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Year
2021
Volume
31
Issue
1
Pages
59-69
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Section Mathematics
Title The absence of residual property for strong exponents of oscillation of linear systems
Author(-s) Stash A.Kh.a
Affiliations Adyghe State Universitya
Abstract In this paper, we study various types of exponents of oscillation (upper or lower, strong or weak) of zeros, roots, hyperroots, strict and non-strict signs of non-zero solutions of linear homogeneous differential systems on the positive semi-axis. On the set of non-zero solutions of autonomous systems the relations between these exponents of oscillation are established. It is proved that all strong exponents of oscillations (unlike Sergeev's frequencies of sign changes, zeros and roots, as well as all the weak exponents of oscillations) considered as functions on the set of solutions to linear homogeneous $n$-dimensional differential systems with continuous coefficients on the semi-line are not residual (i.e. can be changed when changing solution on a finite interval). Besides, at any beforehand given natural $n\ge2$ we give the example of $n$-dimensional differential system, for some solution of which all strong oscillation exponents differ from corresponding weak exponents. In this case, all weak and all strong exponents on the chosen solution coincide with each other, respectively. When proving the results of this work, the case of parity and odd $n$ are considered separately.
Keywords differential equations, linear systems, oscillation, number of zeros, exponents of oscillation, Sergeev's frequencies
UDC 517.926
MSC 34C10
DOI 10.35634/vm210105
Received 11 December 2020
Language Russian
Citation 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, issue 1, pp. 59-69.
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