\section{Introduction}\label{sec1}

Let $(Z,d)$ be a metric space and $CL(Z)$ denotes the collection of non-void closed subsets of $Z$. The set $CB(Z)$ contains all the non-void bounded closed subsets of $Z$. For any $P,Q\in CB(Z)$, 
\begin{equation}
	H(P,Q)=\max\bigg\{\sup_{u\in Q}D(u,P), \sup_{v\in P}D(v,Q)\bigg\}
\end{equation}
is called Hausdorff metric on $CB(Z)$, where $D(u,P)=\inf\{d(u,v):v\in P\}$. For a mapping $F:Z\to CL(Z)$, an element $z^*\in Z$ is known as a fixed point for the map $F$ if $z^*\in Fz^*$. The occurrence of fixed points for set-valued mappings was first deduced by Nadler \cite{nadler} in the year of 1969. Later, Mizoguchi and Takahashi \cite{mizo} improved the Nadler's \cite{nadler} theorem for a set-valued generalized contraction in the year 1989. The authors \cite{mizo} established the below stated theorem.
\begin{theorem}\cite{mizo}\label{mizo_thm}
	Suppose that $(Z,d)$ is complete and a set-valued map $F:Z\to CB(Z)$ in order that for every $y,z\in Z$,
	\begin{equation}\label{mizo_eq}
		H(Fy,Fz)\leq k(d(y,z))d(y,z),
	\end{equation}
	where $k\in W=\{\theta:[0,\infty)\to [0,1)| \limsup_{t\to s^+}\theta(t)<1,~\forall s\in [0,\infty)\}$. Then there is an element $z^*\in Fz^*$. 
\end{theorem} 
In fact, Suzuki \cite{suzuki} provided an example to deduce that every Mizoguchi -Takahashi contraction \cite{mizo} is not necessarily a Nadler contraction \cite{nadler} on arbitrary metric spaces in the year of 2007. However, Eldred-Anuradha-Veeramani \cite{eldred} observed that any Mizoguchi-Takahashi \cite{mizo} contraction follows a Nadler \cite{nadler} contraction, whenever the metric space $(Z,d)$ is metrically convex complete \cite{eldred} in the year 2009. There are several extensions of the Theorem \ref{mizo_thm} due to Mizoguchi and Takahashi \cite{mizo} in the literature, which can be found in \cite{berinde,mam}.

In the year 1961, Edelstein \cite{edel} generalized the famous Banach contraction principle for uniformly local contraction \cite{edel}. Subsequently, Nadler \cite{nadler} improved the Edelstein's \cite{edel} result for set-valued uniformly local contraction \cite{nadler}. Later, Sultana and Vetrivel \cite{mam} extended the Theorem \ref{mizo_thm} due to Mizoguchi and Takahashi \cite{mizo} for set-valued uniformly local contraction. The authors \cite{mam} derived the succeeding theorem.
\begin{theorem}\cite{mam}\label{mizo_local_thm}
	Suppose that $(Z,d)$ is complete $r$-chainable and a set-valued map $F:Z\to CB(Z)$ in order that for each $y,z\in Z$ having $d(y,z)<r$ (where $r>0$) fulfils,
	\begin{equation}\label{mizo_local}
		H(Fy,Fz)\leq k(d(y,z))d(y,z), 
	\end{equation}
	where $k\in W$. Then there is an element $z^*\in Fz^*$.
\end{theorem}  

It is worth to note that every Mizoguchi-Takahashi local contraction \cite{mam} is not a Nadler \cite{nadler} local contraction, see \cite[Example 3.1]{mam_equivalence}. But, Sultana and Qin \cite{mam_equivalence} demonstrated that these two contractions are equivalent, whenever the metric space $(Z,d)$ is metrically convex complete in the year 2019.

On the other hand, Feng and Liu \cite{feng} improved Nadler's \cite{nadler} theorem for the mappings $F$ from $Z$ into $CL(Z)$ in the year of 2006, which is stated below.
\begin{theorem}\cite{feng}\label{feng_thm}
	Suppose that $(Z,d)$ is a complete metric space and a set-valued map $F:Z\to CL(Z)$ in order that for each $y\in Z$, there is $z\in I_{c\in (0,1)}^y=\{x\in Fy:cd(x,y)\leq D(y,Fy)\}$,
	\begin{equation}\label{feng_eq}
		D(z,Fz)\leq \alpha d(y,z)~\textrm{where}~ 0\leq \alpha\leq c<1.
	\end{equation}
	Then there an element $z^*\in Fz^*$ if $h:Z\to \mathbb{R}$ by $h(z)=D(z,Fz)$ is lower semicontinuous.
\end{theorem}

In this manuscript, inspired by Eldred et. al \cite{eldred} and Sultana-Qin \cite{mam_equivalence}, we establish that any set-valued contraction due to Feng-Liu \cite{feng} follows from the Mizoguchi-Takahashi local contraction \cite{mam_equivalence} on a metrically convex metric space. On any arbitrary metric spaces, this result need not be true, an example is provided for this purpose. The fixed points for the set-valued mappings fulfil the equation $(\ref{mizo_local})$ are also studied on metrically convex spaces. Furthermore, the equivalence of Mizoguchi-Takahashi local contraction \cite{mam_equivalence} and Nadler local contraction \cite{nadler} is established in a compact metric space. An invariant approximation result is also discussed through our main theorem.
\section{Notations and Definitions}
This segment contains with some mathematical definitions and notations, which are needful up to the end of this paper. For the metric space $(Z,d)$, let $u,v$ be any two points in $Z$. Then a point $w$ in $Z$, where $u\neq w\neq v$, is called metrically \cite{eldred} between $u$ and $v$ if $d(u,v)=d(u,w)+d(w,v)$. The space $(Z,d)$ is called metrically convex \cite{eldred} if for every pair of elements $u$ and $v$ in $Z$, there is an element $w\in Z$ in order that $w$ is metrically between $u$ and $v$. The below stated lemma is regarding a property of metrically convex space, which is needful for establish our main theorem.

\begin{lemma}\label{metrically}\cite{eldred}
	Assume that $(Z,d)$ is metrically convex complete and $u,v\in Z$ are any arbitrary points. Then there is $[a_1,a_2]\subset \mathbb{R}$ and an isomorphism $\varphi:[a_1,a_2]\to Z$ in order that $\varphi(a_1)=u$, $\varphi(a_2)=v$.
\end{lemma} 

Again for the metric space $(Z,d)$, consider a self map $\Psi$ on $Z$. Then $\Psi$ is defined as uniformly local contractions \cite{edel} if for each $y,z \in Z$ having $d(y,z)<r$ fulfils $d(\Psi (y),\Psi (z))\leq \alpha d(y,z)$, where $0\leq \alpha <1$ and $r>0$. Furthermore, for $y,z\in Z$, a $r$-chain \cite{edel} between $y$ and $z$ means, there is a finite sequence $y=z_0,z_1,\cdots,z_n=z$ in $Z$, such that for every $j\in\{1,2,\cdots,n\}$, $d(z_{j-1},z_j)<r$. The space $(Z,d)$ is stated as $r$-chainable \cite{edel} whenever there is a $r$-chain between every points $y$ and $z$ in $Z$.




\section{Main Result}
Now we commence our main theorem about that any Mizoguchi-Takahashi local contraction \cite{mam}, that is, the mappings that meet the equation (\ref{mizo_local}), is actually a Feng-Liu contraction \cite{feng} on a metrically convex metric space $(Z,d)$.
\begin{theorem}\label{main}
	Suppose that $(Z,d)$ is a complete metric space having metrically convexity property and a set-valued map $F:Z\to CB(Z)$ in order that for $y,z\in Z$ having $d(y,z)<r$ (where $r>0$),
	\begin{equation}\label{MT}
		H(Fy,Fz)\leq k(d(y,z))d(y,z)
	\end{equation}
	where $k\in W$. Then $F$ becomes a Feng-Liu contraction, that is, for each $y^{'}\in Z$, there is $z^{'}\in I_c^{y^{'}}$ fulfils,
	\begin{equation*}
		D(z^{'},Fz^{'})\leq \alpha d(y^{'},z^{'})~\textrm{where}~0\leq \alpha <c<1.
	\end{equation*}
\end{theorem}
\begin{proof}
	Our main goal is to show that there is certain $\alpha$, where $0\leq \alpha<c<1$ in order that for $y\in Z$, $z\in I_c^y$ meets $D(z,Fz)\leq \alpha d(y,z)$. Suppose that
	\begin{equation}
		M=\{m>0: \sup\{k(d(y,z)):d(y,z)\in [0,m]\}=1\}.
	\end{equation}
	Then there are two possibilities that could happen. Firstly, we assume that the set $M$ is void. Then for each positive $m$, the supremum value of $k(d(y,z))$ is strictly less than $1$, where $d(y,z)\in [0,m]$.
	Since $r>0$ is fixed, we assume that 
	\begin{equation}\label{set}
		\sup\{k(d(y,z)):d(y,z)\in [0,r]\}=\alpha<1.
	\end{equation}
	Evidently, $\alpha\in [0,1)$. Let $y^{'}\in Z$ be a fixed element. Then there is an element $z^{'}\in I_c^y{^{'}}$ where $c>\alpha$, due to the fact that $F(y^{'})\subset CL(Z)$. Consequently, for chosen $y^{'}$ and $z^{'}$, an isometry $\varphi_1:[t_1,t_2]\to Z$ exists satisfies $\varphi_1(t_1)=y^{'}$ and $\varphi_1(t_2)=z^{'}$ (where $[t_1,t_2]\subset \mathbb{R}$ ) according to the Lemma \ref{metrically}.
	Now for a fixed positive real number $q<r$, we can always find $L\in \mathbb{N}$ satisfying $t_1+Lq<t_2$ and $t_2\leq t_1+(L+1)q$. Consequently,
	\begin{equation}\label{isomorphism}
		\begin{aligned}
			d(y^{'},z^{'})&=d(\varphi_1(t_1),\varphi_1(t_2))=|t_1-t_2|\nonumber\\
			&=|t_1-(t_1+q)|+|(t_1+q)-(t_1+2q)|+\cdots+|(t_1+Lq)-t_2|\nonumber\\
			&=d(y^{'},\varphi_1(t_1+q))+d(\varphi_1(t_1+q),\varphi_1(t_1+2q))+\cdots+\nonumber\\
			&\qquad d(\varphi_1(t_1+Lq),z^{'}).
		\end{aligned}
	\end{equation}

	
	
	
	
	Again, it is simple to observe that $d(z^{'},\varphi_1(t_1+Lq))\leq q<r$ and $d(\varphi_1(t_1+nq),\varphi_1(t_1+(n+1)q))=q<r$ for every $0\leq n<L$, that is every terms of right hand side of the equation (\ref{isomorphism}) is less than $r$. Therefore for chosen $y^{'}\in Z$ and $z^{'}\in I_c^{y^{'}}$, it yields


		\begin{align*}
			D(z^{'},Fz^{'})&\leq H(Fy^{'},Fz^{'})\\
			&\leq H(Fy^{'},F(\varphi_1(t_1+q)))+\cdots+H(F(\varphi_1(t_1+Lq)),Fz^{'})\\
			&\leq  \alpha[d(y^{'},\varphi_1(t_1+q))+\cdots+d(\varphi_1(t_1+Lq),z^{'})]\\
		
			&\leq \alpha d(y^{'},z^{'}).\qquad[\text{using}~(\ref{isomorphism}) ]
		\end{align*}
	
	
	
	
	
	
	
	
	
	
	
	
	
	Next, we assume that the set $M$ is non-void and $m_0$ is the infimum of $M$. If possible, let $m_0=0$. Then we are able to find two sequences $\{d(y_n,z_n)\}_n$, $\{k(d(y_n,z_n))\}_n$ in order that $d(y_n,z_n)\to 0$ and $k(d(y_n,z_n))\to 1$, which contradicts to the criteria of the map $k\in W$. Hence $m_0\neq 0$. Then for a chosen positive real $q_0<m_0$, we have
	\begin{equation}\label{set1}
		\sup\{k(d(x_1,x_2)):d(x_1,x_2)\in [0,q_0]\}=\alpha_0<1.
	\end{equation} 
	Let $y^{'}\in Z$ be a fixed element. Subsequently, there is $z^{'}\in I_c^{y^{'}}$, where $c>\alpha_0$. Then for that $y^{'}$ and $z^{'}$, there is an isometry $\varphi_2:[s_1,s_2]\to Z$ in order that $\varphi_2(s_1)=y^{'}$ and $\varphi_2(s_2)=z^{'}$, where $[s_1,s_2]\subset \mathbb{R}$. Consequently, for certain fixed positive number $a<\min\{q_0,r\}$, we can find $L^{'}\in \mathbb{N}$ satisfying $s_1+L^{'}a<s_2$ and $s_2\leq s_1+ (L^{'}+1)a$.
	Evidently,
	\begin{equation}\label{isomorphism1}
		\begin{aligned}
			d(y^{'},z^{'})&=d(\varphi_2(s_1),\varphi_2(s_2))=|s_1-s_2|\nonumber\\
			&=|s_1-(s_1+a)|+|(s_1+a)-(s_1+2a)|+\cdots+|(s_1+L^{'}a)-s_2|\nonumber\\
			&=d(y^{'},\varphi_2(s_1+a))+d(\varphi_2(s_1+a),\varphi_2(s_1+2a))+\cdots+\nonumber\\
			&\qquad d(\varphi_2(s_1+L^{'}a),z^{'}).
		\end{aligned}
	\end{equation}
	Again, it is simple to check that $d(z^{'},\varphi_2(s_1+L^{'}a))\leq a$ and $d(\varphi_2(s_1+na),\varphi_2(s_1+(n+1)a))=a$ for every $0\leq n<L^{'}$, that is each terms of right hand side of the equation (\ref{isomorphism1}) is less than or equal to $a$. Therefore for that chosen $y^{'}\in Z$ and $z^{'}\in I_c^{y^{'}}$, it yields	

	
	

	\begin{align*}
		D(z^{'},Fz^{'})\leq& H(Fy^{'},Fz^{'})\\
		\leq&H(Fy^{'},F(\varphi_2(s_1+a)))+\cdots+H(F(\varphi_2(s_1+L^{'}a)),Fz^{'})\\
		\leq & \alpha_0[d(y^{'},\varphi_2(s_1+a))+\cdots+d(\varphi_2(s_1+L^{'}a),z^{'})]\\
	
		\leq&\alpha_0 d(y^{'},z^{'})\qquad[\text{using}~(\ref{isomorphism1})].
	\end{align*}
	Thus $F$ becomes a Feng-Liu contraction. 
\end{proof}

In the succeeding theorem, we establish the fixed points for Mizoguchi-Takahashi local contraction on a metrically convex space. Indeed, this result follows from our above mentioned Theorem \ref{main} and the Theorem \ref{feng_thm} due to Feng and Liu \cite{feng}. Although, this theorem is established by Sultana and Qin \cite[Corollary 3.1]{mam_equivalence} in the year 2019. 
\begin{theorem}\label{cor1}\cite{mam_equivalence}
	Suppose that $(Z,d)$ is metrically convex complete and a set-valued map $F:Z\to CB(Z)$ meets the equation $(\ref{mizo_local})$. Then presence of $z^*\in Z$ in order that $z^*\in Fz^*$ can be assured.
\end{theorem}
\begin{proof}
	Since $F$ meets the equation $(\ref{mizo_local})$ and $Z$ is metrically convex, then by Theorem \ref{main}, the map $F$ fulfils the Feng-Liu contraction. As a consequence, for each $y\in Z$, $z\in I_c^y$ where $0\leq \alpha<c<1$ fulfils
	\begin{eqnarray*}
		D(z,Fz)\leq \alpha d(y,z).
	\end{eqnarray*}
	Now our aim is to establish that $h:Z\to \mathbb{R}$ by $h(z)=D(z,Fz)$ is lower semicontinuous. Let us assume $\{q_n\}_n\in Z$ in order that $q_n\to q$. Then we achieve $l\in \mathbb{N}$ in order that $d(q_n,q)<r$, for each $n\geq l$. Now for every $n\geq 1$ and $z\in Fq_n$,
	\begin{eqnarray}
		D(q,Fq)&\leq&d(q,q_n)+d(q_n,z)+D(z,Fq)\nonumber\\
		&\leq&d(q,q_n)+d(q_n,z)+H(Fq_n,Fq)\nonumber\\
		&\leq&d(q,q_n)+D(q_n,Fq_n)+H(Fq_n,Fq).
	\end{eqnarray}
	As for all $n\geq l$, $d(q_n,q)<r$, then the last inequation leads to 
	\begin{eqnarray*}
		D(q,Fq)&\leq& d(q,q_n)+D(q_n,Fq_n)+k(d(q_n,q))d(q_n,q)\\
		&< & d(q,q_n)+D(q_n,Fq_n)+d(q_n,q).
	\end{eqnarray*}
	Taking $n\to \infty$, we conclude that $D(q,Fq)\leq\liminf_{n\to \infty}D(q_n,Fq_n)$. Now applying the Theorem \ref{feng_thm}, we are able to find $z^*\in Z$ in order that $z^*\in Fz^*$.
\end{proof}

In 2009, Eldred et. al \cite[Theorem 2.4]{eldred} also established that Mizoguchi-Takahashi contraction \cite{mizo} follows Nadler's contraction \cite{nadler}, whenever the metric space is compact. In the below stated theorem, we establish the equivalence between the Mizoguchi-Takahashi local contraction \cite{mam} and the Nadler local contraction \cite{nadler} in a metric space which is compact.
\begin{theorem}
	Let $(Z,d)$ be compact and a set-valued map $F:Z\to CB(Z)$ meets the equation $(\ref{mizo_local})$. Then for each $y,z\in Z$ having $d(y,z)<r$ (where $r>0$), 
	\begin{equation*}
		H(Fy,Fz)\leq \alpha\, d(y,z)\quad \text{where}~~0
		\leq\alpha< 1.
	\end{equation*}
\end{theorem}
\begin{proof}
	Let $0<q<r$ be a fixed real number. Now consider a real number $m$, which is the supremum of $\frac{H(Fy,Fz)}{d(y,z)}$, where the elements $y$ and $z$ lies in $Z$ having $0< d(y,z)\leq q$.

	
	

	
	
	We want to show that $m<1$. If possible let $m=1$, then there is $\{(y_n,z_n)\}_n\in Z\times Z$ having $d(y_n,z_n)\in (0,q]$ fulfils 
	\begin{equation}\label{limit}
		\lim_{n\to\infty}\frac{H(Fy_n,Fz_n)}{d(y_n,z_n)}=1.
	\end{equation}
	On the account of $d(y_n,z_n)\leq q<r$, we have $\frac{H(Fy_n,Fz_n)}{d(y_n,z_n)}\leq k(d(y_n,z_n))<1$. Taking $n\to \infty$, it yields that $k(d(y_n,z_n))\to 1$. If $d(y_n,z_n)$ converges to $0$, then it contradicts the criteria of $k\in W$. Hence we assert that $\{d(y_n,z_n)\}_n$ converges to a positive real number.
	
	As $\{(y_n,z_n)\}_n\in Z\times Z$ and $Z$ is compact, then there is $\{(y_{n_l},z_{n_l})\}_{l\in \mathbb{N}}$ in order that $(y_{n_l},z_{n_l})\to (y^*,z^*)$. Consequently, $d(y_{n_l},z_{n_l})\to d(y^*,z^*)$. Since $d(y_n,z_n)\in (0,q]$, then $d(y^*,z^*)\in (0,q]$, that is $d(y^*,z^*)\leq q<r$. Now
	\begin{eqnarray*}
		\lim_{l\to\infty}\frac{H(Fy_{n_l},Fz_{n_l})}{d(y_{n_l},z_{n_l})}&=&\frac{H(Fy^*,Fz^*)}{d(y^*,z^*)}\\
		&\leq & \frac{k(d(y^*,z^*))d(y^*,z^*)}{d(y^*,z^*)}\qquad [\because~d(y^*,z^*)<r]\\
		&=&k(d(y^*,z^*))<1.
	\end{eqnarray*}
	This contradicts the equation $(\ref{limit})$. Thus we can conclude that $m<1$, that is there is an non-negative $\alpha$ in order that $m=\alpha<1$. Therefore for each $y,z\in Z$ having $d(y,z)<r$, we have
	\begin{equation*}
		H(Fy,Fz)\leq k(d(y,z))d(y,z)\leq \alpha d(y,z)\quad\text{where}~\alpha\in [0,1).
	\end{equation*}
\end{proof}

Many mathematicians are interested to deal the invariant best approximation problems through fixed point, see (\cite{thagafi,pathak}). In \cite{kamran1,kumam}, the authors established invariant best approximation theorems for set-valued mappings. In the following, using the Theorem \ref{feng_thm} due to Feng-Liu \cite{feng} and our main Theorem \ref{main}, we derive a invariant best approximation theorem for a set-valued mapping which meets the criteria $(\ref{mizo_local})$ on a metrically convex space. Let $P$ be a non-void subset of a normed linear space $(Z,\|.\|)$ and $p\in Z$. Then a collection of elements $y\in P$ in order that $D(p,P)=\|y-p\|$ is described as $B_P(p)$ and it is stated as a best $P$-approximates of $p$ over $P$.


\begin{theorem}
	Let $P$ be a subset of a metrically convex normed linear space $(Z,\|.\|)$ and a set-valued map $F:P\to CB(P)$ in order that for every $y,z\in P$ having $d(y,z)<r$ fulfils $(\ref{mizo_local})$. Suppose that the below stated criteria hold:
	\begin{enumerate}
		\item [(i)] the set $B_P(p)$ is a complete subset of $P$,
		\item [(ii)] for every $x\in B_P(p)$, $\sup_{y\in Fx}\|y-p\| \leq \|x-p\|$.
	\end{enumerate}
	Then there is $z^*\in B_P(p)\cap \{z\in Z: z\in Fz\}$ if the function $h:B_P(p)\to \mathbb{R}$ is lower semicontinuous.
\end{theorem}
\begin{proof}
	As $Z$ is metrically convex and the set-valued map $F:P\to CB(P)$ fulfils $(\ref{mizo_local})$ for each $y,z\in P$ having $d(y,z)<r$, then by the Theorem \ref{main}, for each $y\in B_P(p)\subseteq P$, $z\in I_c^y$ meets $D(z,Fz)\leq \alpha d(y,z)$, where $0\leq \alpha<c<1$. Now let $x\in B_P(p)$ and $z\in Fx$. As $x\in B_P(p)$, then $D(p,P)=\|p-x\|$. Again from $(ii)$ we have
	\begin{equation*}
		\|z-p\|\leq \sup_{y\in Fx}\|y-p\| \leq \|x-p\|=D(p,P). 
	\end{equation*}  
	Thus we can conclude that $\|z-p\|=D(p,P)$. Therefore $z\in B_P(p)$. Hence $Fx\subseteq  B_P(p)$ for each $x\in B_P(p)$, that is $F( B_P(p))\subseteq  B_P(p)$. Immediately it observe that $Fx$ are also closed for any $x\in B_P(p)$, on the account of $Fx\subset CL(P)$, for each $x\in P$. Consequently $F|_{ B_P(p)}: B_P(p)\to CL( B_P(p))$. Thus $F$ fulfils all the criteria of the Theorem \ref{feng_thm} on the set $B_P(p)$ and hence
	$$w^*\in \{z\in B_P(p):z\in F|_{ B_P(p)}z\}= \{z\in Z: z\in Fz\}\cap B_P(p).$$
\end{proof}









The succeeding example illustrates that any uniformly local contraction (\ref{mizo_local}) due to Sultana and Vetrivel \cite{mam} does not follow  Feng-Liu contraction \cite{feng} on any arbitrary metric spaces.  

\begin{example}
	Consider $Z=[0,\frac{1}{2}]\cup \{n\in \mathbb{N}: n \geq 2\}$ and $d:Z\times Z\to \mathbb{R}$ is defined by
	\begin{center}
		$d(y,z)=
		\begin{cases}
			~0 &\mbox{if}~y=z,\\
			~|y-z| &\mbox{if}~0\leq y,z\leq \frac{1}{2},\\
			~y+z &\mbox{if one of}~ y,z\notin[0,\frac{1}{2}].
		\end{cases}$
	\end{center}
	Indeed $(Z,d)$ is not a metrically convex metric space. Now define a map $F:Z\to CB(Z)\subseteq CL(Z) $ in order that 	
	\begin{center}
		$Fz=
		\begin{cases}
			~\left\{\frac{1}{2}z^2 \right\} &\mbox{if}~z\in[0,\frac{1}{2}],\\
			~\{7,2z-1\} &\mbox{if}~ z\in\{2,3,4,\cdots\}.
		\end{cases}$
	\end{center}
	Our goal is to show that $F$ is not a Feng-Liu contraction but a Mizoguchi-Takahashi local contraction. Now for $3\in Z$, $F3=\{7,5\}$. Then there are two cases that arise.
	\begin{enumerate}
		\item [(i)] Choose $7\in \{7,5\}$, then $F7=\{7,13\}$. Now we observe that $D(7,F7)=\inf\{d(7,7),d(7,13)\}=\inf\{14,20\}=14$. Again $d(3,7)=10$. Therefore $$D(7,F7)> d(3,7).$$
		
		\item[(ii)] Choose $5\in \{7,5\}$, then $F5=\{7,9\}$. Now we see that $D(5,F5)=\inf\{d(5,7),d(5,9)\}=\inf\{12,14\}=12$. Again $d(3,5)=8$. Therefore $$D(5,F5)> d(3,5).$$
	\end{enumerate} 
	Hence for $y=3\in Z$ and every $z\in F3$, we obtain that 
	$$D(z,Fz)>d(y,z).$$
	This indicates that $F$ is not a Feng-Liu contraction. On the other side, when $d(y,z)<\frac{1}{2}$, then it is easy to visualize that $y,z\in [0,\frac{1}{2}]$. Now 
	\begin{eqnarray*}
		H(Fy,Fz)&=&H\left(\left\{\frac{1}{2}y^2\right\},\left\{\frac{1}{2}z^2\right\}\right)\\
		&\leq& \frac{1}{2}(y+z)d(y,z).
	\end{eqnarray*} 
	Consider a map $k:[0,\infty)\to [0,1)$ by
	\begin{center}
		$k(s)=
		\begin{cases}
			~\frac{3}{4 } &\mbox{if}~s\in[0,1/2)\\
			~0 &\mbox{if}~ s\in[1/2,\infty).
		\end{cases}$
	\end{center}
	Therefore $k\in W$ and hence for $d(y,z)<\frac{1}{2}$, 
	$$H(Fy,Fz)\leq k(d(y,z))d(y,z).$$
\end{example}


\section*{Acknowledgment}

The first author would like to acknowledge the Ministry of Human Resource Development, India for providing financial assistance during the research work.



