\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small 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}