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

We consider the m-point boundary value problem consisting of the equation - u^? = f (u), on (0, 1), where f : R ? R is C¹, with f (0) = 0, together with the boundary conditions u (0) = 0, u (1) = underover(?, i = 1, m - 2) a_i u (?_i), where m = 3, ?_i ? (0, 1) and a_i > 0 for i = 1, ..., m - 2, with underover(?, i = 1, m - 2) a_i < 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 f. © 2006 Elsevier Ltd. All rights reserved.