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\fancyhead[CE]{B. Mohammadi, F. Golkarmanesh and V. Parvaneh} 
\fancyhead[CO]{Some fixed point theorems for nonexpansive... }



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{\noindent Journal of Mathematical Extension \\
Vol. XX, No. XX, (2014), pp-pp (Will be inserted by layout editor)}\\
ISSN: 1735-8299\\
URL: http://www.ijmex.com\\
\vspace*{9mm}

\begin{center}

{\Large \bf 
Some fixed point theorems for nonexpansive self-mappings
and multi-valued mappings in $b$-metric spaces\\}
{\bf  \\ } 


\let\thefootnote\relax\footnote{\scriptsize Received: XXXX; Accepted: XXXX (Will be inserted by editor)}
{\bf B. Mohammadi}\vspace*{-2mm}\\
\vspace{2mm} {\small  Department of
Mathematics, Marand Branch, Islamic Azad University, Marand, Iran.} \vspace{2mm}

{\bf F. Golkarmanesh}\vspace*{-2mm}\\
\vspace{2mm} {\small  Department of Mathematics, Sanandaj Branch, Islamic
Azad University, Sanandaj, Iran.} \vspace{2mm}

{\bf  V. Parvaneh$^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\
\vspace{2mm} {\small   Department of Mathematics, Gilan-E-Gharb Branch, Islamic
Azad University, Gilan-E-Gharb, Iran.} \vspace{2mm}

\end{center}

\vspace{4mm}


{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.} In this paper, motivated by [F. Vetro,  Filomat,  29:9 (2015),
2011-2020] we present some fixed point results for a class of
nonexpansive self-mappings and multi-valued mappings in the
framework of  $b$-metric spaces. Our results generalize and improve
the consequences of [ Khojasteh et al. Abstract and Applied
Analysis, vol. 2014, Article ID 325840, 5 pages, 2014.] and
 [F. Vetro,  Filomat,  29:9 (2015),
2011-2020].  Some examples are provided to illustrate our results.
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:} MSC 47H04; 47H10

\noindent{\bf Keywords and Phrases:}  fixed point, multi-valued mappings,
 nonexpansive self-mapping,  $b$-metric spaces
\end{quotation}}

\section{Introduction}
\label{intro} % It is advised to give each section and subsection a unique label.
Assume that $(X,d)$ be a metric space and $f:X\to X$ be a
single-valued mapping. Then, $f$ is called a $k$-Lipscitz mapping if
$d(fx,fy)\leq kd(x,y)$ for all $x,y\in X$, where $k\geq 0$. If $k\in
[0,1)$, then it is called a contractive mapping and if $k=1$, $f$ is
called a nonexpansive mapping. In \cite{Banach} Banach proved a very
important fixed point result for a contractive self-mapping.
 Obviously, any contractive mapping is
nonexpansive, but the reverse is not true in general. Several
researchers investigated fixed point theory for nonexpansive
mappings such as \cite{Edelstein, Khamsi}. In \cite{Khojasteh}
Khojasteh et al. investigated fixed point results for a new type of
self-mappings and multivalued mappings. Then, Vetro in \cite{Vetro}
extended their results for nonexpansive mappings using a binary
relation on $X$, called $f$-invariant. Also, he established fixed
point results for nonexpansive multi-valued mappings with a new type
of contraction. On the other hand, fixed point theory for mappings
on $b$-metric spaces as a generalization of metric spaces has
attracted many researchers for many years (see \cite{Bakhtin,
Boriceanu1, Boriceanu2, Czerwik1, Czerwik2,
 Mustafa, Nieto, Pacurar, Parvaneh, Ran, Roshan1, 1, 2, 3, Suzana3, Suzana4}).
In this paper, motivated by Vetro \cite{Vetro} we give some fixed
point results for a class of nonexpansive self-mappings and
multi-valued mappings on $b$-metric spaces. Our results generalize
and improve the results of Khojasteh et al. \cite{ Khojasteh} and
Vetro \cite{Vetro}. Some examples are given to illustrate our
results.



\section{Preliminaries}
In this section we give some notions and results that will be needed
in the sequel.
\begin{definition}\cite{Bakhtin}
Let $X$ be a  nonempty set and $s\geq 1$ be a constant real number.
The function $d:X\times X\rightarrow [0,\infty)$ is called a
$b$-metric on
 $X$ if the following conditions hold:
\begin{itemize}
\item[(i)] $d(x,y)=0$ iff $x=y$ for all $x,y\in X$;
\item[(ii)] $d(x,y)=d(y,x)$ for all $x,y\in X$;
\item[(iii)] $d(x,y)\leq s[d(x,z)+d(z,y)]$ for all $x,y,z\in X$.
\end{itemize}
In this case,  $(X,d)$ is called a $b$-metric space with parameter
$s$.
\end{definition}

Denote by $CB(X)$ the set of all nonempty closed bounded subsets of
$X$.
 Assume that $H$ be the  Pompeiu-Hausdorff metric  on $CB(X)$  defined by
$$H(A,B)=\max\{\sup_{x\in A}d(x,B),\sup_{y\in B}d(y,A)\},$$
for all $A, B\in CB(X)$, where $d(x,B)=\inf_{y\in B}d(x,y)$. An
element  $x\in X$ is said to be a  fixed point of a multi-valued
mapping $T:X\to CB(X)$ whenever $x\in Tx$. It is said that  $T:X\to
CB(X)$ is  contractive whenever $H(Tx,Ty)\leq kd(x,y)$ for all
$x,y\in X$, where $k\in [0,1)$. If $k=1$, then $T$ is called a
nonexpansive multivalued mapping. Nadler \cite{Nadler} proved the
existence of fixed point for contractive multivalued mappings.

\begin{lemma} \cite{Vetro}
If $\{a_n\}$ be a nonincreasing sequence of nonnegative real
numbers, then the sequence $\{\frac{a_n + a_{n+1}}{a_n + a_{n+1} +
1}\}$ is nonincreasing too.
\end{lemma}

\begin{cor}\label{c2.2}
Let $(X,d)$ be a $b$-metric space and $f:X\to X$ be a nonexpansive
mapping.
 If $x_0\in X$ and $\{x_n\}$ be a Picard sequence starting with $x_0$,
  that is, $x_n=fx_{n-1}$ for all $n\in \mathbb{N}$, then, the sequence
$$\{\frac{d(x_{n-1},x_n) +d(x_n,x_{n+1})}{d(x_{n-1},x_n) +d(x_n,x_{n+1}) + 1}\}$$
is nonincreasing too.
\end{cor}


\section{Fixed point results for nonexpansive self-mappings}
We prove some results for single-valued mappings defined on a
$b$-metric space endowed with an arbitrary binary relation. Let $X$
be a nonempty set, $f:X\to X$ be a mapping and $\mathcal{R} $ be a
binary relation on $ X$, that is, $\mathcal{R}$  is a subset of $ X
\times X $. Then, $ \mathcal{R} $ is Banach $f$-invariant if $(f x,
f^2x) \in \mathcal{R} $, whenever  $ (x, f x) \in \mathcal{R}$.
Also, a subset $ Y $ of $ X $ is well ordered with respect to
$\mathcal{R}$ if for all $x, y\in Y$ we have $(x,y)\in \mathcal{R}$
or $(y, x)\in \mathcal{R}$. Let $Fix( f ) =\{x\in X : x = f x\}$
denotes the set of all fixed points of $ f $ on $ X $.


\begin{theorem}\label{t3.1}
 Let  $(X,d)$ be a  complete $b$-metric space with parameter $s> 1$ endowed with a binary relation
  $ \mathcal{R} $ on $ X $ and $f : X\to X $ be a nonexpansive mapping such that
 \begin{equation}\label{e1}
d(f x, f y) \leq (\frac{d(x, f y) + d(y, f x)}{s[d(x, f x) + d(y, f
y) + 1]}+ k)d(x, y)
\end{equation}
for all $ (x, y)\in \mathcal{R} $, where $k\in [0, 1)$. Also assume
that
\begin{itemize}
\item[(a)] $ \mathcal{R} $ is Banach $ f $ -invariant,

\item[(b)] if $\{x_n\}$ is a sequence in $ X $ such that $ (x_{n-1}, x_n) \in \mathcal{R} $ for all $ n\in \mathbb{N}$
 and
$ x_n\to z \in X $ as $ n\to \infty $, then $ (x_{n-1}, z)\in
\mathcal{R} $, for all $ n\in \mathbb{N} $;
\item[(c)] $ Fix( f ) $ is well ordered with respect to $ \mathcal{R} $.
\end{itemize}
Let there exists $ x_0\in X $ such that $(x_0, f x_0)\in
\mathcal{R}$ and $\lambda s<1$, where

$$\lambda=\frac{d(x_0, f x_0) + d( fx_0; f^2x_0)}{d(x_0, f x_0) + d( f x_0; f^2x_0) + 1}
+ k .$$ Then,
\begin{itemize}
\item[(i)] $ f $ has at least one fixed point $ z\in X $,
\item[(ii)] the Picard sequence of initial point $ x_0\in X $ converges to a fixed point of $ f $,
\item[(iii)] if $ z,w\in X $ are two distinct fixed points of $ f $, then $ d(z, w)\geq \frac{s- k}{2} $.
\end{itemize}
\end{theorem}


\begin{proof}
Let $ x_0\in X $ be such that $ (x_0, fx_0)\in \mathcal{R}$,
$\lambda s<1$ and let $\{x_n\}$ be a Picard sequence with initial
point $ x_0 $. If $ x_{n-1}=x_n $ for some $ n\in \mathbb{N} $, then
$ x_{n-1} $ is a fixed point of $ f $ and the existence of a fixed
point is proved. Now, we suppose that $ x_{n-1}\neq x_n $ for all $
n\in \mathbb{N} $. From $ (x_0, x_1) = (x_0, f x_0)\in \mathcal{R}
$, since $ \mathcal{R} $ is  Banach $ f $-invariant, we deduce that
$(x_1, x_2) = ( f x_0, f^2x_0)\in \mathcal{R}$. This implies that $
(x_{n-1}, x_n) = ( f^{n-1}x_0, f^nx_0)\in \mathcal{R} $ for all $
n\in \mathbb{N}$. Using the contractive condition (\ref{e1}) with $
x = x_{n-1} $ and $ y = x_n $, we get
 \begin{equation}\label{e2}
 \begin{array}{lll}
d(x_{n}, x_{n+1})=d(f x_{n-1}, f x_n)
&\leq& (\frac{d(x_{n-1}, f x_n) + d(x_n, f x_{n-1})}{s[d(x_{n-1}, f x_{n-1}) + d(x_n, fx_n) + 1]}+ k)d(x_{n-1}, x_n)\\
&=&(\frac{d(x_{n-1},  x_{n+1})}{s[d(x_{n-1}, x_{n}) + d(x_n, x_{n+1}) + 1]}+ k)d(x_{n-1}, x_n)\\
&\leq&(\frac{d(x_{n-1},  x_{n})+d(x_{n},  x_{n+1})}{[d(x_{n-1},
x_{n}) + d(x_n, x_{n+1}) + 1]}+ k)d(x_{n-1}, x_n)
\end{array}
\end{equation}
for all $ n\in \mathbb{N} $. From (\ref{e2}), by Corollary
\ref{c2.2}, we get
\begin{equation}\label{e3}
 \begin{array}{lll}
d(x_{n}, x_{n+1})&\leq& (\frac{d(x_{n-1},  x_{n})+d(x_{n},  x_{n+1})}{[d(x_{n-1}, x_{n}) + d(x_n, x_{n+1}) + 1]}+ k)d(x_{n-1}, x_n)\\
&\leq& (\frac{d(x_0,  x_1)+d(x_1,  x_2)}{[d(x_0, x_1) + d(x_1, x_2) + 1]}+ k)d(x_{n-1}, x_n)\\
&=&\lambda d(x_{n-1}, x_n),
\end{array}
\end{equation}
for all $ n\in \mathbb{N}$. Thus, for any $m,n\in \mathbb{N}$ with
$m>n$, we have
\begin{equation}\label{e4}
 \begin{array}{lll}
d(x_n, x_m)&\leq& s[d(x_n,  x_{n+1})+d(x_{n+1},  x_m)]\\
&\leq& sd(x_n,  x_{n+1})+s^2d(x_{n+1},  x_{n+2})+...+s^{m-n-1}d(x_{m-1},  x_m)\\
&=&\sum_{i=1}^{m-n-1}s^id(x_{n+i-1},  x_{n+i})\leq \sum_{i=1}^{m-n-1}s^i\lambda^{i+n-1}d(x_0,  x_{1})\\
&=&\lambda^{n-1}d(x_0,  x_{1}) \sum_{i=1}^{m-n-1}(\lambda s)^{i}\\
&=&\lambda^{n-1}d(x_0,  x_{1})(\lambda s)^{1-n} \sum_{i=n}^{m-1}(\lambda s)^{i}\\
&=&d(x_0,  x_{1})s^{1-n} \sum_{i=n}^{m-1}(\lambda s)^{i}.
\end{array}
\end{equation}
As $ \lambda s<1 $ and $s>1$, the last term in the above tends to
zero, as $m,n\to \infty$. Thus, $\{x_n\}$ is a Cauchy sequence.
Since $(X,d)$ is complete, there exists $z\in X$ such that $x_n\to
z$. Now we show that $z$ is a fixed point of $f$. By assumption (b),
we deduce that $(x_n,z)\in \mathcal \mathcal{R}$. So, by (\ref{e1}),
we have
\begin{equation}\label{e5}
 \begin{array}{lll}
d(x_{n+1}, fz)=d(f x_{n}, fz)&\leq& (\frac{d(x_{n}, f z) + d(z, f
x_{n})}{s[d(x_{n}, f x_{n})
 + d(z, fz) + 1]}+ k)d(x_{n}, z)\\
&=& (\frac{d(x_{n}, f z) + d(z, x_{n+1})}{s[d(x_{n},  x_{n+1}) +
d(z, fz) + 1]}+ k)d(x_{n}, z).
\end{array}
\end{equation}
Taking limit as $n\to \infty$, in the above inequality, we get
$d(z,fz)\leq 0$. Thus, $d(z,fz)=0$, that is $z=fz$. Thus (i) and
(ii) hold. Now, let $z,w$ are two distinct fixed points of $f$.
Then, we have
\[d(z,w)=d(fz,fw)\leq (\frac{d(z,fw)+d(w,fz)+k}{s})d(z,w),\]
which implies that $d(z,w)\geq \frac{s-k}{2}$.
\end{proof}


Also, we can prove the following result with a weaker contractive
condition.
\begin{theorem}\label{t3.2}
 Let  $(X,d)$ be a  complete $b$-metric space with parameter $s> 1$ endowed with a binary relation
 $ \mathcal{R} $ on $ X $ and $f : X\to X $ be a nonexpansive mapping such that
 \begin{equation}\label{e6}
d(f x, f y) \leq (\frac{d(x, f y) + d(y, f x)}{s[d(x, f x) + d(y, f
y) + 1]}+ k)d(x, y)+Ld(y,fx)
\end{equation}
for all $ (x, y)\in \mathcal{R} $, where $k\in [0, 1)$ and $L$ is a
nonnegative real number. Also, assume that
\begin{itemize}
\item[(a)] $ \mathcal{R} $ is Banach $f$-invariant,
\item[(b)] if $\{x_n\}$ is a sequence in $X$ such that $(x_{n-1}, x_n)\in \mathcal{R}$ for all $n\in \mathbb{N}$ and
$x_n\to z \in X$ as $ n\to \infty$, then $(x_{n-1},z)\in
\mathcal{R}$, for all $n\in \mathbb{N}$;
\item[(c)] $Fix( f )$ is well ordered with respect to $\mathcal{R}$.
\end{itemize}
Let there exists $ x_0\in X $ such that $(x_0, f x_0)\in
\mathcal{R}$ and $\lambda s<1$, where
$$\lambda=\frac{d(x_0, f x_0) + d( fx_0; f^2x_0)}{d(x_0, f x_0) + d( f x_0; f^2x_0) + 1}
+ k.$$ Then,
\begin{itemize}
\item[(i)] $ f $ has at least one fixed point $ z\in X $,
\item[(ii)] the Picard sequence with initial point $ x_0\in X $ converges to a fixed point of $ f $,
\item[(iii)] if $ z,w\in X $ are two distinct fixed points of $ f $, then $d(z, w)\geq \max\{\frac{s(1-L)- k}{2},0\}$.
\end{itemize}
\end{theorem}

\begin{proof}
Let $ x_0\in X $ be such that $ (x_0, fx_0)\in \mathcal{R}$,
$\lambda s<1$ and let $\{x_n\}$ be a Picard sequence with initial
point $ x_0 $. If $ x_{n-1}=x_n $ for some $ n\in \mathbb{N} $, then
$ x_{n-1} $ is a fixed point of $ f $ and the existence of a fixed
point is proved. Now, we suppose that $ x_{n-1}\neq x_n $ for all $
n\in \mathbb{N}$. From $ (x_0, x_1) = (x_0, f x_0)\in \mathcal{R} $,
since $ \mathcal{R} $ is  Banach $ f $-invariant, we deduce $(x_1,
x_2) = ( f x_0, f^2x_0)\in \mathcal{R}$. This implies that $
(x_{n-1}, x_n) = ( f^{n-1}x_0, f^nx_0)\in \mathcal{R} $ for all $
n\in \mathbb{N}$. Using the contractive condition (\ref{e6}) with $
x = x_{n-1} $ and $ y = x_n $, we get
\[
 \begin{array}{lll}
d(x_{n}, x_{n+1})=d(f x_{n-1}, f x_n)&\leq& (\frac{d(x_{n-1}, f x_n)
+ d(x_n, f x_{n-1})}
{s[d(x_{n-1}, f x_{n-1}) + d(x_n, fx_n) + 1]}+ k)d(x_{n-1}, x_n)+Ld(x_n, x_n)\\
&=&(\frac{d(x_{n-1},  x_{n+1})}{s[d(x_{n-1}, x_{n}) + d(x_n, x_{n+1}) + 1]}+ k)d(x_{n-1}, x_n)\\
&\leq&(\frac{d(x_{n-1},  x_{n})+d(x_{n},  x_{n+1})}{[d(x_{n-1},
x_{n}) + d(x_n, x_{n+1}) + 1]}+ k)d(x_{n-1}, x_n)
\end{array}
\]
for all $ n\in \mathbb{N}$. As in the proof of Theorem \ref{t3.1},
$\{x_n\}$ is a Cauchy sequence. Since $(X,d)$ is complete, there
exists $z\in X$ such that $x_n\to z$. Now we show that $z$ is a
fixed point of $f$. By assumption (b), we deduce that $(x_n,z)\in
\mathcal{R}$. So, by (\ref{e6}), we have
\begin{equation}\label{e7}
\begin{array}{lll}
d(x_{n+1}, fz)=d(f x_{n}, fz)&\leq& (\frac{d(x_{n}, f z) + d(z, f
x_{n})}
{s[d(x_{n}, f x_{n}) + d(z, fz) + 1]}+ k)d(x_{n}, z)+Ld(x_{n}, z)\\
&=& (\frac{d(x_{n}, f z) + d(z, x_{n+1})}{s[d(x_{n},  x_{n+1}) +
d(z, fz) + 1]}+ k)d(x_{n}, z)+Ld(x_{n}, z).
\end{array}
\end{equation}
Taking limit as $n\to \infty$ in the above inequality, we get
$d(z,fz)\leq 0$. Thus, $d(z,fz)=0$, that is $z=fz$. Thus, (i) and
(ii) hold. Now, let $z,w$ are two distinct fixed points of $f$.
Then, we have
\[d(z,w)=d(fz,fw)\leq (\frac{d(z,fw)+d(w,fz)+k}{s})d(z,w)+Ld(z,w),\]
which implies that $d(z,w)\geq \frac{s(1-L)-k}{2}$. Thus, (iii)
holds.
\end{proof}




Putting $\mathcal{R}=X\times X$ in Theorems \ref{t3.1} and
\ref{t3.2}, we obtain the following results in $b$-metric spaces:
\begin{theorem}\label{t3.3}
 Let  $(X,d)$ be a  complete $b$-metric space with parameter $s> 1$ and let $f : X\to X $ be a nonexpansive
 mapping such that
 \[d(f x, f y) \leq (\frac{d(x, f y) + d(y, f x)}{s[d(x, f x) + d(y, f y) + 1]}+ k)d(x, y)\]
for all $ x, y\in X $, where $k\in [0, 1)$. Assume that
 there exists $ x_0\in X $ such that  $\lambda s<1$, where
$$\lambda=\frac{d(x_0, f x_0) + d( fx_0; f^2x_0)}{d(x_0, f x_0) + d( f x_0; f^2x_0) + 1}
+ k.$$ Then,
\begin{itemize}
\item[(i)] $ f $ has at least one fixed point $ z\in X $,
\item[(ii)] the Picard sequence with initial point $x_0\in X$ converges to a fixed point of $f$,
\item[(iii)] if $z,w\in X$ are two distinct fixed points of $f$, then $d(z, w)\geq \frac{s- k}{2}$.
\end{itemize}
\end{theorem}



\begin{theorem}\label{t3.4}
 Let  $(X,d)$ be a  complete $b$-metric space with parameter $s> 1$  and let $f : X\to X $ be a nonexpansive
 mapping such that
 \[
d(f x, f y) \leq (\frac{d(x, f y) + d(y, f x)}{s[d(x, f x) + d(y, f
y) + 1]}+ k)d(x, y)+Ld(y,fx)
\]
for all $ (x, y)\in X$, where $k\in [0, 1)$ and $L$ is a nonnegative
real number.

Let there exists $ x_0\in X $ such that  $\lambda s<1$, where
$$\lambda=\frac{d(x_0, f x_0) + d( fx_0, f^2x_0)}{d(x_0, f x_0) + d( f x_0, f^2x_0) + 1}
+ k.$$ Then,
\begin{itemize}
\item[(i)] $ f $ has at least one fixed point $ z\in X $,
\item[(ii)] the Picard sequence with initial point $ x_0\in X $ converges to a fixed point of $ f $,
\item[(iii)] if $ z,w\in X $ are two distinct fixed points of $ f $, then $d(z, w)\geq \max\{\frac{s(1-L)- k}{2},0\}$.
\end{itemize}
\end{theorem}

\begin{example}
Let $X=[0,1]\cup [\frac{5}{2},\infty)$ and $d:X\times X:\to
[0,\infty)$ be defined by $d(x,y)=(x-y)^2$.
 Define $f:X\to X$ by
$ fx= \Bigg\{\begin{array}{lll}
\frac{1}{2}+\frac{1}{2}x\ & if\ x\in [0,1],\\
\frac{5}{4}+\frac{1}{2}x\ & if\ x\in  [ \frac{5}{2},\infty).
\end{array}
$ It is clear that $(X,d)$ is a complete $b$-metric space with
parameter $s=2$ and $f$ is nonexpansive.
 Also, if $x,y\in [0,1]$ or $x,y\in [\frac{5}{2},\infty)$, then
\[d(f x, f y)=\frac{1}{4}(x-y)^2\leq  \frac{1}{2}(x-y)^2=\frac{1}{2}d(x,y).\]
If $x\in [0,1]$ and $y\in [\frac{5}{2},\infty)$, then
\begin{equation}\label{e8}
\frac{d(x, f y) + d(y, f x)}{s[d(x, f x) + d(y, f y) + 1]}\geq
\frac{3}{4}.
\end{equation}
In fact,
\begin{equation}\label{eq1}
\begin{array}{lll}
&&\frac{d(x, f y) + d(y, f x)}{s[d(x, f x) + d(y, f y) + 1]}\geq \frac{3}{4}\\
&&\Longleftrightarrow \frac{(x-\frac{5}{4}-\frac{1}{2}y)^2 +
(y-\frac{1}{2}-\frac{1}{2}x)^2}{[(x-\frac{1}{2}-\frac{1}{2}x)^2 +
(y-\frac{5}{4}-\frac{1}{2}y)^2 + 1]} \geq \frac{3}{2}
\\
&&\Longleftrightarrow
2x^2+2(\frac{5}{4}+\frac{1}{2}y)^2-4x(\frac{5}{4}
+\frac{1}{2}y)+2y^2+2(\frac{1}{2}+\frac{1}{2}x)^2-4y(\frac{1}{2}+\frac{1}{2}x)

\\
&&\geq
3x^2+3(\frac{1}{2}+\frac{1}{2}x)^2-6x(\frac{1}{2}+\frac{1}{2}x)
+3y^2+3(\frac{5}{4}+\frac{1}{2}y)^2-6y(\frac{5}{4}+\frac{1}{2}y)+3\\

&&\Longleftrightarrow 7x^2+7y^2-16xy-10x+17y\geq \frac{77}{4}\\
&&\Longleftrightarrow 7(y-x)^2+17y\geq
\frac{63}{4}+\frac{14}{4}+2x(y+5).
\end{array}
\end{equation}
Since $(y-x)\geq \frac{3}{2}$, thus $7(y-x)^2\geq \frac{63}{4}$.
  It is sufficient  to show that $17y\geq \frac{14}{4}+2x(y+5)$. Now, since
  $x\leq 1$, it is sufficient to show $17y\geq \frac{14}{4}+2(y+5)$ or equally $y\geq \frac{54}{60}$ which is desired.
  Thus, (\ref{e7}) holds. Now
\begin{equation}\label{eq1}
\begin{array}{lll}
d(fx,fy)&=&(\frac{5}{4}+\frac{1}{2}y-\frac{1}{2}-\frac{1}{2}x)^2\\
&=&(\frac{3}{4}+\frac{1}{2}(y-x))^2\leq (\frac{3}{4}+\frac{1}{4})(y-x)^2\\
&\leq & (\frac{d(x, f y) + d(y, f x)}{s[d(x, f x) + d(y, f y) +
1]}+\frac{1}{4})d(x,y).
\end{array}
\end{equation}
Also, for $k=\frac{1}{4}$ and $x_0=\frac{1}{8}$, we have
$$\lambda=\frac{d(x_0, f x_0) + d( fx_0, f^2x_0)}{d(x_0, f x_0) + d( f x_0, f^2x_0) + 1}
+ k=\frac{245}{1269}+\frac{1}{4}.$$ Therefore, $\lambda
s=(\frac{245}{1269}+\frac{1}{4})2=\frac{490}{1269}+\frac{1}{2}<1.$
Thus, all of the conditions of Theorem \ref{t3.3} are satisfied and
so $T$ has a fixed point.
 Here, $z=1$ and $w=\frac{5}{2}$ are two fixed points of $f$. Also,
\[d(z,w)=(1-\frac{5}{2})^2=\frac{9}{4}\geq \frac{7}{8}=\frac{2-\frac{1}{4}}{2}=\frac{s-k}{2}.\]
Note that $f$ is not a contraction. In fact,
$d(f1,f\frac{5}{2})=d(1,\frac{5}{2})$.
\end{example}

Let $(X,d)$ be a $b$-metric space and $\preceq$ be an order on $X$.
Then, the triple $(X,d,\preceq)$ is called a partial ordered
$b$-metric space. Then, two elements $x,y\in X$ are called
comparable if $x\preceq y$ or $y\preceq x$. Also, $(X,d,\preceq)$
is called regular if for any nondecreasing sequence $\{x_n\}$  in $
X $ such that $ x_n\to z \in X $ as $ n\to \infty $, then $
x_n\preceq z $, for all $ n\in \mathbb{N} $. Note that
$\mathcal{R}=\{(x,y):x\preceq y\}$ is a binary relation on $X$.
Also, if $f:X\to X$ be nondecreasing, then $\mathcal{R}$ is Banach
$f$-invariant.
\begin{theorem}\label{t3.5}
 Let  $(X,d,\preceq)$ be a  complete ordered $b$-metric space with parameter $s> 1$
 and $f : X\to X $ be a nonexpansive nondecreasing mapping such that
 \[
d(f x, f y) \leq (\frac{d(x, f y) + d(y, f x)}{s[d(x, f x) + d(y, f
y) + 1]}+ k)d(x, y)
\]
for all comparable elements $ x, y\in X $, where $k\in [0, 1)$. Also
assume that
\begin{itemize}
\item[(a)] $(X,d,\preceq)$ is regular,
\item[(b)] $ Fix( f ) $ is well ordered with respect to $ \preceq $.
\end{itemize}
Let there exists $ x_0\in X $ such that $x_0\preceq f x_0$ and
$\lambda s<1$, where
$$\lambda=\frac{d(x_0, f x_0) + d( fx_0; f^2x_0)}{d(x_0, f x_0) + d( f x_0; f^2x_0) + 1}
+ k.$$ Then,
\begin{itemize}
\item[(i)] $ f $ has at least one fixed point $ z\in X $,
\item[(ii)] the Picard sequence with initial point $ x_0\in X $ converges to a fixed point of $ f $,
\item[(iii)] if $ z,w\in X $ are two distinct fixed points of $ f $, then $ d(z, w)\geq \frac{s- k}{2} $.
\end{itemize}
\end{theorem}


\begin{theorem}\label{t3.6}
 Let  $(X,d,\preceq)$ be a  complete ordered $b$-metric space with parameter $s> 1$  and $f : X\to X $ be a
  nonexpansive nondecreasing mapping such that
 \[
d(f x, f y) \leq (\frac{d(x, f y) + d(y, f x)}{s[d(x, f x) + d(y, f
y) + 1]}+ k)d(x, y)+Ld(x, y)
\]
for all comparable elements $ x, y\in X $, where $k\in [0, 1)$ and
$L$ is a nonnegative real number. Also assume that
\begin{itemize}
\item[(a)] $(X,d,\preceq)$ is regular,
\item[(b)] $ Fix( f ) $ is well ordered with respect to $ \preceq $.
\end{itemize}
Let there exists $ x_0\in X $ such that $x_0\preceq f x_0$ and
$\lambda s<1$, where
$$\lambda=\frac{d(x_0, f x_0) + d( fx_0; f^2x_0)}{d(x_0, f x_0) + d( f x_0; f^2x_0) + 1}
+ k.$$ Then,
\begin{itemize}
\item[(i)] $ f $ has at least one fixed point $ z\in X $,
\item[(ii)] the Picard sequence with initial point $ x_0\in X $ converges to a fixed point of $ f $,
\item[(iii)] if $ z,w\in X $ are two distinct fixed points of $ f $, then  $d(z,w)\geq \frac{s(1-L)-k}{2}$.
\end{itemize}
\end{theorem}

\section{Fixed point results for nonexpansive multi-valued mappings}

In this section, we give some fixed point results for nonexpansive
multi-valued mappings in $b$-metric spaces. Let $(X,d)$ be a
$b$-metric space and $K(X)$ be the set of all nonempty compact
subsets of $X$.
\begin{definition}\label{d4.1}
Let $ \mathcal{R} $ be a binary relation on $ X $. Then, $
\mathcal{R} $ is called Banach $T$-invariant if for any $x\in X$ and
$y\in Tx$ with $(x,y)\in \mathcal{R}$, then we have $(y,z)\in
\mathcal{R}$ for all $z\in Ty$.
\end{definition}

Note that if $T$ be  a single-valued mapping, then Definition
\ref{d4.1} reduces to the definition of $f$-invariant for
 single-valued mappings.
\begin{theorem}\label{t4.1}
 Let $(X,d)$ be a  complete $b$-metric space with parameter $s> 1$ endowed with a binary relation
  $ \mathcal{R} $ on $ X $ and $T:X\to K(X)$ be a nonexpansive  multi-valued mapping such that
 \begin{equation}\label{e9}
H(Tx, Ty) \leq (\frac{d(x, Ty) + d(y, Tx)}{s[d(x, Tx) + d(y, Ty) +
1]}+ k)d(x, y)
\end{equation}
for all $ (x, y)\in \mathcal{R} $, where $k\in [0, 1)$. Also, assume
that
\begin{itemize}
\item[(a)] $ \mathcal{R} $ is Banach $T$-invariant,
\item[(b)] if $\{x_n\}$ is a sequence in $ X $ such that $ (x_{n-1}, x_n) \in \mathcal{R} $
for all $ n\in \mathbb{N} $ and $ x_n\to z \in X $ as $ n\to \infty
$, then $ (x_{n-1}, z)\in \mathcal{R} $, for all $ n\in \mathbb{N}
$;
\end{itemize}
Let there exists $ x_0\in X $ and $x_1\in Tx_0$ such that $(x_0,
x_1)\in \mathcal{R}$, $d(x_0, x_1)=d(x_0, Tx_0)$ and $\lambda s<1$,
where
$$\lambda=\frac{d(x_0, x_1) + d( x_1, Tx_1)}{d(x_0, x_1) + d(x_1, Tx_1) + 1}
+ k.$$ Then, $ T $ has at least one fixed point $ z\in X $.
\end{theorem}
\begin{proof}
Let $ x_0\in X $  and $x_1\in Tx_0$ be such that $(x_0, x_1)\in
\mathcal{R}$, $d(x_0, x_1)=d(x_0, Tx_0)$ and  $\lambda s<1$.  Since
$Tx_1$ is compact, there exists $x_2\in Tx_1$ such that $d(x_1,
x_2)=d(x_1, Tx_1)$.

From $ (x_0, x_1)\in \mathcal{R}$, since $ \mathcal{R} $ is  Banach
$ T $-invariant, we get $(x_1, x_2)\in \mathcal{R}$. Continuing this
process, we have a sequence $\{x_n\}$ in $X$, such that $(x_{n-1},
x_n)\in \mathcal{R}$, $x_n\in Tx_{n-1}$ and $d(x_{n-1},
x_n)=d(x_{n-1}, Tx_{n-1})$ for all $n\in \mathbb{N}$. Since $T$ is
nonexpansine, we have $d(x_{n}, x_{n+1})=d(x_{n}, Tx_{n})\leq
H(Tx_{n-1}, Tx_{n})\leq d(x_{n-1}, x_n)$. Thus, by Corollary
\ref{c2.2},
$$\{\frac{d(x_{n-1},x_n) +d(x_n,x_{n+1})}{d(x_{n-1},x_n) +d(x_n,x_{n+1}) + 1}\}$$
is nonincreasing. Using the contractive condition (\ref{e9})
 with $
x = x_{n-1} $ and $ y = x_n $, we get
 \begin{equation}
 \begin{array}{lll}
d(x_{n}, x_{n+1})&=&d(x_{n},T x_{n})\leq H(T x_{n-1}, T x_n)\\
&\leq& (\frac{d(x_{n-1}, T x_n) + d(x_n, T x_{n-1})}{s[d(x_{n-1}, T x_{n-1}) + d(x_n, Tx_n) + 1]}+ k)d(x_{n-1}, x_n)\\
&=&(\frac{d(x_{n-1},  x_{n+1})}{s[d(x_{n-1}, x_{n}) + d(x_n, x_{n+1}) + 1]}+ k)d(x_{n-1}, x_n)\\
&\leq&(\frac{d(x_{n-1},  x_{n})+d(x_{n},  x_{n+1})}{[d(x_{n-1},
x_{n}) + d(x_n, x_{n+1}) + 1]}+ k)d(x_{n-1}, x_n)
\end{array}
\end{equation}
for all $ n\in \mathbb{N} $. Thus, we get
\begin{equation}\label{e10}
\begin{array}{lll}
d(x_{n}, x_{n+1})&\leq& (\frac{d(x_{n-1},  x_{n})+d(x_{n},
x_{n+1})}{[d(x_{n-1}, x_{n}) + d(x_n, x_{n+1}) + 1]}
+ k)d(x_{n-1}, x_n)\\
&\leq& (\frac{d(x_0,  x_1)+d(x_1,  x_2)}{[d(x_0, x_1) + d(x_1, x_2) + 1]}+ k)d(x_{n-1}, x_n)\\
&=&\lambda d(x_{n-1}, x_n),
\end{array}
\end{equation}
for all $ n\in \mathbb{N}$.  Thus, as in Theorem \ref{t3.1},
$\{x_n\}$ is a Cauchy sequence.
 Since $(X,d)$ is complete, there exists $z\in X$ such that $x_n\to z$.
 Now, we show that $z$ is a fixed point of $T$. By assumption (b), we deduce that $(x_n,z)\in
 \mathcal{R}$. So, by (\ref{e9}), we have
\begin{equation}\label{e11}
 \begin{array}{lll}
d(z,Tz)&\leq& s[d(z,x_{n+1})+d(x_{n+1}, Tz)]\\
&\leq& sd(z,x_{n+1})+sH(Tx_{n}, Tz)\\
&\leq& sd(z,x_{n+1})+(\frac{d(x_{n}, Tz) + d(z, Tx_{n})}{[d(x_{n}, Tx_{n}) + d(z, Tz) + 1]}+ k)d(x_{n}, z)\\
&=&sd(z,x_{n+1}+(\frac{d(x_{n}, Tz) + d(z, x_{n+1})}{[d(x_{n},
x_{n+1}) + d(z, Tz) + 1]}+ k)d(x_{n}, z).
\end{array}
\end{equation}
Taking limit as $n\to \infty$, in the above inequality, we get
$d(z,Tz)\leq 0$. Thus, $d(z,Tz)=0$, that is, $z\in Tz$.
\end{proof}

Also, we can prove the following result for nonexpansive
multi-valued mapping with a weaker contractive condition.
\begin{theorem}\label{t4.2}
 Let $(X,d)$ be a  complete $b$-metric space with parameter $s> 1$ endowed with a binary relation
  $ \mathcal{R} $ on $ X $ and $T:X\to K(X)$ be a nonexpansive multi-valued mapping such that
 \begin{equation}\label{e12}
H(Tx, Ty) \leq (\frac{d(x, Ty) + d(y, Tx)}{s[d(x, Tx) + d(y, Ty) +
1]}+ k)d(x, y)+Ld(x,y)
\end{equation}
for all $ (x, y)\in \mathcal{R} $, where $k\in [0, 1)$ and $L$ is a
nonnegative real number. Also, assume that
\begin{itemize}
\item[(a)] $ \mathcal{R} $ is Banach $T$-invariant,
\item[(b)] if $\{x_n\}$ is a sequence in $ X $ such that $ (x_{n-1}, x_n) \in \mathcal{R} $ for all $ n\in \mathbb{N} $
 and
$ x_n\to z \in X $ as $ n\to \infty $, then $ (x_{n-1}, z)\in
\mathcal{R} $, for all $ n\in \mathbb{N} $;
\end{itemize}
Let there exists $ x_0\in X $, $x_1\in Tx_0$ such that $(x_0,
x_1)\in \mathcal{R}$, $d(x_0, x_1)=d(x_0, Tx_0)$ and $\lambda s<1$,
where
$$\lambda=\frac{d(x_0, x_1) + d( x_1, Tx_1)}{d(x_0, x_1) + d(x_1, Tx_1) + 1}+ k.$$
 Then, $ T $ has at least one fixed point $ z\in X $.
\end{theorem}


Putting $\mathcal{R} =X\times X$ in Theorems \ref{t4.1} and
\ref{t4.2}, we obtain the following results in $b$-metric spaces:
\begin{theorem}\label{t4.3}
 Let  $(X,d)$ be a  complete $b$-metric space with parameter $s> 1$ and $T: X\to K(X) $
 be a nonexpansive multi-valued mapping such that
 \[
H(T x, T y) \leq (\frac{d(x, T y) + d(y, T x)}{s[d(x, T x) + d(y, T
y) + 1]}+ k)d(x, y)
\]
for all $ x, y\in X $, where $k\in [0, 1)$. Assume that
 there exists $ x_0\in X $ and $x_1\in Tx_0$ such that  $d(x_0, x_1)=d(x_0, Tx_0)$ and $\lambda s<1$, where
$$\lambda=\frac{d(x_0, x_1) + d( x_1, Tx_1)}{d(x_0, x_1) + d(x_1, Tx_1) + 1}+ k.$$
Then, $ T $ has at least one fixed point $ z\in X $.
\end{theorem}


\begin{theorem}\label{t4.4}
 Let  $(X,d)$ be a  complete $b$-metric space with parameter $s> 1$ and $T: X\to K(X) $ be a nonexpansive
 multi-valued mapping such that
 \[
H(T x, T y) \leq (\frac{d(x, T y) + d(y, T x)}{s[d(x, T x) + d(y, T
y) + 1]}+ k)d(x, y)+Ld(x, y)
\]
for all $ x, y\in X $, where $k\in [0, 1)$ and $L$ is a nonnegative
real number. Assume that
 there exists $ x_0\in X $ and $x_1\in Tx_0$ such that  $d(x_0, x_1)=d(x_0, Tx_0)$ and $\lambda s<1$, where
$$\lambda=\frac{d(x_0, x_1) + d( x_1, Tx_1)}{d(x_0, x_1) + d(x_1, Tx_1) + 1}+ k .$$
Then, $ T $ has at least one fixed point $ z\in X $.
\end{theorem}

\begin{example}
Let $X=[0,1]\cup [\frac{5}{2},\infty)$ and let $d:X\times X:\to
[0,\infty)$ be defined by $d(x,y)=(x-y)^2$. Define $T:X\to K(X)$ by
$$
Tx=\left\{
\begin{array}{ll}
\left[ \frac{1}{2}+\frac{1}{2}x,1\right] ,  \hbox{if}\,\, x\in [ 0,1] ,\\
\left[\frac{5}{2},\frac{5}{4}+\frac{1}{2}x\right] ,  \hbox{if}\,\,
x\in [ \frac{5}{2},\infty).
\end{array}
\right.
$$
It is clear that $(X,d)$ is a complete $b$-metric space with
parameter $s=2$ and $T$ is nonexpansive. Also, if $x,y\in [0,1]$ or
$x,y\in [\frac{5}{2},\infty)$, then
\[H(Tx, Ty)=\frac{1}{4}(x-y)^2\leq  \frac{1}{2}(x-y)^2=\frac{1}{2}d(x,y).\]
If $x\in [0,1]$ and $y\in [\frac{5}{2},\infty)$, then
\[
\frac{d(x, T y) + d(y, T x)}{s[d(x, T x) + d(y, T y) + 1]}\geq
\frac{3}{4}.
\]
 Now,
\[
\begin{array}{lll}
H(Tx,Ty)&=&\max\{\frac{5}{2}-\frac{1}{2}-\frac{1}{2}x,1+\frac{1}{2}y-1\})^2\\
&=&\max\{2-\frac{1}{2}x,\frac{1}{2}y\})^2\leq (\frac{3}{4}+\frac{1}{4})(y-x)^2\\
&\leq& (\frac{d(x, T y) + d(y, T x)}{s[d(x, T x) + d(y, T y) +
1]}+\frac{1}{4})d(x,y).
\end{array}
\]
Also, for $k=\frac{1}{4}$ and $x_0=\frac{1}{8}$,
$d(x_0,Tx_0)=d(\frac{1}{8},\frac{3}{16})$. Thus, with
$x_1=\frac{3}{16}$ we have
$$\lambda=\frac{d(x_0,  x_1) + d( x_1, Tx_1)}{d(x_0, x_1) + d( x_1, Tx_1) + 1}
+ k=\frac{245}{1269}+\frac{1}{4}.$$ Therefore, $\lambda
s=(\frac{245}{1269}+\frac{1}{4})2=\frac{490}{1269}+\frac{1}{2}<1.$
Thus, all of the conditions of Theorem \ref{t4.3} are satisfied and
so $T$ has a fixed point. Here, $z=1$ and $w=\frac{5}{2}$ are two
fixed points of $T$. Also,
\[d(z,w)=(1-\frac{5}{2})^2=\frac{9}{4}\geq \frac{7}{8}=\frac{2-\frac{1}{4}}{2}=\frac{s-k}{2}.\]
Note that $T$ is not a contraction. In fact,
$H(T1,T\frac{5}{2})=d(1,\frac{5}{2})$.
\end{example}


\begin{theorem}\label{t4.5}
 Let  $(X,d,\preceq)$ be a  complete ordered $b$-metric space with parameter $s> 1$  and $T : X\to K(X) $
 be a nonexpansive multi-valued mapping such that
 \[
H(T x, T y) \leq (\frac{d(x, T y) + d(y, T x)}{s[d(x, T x) + d(y, T
y) + 1]}+ k)d(x, y)
\]
for all comparable elements $ x, y\in X $, where $k\in [0, 1)$.
Also, assume that
\begin{itemize}
\item[(a)] for any $x\in X$ and $y\in Tx$ with $x\preceq y$, then we have $y\preceq z$ for all $z\in Ty$.
\item[(b)] $(X,d,\preceq)$ is regular,
\end{itemize}
Let there exists $ x_0\in X $ and $x_1\in Tx_0$ such that $d(
x_0,Tx_0)=d(x_0,x_1)$, $x_0\preceq x_1$ and
 $\lambda s<1$, where
$$\lambda=\frac{d(x_0,  x_1) + d( x_1, Tx_1)}{d(x_0, x_1) + d( x_1, Tx_1) + 1}
+ k.$$ Then, $ T $ has at least one fixed point $ z\in X $.
\end{theorem}


\begin{theorem}
 Let  $(X,d,\preceq)$ be a  complete ordered $b$-metric space with parameter $s> 1$  and $T : X\to K(X) $
  be a nonexpansive multi-valued mapping such that
 \[
H(T x, T y) \leq (\frac{d(x, T y) + d(y, T x)}{s[d(x, T x) + d(y, T
y) + 1]}+ k)d(x, y)+Ld(x,y)
\]
for all comparable elements $ x, y\in X $, where $k\in [0, 1)$ and
$L$ is a nonnegative real number.
 Also, assume that
\begin{itemize}
\item[(a)] for any $x\in X$ and $y\in Tx$ with $x\preceq y$, then we have $y\preceq z$ for all $z\in
Ty$,
\item[(b)] $(X,d,\preceq)$ is regular.
\end{itemize}
Let there exists $ x_0\in X $ and $x_1\in Tx_0$ such that $d(
x_0,Tx_0)=d(x_0,x_1)$, $x_0\preceq x_1$ and $\lambda s<1$, where
$$\lambda=\frac{d(x_0,  x_1) + d( x_1, Tx_1)}{d(x_0, x_1) + d( x_1, Tx_1) + 1}
+ k.$$ Then, $ T $ has at least one fixed point $ z\in X $.
\end{theorem}

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\end{thebibliography}
\end{center}



{\small

\noindent{\bf B. Mohammadi}

\noindent Department of Mathematics

\noindent Assistant Professor of Mathematics

\noindent Department of
Mathematics, Marand Branch, Islamic Azad University

\noindent  Marand, Iran

\noindent E-mail: bmohammadi@marandiau.ac.ir}\\

{\small
\noindent{\bf  F. Golkarmanesh  }

\noindent  Department of Mathematics

\noindent Assistant Professor of Mathematics

\noindent Department of Mathematics, Sanandaj Branch, Islamic
Azad University

\noindent  Sanandaj, Iran

\noindent E-mail: Fgolkarmanesh@yahoo.com}\\
{\small
\noindent{\bf  V. Parvaneh  }

\noindent  Department of Mathematics

\noindent Assistant Professor of Mathematics

\noindent Department of Mathematics, Gilan-E-Gharb Branch, Islamic
Azad University

\noindent  Gilan-E-Gharb, Iran

\noindent E-mail: zam.dalahoo@gmail.com}\\

\end{document}
