*(English)*Zbl 1029.34015

The author studies a boundary value problem associated to a $2m$th-order ordinary differential equation of the form

where $L$ is a selfadjoint, disconjugate operator on $[0,\pi ]$ and the boundary conditions are separated. It is assumed that $g$ is “superlinear at infinity” and that ${lim}_{\left|\xi \right|\to 0}g(x,\xi )=0$. 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 $Lu=\mu pu$.

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)].