Global bifurcation and multiple results for Sturm–Liouville problems

We consider the nonlinear Sturm–Liouville boundary value problem { ( L u ) ( t ) = λ a ( t ) f ( u ( t ) ) , 0 < t < 1 , R 1 ( u ) = α 1 u ( 0 ) + β 1 u ′ ( 0 ) = 0 , R 2 ( u ) = α 2 u ( 1 ) + β 2 u ′ ( 1 ) = 0 , where L is the linear Sturm–Liouville operator ( L u ) ( t ) = − ( p ( t ) u ′ ( t ) ) ′ + q ( t ) u ( t ) . We obtain a global bifurcation result for a related bifurcation problem. We then use this to obtain multiple (at least eight) solutions of the Sturm–Liouville problem having specified nodal properties.