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Rectifiable oscillations in second-order linear differential equations. (English) Zbl 1168.34027

The paper under review studies the oscillation and rectifiable property of the second order linear differential equation

${y}^{\text{'}\text{'}}\left(x\right)+f\left(x\right)y\left(x\right)=0,\phantom{\rule{1.em}{0ex}}x\in I:=\left(0,1\right),\phantom{\rule{2.em}{0ex}}\left(\mathrm{E}\right)$

where $f$ is a strictly positive continuous map on $I·$

Under the additional hypothesis $f\in {C}^{2}\left(\left(0,1\right]\right)$ and assuming that $f$ satisfies the Hartman-Wintner condition

$\underset{\epsilon \to 0}{lim}{\int }_{\epsilon }^{1}\frac{1}{\sqrt[4]{f\left(x\right)}}|{\left(\frac{1}{\sqrt[4]{f\left(x\right)}}\right)}^{\text{'}\text{'}}|\phantom{\rule{0.166667em}{0ex}}dx<\infty ,\phantom{\rule{2.em}{0ex}}\left(\mathrm{HW}-\mathrm{C}\right)$

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) ${lim}_{\epsilon \to 0}{\int }_{\epsilon }^{1}\sqrt[4]{f\left(x\right)}\phantom{\rule{0.166667em}{0ex}}dx<\infty$ (resp. (IL): ${lim}_{\epsilon \to 0}{\int }_{\epsilon }^{1}\sqrt[4]{f\left(x\right)}\phantom{\rule{0.166667em}{0ex}}dx=\infty$);

(2) Consider the perturbed equation

${y}^{\text{'}\text{'}}\left(x\right)+\left(f\left(x\right)+p\left(x\right)\right)y\left(x\right)=0\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}I,\phantom{\rule{2.em}{0ex}}\left(\mathrm{PE}\right)$

where $\frac{p}{\sqrt{f}}\in {L}^{1}\left(I\right)·$ 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 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 ${dim}_{M}G\left(y\right)$ and the $s$-dimensional upper Minkowski content ${M}^{s}\left(G\left(y\right)\right)$ of the graphs $G\left(y\right)$ 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 ${lim}_{x\to 0}{x}^{\alpha }f\left(x\right)=\lambda ,$ with $\alpha >2$ and $\lambda >0$, it is proved:

(3) On $I,$ all the solutions of Eq. (E) verify ${dim}_{M}G\left(y\right)=1$ and $0<{M}^{1}\left(G\left(y\right)\right)<\infty$ (resp. ${dim}_{M}G\left(y\right)=s\in \left[1,2\right)$ and $0<{M}^{s}\left(G\left(y\right)\right)<\infty ,$ with $s=\frac{3}{2}-\frac{2}{\alpha }$) if $\alpha \in \left(2,4\right)$ (resp. if $\alpha >4$).

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).

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34A30 Linear ODE and systems, general 28A75 Length, area, volume, other geometric measure theory
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