The existence of positive solutions for the Sturm-Liouville boundary value problems

For the Sturm-Liouville boundary value problem (p(t)u′(t))′+λf(t,u(t))=0, 0 t 1, α1u(0)−β1p(0)u′(0)=0, α2u(1)+β2p(1)u′(1)=0, where λ > 0, we shall use a fixed point theorem in a cone to obtain the existence of positive solutions for λ on a suitable interval.