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Zbl 1168.34027
Kwong, Man Kam; Pašić, Mervan; Wong, James S.W.
Rectifiable oscillations in second-order linear differential equations.
(English)
[J] J. Differ. Equations 245, No. 8, 2333-2351 (2008). ISSN 0022-0396

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 {\it 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 $\lambda >0$, it is proved: (3) On $I,$ all the solutions of Eq. (E) verify $\dim_M G(y)=1$ and $0<M^{1}(G(y))<\infty$ (resp. $\dim_M G(y)=s\in [1,2)$ and $0<M^{s}(G(y))<\infty,$ with $s=\frac{3}{2}-\frac{2}{\alpha}$) if $\alpha\in (2,4)$ (resp. if $\alpha>4$). \par Finally, the four result states that is possible to find an appropriate $f$ which allows the coexistence of rectifiable and unrectifiable oscillations within the general solution of Eq. (E).
[Antonio Linero Bas (Murcia)]
MSC 2000:
*34C10 Qualitative theory of oscillations of ODE: Zeros, etc.
34A30 Linear ODE and systems
28A75 Geometric measure theory

Keywords: Linear equation; second order; oscillation; graph; rectifiability; integral criterion; stability; co-existence; fractal dimension; Euler type equation; Liouville transformation; Wronskian; independent solutions

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