A remark on the linearization technique in half-linear oscillation theory

Ondřej Došlý

Abstract. We show that oscillatory properties of the half-linear second order differential equation \[(r(t)\Phi(x'))'+c(t)\Phi(x)=0,\qquad\Phi(x)=|x|^{p-2}x,\quad p\gt 1,\] can be investigated via oscillatory properties of a certain associated second order linear differential equation. In contrast to paper [O. Došlý, S. Peňa, A linearization method in oscillation theory of half-linear differential equations, J. Inequal. Appl. 2005 (2005), 235–245], we do not need to distinguish between the cases \(p\ge 2\) and \(p\in (1,2]\). Our results also improve the oscillation and nonoscillation criteria given in [O. Došlý, A. Lomtatidze, Oscillation and nonoscillation criteria for half-linear second order differential equations, to appear in Hiroshima Math. J.].

Keywords: half-linear oscillation theory, oscillation and nonoscillation criteria, Riccati technique, perturbation principle.

Mathematics Subject Classification: 34C10.

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