# sum-multi-sample.tex

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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}
Faculty of Mathematics and Computer Science\\
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!