Abstract: 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.
Keywords: differential equations, oscillation, number of zeros, exponents of oscillation, Sergeev's frequencies, residual functional, the spectrum of the oscillation exponent
For citation: Stash, A. Kh. On Some Properties of Strong Oscillation Exponents of Solutions of Linear Homogeneous Differential Equations, Vladikavkaz Math. J., 2024, vol. 26, no. 2, pp. 122-132 (in Russian).
DOI 10.46698/x2543-2938-8548-c
1. Sergeev, I. N. Definition and Properties of Characteristic
Frequencies of a Linear Equation, Journal of Mathematical
Sciences, 2006, vol. 135, no. 1, pp. 2764-2793.
DOI: 10.1007/s10958-006-0142-6.
2. Sergeev, I. N. Oscillation and Wandering Characteristics of
Solutions of a Linear Differential Systems, Izvestiya:
Mathematics, 2012, vol. 76, no. 1, pp. 139-162.
DOI: 10.1070/IM2012v076n01ABEH002578.
3. Sergeev, I. N. The Remarkable Agreement Between the
Oscillation and Wandering Characteristics of Solutions of
Differential Systems, Sbornik: Mathematics, 2013, vol. 204,
no. 1, pp. 114-132. DOI: 10.1070/SM2013v204n01ABEH004293.
4. Sergeev, I. N. Oscillation, Rotation, and Wandering Exponents
of Solutions of Differential Systems, Mathematical
Notes, 2016, vol. 99, no. 5, pp. 729-746. DOI: 10.1134/S0001434616050114.
5. Bykov, V. V. On the Baire Classification of Sergeev Frequencies of
Zeros and Roots of Solution of Linear Differential Equation,
Differential Equation, 2016, vol. 52, no. 4, pp. 413-420. DOI:
10.1134/S0012266116040029.
6. Barabanov, E. A. and Voidelevich, A. S. Remark on the Theory of Sergeev
Frequencies of Zeros, Signs and Roots for Solution of Linear
Differential Equation: I, Differential Equation, 2016, vol. 52,
no. 10, pp. 1249-1267. DOI: 10.1134/S0012266116100013.
7. Barabanov, E. A. and Voidelevich, A. S. Remark on the Theory of Sergeev Frequencies of Zeros,
Signs and Roots for Solution of Linear Differential Equation: II,
Differential Equation, 2016, vol. 52, no. 12,
pp. 1523-1538. DOI: 10.1134/S0012266116120016.
8. Barabanov, E. A. and Voidelevich, A. S. Spectra of the Upper Sergeev Frequencies of Zeros and Signs of
Linear Differential Equation, Doklady Nacional'noj akademii nauk Belarusi [Doklady of the National Academy of Sciences of Belarus], 2016, vol. 60, no. 1, pp. 24-31 (in Russian).
9. Voidelevich, A. S. On Spectra of Upper Sergeev Frequencies of Linear Differential
Equation, Journal of the Belarusian State University. Mathematics and Informatics, 2019, no. 1, pp. 28-32 (in Russian).
10. Kuratovsky, K. Topology, vol. 1, New York, London/Warszawa, Academic Press/Panstwowe Wydawnictwo Naukowe, 1966.
11. Sergeev, I. N. A Contribution to the Theory of Lyapunov Exponents
for Linear Systems of Differential Equations, Journal of
Mathematical Sciences, 1986, vol. 33, no. 6, pp. 1245-1292.
DOI: 10.1007/BF01084752.
12. Burlakov, D. S. and Tsoii, S. V. Coincidence of Complete and Vector Frequencies
of Solutions of a Linear Autonomous System,
Journal of Mathematical Sciences, 2015, vol. 210, no. 2,
pp. 155-167. DOI: 10.1007/s10958-015-2554-7.
13. Stash, A. Kh. Properties of Exponents of Oscillation of Linear
Autonomous Differential System Solutions, Vestnik Udmurtskogo Universiteta.
Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, no. 4, pp. 558-568 (in Russian).
14. Stash, A. Kh. Some Properties of Oscillation Indicators of Solutions to a Two-Dimensional System,
Moscow University Mathematics Bulletin, 2019, vol. 74, no. 5, pp. 202-204.
DOI: 10.3103/S0027132219050061.
15. Stash, A. Kh. The Absence of Residual Property for Strong Exponents
of Oscillation of Linear Systems, Vestnik Udmurtskogo
Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021,
vol. 31, no. 1, pp. 59-69 (in Russian). DOI: 10.35634/vm210105.
16. Stash, A. Kh. The Absence of Residual Property for Total Hyper-Frequencies of
Solutions to Third Order Differential Equations, Moscow
University Mathematics Bulletin, 2017, vol. 72, pp. 81-83.
DOI: 10.3103/S0027132217020085.
17. Stash, A. Kh. and Loboda, N. A. On the Question of Residual of Strongs Exponents of
Oscillation on the Set of Solutions of Third-Order Differential
Equations, Izvestiya of Saratov University. Mathematics.
Mechanics. Informatics, 2023, vol. 23, no. 3, pp. 348-356 (in
Russian). DOI: 10.18500/1816-9791-2023-23-3-348-356.
18. Polya, G. and Szego, G. Aufgaben und Lehrsatze aus der Analysis, vol. 2, Hardcover, 1945.
19. Sergeev, I. N. Controlling Solutions to a Linear Differential
Equation, Moscow University Mathematics Bulletin, 2009, vol. 64, pp. 113-120.
DOI: 10.3103/S0027132209030048.
20. Stash, A. Kh. On the Control of the Spectra of Upper Strong Oscillation Exponents of Signs,
Zeros, and Roots of Third-Order Differential Equations, Differential Equation,
2023, vol. 59. no. 5, pp. 597-605. DOI: 10.1134/S0012266123050038.
21. Stash, A. Kh. Properties of Complete and Vector Sign Frequencies of Solutions
of Linear Autonomous Differential Equations, Differential
Equation, 2014, vol. 50, no. 10, pp. 1413-1417. DOI: 10.1134/S0012266114100206.