On Some Properties of Strong Oscillation Exponents of Solutions to Homogeneous Linear Differential Equations

Within the theory of Lyapunov exponents and the oscillation theory, we studyvarious types of oscillation exponents (upper or lower, strong orweak) of strict signs, nonstrict signs, zeros, roots, and hyperrootsof nonzero solutions to homogeneous linear differential equationswith continuous coefficients onthe positive semiaxis.We construct some exampleof a homogeneous linear differential equation of order greater than 2whose spectra of the upper strongoscillation exponents of strict signs, zeros and rootscoincide with a given Suslin set of the nonnegative semiaxis of the extended real axiswhich contains zero.At the same time, alllisted oscillation exponents are absolute on the set of solutions of the equation.We use the analytical methods of the qualitative theory ofdifferential equations, in particular, the author’s technique forcontrolling the fundamental system of solutions to these equations inone particular case.We then prove that the strong oscillation exponents of nonstrict signs, zeros,roots, and hyperroots are not residualon the set of solutions of equations of order greater than 2.As a consequence, we demonstrate the existence of a function from the set with the followingproperties: All listed oscillation exponents areaccurate, but not absolute.In this event, all strong exponent as well as weak exponents are equal to each other.