Problem with numbering theorem by section

Actually I have tried the solution of egreg's, and then find out that you don't need to create the command \spnewtheorem{xx}{XX}[section]

It is the option envcountsect in the svjour3 class that really helps.

\documentclass[envcountsect]{svjour3}

If you look into the file svjour.cls, and seach \spnewtheorem{theorem}{Theorem}, around line 888, you would find things like @envcntsect is prerequisite of the numbering by section.

And then, you just use

\begin{theorem}
    your theorem here.
\end{theorem}

to wrap your theorem. Proof, lemma, definition, case... are likewise.

Theorem-like environments should be defined, with the svjour3 class, by using

\spnewtheorem

and not \newtheorem. Here's how you should do:

\documentclass[smallextended,referee,envcountsect]{svjour3}

\usepackage{amsmath}
\usepackage{bm}

\smartqed
\journalname{foo}

% Theorem-like environments are defined with \spnewtheorem
%
% Usage:
%
%     \spnewtheorem{env_nam}{caption}[within]{cap_font}{body_font}
% or  \spnewtheorem{env_nam}[numbered_like]{caption}{cap_font}{body_font}
% or  \spnewtheorem*{env_nam}{caption}{cap_font}{body_font}

\spnewtheorem{thm}{Theorem}[section]{\bfseries}{\itshape}
\spnewtheorem{cor}[theorem]{Corollary}{\bfseries}{\itshape}
\spnewtheorem{lem}[theorem]{Lemma}{\bfseries}{\itshape}
\spnewtheorem{ntn}{Notations}[section]{\bfseries}{\itshape}
\spnewtheorem{pro}{Proposition}[section]{\bfseries}{\itshape}
\spnewtheorem{dfn}{Definition}[section]{\bfseries}{\itshape}% maybe \upshape?
\spnewtheorem{as}{Assumption}[section]{\bfseries}{\itshape}
\spnewtheorem{rem}{Remark}[section]{\bfseries}{\itshape}
\spnewtheorem{ob}{Observation}[section]{\bfseries}{\itshape}

\DeclareMathOperator{\Div}{div} % \div is already defined
\DeclareMathOperator{\D}{\mathbf{D}}
\DeclareMathOperator{\FP}{p}
\DeclareMathOperator{\Span}{Span}

\newcommand{\boldalpha}{\bm{\alpha}}
\newcommand{\boldpi}{\bm{\pi}}
\newcommand{\boldzeta}{\bm{\zeta}}
\newcommand\boldtheta{\bm{\theta}}


\begin{document}

\section{Solution}\label{Solution}

\begin{lem}\label{conditions}
 Words for lemma
\end{lem}

\begin{thm}\label{mainthm}
 Words for theorem
\end{thm}

\begin{rem}\label{remone}
Words for remark
\end{rem}

\section{Analysis}\label{Analysis}

\begin{pro}\label{observation}
Words for proposition
\end{pro}

\begin{rem}\label{remtwo}
Words for remark
\end{rem}

\begin{thm}\label{secondthm}
Words for remark
\end{thm}

\begin{pro}\label{proposition}
Words for proposition
\end{pro}

\end{document}

Note that I left only the packages necessary for the example to run. Note also how I redefined your personal commands, in the correct way: in particular, don't redefine commands you don't know about (\div, in this case).

About the packages you load in the example:

• amsfonts is already loaded by amssymb
• amsthm is not compatible with svjour3
• epsfig must not be loaded in new document (it's just for back compatibility)

• subfigure has been obsolete for 10+ years, call

  \usepackage[caption=false]{subfig}
  

  instead; the \subfigure command becomes \subfloat.

About the theorem numbering: having separate counters for all kinds of statements is unnecessarily confusing for the reader, who will have hard times in finding the statements when referenced at: is lemma 1.1 before or after proposition 1.1?