\documentclass[10pt,twoside]{article}
\usepackage{amssymb}
\setlength{\textwidth}{5.0in}
\setlength{\textheight}{7.5in}
%\setcounter{page}{1}
%\pagestyle{myheadings}
%\markboth{Author}{Title}
\usepackage{latexsym}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{example}{Example}[section]
\newtheorem{corollary}{Corollary}[section]
\def\proof{\mbox {\it Proof.~}}

\makeatletter\def\theequation{\arabic{section}.\arabic{equation}}\makeatother
\begin{document}
\title{
%\begin{flushleft}
%{\small{\it Advanced Nonlinear Studies} {\bf 1} (2001), xxx--xxx}
%\end{flushleft}
\vspace{0.5in}
{\bf\Large  The Title of the Paper}}
\author{{\bf\large Name}\footnote{The author acknowledges the support...}\hspace{2mm}
{\bf\large Name}\vspace{1mm}\\
{\it\small Department of Mathematics}\\ {\it\small University of Texas}, 
{\it\small San Antonio, Texas 78249}\\
{\it\small e-mail: address}\vspace{1mm}\\
%\date{\small{Received ---------------} \\
%{\it\small Communicated by ----------}}

\maketitle
\begin{center}
{\bf\small Abstract}

\vspace{3mm}
\hspace{.05in}\parbox{4.5in}
{{\small The abstract should be around 300 words, but an attempt should be made not  
\indent
to exceed 400 words.}}
\end{center}

\noindent
{\it \footnotesize 1991 Mathematics Subject Classification}. {\scriptsize 12345, 54321}.\\
{\it \footnotesize Key words}. {\scriptsize Differential operator, Hilbert space}

\section{\bf Introduction}
\def\theequation{1.\arabic{equation}}\makeatother
\setcounter{equation}{0}
\begin{definition}
{\rm The texts of Definition, Remark, and Examples must be in roman.} 
\end{definition}

\begin{proposition}
This is the proposition.
\end{proposition}

\noindent
\proof

\noindent
This is the end of the proof.

\begin{lemma}
This is the lemma.
\end{lemma}

\begin{corollary}
This is the corollary.
\end{corollary}

\section{\bf Title of this section}
\def\theequation{2.\arabic{equation}}\makeatother
\setcounter{equation}{0}
\begin{eqnarray}
A+B=c+d
\end{eqnarray}
$$
\alpha+\beta=\gamma+\delta
$$
The references are cited as \cite{BO89} and \cite{RA83}. {\bf Note that the text and the 
mathematical expressions should be adjusted or split so that they do not exceed the right margin}.


\section{\bf Title of this section}
\def\theequation{3.\arabic{equation}}\makeatother
\setcounter{equation}{0}
\begin{definition}
{\rm The texts of Definition, Remark, and Examples must be in roman.} 
\end{definition}

\begin{theorem}
This is the theorem.
\end{theorem}


\begin{theorem}
This is another theorem in the same section.
\end{theorem}


\begin{thebibliography}{777}
\bibitem{BO89} N. Bourbaki,
Algebra I,
Springer-Verlag, Berlin, 1989.
\bibitem{RA83} J. Rawnsley, W. Schmid, and J.A. Wolf, 
{\it Singular unitary representations and indefinite harmonic theory},
J. Funct. Anal. {\bf 51} (1983), 1--114.


\end{thebibliography}

\enddocument