Opuscula Math. 26, no. 2 (2006), 305-315

Opuscula Mathematica

# 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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• Ondřej Došlý
• Masaryk University, Department of Mathematics, Janáčkovo nám. 2a, CZ-662 95 Brno

Ondřej Došlý, A remark on the linearization technique in half-linear oscillation theory, Opuscula Math. 26, no. 2 (2006), 305-315

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