Oscillation and nonoscillation of Emden-Fowler type equations of second order. (English) Zbl 0908.34023

Lomtatidze, A.

Arch. Math., Brno 32, No. 3, 181-193 (1996).

The author considers oscillatory properties of solutions to the equation \[ u''+p(t)| u | ^{\alpha}| u'| ^{1-\alpha}\text{sgn } u = 0,\tag{1} \] with \(\alpha \in [0,1]\) and \(p:[0,\infty)\to [0,\infty)\) is locally integrable. The equation (1) is defined to be nonoscillatory if it possesses at least one solution \(u\) on \([0,\infty)\) such that there is a neighbourhood \(\mathcal V\) of \(+\infty \) such that \(u(t)\neq 0\) on \(\mathcal V,\) while there exists an \(a>0\) such that the set \(\{t>a: u'(t)= 0\}\) has a zero measure, and it is said to be oscillatory, otherwise. Several criteria assuring that (1) is oscillatory and a criterion assuring that (1) is nonoscillatory are given.

Reviewer: M.Tvrdý (Praha)

MSC:

34C15

Nonlinear oscillations and coupled oscillators for ordinary differential equations

Keywords:

Emden-Fowler type equations of second order; oscillatory solutions; nonoscillatory solutions; oscillatory equations; nonoscillatory equations