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Abstract <p> Various types of oscillation indices for solutions of linear homogeneous differentialequations and systems with continuous coefficients on the positive half-line are investigated. Theabsence of the residual property (i.e., invariance with respect to a change in the solution ona finite-length interval) is proved for strong oscillation indices on the solution sets of differentialequations of order higher than <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10625_2025_2814_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">\(2\)</EquationSource> </InlineEquation>. The sharpness andabsoluteness of all oscillation indices are established on the solution sets of autonomous differentialsystems, their spectra are found, and their principal values are described. The sets of values ofoscillation indices for individual classes of nonstationary differential equations and systems arepresented as well. In addition, a study of the spectra of oscillation indices in the firstapproximation is carried out; namely, we show that the cardinalities of the spectra of oscillationindices of a two-dimensional nonlinear system and its first approximation system may not be thesame.</p>

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On the Properties of Oscillation Indices of Solutions to Differential Systems

  • 期刊论文

  • 发表日期：

    2025年6月1日

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  • A. Kh. Stash

摘要

Abstract

Various types of oscillation indices for solutions of linear homogeneous differentialequations and systems with continuous coefficients on the positive half-line are investigated. Theabsence of the residual property (i.e., invariance with respect to a change in the solution ona finite-length interval) is proved for strong oscillation indices on the solution sets of differentialequations of order higher than \(2\) . The sharpness andabsoluteness of all oscillation indices are established on the solution sets of autonomous differentialsystems, their spectra are found, and their principal values are described. The sets of values ofoscillation indices for individual classes of nonstationary differential equations and systems arepresented as well. In addition, a study of the spectra of oscillation indices in the firstapproximation is carried out; namely, we show that the cardinalities of the spectra of oscillationindices of a two-dimensional nonlinear system and its first approximation system may not be thesame.