Spectral properties and nodal solutions for second-order, mm-point, boundary value problems

We consider the mm-point boundary value problem consisting of the equation −u″=f(u),on (0,1), where f:R→Rf:R→R is C1C1, with f(0)=0f(0)=0, together with the boundary conditions u(0)=0,u(1)=∑i=1m−2αiu(ηi), where m≥3m≥3, ηi∈(0,1)ηi∈(0,1) and αi>0αi>0 for i=1,…,m−2i=1,…,m−2, with ∑i=1m−2αi<1. We first show that the spectral properties of the linearisation of this problem are similar to the well-known properties of the standard Sturm–Liouville problem with separated boundary conditions (with a minor modification to deal with the multi-point boundary condition). These spectral properties are then used to prove a Rabinowitz-type global bifurcation theorem for a bifurcation problem related to the above problem. Finally, we use the global bifurcation theorem to obtain nodal solutions (that is, sign-changing solutions with a specified number of zeros) of the above problem, under various conditions on the asymptotic behaviour of ff.