Rectifiable oscillations in second-order half-linear differential equations
The authors continue their investigation of geometric aspects of oscillation theory of various second order differential equations [see, e.g. \textit{M. Pašić} and \textit{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.
scientific article; zbMATH DE number 5610638
scientific article