# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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