Oscillation and nonoscillation of certain second order quasilinear dynamic equations. (English) Zbl 1266.34145

Xu, Zhiting; Wang, Yuanfeng

Hiroshima Math. J. 42, No. 3, 385-409 (2012).

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 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 S. R. Grace et al. [J. Appl. Math. Comput. 32, No. 1, 205–218 (2010; Zbl 1198.34194)].

Reviewer: Roman Šimon Hilscher (Brno)

MSC:

34N05	Dynamic equations on time scales or measure chains
34K11	Oscillation theory of functional-differential equations

Keywords:

oscillation; nonoscillation; quasilinear dynamic equations; time scale; second order

Citations:

Zbl 1140.34356; Zbl 1198.34194

References:

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