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Rectifiable oscillations in second-order half-linear differential equations. (English) Zbl 1184.34043

The authors continue their investigation of geometric aspects of oscillation theory of various second order differential equations [see, e.g. M. Pašić and J. S. W. Wong, Differ. Equ. Appl. 1, No. 1, 85–122 (2009; Zbl 1160.26304)]. In this paper, the main attention is devoted to the half-linear differential equation

${\left({\Phi }\left({y}^{\text{'}}\right)\right)}^{\text{'}}+f\left(x\right){\Phi }\left(y\right)=0,\phantom{\rule{1.em}{0ex}}{\Phi }\left(y\right):={|y|}^{p-2}y,\phantom{\rule{4pt}{0ex}}p>1,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $x\in I:=\left(0,1\right]$, $f\in {C}^{2}\left(I\right)$, $f\left(x\right)>0$, and ${lim}_{x\to 0-}f\left(x\right)=\infty$. Integral criteria are established which guarantee that (1) is oscillatory for $x\to 0-$ and that the graph of oscillatory solutions has finite/infinite arclength. Some other geometrical aspects of oscillation of (1), like the fractal dimension of graphs of its solutions, are investigated as well.

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
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