On some properties of strong oscillation exponents of solutions of linear homogeneous differential equations

A. Kh. Stash

Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 2, pp. 122-132

Résumé

The research topic of this work is at the junction of the theory of Lyapunov exponents and of the theory of oscillation. We study various types of exponents of oscillation (upper or lower, strong or weak) of strict signs, non-strict signs, zeros, roots and hyperroots of non-zero solutions of linear homogeneous differential equations higher than third order with continuous and bounded coefficients on the positive semi-axis. In the first part of this paper, an example of a linear homogeneous differential equation of order higher than the second is constructed, the spectra of the upper strong oscillation exponents of strict signs, zeros and roots of which coincide with a given Suslin set of a non-negative semi-axis of an extended numerical line containing zero. At the same time, all the listed exponents of oscillation on the set of solutions of the constructed equation are absolute. When constructing the indicated equation, analytical methods of the qualitative theory of differential equations, in particular, the author's technique for controlling the fundamental system of solutions of such equations in one particular case. In the second part of the paper it is proved that on the set of solutions of equations of order higher than the second, strong oscillation exponents of non-strict signs, zeros, roots and hyperroots are not residual. As a consequence, the existence of a function from the specified set with the following properties is proved: all the listed exponents of oscillation are accurate, but not absolute. At the same time, all strong exponents, as well as all weak ones, are equal to each other.

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