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Zbl 1184.34043
Pašić, Mervan; Wong, James S.W.
Rectifiable oscillations in second-order half-linear differential equations.
(English)
[J] Ann. Mat. Pura Appl. (4) 188, No. 3, 517-541 (2009). ISSN 0373-3114; ISSN 1618-1891/e

The authors continue their investigation of geometric aspects of oscillation theory of various second order differential equations [see, e.g. {\it M. Pašić} and {\it 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 $$(\Phi(y'))'+f(x)\Phi(y)=0,\quad \Phi(y):=|y|^{p-2}y,\ p>1, \tag{1}$$ where $x\in I:=(0,1]$, $f\in C^2(I)$, $f(x)>0$, and $\lim_{x\to 0-} f(x) =\infty$. 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.
[Ondřej Došlý (Brno)]
MSC 2000:
*34C10 Qualitative theory of oscillations of ODE: Zeros, etc.

Keywords: oscillations; nonlinear equations; graph; rectifiability; fractal dimension; Minkowski content; asymptotics; perturbation

Citations: Zbl 1160.26304

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