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\fancyhead[CE]{ H. Aydi and E. Karap{\i}nar}
\fancyhead[CO]{Generalized partial metric spaces with a fixed point theorem}



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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}
‎
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{\Large \bf Generalized partial metric spaces with a fixed point theorem\\}
%{\bf Do You Have a Subtitle? \\ If so, Write It Here}


\let\thefootnote\relax\footnote{\scriptsize Received: XXXX; Accepted: XXXX (Will be inserted by editor)}

{\bf Hassen AYDI}\vspace*{-2mm}\\
\vspace{2mm} {\small  Department of Mathematics, College of Education- Jubail, \\
Imam Abdulrahman Bin Faisal University, \\
P.O. 12020, Industrial Jubail 31961, Saudi Arabia.} \vspace{2mm}

{\bf  Erdal KARAPINAR$^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\
\vspace{2mm} {\small   Atilim University,  
Department of Mathematics, 06836, \.Incek, Ankara, Turkey,\newline
\noindent and  Department of Medical Research, China Medical University, Taichung, Taiwan} \vspace{2mm}

\end{center}

\vspace{4mm}


{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.} In this paper, we introduce the notion of extended partial metric space and we present some fixed point theorems in generalized partial metric spaces involving linear and nonlinear contractions.
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:}   46T99; 47H25 ; 54H10

\noindent{\bf Keywords and Phrases:} Partial metric, generalized partial metric, nonlinear contraction, fixed point
\end{quotation}}






\section{Introduction and Preliminaries}
Very recently, Aydi and Czerwik \cite{AC} proposed a new notion, generalized $b$-metric space and investigated the existence and uniqueness of a fixed point of certain mappings on this new space. In this paper, we introduce the generalized partial metric space inspired of the notion of
a partial metric space was introduced by Matthews \cite{Ma} in 1994 as a part to study the denotational semantics of dataflow networks which play an
important role in constructing models in the theory of computation (see also e.g. \cite{pm1},\cite{AC},
\cite{pm21a}-\cite{pm4},\cite{Rom}).

\begin{definition}(cf. \cite{Ma})
A generalized partial metric on a nonempty set $X$ is a function $p:X\times X\rightarrow\mathbb [0,\infty]$ such that for all $x,y,z \in X$
\begin{enumerate}
\item[(PM1)] $p(x,x) = p(x,y) = p(y,y)$, then $x=y$;

\item[(PM2)] $p(x,x) \leq p(x,y)$;

\item[(PM3)] $p(x,y) = p(y,x)$;

\item[(PM4)] $p(x,z) + p(y,y) \leq p(x,y) + p(y,z)$.
\end{enumerate}
The pair $(X,p)$ is then called a generalized partial metric space (gpms).
\end{definition}


As usual, by $\mathbb{N},\ \mathbb{N}_0,\ \mathbb{R}_{+}$ we denote the set of all natural numbers, the set of all nonnegative integers or the set of all nonnegative real numbers, respectively.


If $f\colon X\to X$, by $f^n$ we denote the $n$-th iterate of $f$:
$$f^0(x)=x,\quad x\in X;\quad f^{n+1}=f\circ f^n.$$
Here the symbol $\varphi \circ f$ denotes the function $\varphi[f(x)]$ for $x\in X$.

As in \cite{Ma}, we may state the following definitions and remarks. If $p$ is a generalized partial metric on $X,$ then the function $d_p : X \times X\to [0,\infty]$ defined by
$$
d_p(x, y) = 2p(x, y)-p(x, x)-p(y, y)
$$
for all $x,y\in X,$ is a generalized metric on $X$  (defined in \cite{AC} with $s=1$). More precisely,
for a nonempty set $X$, a function $d_p: X \times X \to [0, \infty ] $ is called a generalized metric space if and only if for $x, y, z \in X$ the conditions are satisfied:
\begin{enumerate}
\item[(${d}_1$)] $d_p(x,y) = 0$ if and only if $x = y$, (self-distance)
\item[(${d}_2$)] $d_p(x,y) = d_p(y,x)$, (symmetry)
\item[(${d}_3$)] $d_p(x,y) \leq   d_p(x,z) + d_p(z,y) $ (triangle inequality).
\end{enumerate}





Note that if a sequence converges in a generalized partial metric space $(X, p)$ with respect to  the topology of $d_p$, then it converges with respect to the topology of $p$.

Also, a sequence $\{x_n\}$  is Cauchy in a generalized partial metric space $(X, p)$ if and only if it is  Cauchy  in the generalized metric space $(X, d_p).$ Consequently, a generalized partial metric space $(X, p)$ is complete if
and only if the generalized metric space $(X, d_p)$ is complete. Moreover, if $\{x_n\}$ is a sequence in a generalized partial metric space $(X, p)$ and $x\in X,$ one has that
$$
\lim_{n\to\infty}d_p(x_n,x)=0\Leftrightarrow p(x,x)=\lim_{n\to\infty}p(x_n,x)=\lim_{n,m\to\infty}p(x_n,x_m).
$$

\begin{definition}
 Let $(X,p)$ be a generalized partial metric space. We say that $T: X\rightarrow X$ is (sequentially) continuous if $p(x_n,x)\rightarrow p(x,x)$, then
 $p(Tx_n,Tx)\rightarrow p(Tx,Tx)$ as $n\rightarrow \infty$.
 \end{definition}
\begin{lemma}\label{lem1}
 Let $(X,p)$ be a generalized partial metric space. Then\\
$(1)$ if $p(x,y)=0$, we have $x=y$,\\
$(2)$ if $x\neq y$, we have $p(x,y)>0.$
\end{lemma}
\section{Linear quasi-contractions}
\indent We start with the following theorem
\begin{theorem}\label{t1}
Let $(X,d)$ be a complete generalized partial metric space. Assume that $T\colon X \to X$ is continuous on $(X,d_p)$. If there exists an $\alpha \in [0,1)$  such that
\begin{equation}\label{1}
p(T(x), T^2(x))\leq \alpha p(x,T(x)),
\end{equation}
for $x \in X$  with$p(x,T(x))<\infty $, then, for an arbitrary fixed $x\in X$, one of  the following alternative holds : either
 \begin{itemize}
  \item [(A)]\  for every nonnegative integer $n\in \mathbb{N}_0$, $$p(T^n(x),T^{n+1}(x))=\infty,$$ or
  \item [(B)]\  there exists an $k\in \mathbb{N}_0$ such that
  $$  p(T^k(x), T^{k+1}(x))<\infty.$$
 \end{itemize}
If (B) holds, then, we also conclude the followings:
 \begin{itemize}
\item [(i)] \ the sequence $\{T^m(x)\}$ is a Cauchy sequence in $(X,p)$;
\item [(ii)] \ there exists a point $u\in X$ such that $$\lim_{m\to \infty} d_p(T^m(x),u)=0\quad and \quad T(u)=u.$$
\end{itemize}
\end{theorem}
\begin{proof} From (\ref{1}) we get (in case (B))
$$p(T^{k+1}(x),T^{k+2}(x))\leq\alpha p(T^{k}(x),T^{k+1}(x))<\infty$$
and by induction
\begin{equation}\label{3}
p(T^{k+n}(x),T^{k+n+1}(x))\leq \alpha^n p(T^{k}(x),T^{k+1}(x)) ,\quad n=0,1,2,\ldots\ .
\end{equation}
Consequently, for $n, v \in \mathbb{N}_0$, by (\ref{3}) we obtain
\begin{eqnarray*}
p(T^{k+n}(x),T^{k+n+v}(x))&\leq&p(T^{k+n}(x),T^{k+n+1}(x))+\ldots+p(T^{k+n+v-2}(x),T^{k+n+v-1}(x))\\&+&p(T^{k+n+v-1}(x),T^{k+n+v}(x))\\
&\leq&\alpha^n p(T^{k}(x),T^{k+1}(x))+\ldots+\alpha^{n+v-2}p(T^{k}(x),T^{k+1}(x))\\&+&\alpha^{n+v-1}p(T^{k}(x),T^{k+1}(x))\\
&\leq&\alpha^n[1+s\alpha+\ldots + (\alpha)^{v-1}]p(T^{k}(x),T^{k+1}(x))\\
&\leq&\alpha^n\sum_{m=0}^{\infty}(\alpha)^{m}p(T^{k}(x),T^{k+1}(x))\\
&\leq&\frac{\alpha^n}{1-\alpha}p(T^{k}(x),T^{k+1}(x)).
\end{eqnarray*}
Finally, we derive that 
\begin{equation}\label{4}
p(T^{k+n}(x),T^{k+n+v}(x))\leq\frac{\alpha^n}{1-\alpha}p(T^{k}(x),T^{k+1}(x))
\end{equation}
for $n, v \in \mathbb{N}_0$. By (\ref{4}) it follows that $\{T^n(x)\}$ is a Cauchy sequence in  $(X,p)$, which is complete, so there exists $u\in X$ such that
$$
\lim_{n\to \infty} p(T^n(x),u)=p(u,u)=\lim_{n,m\to \infty} p(T^n(x),T^m(x))=0.
$$
We have $\lim_{n\to \infty} d_p(T^n(x),u)=0$. Since $T$ is continuous on $(X,d_p)$,  we have
$$\lim_{n\to \infty}d_p(T^{n+1}(x),Tu)=\lim_{n\to \infty}d_p(T(T^{n}(x)),Tu)=0.$$
Moreover, $\lim_{n\to \infty}d_p(T^{n+1}(x),Tu)=d_p(u,Tu)$. By uniqueness of limit, we get $T(u)=u$.
and $u$ is a fixed point of $T$, which ends the proof.
\end{proof}
\begin{remark}
Theorem \ref{t1} extends the results  of Aydi and Czerwik (\cite{AC} with $s=1$), Diaz and Margolis \cite{Diaz}, Luxemburg \cite{Lux1,Lux2} and  Banach (\cite{Ban} to generalized partial metric spaces.
\end{remark}
\section{Nonlinear contractions}
In this section, we present the following result.
\begin{theorem}\label{t2}
 Assume that $(X,p)$ is a complete generalized partial space. Suppose that $T\colon X \to X$ satisfies the condition
\begin{equation}\label{7}
p(T(x), T(y))\leq \varphi[p(x,y)]
\end{equation}
for $x,y\in X,\ p(x,y)<\infty$, where $\varphi\colon [0,\infty)\to [0,\infty)$ is nondecreasing and
\begin{equation}\label{8}
\lim_{n\to \infty}\varphi^n(z)=0 \quad for\ z>0.
\end{equation}
Let $x\in X$ be arbitrarily fixed. Then the following alternative holds: either
 \begin{itemize}
  \item [(C)] \quad for every nonnegative integer $n\in \mathbb{N}_0$ $$p(T^n(x),T^{n+1}(x))=\infty,$$      or
  \item [(D)] \quad there exists an $k\in\mathbb{N}_0$ such that $$p(T^k(x),T^{k+1}(x))<\infty.$$
 \end{itemize}
In (D), $T$ has a unique fixed point in   $A:=\{t\in X \colon d_p(T^{k}(x),t)<\infty\}$.
\end{theorem}
\begin{proof}

First, take $x\in X$ and $\varepsilon > 0$. Take $n\in \mathbb{N}$ such that $$\varphi^n(\varepsilon)<\frac{\varepsilon}{2}.$$
Put $\alpha=\varphi^n$ and $x_m=T^{m+n}(x)$ for $m\in \mathbb{N}$. Then for all $x,y\in X$ such that $p(x,y)<\infty$, one gets
\begin{equation}\label{10}
p(T^{n}(x),T^{n}(y))\leq \varphi^n[p(x,y)]=\alpha[p(x,y)].
\end{equation}
Consider the following set
$$B:=\{t\in X \colon p(T^{k}(x),t)<\infty\}.$$
%$2^0$\big) One can prove that $(B,p)$ is a complete partial space.
Clearly, $B\subset A$  and $T^k(x),\ T^{k+1}(x)\in B$.

Now we observe that $T\colon B\to B$. Indeed,  if $t\in B$, i.e., $p(T^k(x),t)<\infty$, then

\begin{eqnarray*}
p(T^{k}(x),T(t))&\leq&p(T^{k}(x),T^{k+1}(x))+p(T^{k+1}(x),T(t))]\\
&\leq&\varepsilon_1+\varphi[p(T^{k}(x),t)]\\
&\leq&\varepsilon_1+\varepsilon_2<\infty,
\end{eqnarray*}
where $\varepsilon_1$ and $\varepsilon_2$ are some positive numbers. Consequently,  $T^{n}\colon B\to B$. Put $T^n=F$. We have $F\colon B\to B$. We rewrite (\ref{10}) as
\begin{equation}\label{h1}
p(F(x),F(y))\leq \varphi^n[p(x,y)]=\alpha[p(x,y)].
\end{equation}
For $t\in B$, we have $\{F^{m}(t)\}\subset B$, for all $m\in \mathbb{N}_0$. We verify that $\{F^{m}(t)\}$ is a Cauchy sequence. In fact, putting $y_m=F^{m}(t)$, $m\in \mathbb{N}_0$, we get
$$p(F(t),F^2(t))=p(T^{n}(t),T^{n+1}(t))\leq \alpha[p(t,T^{n}(t))].$$
By induction, we get
$$p(F^{m}(t),F^{m+1}(t))\leq \alpha^m[p(t,F(t))],$$
that is equivalent to
$$p(y_m,y_{m+1})\leq \alpha^m[p(t,F(t))].$$
Consequently, $p(y_m,y_{m+1})\to 0$ as $m\to \infty$. Let $m$ be such that $$p(y_m,y_{m+1})<\frac{\varepsilon}{2}.$$
Then for every $z\in K(y_m,\varepsilon):=\{y\in X \colon p(y_m,y)\leq\varepsilon\}$,
we obtain
$$p(F(z),F(y_m))\leq\alpha[p(z,y_m)]\leq\alpha(\varepsilon)=\varphi^n(\varepsilon)<\frac{\varepsilon}{2}.$$
Also, we know that
$$p(F(y_m),y_m)=p(y_{m+1},y_m)<\frac{\varepsilon}{2}.$$
Thus we have
 $$p(T^{n}(z),y_m)=p(F(z),y_m)\leq p(F(z),F(y_m))+p(F(y_m),y_m)<\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon,$$
which means that $F=T^{n}$ maps $K(y_m,\varepsilon)$ into itself. Therefore
$$p(y_r,y_l)\leq 2\varepsilon\quad for\quad r,l\geq m,$$
so $\{y_r\}=\{F^{r}(t)\}_r$ is a Cauchy sequence in $B$. Since $B\subset A$, $\{y_r\}=\{F^{r}(t)\}_r$ is a Cauchy sequence in $A$.
Since $(X,p)$ is complete, $(X,d_p)$ is also complete. Clearly, $(A,d_p)$ is closed, so it is complete.
%On the other hand, the function $t\longmapsto d_p(T^k(x),t)$  is continuous on $(X,d_p)$, so  obviously $(A,d_p)$ is complete.
Hence there exists $u\in A\subset X$ such that %$y_r\to u$ as $r\to \infty$, that is,
 $$\lim_{r\rightarrow \infty} d_p(y_r,u)=0.$$
 We deduce that
 \begin{equation}
 \label{h2}
 p(u,u)=\lim_{r\rightarrow \infty} p(y_r,u)=\lim_{r,j\rightarrow \infty} p(y_r,y_j)=0.
 \end{equation}
 Thus, for a large $r$,
 \begin{equation}
 \label{h8}
 p(y_r,u)<\infty.
 \end{equation}
 Also, we have
 \begin{equation}
 \label{h2+}
 \lim_{r\rightarrow \infty} d_p(y_{r+1},Fu)=d_p(u,F(u)).
 \end{equation}
 Moreover, by (\ref{h1}) and (\ref{h8}),
 \begin{equation}
 \label{h3}
 p(y_{r+1},F(u))=p(F(y_r),F(u))\leq \alpha[p(y_r,u)]
 \end{equation}
 letting $r\rightarrow \infty$ in (\ref{h3}), due to (\ref{h2}), we get
 \begin{equation}
 \label{h4}
 \lim_{r\rightarrow \infty} p(y_{r+1},F(u))=0.
 \end{equation}
 Consequently, we find
 \begin{equation}
 \label{h5}
 \lim_{r\rightarrow \infty} d_p(y_{r+1},F(u))=0.
 \end{equation}
 Comparing (\ref{h2+})  to (\ref{h5}) yields that $d_p(u, F(u))=0$, i.e., $u=F(u)$, that is, $u$ is a fixed point of $F$. Suppose there are two different fixed points  $u$ and $v$ of $F$ in $A$. Then
   \[
   d_p(u,v)\leq d_p(u,T^{n}(x))+d_p(T^{n}(x),v)<\infty.
   \]
   Now, applying (\ref{7}),
   \[
   p(u,v)=p(F(u),F(v))\leq \alpha[p(u,v)].
   \]
   Taking into consideration that $\alpha(t)=\varphi^n(t)<t$ for any $t>0$, we get a contradiction. Thus, $F$ has exactly one fixed point in $A$.
   Now, we shall show that $u$ is also a fixed point of $T$.
   Applying (\ref{7}) and (\ref{h8}),
   \[
   p(T(u),T(y_r))\leq \varphi(p(y_r,u).
   \]
   In view of (\ref{h2}),
   \begin{equation}
 \label{h7}
 \lim_{r\rightarrow \infty} p(T(u),T(y_{r}))=0.
 \end{equation}
   On the other hand,
   \[
   p(T(u),Ty_r)=p(T(u),T(F^r(t)))=p(T(u),F^r(T(t)))\rightarrow p(T(u),u).
   \]
 By comparison, we deduce that $p(u,T(u))=0$, so $u=T(u)$, hence $u$ is a fixed point of $T$. Again, obviously by (\ref{7}) such point is the unique fixed point of $T$ in $A$.
%Eventually, since for every $t\in B$ and every $r=0,1,\ldots, n-1$,
%$$T^m(t)=T^{nl+r}(t)=F^l[T^r(t)]\to u \quad as\quad l\to \infty,$$
%so
%$$d(T^m(t),u)\to 0\quad as\quad m\to \infty$$
%for every $t\in B$, whence we get \emph{(vi)}.
\end{proof}

If $X$ is a partial metric space, then $B=A=X$ and we have from Theorem \ref{t2}.
\begin{corollary}
\label{cor1}
Let $(X,d)$ be a complete partial space. Suppose that $T\colon X\to X$ satisfies
$$p(T(x), T(y))\leq \varphi[p(x,y)],\quad x,y\in X,$$
where $\varphi\colon \mathbb{R}_{+}\to \mathbb{R}_{+}$ is nondecreasing function such that $\lim_{n\to \infty}\varphi^n(t)=0$ for each $t>0$. Then $T$ has exactly one fixed point $u \in X$.
\end{corollary}
\begin{remark}
Corollary \ref{cor1} corresponds to Corollary 1 of Romaguera \cite{Rom}, which a Matkowski type result \cite{Matk}. Theorem \ref{t1} extended the main result of Aydi and Czerwik \cite{AC} to generalized partial matric spaces.
\end{remark}

\vspace{4mm}\noindent{\bf Competing interests}\\  
The authors declare that they have no competing interests.


\vspace{4mm}\noindent{\bf Authors contributions}\\    
All authors contributed equally and significantly in writing this
article. All authors read and approved the final manuscript.


\vspace{4mm}\noindent{\bf Acknowledgements}\\
The authors thanks to anonymous referees for their remarkable comments, suggestion and ideas
that helps to improve this paper.


\begin{center}


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\end{thebibliography}
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{\small

\noindent{\bf Hassen AYDI}

\noindent Department of Mathematics

\noindent  Professor of Mathematics

\noindent Imam Abdulrahman Bin Faisal University

\noindent Jubail, Saudi Arabia

\noindent E-mail: hmaydi@uod.edu.sa}\\

{\small
\noindent{\bf  Erdal Karap{\i}nar}

\noindent  Department of Mathematics

\noindent Full Professor of Mathematics

\noindent Atilim University

\noindent Ankara, Turkey 

\noindent  Department of Medical Research, 
\noindent China Medical University, 
\noindent Taichung, Taiwan 

\noindent E-mail:erdalkarapinar@gmail.com}\\



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