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2005-Oujda International Conference on Nonlinear Analysis.
\newline {\em Electronic Journal of Differential Equations},
Conference 14, 2006, pp. 223--225.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{223}
\begin{document}
\title[\hfilneg EJDE/Conf/14 \hfil Common fixed points]
{Common fixed points for lipschitzian semigroups}
\author[S. Lahrech, A. Mbarki, A. Ouahab \hfil EJDE/Conf/14 \hfilneg]
{Samir Lahrech, Abderrahim Mbarki, Abdelmalek Ouahab}
\address{Samir Lahrech \newline
D\'epartement de Math\'ematiques,
Universit\'e Oujda, 60000 Oujda, Morocco}
\email{lahrech@sciences.univ-oujda.ac.ma}
\address{Abderrahim Mbarki \newline
Current address: National school of Applied Sciences, P.O. Box 669,
Oujda University, Morocco}
\email{ambarki@ensa.univ-oujda.ac.ma}
\address{Abdelmalek Ouahab \newline
D\'epartement de Math\'ematiques,
Universit\'e Oujda, 60000 Oujda, Morocco}
\email{ouahab@sciences.univ-oujda.ac.ma}
\date{}
\thanks{Published September 20, 2006.}
\subjclass[2000]{47H09, 47H10}
\keywords{Left reversible uniformly $k$-Lipschitzain
semigroups; \hfill\break\indent
common fixed point; uniform structure;
convexity structure; metric space}
\begin{abstract}
Lim and Xu \cite{l1} established a fixed point theorem for
uniformly Lipschitzian mappings in metric spaces with uniform
normal structure. Recently, Huang and Hong \cite{h1} extended
hyperconvex metric space version of this theorem, by showing a
common fixed point theorem for left reversible uniformly
$k$-Lipschitzian semigroups. In this paper, we extend Huang and
Hong's theorem to metric spaces with uniform normal structure.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\section{Introduction and main results}
Throughout this paper, $(X, d)$ stands for a metric space, a
nonempty family $\mathcal{F}$ of subsets of $X$ is said to define
a convexity structure on $X$ if it is stable by intersection.
Recall that a subset of $X$ is said admissible if it is an
intersection of closed balls. We denote, by $\mathcal{A}(X)$ the
family of all admissible subsets of $X$. Obviously,
$\mathcal{A}(X)$ defines a convexity structure on $X$. In this
paper any convexity structure $\mathcal{F}$ on $X$ is always
assumed to contain $\mathcal{A}(X)$. For
$r\geq 0$ and $x$ in $X$ and a bounded subset $M$ of $X$, we
adopt the following notation:
\begin{gather*}
B(x, r) \mbox{ is the closed ball centered at $x$ with radius $r$},\\
r(x, M ) = \sup\{d(x, y) : y \in M\},\\
\delta(M) = \sup\{ r(x, M) : x \in M\},\\
R(M) = \inf\{r (x, M) : x \in M\}.
\end{gather*}
\begin{definition}[\cite{k1}] \label{def1.1}\rm
A metric space ($X,d$) is said to have normal (resp. uniform normal) structure if
there exists a convexity structure $\mathcal{F}$ on $X$ such that
$R(A) < \delta (A)$ (resp. $R(A) \leq c\delta (A)$ for some
constant $c \in (0,1)$) for all $A$ in $\mathcal{F}$
which is bounded and $\delta (A) > 0 $. It is also said that
$\mathcal{F}$ is normal and (resp. uniformly normal).
\end{definition}
The uniform normal structure coefficient $N(X)$ of $X$ relative to $\mathcal{F}$ is
the number
$$
\sup\{ \frac{R(A)}{\delta(A)}: A\in \mathcal{F} \mbox{ is bounded and }
\delta(A)>0 \}.
$$
\begin{definition}[\cite{k2}] \label{def1.2} \rm
Let ($X,d$) a metric space and $\mathcal{T}$ is the family of subsets
of $X$ consisting of $X$ and sets which are
complements of closed balls of $X$. The weak topology
(also called ball topology) on
$X$ is the topology whose open sets are generated by $\mathcal{T}$.
\end{definition}
It is clear that $X$ is compact in the weak topology if and only
if every subfamily of $\mathcal{A}(X)$ with the finite intersection
property has nonempty intersection.
Kulesza and Lim proved the following result.
\begin{lemma}[\cite{k2}] \label{lem1.1}
Every complete metric space with uniform
normal structure is compact in the weak topology.
\end{lemma}
For a bounded subset $A$ of $X$, the admissible hull of $A$,
denoted by $ad(A)$, is the set
$$
\cap\{ B: A\subseteq B\subseteq X \mbox{ with } B \mbox{ admissible}\}.
$$
The following definition is a net version of \cite[def. 5]{l1}.
\begin{definition}[\cite{h1}] \label{def1.3} \rm
A metric space ($X, d$) is said to have
the property (P) if given any two bounded nets $\{x_i\}_{i\in I}$
and $\{z_i\}_{i\in I}$ in $X$, one can find some
$z \in \cap_{i\in I} \mathop{\rm ad}\{z_j: j \geq i\}$ such that
$$
\overline{\lim}_{i\in I}d(z, x_i)
\leq \overline{\lim}_{j\in I}\overline{\lim}_{i\in I}d(z_j, x_i),
$$
where $\overline{\lim}_{i\in I}d(z, x_i)
= \inf_{\beta\in I }\sup_{i\geq\beta }d(z,x_i)$.
\end{definition}
\begin{remark} \label{rmk1.1}\rm
If $X$ has uniform normal structure, then $\cap_{i\in I}
\mathop{\rm ad}\{z_j: j \geq i\}\neq \emptyset$
(by Lemma \ref{lem1.1}). Also, if $X$ is a weakly
compact convex subset of a normed linear space, then admissible hulls
are closed convex and $\cap_{i\in I} \mathop{\rm ad}\{z_j: j \geq i\}\neq \emptyset$ by weak compactness
of $X$ and that
$X$ possesses property (P) follows directly from the weak lower
semicontinuity of the function
$ x \longmapsto \overline{\lim}_{i\in I}\| x_i -x\| $.
\end{remark}
The following Lemma is a net version of \cite[lemma. 5]{l1}.
\begin{lemma} \label{lem1.2}
Let ($X, d$) be a complete bounded metric space with both
property (P) and uniform normal structure. Then for any net $\{x_i\}_{i\in I}$ in
$X$ and any $\overline{c} > N(X)$, the normal structure coefficient with respect to
the given convexity structure $\mathcal{F}$, there exists a point $z\in X$
satisfying the properties:
\begin{itemize}
\item[(i)] $\overline{\lim}_{i\in I}d(z, x_i)\leq \overline{c}
\delta(\{x_i\}_{i\in I})$;
\item[(ii)] $d(z, y) \leq \overline{\lim}_{i\in I}d(x_i, y)$
for all $y \in X$.
\end{itemize}
\end{lemma}
\begin{proof} Using the Lemma \ref{lem1.1} to conclude that $\cap_{i\in I}
A_{i}\neq\emptyset$ for any deceasing net $\{ A_i\}_{i\in I}$ of
admissible subsets of $X$, the rest of the proof of lemma is the
same as that in Lim et al. \cite{l1}.
\end{proof}
Let $S$ be a semigroup of selfmaps on a metric space ($X,d$). For any
$x\in X$ (resp. $ b\in S $), we denote by $Sx$ (resp. $bS$) the
subset $\{gx: g\in S\}$ (resp $\{bg: g\in S\}$ ) of $X$ (resp. of $S$).
Recall that a semigroup $S$ is said to be left reversible if, for any
$f, g$ in $S$, there are $a, b$ in $S$ such that $fa=gb$.
Examples of left reversible semigroups include all commutative
semigroups and all groups.
Let $S$ be a left semigroup. For $a, b$ in $S$ we say
that $a \geq b$ if $a \in bS\cup\{b\}$. Then ($S,\geq$)
is a directed set. In what follows in this paper, we deal only
with ``$\geq$''.
\begin{definition}[\cite{h1}] \label{def1.4:}\rm
A semigroup $S$ acting on a metric space ($X, d$) is said to be a uniformly
$k$-Lipschitzian semigroup if
$$
d(tx, ty) \leq k d(x, y)
$$
for all $t$ in $S$ and all $x, y$ in $X$.
\end{definition}
If $S$ is a left reversible semigroup, then ($S, \geq$) is a linearly directed set
if any $a, b$ in $S$ satisfy either $a \geq b$ or $b\geq a$. For example, if $\Delta
=\{ T_s : s\in [0, \infty) \}$ is a family of selfmaps on $\mathbb{R}$ such that
$T_{h+t}(x) =T_{h}T_{t}(x)$ for all $h, t$ in $[0, \infty)$ and $x\in \mathbb{R}$,
then ($\Delta,\geq$) is a linearly directed left reversible semigroup.\\[2mm] {\bf Our
main result is as follows.}
\begin{theorem} \label{thm1.1}
Let ($X,d$) be a complete bounded metric space with both
property (P) and uniform normal structure and let $S$ be a left
reversible uniformly
$k$-Lipschitzian semigroup of selfmaps on $X$ such that
$k< N(X)^{-1/2}$ and ($S,\geq$) is a linearly directed set.
Then $S$ has a common fixed point $z$ in $X$.
\end{theorem}
\begin{proof}
Choose a constant $\overline{c}$, $1> \overline{c} > N(X)$, such that
$k <\overline{c} ^{-1/2}$. Fix an $x_0 \in X$. For $t \in S$, denote
$tx_0$ by $x_{0, t}$. Then $\{x_{o,t}\}$ is a net in $X$. By
Lemma \ref{lem1.2},
we can inductively construct a sequence $\{x_j\}\subset X$ such that
for each integer $j\leq0$,
\begin{itemize}
\item[(a)] $\overline{\lim}_{t\in S}d(x_{j+1}, x_{j, t})
\leq \overline{c}\delta(Sx_j)$;
\item[(b)] $d(x_{j+1}, y)\leq \overline{\lim}_{t\in S}d(x_{j, t}, y)$
for all $y$ in $X$.
\end{itemize}
Write
$$
D_{j}= \overline{\lim}_{t\in S}d(x_{j+1}, x_{j, t})
\mbox{ and } h=\overline{c}k^2 < 1.
$$
The rest of the proof of Theorem is the same as that in Huang and
Hong \cite{h1}.
\end{proof}
\begin{remark} \label{rmk1.2}\rm
It can be seen from the above that the conclusion of main
theorem is still valid if we only assume that $\mathcal{A}(X)$,
the family of all admissible subsets of $X$, is uniformly normal.
\end{remark}
The following corollary follows immediately from the main Theorem.
\begin{corollary}[Huang and Hong \cite{h1}] \label{coro1.1}
Let ($X,d$) be a bounded hyperconvex metric
space with both property (P) and let $S$ be a left reversible uniformly
$k$-Lipschitzian semigroup of selfmaps on $X$ such that $k< \sqrt{2}$
and ($S,\geq$) is a linearly directed set. Then $S$ has a common
fixed point $z$ in $X$.
\end{corollary}
\begin{thebibliography}{00}
\bibitem{h1} Y. Y. Huang, C. C. Hong;
\emph{Common fixed point theorems for semigroups on metric spaces,} Internat. J.
Math. \& Math. Sci. \textbf{22}, no. 2, (1999), 377- 386.
\bibitem{k1} M. A.Khamsi; \emph{On metric spaces with uniform normal structure,}
Proc. Amer. Math. Soc. \textbf{106} (1989), no. 3, 723-726.
\bibitem{k2} J. Kulesza, T. C. Lim;
\emph{On Weak compactness and countable weak compactness in fixed
point theory,} Proc. Amer. Math. Soc. \textbf{124} (1996), no. 11, 3345-3349.
\bibitem{l1} T. C. Lim, H. K. Xu;
\emph{Uniformly Lipschitzian mappings in metric sapces with uniform
normal structure,} Nonlinear Anal. \textbf{25} (1995), no. 11, 1231-1235.
\end{thebibliography}
\end{document}