On Oscillation of Solutions of Linear Differential Equations

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• 作者：Janne Gröhn ; Jouni Rättyä
• 关键词：Linear differential equations ; Oscillation theory ; Schwarzian derivative
• 刊名：The Journal of Geometric Analysis
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：27
• 期：1
• 页码：868-885
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Differential Geometry; Convex and Discrete Geometry; Fourier Analysis; Abstract Harmonic Analysis; Dynamical Systems and Ergodic Theory; Global Analysis and Analysis on Manifolds;
• 出版者：Springer US
• ISSN：1559-002X
• 卷排序：27

文摘

An interrelationship is found between the accumulation points of zeros of non-trivial solutions of \(f''+Af=0\) and the boundary behavior of the analytic coefficient A in the unit disc \(\mathbb {D}\) of the complex plane \(\mathbb {C}\). It is also shown that the geometric distribution of zeros of any non-trivial solution of \(f''+Af=0\) is severely restricted if $$\begin{aligned} |A(z)| \left( 1-|z|^2\right) ^2 \le 1 + C \left( 1-|z|\right) , \quad z\in \mathbb {D}, \end{aligned}$$ (*)for any constant \(0<C<\infty \). These considerations are related to the open problem of whether (*) implies finite oscillation for all non-trivial solutions.KeywordsLinear differential equationsOscillation theorySchwarzian derivativeMathematics Subject ClassificationPrimary 34C1034M10References1.Chuaqui, M., Duren, P., Osgood, B.: Schwarzian derivatives and uniform local univalence. Comput. Methods Funct. 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A I Math. 8(2), 325–332 (1983)MathSciNetCrossRefMATHGoogle ScholarCopyright information© Mathematica Josephina, Inc. 2016Authors and AffiliationsJanne Gröhn1Email authorJouni Rättyä11.Department of Physics and MathematicsUniversity of Eastern FinlandJoensuuFinland About this article CrossMark Publisher Name Springer US Print ISSN 1050-6926 Online ISSN 1559-002X About this journal Reprints and Permissions Article actions .buybox { margin: 16px 0 0; position: relative; } .buybox { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; } .buybox { zoom: 1; } .buybox:after, .buybox:before { content: ''; display: table; } .buybox:after { clear: both; } /*---------------------------------*/ .buybox .buybox__header { border: 1px solid #b3b3b3; border-bottom: 0; padding: 8px 12px; position: relative; background-color: #f2f2f2; } .buybox__header .buybox__login { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; 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