Rectifiable oscillations in second-order linear differential equations
type: Item
instance of: scholarly article
title: Rectifiable oscillations in second-order linear differential equations
zbMATH Open document ID: 1168.34027
author: Man Kam Kwong
author: Mervan Pašić
author: James S. W. Wong
published in: Journal of Differential Equations
publication date: 2008-10-21T00:00:00Z
review text: The paper under review studies the oscillation and rectifiable property of the second order linear differential equation \[ y''(x)+f(x)y(x)=0, \quad x\in I:=(0,1), \tag{E} \] where \(f\) is a strictly positive continuous map on \(I.\) Under the additional hypothesis \(f\in C^2((0,1])\) and assuming that \(f\) satisfies the Hartman-Wintner condition \[ \lim_{\varepsilon\rightarrow 0} \int_{\varepsilon}^{1} \frac{1}{\root {4}\of{f(x)}} \bigg|\bigg(\frac{1}{\root {4}\of{f(x)}}\bigg)''\bigg|\,dx < \infty, \tag{HW-C} \] in the first part of the paper the authors establish: (1) Equation (E) is rectifiable oscillatory (resp. unrectifiable oscillatory) on \(I\) if and only if (FL) \(\lim_{\varepsilon\rightarrow 0}\int_{\varepsilon}^{1}\root {4}\of{f(x)}\,dx<\infty\) (resp. (IL): \(\lim_{\varepsilon\rightarrow 0}\int_{\varepsilon}^{1}\root {4}\of{f(x)}\,dx=\infty\)); (2) Consider the perturbed equation \[ y''(x)+(f(x)+p(x))y(x)=0 \text{ on } I, \tag{PE} \] where \(\frac{p}{\sqrt{f}}\in L^{1}(I).\) Equation (PE) is rectifiable oscillatory (resp. unrectifiable oscillatory) on \(I\) if and only if condition (FL) (resp. condition (IL)) holds. Recall that an equation is rectifiable oscillatory (resp. unrectifiable oscillatory) on \(I\) if all its solutions are oscillatory and the corresponding graphs have a finite length (resp. an infinite length). It is worth mentioning that the above result \(1.\) improves Theorem 2 of \textit{Wong} [Electronic Journal of Qualitative Theory of Differential Equations 20, 1--12 (2007)]. In the second part of the paper the authors study the values of the upper Minkowski-Bouligand dimension \(\dim_M G(y)\) and the \(s\)-dimensional upper Minkowski content \(M^{s}(G(y))\) of the graphs \(G(y)\) associated to solutions of Eq. (E). Assuming again that \(f\) is of class \(C^2,\) with \(f>0\) on \(I\) and satisfying condition (HW-C), and adding the asymptotical condition \(\lim_{x\rightarrow 0}x^{\alpha}f(x)=\lambda,\) with \(\alpha >2\) and \(\lambda >…
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 28A75
zbMATH DE Number: 5354965
zbMATH Keywords: Linear equation
zbMATH Keywords: second order
zbMATH Keywords: oscillation
zbMATH Keywords: graph
zbMATH Keywords: rectifiability
zbMATH Keywords: integral criterion
zbMATH Keywords: stability
zbMATH Keywords: co-existence
zbMATH Keywords: fractal dimension
zbMATH Keywords: Euler type equation
zbMATH Keywords: Liouville transformation
zbMATH Keywords: Wronskian
zbMATH Keywords: independent solutions
reviewed by: Antonio Linero Bas
MaRDI profile type: Publication
full work available at URL: j.jde.2008.05.016
OpenAlex ID: W1965684788
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DOI: 10.1016/J.JDE.2008.05.016
description: scientific article; zbMATH DE number 5354965