The author studies a boundary value problem associated to a th-order ordinary differential equation of the form
where is a selfadjoint, disconjugate operator on and the boundary conditions are separated. It is assumed that is “superlinear at infinity” and that . It is proved the existence of infinitely many solutions having specified nodal properties.
The main result represents a generalization to higher-order problems of a result by P. Hartman [J. Differ. Equations 26, 37-53(1997; Zbl 0365.34032)]. In the proof, it is used a generalization of the Rabinowitz global bifurcation theorem together with general results on the nodal properties of the solutions to the linear eigenvalue problem .
Related results for fourth-order problems have been given, among others, by M. Conti, S. Terracini and G. Verzini [Infinitely many solutions to fourth order superlinear periodic problems (preprint)] and by M. Henrard and F. Sadyrbaev [Nonlinear Anal., Theory Methods Appl. 33, 281-302 (1998; Zbl 0937.34020)].