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%%%%%%%%%%%% A fixed point theorem for a sum of two %%%%%%%%%%%% multifunctions (fragment of a paper published in %%%%%%%%%%%% the International Journal of Evolution Equations) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Feel free to use this simple template (derived from an %% %% actual paper) for your own papers. However, it might be %% %% also a good idea to _learn_ from it a bit: try to change %% %% some things (e.g., as suggested in the comments below) and %% %% see what happens. %% %% %% %% Any comments regarding this template will be appreciated. %% %% %% %% Marcin Borkowski %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % We will use the ``amsart'' document class, provided by the AMS. It % is very well suited to mathematical papers (what a surprise!). It % is a part of a larger bundle, together with ``amsthm'', ``amsrefs'' % and some other packages. \documentclass[12pt]{amsart} % We load the ``amsrefs'' package so that we will be able to make % bibliography in an easy and portable way (in particular, it does not % depend on external programs like BibTeX). \usepackage{amsrefs} % The ``amsrefs'' package supports only basic types of % entries---articles, books and a few others. Here we will need an % entry type for an article in a conference proceedings. Adding % customary entry types with amsrefs is very easy (unlike % BibTeX)---the process is also well documented in amsrefs' user's % manual. \BibSpec{proceedings.article}{% +{} {\PrintAuthors} {author} +{,} { \textit} {title} +{.} { } {part} +{:} { \textit} {subtitle} +{,} { } {conference} +{} { (} {series} +{} { } {volume} +{).}{ \PrintEditorsC} {editor} +{,} { \PrintDateB} {date} +{,} { pp.~} {pages} +{.} {} {transition} } % \B and \K will be operators like \sin, \log etc., in particular % they will have proper spacing and will be typeset in roman font % (properly scaled in sub- and superscripts). \DeclareMathOperator{\B}{Bd} \DeclareMathOperator{\K}{Komp} % The \mapping macro takes two arguments: the domain and the codomain. % Notice the use of \colon instead of ``:'', which has a wrong spacing % in this context. \newcommand{\mapping}[2]{\colon #1\to #2} % As usual, we will use ``blackboard bold'' for the set of natural % numbers. Extra braces are here in case we use something like % $2^\Nset$, which would fail without them. \newcommand{\Nset}{{\mathbb{N}}} % The \complement macro takes two arguments, the first one being % optional (and having a default value of ``X''). We discard the % first one completely, but if we changed our mind and decided to % write ``X\A'' instead of ``A' '' to denote the complement, we could % easily say something like: % \newcommand{\complement}[2][X]{#1\setminus #2} % and we wouldn't have to change anything except this single line in % the preamble. Also, making the first argument optional with a % reasonable default value saves lots of keystrokes when typing the % paper. \newcommand{\complement}[2][X]{#2'} % We want a macro for norm, so that we can write ``\norm[x]'' to % obtain ``||x||'' and ``\norm'' to obtain ``||.||''. Also notice % that it's better to use \lVert ... \rVert than \| ... \"; the former % gives TeX an additional clue that the bars are a left and a right % delimiter respectively, which might help TeX determine proper % spacing in some cases. \newcommand{\norm}[1][\cdot]{\lVert#1\rVert} % We prefer the \varepsilon over the \epsilon;) \newcommand{\eps}{\varepsilon} % Finally, we will use \textup rather frequently, so why don't we make % a shorter alias for it? \let\tu=\textup % Now we need some theorem environments. Please refer to the amsthm % package documentation for details (in particular, for a list showing % which pieces of mathematical text are usually typeset using which % \theoremstyle (which isn't at all obvious, at least for me). \usepackage{amsthm} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \theoremstyle{definition} \newtheorem{definition}{Definition} % OK, enough definitions---let's get to the paper itself! \begin{document} % The first version (in brackets: [...]) will show up in the running % head. A \\ in the second argument is a linebreak. \title [Krasnoselskii-type theorem for multifunctions] {A~Krasnoselskii-type fixed point theorem\\for multifunctions\\defined on a~hyperconvex space} % These commands should be rather obvious. Refer to the AMS' % ``Instructions for Preparation of Papers and Monographs'' for a % detailed list of possible commands in the preamble (and how to use % them when there is more than one author). \author{Marcin Borkowski} \address{Optimization and Control Theory Department\\ Faculty of Mathematics and Computer Science\\ Adam Mickiewicz University\\ ul.\ Umultowska 87\\ 61\nobreakdash-614 Pozna\'n\\ Poland} \email{fake.email@to.fight.spam} % Remember to put the abstract _before_ the \maketitle command! \begin{abstract} We present a~fixed point theorem for a~sum of two convex-valued multifunctions acting on a~weakly compact, hyperconvex subset of a~normed space. The theorem is a~multivalued version of a~result of D.~Bugajewski. \end{abstract} % OK, now we can actually set the above information on the page. \maketitle \section{Introduction} \label{sec:intro} % Notice the usage of the amsrefs' \cite command, which is very % different from the usual LaTeX form. See its documentation for the % rationale and details on usage. In~\cite{Bugajewski}*{s.~1458, Theorem~2}, D.~Bugajewski proved the following Krasnoselskii-type theorem in a~hyperconvex setting. % Notice how we use \tu (an earlier-defined shorter form of \textup) % to make punctuation upright, which is suggested by AMS' style % guide. Notice also that we _don't_ do this within a math formula, % where all the punctuation is already set in roman type. \begin{theorem} \label{th:bugajewski} Let~$K$ be a~bounded hyperconvex subset of a~normed space $(X,\norm)$ such that $\lambda K\subset K$ for every $\lambda\in(0,1]$. Assume that \begin{enumerate} \item $f_1\colon K\to X$ is nonexpansive\tu; \item $f_2\colon K\to X$ is completely continuous\tu; \item $f_1(x)+f_2(y)\in K$ for any $x,y\in K$\tu; \item every sequence $(x_n)$ such that $x_n\in K$ for $n\in\Nset$ and \begin{equation*} \lim_{n\to\infty}\bigl(x_n-f(x_n)\bigr)=0, \end{equation*} where $f:=f_1+f_2$, has a~limit point. \end{enumerate} Then\tu, $f$~has a~fixed point. \end{theorem} Recall that the assumption that $\lambda H\subset H$ can be released, as it was shown in~\cite{BE}. Recently, M.~\"Ozdemir and S.~Akbulut published the paper~\cite{OA} with a~multivalued version of Bugajewski's theorem. Unfortunately, their proof contains some errors. We will state a~slightly different version of this theorem and then discuss the errors in~\cite{OA}. \section{Preliminaries} \label{sec:prelims} Let $X$ be a~metric space. By $\B X$ we denote the family of nonempty, bounded and closed subsets of~$X$ and by $\K X$ the set of nonempty compact subsets of~$X$. In what follows, we will use the symbol~$d_X$ for a~metric in the space~$X$ and $H_X$ for the Hausdorff metric in the hyperspace~$\B X$; we will write~$d$ and~$H$ if the underlying space is obvious from the context. By $\complement{A}$ we will denote the complement of the subset~$A$ of some space~$X$, i.e., the set $X\setminus A$. \begin{definition} We call a~mapping $f\mapping{X}{Y}$ between metric spaces \emph{nonexpansive}, if $d(f(x),f(y))\le d(x,y)$ for each $x,y\in X$. \end{definition} \begin{definition} Let $A$ be any subset of a~metric space~$X$. The \emph{Kuratowski measure of noncompactness} of the set~$A$, denoted by~$\alpha(A)$, is the greatest lower bound of the numbers~$\eps>0$ such that $A$ can be covered by a~finite family of sets of diameter not greater than~$\eps$. (We put $\alpha(A)=+\infty$ for unbounded sets.) A~mapping $f\mapping{X}{Y}$ between metric spaces is called \emph{$\alpha$-condensing} if $\alpha(f(A))\le\alpha(A)$ for each nonempty $A\subset X$ and $\alpha(f(A))<\alpha(A)$ provided that $\alpha(A)>0$. \end{definition} % Of course, the actual paper is a bit longer;). But we will finish % here due to copyright reasons;). Now it's time for the % bibliography. Using an \ndash instead of ``--'' is a nicety; it % cannot be guaranteed that ``--'' will typeset an n-dash in fonts % other than the default Computer Modern family. \begin{bibdiv} \begin{biblist} \bib{Bugajewski}{article}{ author={Bugajewski, D.}, title={Fixed-point theorems in hyperconvex spaces revisited}, journal={Math. Comput. Modelling}, volume={32}, number={13}, date={2000}, pages={1457\ndash 1461}, } \bib{BE}{proceedings.article}{ author={Bugajewski, D.}, % The last thing worth mentioning: \i stands for a ``dotless i'', % which is used to put accents on the letter ``i''. author={Esp\'\i nola, R.}, title={Remarks on some fixed point theorems for hyperconvex spaces and absolute retracts}, conference={Function Spaces the Fifth Conference}, series={Lecture Notes in Pure and Applied Mathematics Series}, volume={213}, editor={Hudzik, H.}, editor={Skrzypczak, L.}, year={2000}, pages={85\ndash92}, } \bib{OA}{article}{ author={\"Ozdemir, M.}, author={Akbulut, S.}, title={A~fixed point theorem for multivalued maps in hyperconvex spaces}, journal={Appl. Math. Comput.}, volume={157}, date={2004}, pages={637\ndash 642}, } \end{biblist} \end{bibdiv} \end{document} % That's all, folks!