On the question of the residual of strong exponents of oscillation on~the~set of solutions of third-order differential equations

A. Kh. Stash ; N. A. Loboda

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 23 (2023) no. 3, pp. 348-356

Résumé

In this paper, we study various types of exponents of oscillation (upper or lower, strong or weak) of non-strict signs, zeros, and roots of non-zero solutions of linear homogeneous differential equations of the third order with continuous and bounded coefficients on the positive semi-axis. A nonzero solution of a linear homogeneous equation cannot be zeroed due to the existence and uniqueness theorem. Therefore, the spectra of all the listed exponents of oscillation (i.e. their sets of values on nonzero solutions) consist of one zero value. In addition, it is known that the spectra of the oscillation exponents of linear homogeneous equations of the second order also consist of a single value. Consequently, on the set of solutions of equations up to the second order there is a residual of all exponents of oscillation. On the set of solutions of third-order equations, strong exponents vibrations of hyper roots are not residual, i.e. are not invariant with respect to the change in the solution at any finite section of the half-axis of time. In this article, it is proved that on the set of solutions of third-order equations, strong oscillation indices of non-strict signs, zeros, and roots are not residual. In parallel, the existence of a function from the specified set with the following properties is proved: all listed exponents of oscillation are accurate, but not absolute. At the same time, all strong exponents like all weak ones, are equal to each other.