}$. Further, this reflexive graph $G$ does not possess any parallel edges.



\section{Main results}
Up to the end of this part, we consider a metric space $(W,d)$ along with a directed graph $G(V(G),E(G))$, whereas $V(G)=W$, $\Delta \subseteq E(G)$ and $G$ has no parallel edges. In the below stated theorem, we scrutinize common fixed points on a metric space having a graph $G$ for a set-valued generalized contraction, which is assumed to be only closed-valued. 
\begin{theorem}\label{thm3.2}
	For the metric space $(W,d)$, the maps $F:W \to CL(W)$, $f:W\to W$ in order that for each $v\in W$, $fw\in Fv$ with $(fv,fw)\in E(G)$ fulfils,
	\begin{enumerate}\label{main}
		\item[(i)] $D(fw,Fw)\leq k(d(fv,fw))d(fv,fw)$, where $k\in U$,
		\item[(ii)] for any $fp\in Fw$ with $d(fw,fp)\leq d(fv,fw)$, indicates $(fw,fp)\in E(G)$.
	\end{enumerate} 
	Suppose that $w_0\in W$, $p_0\in F(w_0)$ with $(fw_0,p_0)\in E(G)$. Also assume that if any $\{fu_n\}_{n}\subset W$ with $(fu_{n-1},fu_{n})\in E(G)$, $fu_n\in Fu_{n-1}$ for every $n\in \mathbb{N}$ and $fu_n\to fu$, then there is $\{fu_{n_r}\}\subseteq\{fu_n\}$ in order that $D(fu,Fu)\leq \liminf_{r\to \infty}D(fu_{n_r},Fu_{n_r})$. Then there is a sequence $\{fw_n\}_{n}$ with $fw_n\in Fw_{n-1}$ converges to $fw^*$, where $fw^*\in Fw^*$ 
	is guaranteed if for each $u\in W$, $Fu\subset fW$ and $(fW,d)$ is complete.
	
	Moreover, $f$ and $F$ possess a common fixed point whenever the map $f$ is weakly commuting at $w^*$ along with $fw^*=ffw^*$.    
	
	
\end{theorem}

















	

	\begin{proof}
		As $p_0\in Fw_0$ and $Fw_0\subset fW$, then there is $w_1\in W$ in order that $p_0=fw_1$. Consequently, we obtain $(fw_0,fw_1)\in E(G)$ and $fw_1\in F(w_0)$.
	
	
		Since $D(fw_1,Fw_1)=\inf\{d(fw_1,y): y\in Fw_1\}$, then for $\frac{1}{\sqrt{k(d(fw_0,fw_1))}}>1$ there is $w_2\in W$ with $fw_2\in F(w_1)$ fulfils
		\begin{eqnarray}\label{eqn_1st}
			d(fw_1,fw_2)&\leq& D(fw_1,Fw_1)+ \left(\frac{1}{\sqrt{k(d(fw_0,fw_1))}}-1\right)D(fw_1,Fw_1)\nonumber\\&\leq&\frac{1}{\sqrt{k(d(fw_0,fw_1))}} k(d(fw_0,fw_1))d(fw_0,fw_1)\nonumber\\
		
			&\leq&\sqrt{k(d(fw_0,fw_1))}d(fw_0,fw_1)< d(fw_0,fw_1).
		\end{eqnarray}
	
	
		
		
		
		
		Due to the fact that $F$ meets the hypothesis $(ii)$ and $(fw_0,fw_1)\in E(G)$ with $fw_1\in Fw_0$, then the previous inequation $(\ref{eqn_1st})$ leads to $(fw_1,fw_2)\in E(G)$.  
	
	
	
		Repeating the above methodology, for $\frac{1}{\sqrt{k(d(fw_1,fw_2))}}>1$ we get $fw_3\in F(w_2)$ fulfilling
		\begin{eqnarray*}
			d(fw_2,fw_3)&\leq& D(fw_2,Fw_2)+\left( \frac{1}{\sqrt{k(d(fw_1,fw_2))}}-1\right)D(fw_2,Fw_2)\nonumber\\
			&\leq& \frac{1}{\sqrt{k(d(fw_1,fw_2))}}k(d(fw_1,fw_2))d(fw_1,fw_2)< d(fw_1,fw_2
		
		
		\end{eqnarray*}
		Hence by the same argument as before we have $(fw_2,fw_3)\in E(G)$. Following this approach, we are able to construct two sequences $\{f(w_n)\}_{n}$, $\{F(w_n)\}_{n}$ in order that for every $n$, $(fw_{n-1},fw_n)\in E(G)$ and $f(w_n)\in F(w_{n-1})$ fulfils
		\begin{equation}\label{eqn2}
			d(fw_n,fw_{n+1})
		
		
			\leq\sqrt{k(d(fw_{n-1},fw_n))}d(fw_{n-1},fw_n)<d(fw_{n-1},fw_n).
		
		\end{equation}
		Now for $n\geq 1$, we consider $s_n=d(fw_n,fw_{n+1})$. Consequently from $(\ref{eqn2})$, it is clearly observed that for each $n$, $s_n$ are non-negative and $s_{n+1}<s_n$.
	
		Therefore $\{s_n\}_n$ is convergent and assume $s_n \to q$, where $q\geq 0$. On the account of $\limsup_{t\to q^+}k(t)<1$, there is $0\leq \alpha<1$ and $M\geq 1$ meeting the condition $k(s_n)\leq \alpha$ for each $n\geq M$. Now for every $n> M$, the inequation $(\ref{eqn2})$ leads to
		\begin{eqnarray}\label{bounded}
			d(fw_n,fw_{n+1})&\leq&\alpha^{(n-M)/2}\prod_{j=1}^{M-1}\sqrt{k(d(fw_{j-1},fw_{j}))}d(fw_{0},fw_1)\nonumber\\
		
			&<&B\alpha^{n/2}d(fw_{0},fw_1),
		\end{eqnarray}
		where $B=\alpha^{-{M/2}}\prod_{j=1}^{M-1}\sqrt{k(d(fw_{j-1},fw_{j}))}$.
	
	
		
		
	
	
		
		
		
		
		
		
	
	
		
		
		
		
		Now using $(\ref{bounded})$ and for any arbitrary positive integer $m$ and $n>M$,
		\begin{equation*}
			\begin{aligned}
				d(fw_n,fw_{n+m})&\leq d(fw_{n},fw_{n+1})+d(fw_{n+1},fw_{n+2})+\cdots+\\
				& \qquad d(fw_{n+m-1},fw_{n+m})\\
			
				&\leq B\alpha^{n/2}\left[1+\alpha^{1/2}+ \cdots+ \alpha^{(m-1)/2}\right]d(fw_{0},fw_1)\\
			
				&\leq B\frac{\alpha^{n/2}}{1-\alpha^{1/2}}d(fw_{0},fw_1).
			\end{aligned}
		\end{equation*}
		Thus the sequence $\{fw_n\}_{n}\subset fW$ turns out to be Cauchy, on the account of $\alpha^{n/2}\to 0$ as $n\to \infty$. 
	
		Since $fW$ is complete, then $\{fw_n\}_{n}$ is converges to $fw^*$, for some $w^*\in W$.
		Again for every $n$, $(fw_{n-1},fw_n)\in E(G)$ and $fw_n\in Fw_{n-1}$, it yields from $(i)$ that $D(fw_n,Fw_n)\leq k(d(fw_{n-1},fw_n))d(fw_{n-1},fw_n)<d(fw_{n-1},fw_n)$. Hence $\lim_{n \rightarrow \infty}D(fw_n,Fw_n)=0$. 
		
		
		On the account of $(fw_{n-1},fw_n)\in E(G)$ with $fw_{n}\in Fw_{n-1}$ for each $n$ and $fw_n\to fw^*$, consequently by the hypothesis there is a subsequence $\{fw_{n_r}\}_{r\in \mathbb{N}}$ having  $D(fw^*,Fw^*)\leq \liminf_{r\to \infty}D(fw_{n_r},F(w_{n_r}))$. Thus we can conclude that $D(fw^*,Fw^*)=0$. Due to the fact that $Fw^*$ is closed, therefore $fw^*\in Fw^*$.
		
		Now let $a=fw^*$, then the hypothesis $fw^*=ffw^*$ indicates that $fa=a$. Again $f$ and $F$ are weakly commuting at $w^*$, hence $ffw^*\in Ffw^*$ leads to $a=fa\in Fa$. 
	\end{proof}


	\begin{remark}
		Take a graph $G$ on the set $W$ in order that $V(G)=W$ and $E(G)= W\times W$. Consequently, the aforementioned Theorem \ref{feng} due to Feng-Liu \cite{feng} is followed from our main Theorem \ref{thm3.2} by considering the map $f$ is an identity map on $W$ and for each $0\leq t<\infty$, $k(t)=\theta$, where $\theta\in [0,1)$.
	\end{remark}

	
	The succeeding corollary deals with common fixed points for set-valued generalized contraction in a metric space, which is indeed an extension of the Theorem \ref{feng} due to Feng-Liu \cite{feng}. Moreover, this result is followed from the aforementioned Theorem \ref{thm3.2} by choosing a graph $G$ on $W$ in order that $V(G)=W$ and $E(G)=W\times W$.  
	\begin{corollary}\label{cor}
		For the metric space $(W,d)$, the maps $F:W \to CL(W)$, $f:W\to W$ in order that for each $v\in W$, $fw\in Fv$ fulfils,
		\begin{equation*}
			D(fw,Fw)\leq k(d(fv,fw))d(fv,fw), ~\textrm{where}~ k\in U.
		\end{equation*}
		Suppose that the function $h: W\to \mathbb{R}$ by $h(v)=D(fv,Fv)$ is lower semi-continuous. Then occurrence of $w^*$ having $fw^*\in Fw^*$ is guaranteed if for each $u\in W$, $Fu\subset fW$ and $(fW,d)$ is complete.
		
		Moreover, $f$ and $F$ possess a common fixed point whenever the map $f$ is weakly commuting at $w^*$ along with $fw^*=ffw^*$.
	\end{corollary}
	

	
	The below-stated example illustrates the preceding Theorem \ref{thm3.2}.




	
	
		
		
		
		
	


	
	
		
		
		
	


	
	


	
	
		
		
		
		
	






	\begin{example}\label{example}
		Consider a set $W=\{0,1\}\cup\left\{\frac{1}{3^n}:n\in \mathbb{N}\right\}$ together with usual metric. Suppose that a single valued map $f:W\to W$ by
		\begin{center}
			$f(w)=
			\begin{cases}
				~\frac{1}{3^{n+1}} &\mbox{if}~~w=\frac{1}{3^n},~n\in \mathbb{N}\cup\{0\}\\
				~0~&\mbox{if}~~w=0.
			\end{cases}$
		\end{center}
		and set-valued map $F: W\to CL(W)$ is defined as
		\begin{center}
			$F(w)=
			\begin{cases}
				~\left\{0,\frac{1}{3}\right\} &\mbox{if}~~w=0,\\
				~\left\{\frac{1}{3},\frac{1}{3^{n+2}}\right\} &\mbox{if}~~w=\frac{1}{3^n},~\text{where}~ n\in \mathbb{N},\\
				~\left\{0\right\}~&\mbox{if}~~w=1,
			\end{cases}$
		\end{center} 
		Take a graph $G$ on $W$ in order that $V(G)=W$ and $E(G)=\{(v,w)\in W\times W: d(v,w)< \frac{1}{9}\}$. Then, $G$ does not have any parallel edges and $E(G)$ contains the set $\Delta$. Now for $v=\frac{1}{3^n}$ and $fw=\frac{1}{3^{n+2}}\in Fv$, we see that for each $n$, $(fv,fw)\in E(G)$. Further,
		\begin{equation*}
			D(fw,Fw)=D\left(\frac{1}{3^{n+2}},\left\{\frac{1}{3},\frac{1}{3^{n+3}}\right\}\right)=\frac{1}{3^{n+2}}-\frac{1}{3^{n+3}}\leq \frac{1}{3}d(fv,fw).
		\end{equation*} 
		Again, for $v=0$ and $fw=0\in Fv$, the pair $(fv,fw)\in E(G)$. Also, it is easy to check that $D(fw,Fw)\leq \frac{1}{3}d(fv,fw)$. Thus we conclude that for each $v\in W$, $fw\in Fv$ with $(fv,fw)\in E(G)$ fulfils $D(fw,Fw)\leq k(d(fv,fw))d(fv,fw)$, where $k(t)=\frac{1}{3}$ for each $t\in [0,\infty)$. Furthermore, if $fp\in Fw$ with $d(fw,fp)\leq d(fv,fw)$, then $(fw,fp)\in E(G)$.
		Now for choosing $w_0=\frac{1}{3}$, we observe that $\frac{1}{27}\in Fw_0$ and $(fw_0,\frac{1}{27})\in E(G)$. It is clear to observe that the map $h:W\to \mathbb{R}$ by $h(w)=D(fw,Fw)$ is lower semi-continuous. Further, $(fW,d)$ is complete and for each $u\in W$, $Fu\subset fW$. Thus all the criteria of the Theorem \ref{thm3.2} is fulfilled, which confirm the occurrence of coincidence point $w^*$. We visualize that $0$ is a coincidence point of $f$ and $F$, that is, $f0\in F0$. Again $ff0=f0$ and $ff0\in Ff0$. We note that $0$ is a common fixed point of $f$ and $F$.
		
		On the other side, if we choose $v=\frac{1}{3^n}$ for each $n\geq 1$ and $fw=\frac{1}{3}\in Fw$, then $d(fv,fw)=\frac{1}{3}-\frac{1}{3^{n+1}}$ and $D(fw,Fw)=\frac{1}{3}$. Therefore for such chosen points $D(fw,Fw)>d(fv,fw)$. This indicates that the Theorem \ref{thm3.2} is indeed an extension of the Corollary \ref{cor}.  
	\end{example}	

	\begin{remark}
		It is worth to observe that the preceding Theorem \ref{thm3.2} is an extension of the Theorem \ref{thm1} due to Kamran \cite{kamran}. Indeed, consider $W$ incorporates with a graph $G$ having $V(G)=W$ and $E(G)=W\times W$. Let $v\in W$, $fw\in Fv$ with $(fv,fw)\in E(G)$, then it follows
		\begin{eqnarray*}
			D(fw,Fw)\leq H(Fv,Fw)&\leq& k(d(fv,fw))d(fv,fw)+M\,D(fw,Fv)\\
			&=& k(d(fv,fw))d(fv,fw),~\textrm{where}~ k\in U.
		\end{eqnarray*}
		Moreover, for the chosen graph $G$, it always follows that $(fw,fp)\in E(G)$.
		Also we are able to find $w_0,p\in W$ in order that $(fw_0,p)\in E(G)$ and $p\in Fw_0$. Now assume a $\{fw_n\}_n\in W$ in order that $fw_n\in Fw_{n-1}$ for every $n$, and $fw_n\to fw$. Evidently for each $n$, $(fw_n,fw_{n+1})\in E(G)$. Consequently for each $n$ and $fz\in Fw_n$,
		\begin{equation*}
			\begin{aligned}
				D(fw,Fw
				&\leq  d(fw,fw_n)+d(fw_n,fz)+H(Fw_n,Fw)\\
				&\leq d(fw,fw_n)+ D(fw_n,Fw_n)+ d(fw,fw_n)+ M\,D(fw,Fw_n)\\
			
				&\leq d(fw,fw_n)+ D(fw_n,Fw_n)+ d(fw,fw_n).~ \textrm{[as $fw_n\in F(w_{n-1})]$}
			\end{aligned} 
		\end{equation*}
	
		Now taking $n\to \infty$ it yields, $D(fw,Fw)\leq\liminf_{n\to \infty} D(fw_n,Fw_n)$. Hence all the criteria of the Theorem \ref{thm3.2} is fulfilled.  
	\end{remark}

	
	The below-mentioned example demonstrates that, the Theorem \ref{thm3.2} indeed an extension of Theorem \ref{thm1} due to Kamran \cite{kamran}.

	\begin{example}
		In the Example \ref{example}, choose two points $0$ and $1$ in $W$. Then we see that $H(F0,F1)=\frac{1}{3}$, $d(f0,f1)=\frac{1}{3}$ and $D(f1,F0)=0$. Therefore for all $k\in U$ and $M\geq 0$, $H(F0,F1)=d(f0,f1)> k(d(f0,f1))d(f0,f1)+M\,D(f1,F0)$. This implies that $F$ does not fulfils the equation (\ref{eqn1})  of the Theorem \ref{thm1} due to Kamran. 
	\end{example}

	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	

	\begin{remark}
		The Theorem \ref{mam} due to Sultana and Vetrivel \cite{mam_feng} follows
		from our Theorem \ref{thm3.2} by choosing $f$ to be an identity map.
	\end{remark}

	
	Let the set $W$ be endowed with a norm $\|.\|$ and $Q\subseteq W$ with $Q\neq \emptyset$. Now for $z\in W$, a set $B_Q(z)$ contains $y\in Q$ in order that $D(z,Q)=\|y-z\|$. Then the set $B_Q(z)$ is called as a best $Q$-approximates of $z$ over $Q$. Invariant best approximation problems can be solved through the perception of common fixed points for single valued mappings, as one may see in \cite{thagafi,pathak}. Also in \cite{kamran1,kumam}, the authors established invariant best approximation results for a couple of single valued and set-valued mappings. In the succeeding theorem, we investigate invariant best approximation for single valued as well as set-valued maps meeting the condition $(i)$ of the Theorem \ref{thm3.2} on a normed linear space without considering the Hausdorff distance function. 

	\begin{theorem}
		Suppose that $(W,\|.\|)$ is a normed linear space and $Q\subseteq W$ such that $Q\neq \emptyset$. Assume that the mappings $f:Q\to Q$ and $F:Q \to CL(Q)$ fulfils the following:
		\begin{enumerate}
			\item[(i)] for every $v\in B_Q(z)$ and $fw\in Fv$ fulfils $D(fw,Fw)\leq k(\|fv-fw\|)\|fv-fw\|$ where $k \in U$, 
			
			\item[(ii)] $f(B_Q(z))=B_Q(z)$ and $B_Q(z)$ is complete,
			
		
			\item[(iii)] $\sup_{u\in Fp}\|u-z\|\leq \|fp-z\|$ for each $p\in B_Q(z)$. 
		\end{enumerate}
		Then $w^*\in Coin(f,F)\cap B_Q(z)$ if for any $\{fw_n\}_n\in B_Q(z)$ with $fw_n\in Fw_{n-1}$ and $fw_n\to fw$ implies $D(fw,Fw)\leq \liminf_{n\to \infty}D(fw_n,Fw_n)$. Furthermore, $Fix(f,F)\cap B_Q(z)\neq \emptyset$ if $ffw^*=fw^*$ and $ffw^*\in Ffw^*$. 
	\end{theorem}

	\begin{proof}
		Consider a graph $G$ on $B_Q(z)$ in order that $V(G)=B_Q(z)$ and $E(G)=B_Q(z)\times B_Q(z)$. Now assume an element $p$ lies in $B_Q(z)$ and $w\in Fp$. Then we have $fp\in B_Q(z)$ for every $p\in B_Q(z)$, on the account of $f(B_Q(z))=B_Q(z)$.
		Therefore we get $\|fp-z\|=D(z,Q)$, by using the definition of $B_Q(z)$. Again the condition $(iii)$ yields that $$\|w-z\|\leq \sup_{u\in Fp}\|u-z\|\leq \|fp-z\|=D(z,Q).$$	
	
		Hence $w\in B_Q(z)$. This indicates that  for each $p \in B_Q(z)$,  $Fp\subseteq B_Q(z)$. As the set $Fp$ is closed for each $p\in Q$, then $Fp$ is also closed for any $p \in B_Q(z)$. Subsequently, $F|_{ B_Q(z)}$ is a set-valued mapping from $B_Q(z)$ into $CL( B_Q(z))$ and $f|_{ B_Q(z)}: B_Q(z)\to  B_Q(z)$. Then eventually 
		$$Fix(f|_{ B_Q(z)},F|_{ B_Q(z)})=Fix(f,F)\cap B_Q(z).$$
		Thus this theorem is followed by applying our main Theorem \ref{thm3.2} under the assumption $W=B_Q(z)$. 	
	\end{proof}
	
	
	

	Jachymski \cite[Theorem 4.1]{Jach}, established the convergence of iterates for some linear operator with the perception of fixed point theory on a normed linear space, which is indeed an extension of Kelisky-Rivlin \cite[Theorem 1]{KR8} theorem about the convergence of iterates for Bernstein operator \cite{KR8}. Later, Sultana and Vetrivel \cite[Theorem 5]{mam_feng} improved this result on a complete normed linear space. The coming theorem deals with the convergence of iterates for certain operators in a normed linear space, where the operators need not to be linear. Essentially, our theorem generalizes the Sultana and Vetrivel's \cite[Theorem 5]{mam_feng} result.   
	

	\begin{theorem}\label{iterates}
		For the complete normed linear space $(W,\|.\|)$, let $W_0\subset W$ be a closed subspace. A map $T:W\to W$ with $(I-T)(W)\subseteq W_0$ and for $w-Tw\in W_0$,
		\begin{equation}\label{convergence}
			\|Tw-T^2w\|\leq k(\|w-Tw\|)\|w-Tw\|\quad\text{where}~k\in U.
		\end{equation} 
		Then for every $w\in W$, $\lim_{n \rightarrow \infty}T^nw\in \{w^*\in W: w^*=Tw^*\}\cap (w+W_0)$ if $h:W\to \mathbb{R}$ by $h(u)=D(u,Tu)$ is lower semicontinuous.
	\end{theorem}

	\begin{proof}
		Take a graph $G$ on $W$ in order that $V(G)=W$ and $E(G)$ is the collection of $(v,w)\in W\times W$ having $v-w\in W_0$. Consider $f$ is an identity map on $W$ and a set valued map $F$ is chosen by $Fw=\{Tw\}$, for each $w\in W$. Let $v$ be any arbitrary point in $W$ and $w=fw\in Fv=\{Tv\}$ with $(fv,fw)=(v,w)\in E(G)$. Hence $(v,Tv)\in E(G)$, that is $v-Tv\in W_0$. Consequently by (\ref{convergence}), $\|fw-Fw\|=\|Tv-T(Tv )\|=\|Tv-T^2v\|\leq k(\|v-Tv\|)\|v-Tv\|$ where $k\in U$. On the account of $(I-T)(W)\subseteq W_0$, it yields that for $p=fp \in Fw=\{Tw\}$, we have $w-p=w-Tw\in W_0$, that is $(w,p)=(fw,fp)\in E(G)$. Hence condition (i) and (ii) of Theorem \ref{thm3.2} is fulfilled. Choose an element $w\in W$. Moreover, for $w=w_0\in W$, there is $p_0\in F(w_0)$ with $(fw_0,fp_0)\in E(G)$ due to the fact that $(I-T)(W)\subseteq W_0$.
		Thus for $w\in W$, applying the previous Theorem \ref{thm3.2} we obtain that $\lim_{n\to \infty}T^nw\in \{w^*\in W: w^*=Tw^*\}$.
		
		As $(I-T)W\subseteq W_0$, then it is easy observe that $w-T^nw\in W_0$ for each $n \geq 1$. On the account of $W_0$ is closed, we obtain $\lim_{n \rightarrow \infty}T^nw\in (w+W_0)$. Hence we conclude that  $\lim_{n \rightarrow \infty}T^nw\in \{w^*\in W: w^*=Tw^*\}\cap (w+W_0)$. 
	\end{proof}	
	

	\section{Applications}
	In this part we provide some applications of our main Theorem \ref{thm3.2}. First we discuss about the occurrence of solution of a fractional differential equation and at the end of this article, we deduce the convergence of iterates for a nonlinear $q$-analogue Bernstein operator. 
	
	A fractional differential equation is a generalization of a differential equation, whose order lies in the set of natural numbers. A fractional derivative looks like $D^\beta$, where $\beta>0$ and the symbol ${}^cD^\beta$ denotes the Caputo fractional derivative \cite{baleanu} with order $\beta$. In fact if $\beta \in \mathbb{N}$, then fractional differential equation turns out to be a normal differential equation. The presence of solution for fractional differential equations are deduced through the perception of fixed point theory, (see \cite{baleanu,khan,gopal}).   
	In this article, we assume a generalized fractional differential equation
	\begin{equation}\label{eqn6}
		{}^cD^\beta w(b)+g(b,f(w(b)))=0~ \textrm{whenever}~~ w\in C[0,1],~ 0<b<1~\textrm{and}~1<\beta,
	\end{equation}
	having boundary criteria $w(0)=0$ and $w(1)=0$, where the maps $f:C[0,1]\to C[0,1]$ and $g:[0,1]\times C[0,1]\to \mathbb{R}$ are both continuous. The Green function $G(b,a)$ corresponding to the equation $(\ref{eqn6})$ takes the value $(b(1-a))^{\beta-1}-(b-a)^{\beta-1}$ whenever $0\leq a\leq b\leq 1$, and it attains the value $\frac{(b(1-a))^{\beta-1}}{\Gamma(\beta)}$ whenever $0\leq b\leq a\leq1$.  



	
	
	

	The collection of every continuous real-valued mappings on $[0,1]$ is represented by $C([0,1])$, which incorporates with $\|.\|=\max_{b\in [0,1]}|w(b)|$. By applying our main Theorem \ref{thm3.2}, in the succeeding result we establish the sufficient criteria for ensuring a solution of the above mentioned fractional differential equation $(\ref{eqn6})$.
	

	\begin{theorem}\label{thm4.1}
		Take the fractional differential equation as stated in $(\ref{eqn6})$. Assume that the succeeding criteria happen:
		\begin{enumerate}
			\item[(i)] for every $v,w\in C[0,1]$ and an element $b$ lies in $[0,1]$,
			\begin{equation*}
				\left|g(b,f(v(b)))-g(b,f(w(b)))\right|\leq k\left(\left\|fv-fw\right\|\right)\left|fv(b)-fw(b)\right|
			\end{equation*} 
		    where $k\in U$,
			\item[(ii)] for each $w\in C[0,1]$, $Fw\subseteq f\left(C[0,1]\right)$ where $F$ is a single valued mapping on $C[0,1]$, which is stated as
			\begin{equation*}
				F(w(b))=\int_0^1G(b,a)g(a,(fw)(a)) da.
			\end{equation*}
		\end{enumerate}
		Then equation $(\ref{eqn6})$ admits a solution if  $ f(C[0,1])$ is closed set in $C[0,1]$.
	\end{theorem}

	\begin{proof}
		We assume that $W=C[0,1]$ incorporate with a graph $G$ having $V(G)=W$ and $E(G)=\{(v,w):(v,w)\in W\times W\}$. Now we visualize that, a function $w$ lies in $W$ meets the equation $(\ref{eqn6})$, if and only if it fulfils the succeeding equation,
		\begin{equation}\label{eqn7}
			w(b)=\int_0^1G(b,a)g(a,(fw)(a))da.
		\end{equation}
		Let $v\in W$ and $fw\in Fv=\int_0^1G(b,a)g(a,(fv)(a))da$. Now
		\begin{eqnarray*}
			|fw(b)-Fw(b)|&= & \left|\int_0^1G(b,a)g(a,(fv)(a))da-\int_0^1G(b,a)g(a,(fw)(a))da\right|\\
			&\leq &\int_0^1\left|G(b,a)\right|\left|g(a,(fv)(a))-g(a,(fw)(a))\right|da\\
			&\leq &k\left(\left\|fv-fw\right\|\right)\int_0^1\left|G(b,a)\right|\left|fv(a)-fw(a)\right|da~\text{[from $(i)$]}\\ 
			&\leq &k\left(\left\|fv-fw\right\|\right)\|fv-fw\|\int_0^1\left|G(b,a)\right|da\\
			&\leq & k\left(\left\|fv-fw\right\|\right)\|fv-fw\|.
		\end{eqnarray*} 
		Therefore $\|fw-Fw\|\leq k(\|fv-fw\|)\|fv-fw\|$. Assume $\{fu_n\}_n \in W$ in order that $fu_n\to fu$. Consequently for each $n$,
		\begin{eqnarray*}
			|fu-Fu|&\leq &|fu-fu_n|+|fu_n-Fu_n|+|Fu_n-Fu|\\
			&\leq & \left|\int_0^1G(b,a)g(a,(fu_n)(a))da-\int_0^1G(b,a)g(a,(fu)(a))da\right|\\  & &~ + |fu-fu_n|+|fu_n-Fu_n|\\
			&\leq & |fu-fu_n|+|fu_n-Fu_n|+k(\|fu_n-fu\|)\|fu_n-fu\|\textrm{[by~ $(i)$]}.
		\end{eqnarray*}
		Therefore for each $n$, $\|fu-Fu\|\leq \|fu-fu_n\|+\|fu_n-Fu_n\|+\|fu_n-fu\|$. Hence $\|fu-Fu\|\leq\liminf_{n\to \infty}\|fu_n-Fu_n\|$. Then using Theorem \ref{thm3.2} we can conclude that $F$ and $f$ has a function $w^*(b)\in W$ in order that $Fw^*(b)=f(w^*(b))$. Hence $f(w^*(b))$ fulfils the equation $(\ref{eqn7})$, therefore the given equation $(\ref{eqn6})$ has a solution.
	\end{proof}

	
	For $n\in\mathbb{N}$ and $q>0$, Lupa\c{s} \cite{lupas} first presented the $q$-analogue Bernstein operator $L_{n,q}$ on the space $C[0,1]$, in the year $1987$. In fact if $q=1$, then this operator reduces to the classical Bernstein operator \cite{KR8}. For $q>0$ and any $\phi$ lies in $C[0,1]$, we consider a nonlinear $q$-analogue Bernstein operator,
	\begin{equation}\label{eq1_Bern}
		\tilde {L}_{n,q}(\phi(a))=\sum_{i=0}^n \left|\phi\bigg(\frac{[i]_q}{[n]_q}\bigg)\right|b_{n,i}(q,a)\quad\textrm{for}~a\in[0,1],
	\end{equation}
	where for each $i\in \mathbb{Z}^{+}$,
	\begin{equation*}
		b_{n,i}(q,a)=\frac{\left[\begin{array}{l}{n}\\{i}\end{array}\right]_q q^{\frac{i(i-1)}{2}}a^i(1-a)^{n-i}}{\prod_{j=0}^{n-1}\{1-a+q^ja\}}, 
		\hspace*{0.15cm} 
		\left[\begin{array}{l}{n}\\{i}\end{array}\right]_q=\frac{[n]_q!}{[n-i]_q![i]_q!},
		\hspace*{0.15cm}
		[n]_q!=\prod_{i=1}^{n}[i]_q,
	\end{equation*}
	and $[0]_q!=1$, where $[i]_q=\sum_{k=0}^{i-1}q^{k}$ for $i\geq 1$ and $[0]_q=0$.
	Agratini \cite{Agra} first studied the convergence of iterates of $q$-analogue Bernstein operator $L_{n,q}$ on the space $C[0,1]$ in the year 2008. In the upcoming corollary we derive the the convergence of iterates for nonlinear  $q$-analogue Bernstein operator $\tilde{L}_{n,q}$ through Theorem \ref{iterates}.
	

	\begin{corollary}
		Suppose that $\tilde{L}_{n,q}~(q>0)$ is an operator as mentioned in $(\ref{eq1_Bern})$. Then for each $\varphi\in C[0,1]$ and fixed $n\in \mathbb{N}$, $\lim_{j\to \infty}\tilde{L}_{n,q}^j\varphi \in \{\varphi^{*}\in C[0,1]:\tilde{L}_{n,q}\varphi^{*}=\varphi^{*}\}$. Furthermore, 
		for each $\varphi\in W^{'}=\{\varsigma\in C[0,1]$: $\varsigma(1)\geq 0$ and $\varsigma(0)\geq 0$\},
		\begin{equation*}
			\lim_{j\to \infty}(\tilde{L}_{n,q}^j\varphi)(a)=\varphi(0)(1-a)+\varphi(1)a \quad \textrm{whereas}~0\leq a\leq 1.
		\end{equation*}
	\end{corollary}

	\begin{proof}
		Consider $W=C[0,1]$ with supremum norm. Let $W_0\subseteq W$ consist of all $\varphi\in W$ whose zeros are $0$ and $1$. Take a function $\phi$ in $W$ in order that $\phi-\tilde{L}_{n,q}\phi\in W_0$. Then for each $a$ lies in $[0,1]$ we have,
		\begin{eqnarray}
		
			\big|\tilde{L}_{n,q}\phi(a)-\tilde{L}_{n,q}^2\phi(a)\big|&\leq&\sum_{i=1}^{n-1} b_{n,i}(q,a) \left|(\phi-\tilde{L}_{n,q}\phi)\left(\frac{[i]_q}{[n]_q}\right)\right|\nonumber\\
			&\leq&\left[1-b_{n,n}(q,a)-b_{n,0}(q,a)\right]\|\phi-\tilde{L}_{n,q}\phi\|.\label{eq10_application2}
		
		\end{eqnarray}
		Consider $h_{n}(a)=q^{n(n-1)/2}a^n+(1-a)^n$, then for $n=1$, the minimum value of $h_{n}(a)$ is $1$. For $n \geq 2$, the minimum value of $h_{n}(a)$  is $\left(q^{n/2}/(1+q^{n/2})\right)^{n-1}$, which occurs at $a=(1+q^{n/2})^{-1}$. 
		On the other hand consider $g_{n}(a)=\prod_{s=0}^{n-1}(1-a+q^s a)$, then one can easily verify that maximum of $g_{n}(a)$ over $[0,1]$ is less than or  equal to $\max\{1,q^{(n-1)^2}\}$. Consequently, 
		\begin{equation}
			\frac{1}{\max\{q^{(n-1)^2},1\}}\bigg(\frac{q^{n/2}}{1+q^{n/2}}\bigg)^{n-1} \leq \displaystyle{\frac{h_{n}(a)}{g_{n}(a)}}= b_{n,n}(q,a)+b_{n,0}(q,a).
		
			\label{eq9_application2}
		\end{equation}
		Again we observe that $0<b_{n,q}=\frac{1}{\max\{1,q^{(n-1)^2}\}}\bigg(\frac{q^{n/2}}{1+q^{n/2}}\bigg)^{n-1} \leq 1$. Subsequently from (\ref{eq10_application2}) and (\ref{eq9_application2}), it follows that
		$||\tilde{L}_{n,q} \phi-\tilde{L}_{n,q}^2\phi||\leq k(||\phi-\tilde{L}_{n,q} \phi||)||\phi-\tilde{L}_{n,q} \phi||$, where $\forall\,t \geq 0$, $k(t)=(1-b_{n,q})\in U$.
		Consider a sequence $\{\eta_j\}_j\in W$ in order that $\eta_j\to \eta$. Then for every $a\in[0,1] $ and $j\in \mathbb{Z}^{+}$,
		\begin{equation*}
			\begin{aligned}
			|\eta(a) -\tilde{L}_{n,q}\eta(a)|&\leq|\eta(a)-\eta_j(a)|+|\eta_j(a) -\tilde{L}_{n,q}\eta_j(a)|+|\tilde{L}_{n,q}\eta_j(a)-\tilde{L}_{n,q}\eta(a)|\\
			&\leq \|\eta-\eta_j\|+\|\eta_j -\tilde{L}_{n,q}\eta_j\|+\|\eta_j-\eta\|.
			\end{aligned}
		\end{equation*}
		Therefore $\|\eta -\tilde{L}_{n,q}\eta\|\leq\|\eta-\eta_j\|+\|\eta_j -\tilde{L}_{n,q}\eta_j\|+\|\eta_j-\eta\|$. Now taking $j\to \infty$, we have $\|\eta -\tilde{L}_{n,q}\eta\|\leq\liminf_{j\to \infty}\|\eta_j -\tilde{L}_{n,q}\eta_j\|$. Again it is simple to observe that $(I-\tilde{L}_{n,q})(W)\subseteq W_0$. Then using the Theorem $\ref{iterates}$, for any $\varphi\in W$, 
		$\displaystyle{\left\{\lim_{j\to \infty}\tilde{L}_{n,q}^j\varphi\right\}}\subseteq(\varphi+W_0)\cap Fix\,\tilde{L}_{n,q}$. It is simple to verify that the set $(\varphi+W_0)\cap Fix\,\tilde{L}_{n,q}$ is singleton, also $a$ and $1-a$ are fixed points of $\tilde{L}_{n,q}$. Again for each $\varphi\in W^{'}$, $\varphi(0)(1-a)+\varphi(1)a \in Fix\,\tilde{L}_{n,q} $. Furthermore, $\varphi(0)(1-a)+\varphi(1)a\in \varphi+W_0$. Consequently, we conclude that 	$\lim_{j\to \infty}(\tilde{L}_{n,q}^j\varphi)(a)=\varphi(0)(1-a)+\varphi(1)a$ for each $\varphi\in W^{'}$.
	\end{proof}
	
	
	
	
	

	























	
	\section*{Acknowledgment}
	The first author is thankful to the Ministry of Human Resource Development (MHRD), India for the financial support.

	
	
	