\documentclass[11pt,twoside]{article} \usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, color} \usepackage[bookmarksnumbered, colorlinks]{hyperref} \usepackage{float} \usepackage{lipsum} \usepackage{afterpage} \usepackage[labelfont=bf]{caption} \usepackage[nottoc,notlof,notlot]{tocbibind} %\renewcommand\bibname{References} \def\bibname{\Large \bf References} \usepackage{lipsum} \usepackage{fancyhdr} \pagestyle{fancy} \fancyhf{} \renewcommand{\headrulewidth}{0pt} \fancyhead[LE,RO]{\thepage} \thispagestyle{empty} %\afterpage{\lhead{new value}} \fancyhead[CE]{ H. Aydi and E. Karap{\i}nar} \fancyhead[CO]{Generalized partial metric spaces with a fixed point theorem} %\topmargin=-1.6cm \textheight 17.5cm% \textwidth 12cm % \topmargin 8mm % \oddsidemargin 20mm % \evensidemargin 20mm % \footskip=24pt % \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} %\theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \renewenvironment{proof}{{\bfseries \noindent Proof.}}{~~~~$\square$} \makeatletter \def\th@newremark{\th@remark\thm@headfont{\bfseries}} \makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % If you want to insert other packages. Insert them here %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\long\def\symbolfootnote[#1]#2{\begingroup% %\def\thefootnote{\fnsymbol{footnote}}\footnote[#1]{#2}\endgroup} \def \thesection{\arabic{section}} \begin{document} %\baselineskip 9mm %\setcounter{page}{} \thispagestyle{plain} {\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 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 $\alpha(t)=\varphi^n(t)0$, 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} \begin{thebibliography} {300} \bibitem{pm1} T.Abdeljawad, E. Karap\i nar, E. and K.Tas, Existence and Uniqueness of a Common Fixed Point on Partial Metric Spaces, Appl.Math.Lett., 24 (2011), no:11,1894-1899. \bibitem{AC} H.~Aydi and S.~Czerwik, Fixed point theorems in generalized $b$-metric spaces, Modern Discrete Mathematics and Analysis (Springer), (2017). \bibitem{Ban} S. 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