Abstract: In this paper, the following nonlinear Sturm-Liouville problem
\[
\left\{
\begin{array}{lcl}
-u''=\lambda f(x, u),\\
\alpha_0 u(0)+\beta_0 u'(0)=0,\ \ \alpha_1 u(1)+\beta_1 u'(1)=0;
\end{array}
\right.
\]
is discussed by topological methods.
In the case that the nonlinear term is non-singular or singular,
global structure of the positive solution set of the above problem is obtained,
and the existence of positive solutions
of the above problem is proved under the condition that the nonlinear term $f(x,u)$ does not satisfy f(x,u)≥0(u≥0).

Key words: Nonlinear Sturm-Liouville problem, Positive solution, Global structure,
Topological methods