###
*Abstract*

Positive solutions to the boundary value problem,
$$
\left\{ {\begin{array}{*{20}l}
{{{y}\ifmmode{''}\else$''$\fi = - f{\left( {x,y{\left( {w{\left( x \right)}} \right)}} \right)},} \hfill} & {{0 < x < 1,} \hfill} \\
{{\alpha y{\left( x \right)} - \beta {y}\ifmmode{'}\else$'$\fi{\left( x \right)} = \xi {\left( x \right)},} \hfill} & {{a \leqslant x \leqslant 0,} \hfill} \\
{{\gamma y{\left( x \right)} + \delta {y}\ifmmode{'}\else$'$\fi{\left( x \right)} = \eta {\left( x \right)},} \hfill} & {{1 \leqslant x \leqslant b,} \hfill} \\
\end{array} } \right.
$$
are obtained by applying the Schauder fixed point theorem, where *w*(*x*) is a continuous function defined on [0, 1] and *f*(*x, y*) is a function defined on (0, 1)×(0, ∞), which satisfies certain restrictions and may have singularity at *y*=0. The result corrects and improves an existence theorem due to Erbe and Kong^{[1]}.