In the present paper the separation variables method is applied to finding an exact solution of the sixth-order mathematical model with nonsmooth solutions. Analyzing arising difficulties, in particular the spectral problem, we use a pointwise method of interpretation of solutions proposed by Yu.V. Pokornyi. This method showed its effectiveness in constructing of an exact parallel to the classical theory of differential equations, including oscillation theorems, both second and fourth orders.

The sixth-order nonlinear spectral problem with nonsmooth solutions is studied. It is proved that the set of non-negative values for which the nonlinear spectral problem has at least one non-trivial non-negative solution is nonempty and coincides with a certain interval. We use the pointwise approach proposed by Yu. V. Pokorny analyzing solutions to a boundary value problem. This approach shown to be effective in the study of the second-order problems. Based on the previously obtained estimates of the Green’s function of the boundary-value problem, it was possible to show that the operator inverting the studied nonlinear problem, representable as a superposition of completely continuous and continuous operators, acts from the cone of nonnegative continuous functions to a narrower set. The last fact allows us to prove the uniqueness of a solution of a nonlinear boundary value problem using the theory of spaces with a cone.

В статье уточнена скорость роста собственных значений одной спектральной задачи четвертого порядка с негладкими решениями. Анализ задачи опирается на предложенный Ю. В. Покорным поточечный подход, показавший свою эффективность при изучении линейных граничных задач второго и четвертого порядков с непрерывными решениями.