On Infinite Spectra of Oscillation Exponents of Third-Order Linear Differential Equations

Abstract

The research topic of this work is at the junction of the theory of Lyapunov exponents and oscillation theory. In this paper, we study the spectra (that is, the sets of different values on nonzero solutions) of the exponents of oscillation of signs (strict and nonstrict), zeros, roots, and hyperroots of linear homogeneous differential equations with coefficients continuous on the positive semiaxis. In the first part of the paper, we build a third-order linear differential equation with the following property: the spectra of all upper and lower strong and weak exponents of oscillation of strict and nonstrict signs, zeros, roots, and hyperrots contain a countable set of different essential values, both metrically and topologically. Moreover, all these values are implemented on the same sequence of solutions of the constructed equation, that is, for each solution from this sequence, all of the oscillation exponents coincide with each other. In the construction of the indicated equation and in the proof of the required results, we used analytical methods of the qualitative theory of differential equations and methods from the theory of perturbations of solutions to linear differential equations, in particular, the author’s technique for controlling the fundamental system of solutions of such equations in one special case. In the second part of the paper, the existence of a third-order linear differential equation with continuum spectra of the oscillation exponents is established, wherein the spectra of all oscillation exponents fill the same segment of the number axis with predetermined arbitrary positive incommensurable ends. It turned out that for each solution of the constructed differential equation, all of the oscillation exponents coincide with each other. The results are theoretical in nature; they expand our understanding of the possible spectra of oscillation exponents of linear homogeneous differential equations.