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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

(Φ(y ' )) ' +f(x)Φ(y)=0,Φ(y):=|y| p-2 y,p>1,(1)

where xI:=(0,1], fC 2 (I), f(x)>0, and lim x0- f(x)=. Integral criteria are established which guarantee that (1) is oscillatory for x0- 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:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
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