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\documentclass[12pt]{article}
\begin{document}
\title{Singular boundary value problems}
\author{ S. Stan\v ek}
\maketitle
Let $A$ and $T$ be positive numbers. The singular
differential equation
\begin{equation}
(r(x(t))x'(t))'=\mu q(t) f(t,x(t))
\end{equation}
depending on the positive parameter $\mu$ is considered. Here
$q>0$ a.e. on $[0,T]$, $r>0$ on $(0,A]$ may be singular at $x=0$
and $f(t,x) \le 0$ may be singular at $x=0$ and $x=A$ of the
phase variable $x$. Effective
sufficient conditions imposed on $r,\, \mu,\,q$ and $f$ are given
for the existence of a solution $x$ of (1) satisfying either the
Dirichlet boundary conditions $x(0)=x(t)=0$ or the periodic
boundary conditions $x(0)=x'(T)=0$, $x(T)=x'(T)=0$.
Solutions of our boundary value problems are considered in the sets
\begin{eqnarray*}
\Delta_A^0 &=&\Big\{x: x \in C^0(J) \cap C^1((0,T)), \,
\int_0^{x(\cdot)} r(u)\,du \in C^1(J) \cap C^2((0,T)), \\
&& 0 < x(t) < A \ \mbox{for} \ t \in (0,T)\Big\}
\end{eqnarray*}
and
\begin{eqnarray*}
\Delta_A^1 &=&\Big\{x: x \in C^1(J), \,
\int_0^{x(\cdot)} r(u)\,du \in C^1(J) \cap C^2((0,T)), \\
&& 0 < x(t) < A \ \mbox{for} \ t \in (0,T)\Big\}.
\end{eqnarray*}
The proofs are based on the regularity, sequential and truncation
techniques and use the topological transversality theorem.

\begin{thebibliography}{99}
\bibitem{S} S. Stan\v ek, Positive solutions of singular
Dirichlet and periodic boundary value problems, \textit{Math.
Computer Modelling} (accepted for publication)
\end{thebibliography}
\end{document}