To cite this article: T. Garbuza (2008) Abstract. A special technique based on the analysis of oscillatory behaviour of linear equations is applied to investigation of a nonlinear boundary value problem of sixth order. We get the estimation of the number of solutions to boundary value problems of the type, where f is continuous together with the partial derivative f ′ x which is supposed to be positive. We assume also that at least one solution of the problem under consideration exists.

We consider positively homogeneous the sixth order differential equations of the type x (6) = h(t, x), where h possesses the property that h(t, cx) = ch(t, x) for c ≥ 0. This class includes the linear equations x (6) = p(t)x and piece-wise linear ones x (6) = k2 x + − k1 x − . We consider conjugate points and angles associated with extremal solutions and prove some comparison results.Key words: differential equations of 6-th order, equations with asymmetric nonlinearities, conjugate points, positive homogeneous equations.

We provide explicit formulas for the Pučik spectra of boundary value problems with the Sturm‐Liouville boundary conditions.