Oscillation and nonoscillation of certain second order quasilinear dynamic equations (Q1940037)

The paper studies the oscillation and nonoscillation of the second order quasilinear dynamic equation \[ \big( r(t)\, |y^\Delta(t)|^{\alpha-1} y^\Delta(t) \big)^\Delta +f(t,y^\sigma(t))=0, \qquad t\in[t_0,\infty)_{\mathbb T}, \] where \({\mathbb T}\) is a time scale (any nonempty closed subset of \({\mathbb R}\)) with \(t_0\in{\mathbb T}\) and \([t_0,\infty)_{\mathbb T}:=[t_0,\infty)\cap{\mathbb T}\). The results extend the case of differential equations by \textit{J. Wang} [Funkc. Ekvacioj, Ser. Int. 41, No. 1, 25--54 (1998; Zbl 1140.34356)] to arbitrary time scales, and also to the discrete case. The results also extend the case of \(f(t,y)=q(t)\,g(y)\) considered by \textit{S. R. Grace} et al. [J. Appl. Math. Comput. 32, No. 1, 205--218 (2010; Zbl 1198.34194)].