Abstract: Let n be an arbitrary natural number. The existence of n positive solutions is proved for a singular beam equation fixed at both ends, where the nonlinear term is a Caratheodory function. Main tools are height functions concerned
with nonlinear term and Guo-Krasnoselskii fixed point theorem of cone expansion-compression type. Further research shows that the equation may have positive solution if the growth limits of nonlinear term at zero and infinity are unbounded functions.

Key words: Nonlinear ordinary differential equation, Boundary value problem, Singularity, Positive solution, Existence, Multiplicity